Block patterns in generalized Euler Permutations

TL;DR

This paper extends the cluster method to analyze the distribution of block patterns in generalized Euler permutations, broadening the scope of pattern analysis in restricted combinatorial objects.

Contribution

It introduces the generalized cluster method, an extension of Goulden and Jackson's approach, for studying pattern distributions in various restricted combinatorial structures.

Findings

Developed the generalized cluster method for restricted objects

Applied the method to block patterns in generalized Euler permutations

Enhanced understanding of pattern distributions in combinatorial classes

Abstract

Goulden and Jackson introduced a very powerful method to study the distributions of certain consecutive patterns in permutations, words, and other combinatorial objects which is now called the cluster method. There are a number of natural classes of combinatorial objects which start with either permutations or words and add additional restrictions. These include up-down permutations, generalized Euler permutations, words with no consecutive repeated letters, Young tableaux, and non-backtracking random walks. We develop an extension of the cluster method which we call the {\em generalized cluster method} to study the distribution of certain consecutive patterns in such restricted combinatorial objects. In this paper, we focus on block patterns in generalized Euler permutations.

Equations

$$Des(\sigma )$$

$$Ris(\sigma )$$

$$\mathrm{inv}(\sigma )$$

$$\mathrm{coinv}(\sigma )$$

$$[n{]}_q=\frac{1-q^n}{1-q}=1+q+\cdots +q^{n-1},$$

$$[n{]}_q!=[n{]}_q[n-1{]}_q\cdots [1{]}_q,$$

$${\left[\genfrac{}{}{0ex}{}{n}{k}\right]}_q=\frac{[n{]}_q!}{[k{]}_q![n-k{]}_q!},\text{and}$$

$${\left[\genfrac{}{}{0ex}{}{n}{{\lambda }_1,\dots ,{\lambda }_{\ell }}\right]}_q=\frac{[n{]}_q!}{[{\lambda }_1{]}_q!\cdots [{\lambda }_{\ell }{]}_q!}$$

$$Ri{s}_{i,k}(\sigma )=\{s:0\le s\le n\text{ and }{\sigma }_{i+sk}<{\sigma }_{i+sk+1}\}$$

$$1+\sum _{n\ge 0}\frac{E_{2n+2}^{0,2,2}}{(2n+2)!}{t}^{2n+2}=\sec t.$$

$$\sum _{n\ge 0}\frac{E_{2n+1}^{0,1,2}}{(2n+1)!}{t}^{2n+1}=\tan t.$$

$$\sum _{n\ge 0}\frac{t^{i+kn+j}}{[i+kn+j{]}_q!}\sum _{\sigma \in {\mathcal{C}}_{i+kn+j}^{i,j,k}}{x}^{ri{s}_{i,k}(\sigma )}{q}^{\mathrm{inv}(\sigma )}.$$

$$\sum _{n\ge 0}\frac{t^{i+kn+j}}{[i+kn+j{]}_q!}\sum _{\sigma \in {\mathcal{E}}_{i+kn+j}^{i,j,k}}{q}^{\mathrm{inv}(\sigma )}.$$

$$\sigma =256891710411123$$

$$Q\text{-mch}(F)=|\{i:\mathrm{red}\left(\begin{array}[]{|c|c|}\hline\cr F(i,2)&F(i+1,2)\\ \hline\cr F(i,1)&F(i+1,1)\\ \hline\cr\end{array}\right)=Q\}|.$$

$$G_{{\Gamma }_1,\dots ,{\Gamma }_s,\mathcal{R}}^{i,j,k}(x_1,\dots ,x_s,t)=\sum _{n\ge 0}\frac{t^{i+kn+j}}{(i+kn+j)!}\sum _{F\in {\mathcal{P}}_{i+kn+j}^{i,j,k}}\prod _{r=1}^s{x}_r^{{\Gamma }_r\text{-}\mathrm{mch}(F)}.$$

$$G_{P,\mathcal{R}}^{i,j,k}(x,t)=\sum _{n\ge 0}\frac{t^{i+kn+j}}{(i+kn+j)!}\sum _{F\in {\mathcal{P}}_{i+kn+j}^{i,j,k}}{x}^{\text{P-mch}(F)}.$$

$$\begin{array}[]{cc}15478263&\overset{x}{1}5478263\\ 15478\overset{x}{2}63&\overset{x}{1}5478\overset{x}{2}63\end{array}$$

$$C_{n,\tau }(x)=\sum _{\sigma \in \mathcal{CM}{S}_{n,\tau }}{x}^{m_{\tau }(\sigma )}.$$

$$m{p}_{2n+1,132}=(2n-1)m{p}_{2n-1,132}=\prod _{i=0}^{n-1}(2i+1).$$

$$1+\sum _{n\ge 1}\frac{t^n}{n!}\sum _{\sigma \in S_n}{x}^{\text{τ-mch}(\sigma )}=\frac{1}{1-t-{\sum }_{n\ge 2}\frac{t^n}{n!}{C}_{n,\tau }(x-1)}.$$

$$C_{kn,\Gamma }^{0,0,k}(x)=\sum _{F\in {\mathcal{CM}}_{kn,\Gamma }^{0,0,k}}{x}^{m_{\Gamma }(F)}$$

$$1+\sum _{n\ge 1}\frac{t^n}{(kn)!}\sum _{F\in {\mathcal{P}}_{kn}^{0,0,k}}{x}^{\text{Γ-mch}(F)}=\frac{1}{1-{\sum }_{n\ge 1}\frac{t^{kn}}{(kn)!}{C}_{kn,\Gamma }^{0,0,k}(x-1)}.$$

$$1+\sum _{n\ge 1}\frac{t^n}{(kn)!}\sum _{F\in {\mathcal{P}}_{kn}^{0,0,k}}(x+1{)}^{\text{Γ-mch}(F)}=1+\sum _{n\ge 1}\frac{t^n}{(kn)!}\sum _{F\in {\mathcal{M}\mathcal{P}}_{kn,\Gamma }^{0,0,k}}{x}^{m_{\Gamma }(F)}.$$

$$1+\sum _{n\ge 1}\frac{t^n}{(kn)!}\sum _{F\in {\mathcal{M}\mathcal{P}}_{kn,\Gamma }^{0,0,k}}{x}^{m_{\Gamma }(F)}=\frac{1}{1-{\sum }_{n\ge 1}\frac{t^{kn}}{(kn)!}{C}_{kn,\Gamma }^{0,0,k}(x)}.$$

$$\frac{1}{1-{\sum }_{n\ge 1}\frac{t^{kn}}{(kn)!}{C}_{kn,\Gamma }^{0,0,k}(x)}=1+\sum _{m\ge 1}{\left(\sum _{n\ge 1}\frac{t^{kn}}{(kn)!}{C}_{kn,\Gamma }^{0,0,k}(x)\right)}^m.$$

$$\sum _{F\in {\mathcal{M}\mathcal{P}}_{sn,\Gamma }^{0,0,k}}{x}^{m_{\Gamma }(F)}$$

$$G{C}_{kn,\Gamma ,\mathcal{R}}^{0,0,k}(x)=\sum _{Q\in {\mathcal{G}\mathcal{C}}_{kn,\Gamma ,\mathcal{R}}^{0,0,k}}{w}_{\Gamma ,\mathcal{R}}(Q).$$

$$1+\sum _{n\ge 1}\frac{t^{kn}}{(kn)!}\sum _{F\in {\mathcal{P}}_{kn,\mathcal{R}}^{0,0,k}}{x}^{\text{Γ-mch}(F)}=\frac{1}{1-{\sum }_{n\ge 1}\frac{t^{kn}}{(kn)!}G{C}_{kn,\Gamma ,\mathcal{R}}^{0,0,k}(x-1)}.$$

$$1+\sum _{n\ge 1}\frac{t^{kn}}{(kn)!}\sum _{F\in {\mathcal{P}}_{kn,\mathcal{R}}^{0,0,k}}(x+1{)}^{\text{Γ-mch}(F)}=1+\sum _{n\ge 1}\frac{t^{kn}}{(kn)!}\sum _{F\in {\mathcal{M}\mathcal{P}}_{kn,\Gamma ,\mathcal{R}}^{0,0,k}}{x}^{m_{\Gamma }(F)}.$$

$$1+\sum _{n\ge 1}\frac{t^{kn}}{(kn)!}\sum _{F\in {\mathcal{M}\mathcal{P}}_{kn,\Gamma ,\mathcal{R}}^{0,0,k}}{x}^{m_{\Gamma }(F)}=\frac{1}{1-{\sum }_{n\ge 1}\frac{t^{kn}}{(kn)!}G{C}_{kn,\Gamma ,\mathcal{R}}^{0,0,k}(x)}.$$

$$\frac{1}{1-{\sum }_{n\ge 1}\frac{t^{kn}}{(kn)!}G{C}_{kn,\Gamma ,\mathcal{R}}^{0,0,k}(x)}=1+\sum _{m\ge 1}{\left(\sum _{n\ge 1}\frac{t^{kn}}{(kn)!}G{C}_{kn,\Gamma ,\mathcal{R}}^{0,0,k}(x)\right)}^m.$$

$$\sum _{F\in {\mathcal{M}\mathcal{P}}_{sn}^{0,0,k}}{x}^{m_{\Gamma }(F)}$$

Full text

1 Introduction

Goulden and Jackson [14] introduced a very powerful method to study the distributions of certain consecutive patterns in permutations, words, and other combinatorial objects which is now called the cluster method. There are now numerous applications of the cluster method to study consecutive patterns in permutations and words, see, for example, [5, 10, 17, 23, 24].

The main goal of this paper is to give an extension of cluster method which we call the generalized cluster method which is suitable for studying the distribution of consecutive patterns in permutations and words that satisfy additional restrictions. These include various subsets of permutations with regular descent patterns such as up-down permutations, generalized Euler permutations, permutations and words associated with various classes of Young tableaux, and various subsets of words such as words with no consecutive repeated letters and words associated with non-backtracking random walks. Our work was motivated by a paper from the second author [25] who studied consecutive patterns in up-down permutations. In particular, [25] introduced the notion of generalized maximum packings which is an extension of the notion of maximum packings introduced by Duane and Remmel [6]. We realized that one could reformulate the results of [6] in terms of the cluster method and that a similar reformulation of the results [25] led to a natural generalization of the cluster method which we call the generalized cluster method.

In this paper, we will illustrate our method by studying the distribution of certain consecutive patterns permutations in permutations which have regular descent patterns. Let $\sigma=\sigma_{1}\ldots\sigma_{n}$ be a permutation in the symmetric group $S_{n}$. Then we let

[TABLE]

where for any statement $A$, $\chi(A)$ equals 1 if $A$ is true and $\chi(A)=0$ if $A$ is false.

For any $n>0$, we let

[TABLE]

be the usual $q$-analogues of $n$, $n!$, $\binom{n}{k}$, and $\binom{n}{\lambda_{1},\ldots,\lambda_{\ell}}$, respectively.

Given integers $k\geq 2$ and $i,j,n\geq 0$, we let $\mathcal{C}^{i,j,k}_{i+kn+j}$ denote the set of permutations $\sigma=\sigma_{1}\ldots\sigma_{i+kn+j}$ in $S_{i+kn+j}$ such that $Des(\sigma)\subseteq\{i,i+k,\ldots,i+nk\}$ and $C^{i,j,k}_{i+kn+j}=|\mathcal{C}^{i,j,k}_{i+kn+j}|$. Thus permutations in $\mathcal{C}^{i,j,k}_{i+kn+j}$ start with an increasing block of size $i$ followed by $n$ increasing blocks of size $k$ and end with an increasing block of size $j$.

We let $\mathcal{E}^{i,j,k}_{i+kn+j}$ denote the set of permutations $\sigma\in S_{i+kn+j}$ with $Des(\sigma)=\{i,i+k,\ldots,i+nk\}$ and $E^{i,j,k}_{i+kn+j}=|\mathcal{E}^{i,j,k}_{i+kn+j}|$. For any $\sigma\in S_{i+kn+j}$, let

[TABLE]

and $ris_{i,k}(\sigma)=|Ris_{i,k}(\sigma)|$. Then $E^{i,j,k}_{i+kn+j}$ is the number of $\sigma\in\mathcal{C}^{i,j,k}_{i+kn+j}$ such that $Ris_{i,k}(\sigma)=\varnothing$. Thus permutations $\sigma$ in $\mathcal{E}^{i,j,k}_{i+kn+j}$ have the same block structure as permutations in $\mathcal{C}^{i,j,k}_{i+kn+j}$, but we require the additional restriction that for any two consecutive blocks $B$ and $C$ in $\sigma$, the last element of block $B$ must be larger than the first element of block $C$.

In the special case where $k=2$, $i=0$, and $j=2$, $E^{0,2,2}_{2n+2}$ is the number of permutations in $S_{2n+2}$ with descent set $\{2,4,\dots,2n\}$. These are the classical up-down permutations. André [1, 2] proved that

[TABLE]

Similarly, $E^{0,1,2}_{2n+1}$ counts the number of odd up-down permutations and André proved that

[TABLE]

These numbers are also called the Euler numbers. When $k>2$, $E^{0,j,k}_{kn+j}$ are called generalized Euler numbers [18]. There are well-known generating functions for $q$-analogues of the generalized Euler numbers; see Stanley’s book [27], page 148. Various divisibility properties of the $q$-Euler numbers have been studied in [3, 4, 11] and of the generalized $q$-Euler numbers in [12, SZ]. Prodinger [22] also studied $q$-analogues of the number $E^{1,2,2}_{2n+1}$ and $E^{1,1,2}_{2n+2}$.

Mendes, Remmel, and Riehl [20] computed the generating functions

[TABLE]

Setting $x=0$ in (1) gives the generating function

[TABLE]

For our purposes, we shall picture a $\sigma$ in $\mathcal{C}^{i,j,k}_{i+kn+j}$ as an array $F(\sigma)$ starting with a column of size $i$, followed by $n$ columns of size $k$, and ending with a column of size $j$ filled with the permutation $\sigma$ so that one recovers $\sigma$ by reading the elements in each column from bottom to top and reading the columns from left to right. This means that in each column, the numbers are increasing when read from bottom to top. For example, the array associated with the permutation

[TABLE]

in $\mathcal{C}^{2,1,3}_{12}$ is pictured in Figure 1. Elements of $\mathcal{E}^{i,j,k}_{i+kn+j}$ can be viewed as restricted arrays where the top element of each column has to be bigger than the bottom element in the column immediately to its right.

More generally, let $D^{i,j,k}_{i+kn+j}$ denote the diagram which consists of a column of height $i$, followed by $n$ columns of height $k$, and ending with a column of height $j$. We let $(s,t)$ denote the cell which is in the $s^{th}$ column reading from left to right, and the $t^{th}$ row reading from bottom to top. For example, for the filling of $D^{2,1,3}_{12}$ is pictured in Figure 1, the number $7$ is in cell $(3,2)$. Let $F^{i,j,k}_{i+kn+j}$ denote the set of all fillings of $D^{i,j,k}_{i+kn+j}$ with the elements $1,\ldots,i+kn+j$. The word $w(F)$ of any filling $F\in F^{i,j,k}_{i+kn+j}$ is the word obtained by reading the columns from bottom to top and the columns from left to right. We let $\mathcal{P}^{i,j,k}_{i+kn+j}$ denote the set of all fillings of $D^{i,j,k}_{i+kn+j}$ with the elements of $1,\ldots,i+kn+j$ such that the elements are increasing reading from bottom to top in each column. Thus for any $F\in\mathcal{P}^{i,j,k}_{i+kn+j}$, $w(F)\in\mathcal{C}^{i,j,k}_{i+kn+j}$.

In this paper, we will be mostly interested in patterns that occur between columns of height $k$ for elements of $\mathcal{P}^{i,j,k}_{i+kn+j}$. These types of patterns were first studied by Harmse and Remmel [15] for elements in $\mathcal{P}^{0,0,k}_{nk}$.

If $F$ is any filling of a $k\times n$-rectangle with pairwise distinct positive integers, then we let $\mathrm{red}(F)$ denote the filling which results from $F$ by replacing the $i$-th smallest element of $F$ by $i$. For example, Figure 2 we picture a filling, $F$, with its corresponding reduced filling, $\mathrm{red}(F)$.

If $F\in\mathcal{F}^{i,j,k}_{i+kn+j}$ and $2\leq c_{1}<\cdots<c_{s}\leq n+1$, then we let $F[c_{1},\ldots,c_{s}]$ be the filling of the $k\times j$ rectangle where the elements in column $a$ of $F[c_{1},\ldots,c_{s}]$ equal the elements in column $c_{a}$ in $F$ for $a=1,\ldots,j$. We can then extend the usual pattern matching definitions from permutations to elements of $\mathcal{F}^{i,j,k}_{i+kn+j}$ as follows.

Definition 1**.**

Let $\Gamma$ be a set of elements of $\bigcup_{r\geq 2}\mathcal{F}^{0,0,k}_{kr}$ and $F\in\mathcal{F}^{i,j,k}_{i+kn+j}$. Then we say

$\Gamma$* occurs in $F$ if there is an $r\geq 1$ and $2\leq i_{1}<i_{2}<\cdots<i_{r}\leq n+1$ such that $\mathrm{red}(F[i_{1},\ldots,i_{r}])\in\Gamma$,* 2. 2.

$F$* avoids $\Gamma$ if there is no occurrence of $\Gamma$ in $F$, and* 3. 3.

*there is a $\Gamma$-match in $F$ starting at column $i$ if there is an $r\geq 1$ such that *

$\mathrm{red}(F[i,i+1,\ldots,i+r-1])\in\Gamma$.

We let \text{\Gamma$-$\mathrm{mch}}(F) denote the number of $\Gamma$-matches in $F$. In the case where $\Gamma=\{P\}$ is a singleton, we shall simply write $P$ occurs in $F$ ($P$ avoids $F$, $F$ has a $P$-match starting at column $i$) for $\Gamma$ occurs in $F$ ($\Gamma$ avoids $F$, $F$ has a $\Gamma$-match starting at column $i$). When $i=j=0$ and $k=1$, then $\mathcal{P}^{0,0,1}_{n}=S_{n}$, where $S_{n}$ is the symmetric group, and our definitions reduce to the standard definitions that have appeared in the pattern matching literature. We note that Kitaev, Mansour, and Vella [16] have studied pattern matching in matrices which is a more general setting than the one we are considering for $i=j=0$ in this paper.

We will illustrate our methods by finding generating functions for the distribution of such $\Gamma$-matches in various restricted arrays in $\mathcal{F}^{i,j,k}_{i+kn+j}$. To illustrate our methods in this paper, we will restrict ourselves to studying restrictions that arise by specifying some binary relation $\mathscr{R}$ between pairs of columns of integers. Then we let $\mathcal{P}^{i,j,k}_{i+kn+j,\mathscr{R}}$ denote the set of all elements $F$ in $\mathcal{P}^{i,j,k}_{i+kn+j}$ such that for all $1\leq i\leq n+1$, $(F[i],F[i+1])\in\mathscr{R}$. For example, consider the following relations $\mathscr{R}$.

Let $\mathscr{R}$ be the relation that holds between a pair if columns of integers $(C,D)$ if and only if the top element of $C$ is greater than the bottom element of $D$. Then it is easy to see that $\mathcal{E}^{i,j,k}_{i+kn+j}$ equals $\mathcal{P}^{i,j,k}_{i+kn+j,\mathscr{R}}$. 2. 2.

Let $\mathscr{R}$ be the relation that holds between a pair of columns of integers $(C,D)$ if and only if in the array $CD$, the rows are strictly increasing. Then it is easy to see that set of standard tableaux of shape $n^{k}$ equals $\mathcal{P}^{0,0,k}_{kn,\mathscr{R}}$. 3. 3.

Let $\mathscr{R}$ be the relation that holds between a pair of columns of integers $(C,D)$ of integers if and only if the bottom element in column $C$ is less than the bottom element of column $D$. It is easy to see that if $i,j\geq 1$ and $k\geq 2$, then $\mathcal{P}^{i,j,k}_{i+kn+j,\mathscr{R}}$ is the set of all elements $F\in\mathcal{P}^{i,j,k}_{i+kn+j}$ such that $F(1,1)<F(2,1)<\cdots<F(n+2,1)$.

We should note that our methods can be applied to study more complicated restrictions than those that are determined by binary relations on columns of integers, but we shall not study such restrictions in this paper.

Our methods can also be applied to study more general types of patterns in such fillings. For example, suppose that we wanted to find the distribution of the pattern Q=\begin{array}[]{|c|c|}\hline\cr 3&4\\ \hline\cr 1&2\\ \hline\cr\end{array} in the first two rows of fillings in $\mathcal{P}^{2,3,4}_{25}$. Then for any $F\in\mathcal{P}^{2,3,4}_{25}$, we would let

[TABLE]

For example, for the $F\in\mathcal{P}^{2,3,4}_{25}$ pictured in Figure 3, $Q\text{-mch}(F)=3$ as there are $Q$-matches starting in columns 1, 3, and 6.

The main goal of the paper is to study the generating functions of the distribution of $\Gamma$-matches in restricted arrays such at the ones described above. For example, suppose that $\mathscr{R}$ is some binary relation on pairs of columns whose entries strictly increase, reading from bottom to top. Suppose that we are given a sequence of pairwise disjoint sets of patterns $\Gamma_{1},\ldots,\Gamma_{s}$ such that for each $i$, $\Gamma_{i}\subseteq\mathcal{P}^{0,0,k}_{r_{i}k}$ where $r_{i}\geq 2$. Then we will be interested in computing generating functions of the form

[TABLE]

In the case where $s=1$ and $\Gamma_{1}=\{P\}$, then we shall let

[TABLE]

The outline of this paper is as follows, In Section 2, we shall recall the cluster method and extend it to cover the types of filled arrays described above. We will also introduce the generalized cluster method to study generating functions of the form $G^{0,0,k}_{P,\mathscr{R}}(x,t)$. In section 3, we shall describe how one can extend the methods of Section 2 to study generating functions of the form $G^{i,0,k}_{P,\mathscr{R}}(x,t)$, $G^{0,j,k}_{P,\mathscr{R}}(x,t)$, and $G^{i,j,k}_{P,\mathscr{R}}(x,t)$ where $i,j>0$ and $k\geq 2$. In section 4, we shall introduce the notion of joint clusters and joint generalized clusters to study generating functions of the form $G^{i,j,k}_{\Gamma_{1},\ldots,\Gamma_{s},\mathscr{R}}(x_{1},\ldots,x_{s},t)$.

2 Clusters and Generalized Clusters for $\mathcal{P}^{0,0,k}_{kn}$.

In this section, we shall describe the cluster method and the generalized cluster method for elements in $\mathcal{P}^{0,0,k}_{kn}$.

We start by recalling the definition of clusters for permutations. Let $\tau\in S_{j}$ be a permutation. Then for any $n\geq 1$, we let $\mathcal{M}S_{n,\tau}$ denote the set of all permutations in $\sigma\in S_{n}$ where we have marked some of the $\tau$-matches in $\sigma$ by placing an $x$ at the start of $\tau$-match in $\sigma$. Given a $\sigma\in\mathcal{M}S_{n,\tau}$, we let $m_{\tau}(\sigma)$ be the number of marked $\tau$-matches in $\sigma$. For example, suppose that $\tau=132$ and $\sigma=15478263$. Then there are two $\tau$-matches in $\sigma$, one starting at position 1 and one starting at position 6. Thus $\sigma$ gives rises to four elements of $\mathcal{M}S_{8,\tau}$.

[TABLE]

A $\tau$-cluster is an element of $\sigma=\sigma_{1}\ldots\sigma_{n}\in\mathcal{M}S_{n,\tau}$ such that

every $\sigma_{i}$ is an element of a marked $\tau$-match in $\sigma$ and 2. 2.

any two consecutive marked $\tau$-matches share at least one element.

We let $\mathcal{CM}S_{n,\tau}$ denote the set of all $\tau$-clusters in $\mathcal{M}S_{n,\tau}$.

For each $n\geq 2$, we define the cluster polynomial

[TABLE]

By convention, we set $C_{1,\tau}(x)=1$.

We say that a permutation $\tau\in S_{j}$ is minimal overlapping if the smallest $n$ such that there exists a $\sigma\in S_{n}$ where \text{\tau$-$\mathrm{mch}}(\sigma)=2 is $2j-1$. This means that two consecutive $\tau$-matches in a permutation $\sigma$ can share at most one element which must be the element at the end of the first $\tau$-match and the element which is at the start of the second $\tau$-match. In such a situation, the smallest $m$ such that there exists a $\sigma\in S_{m}$ such that \text{\tau$-$\mathrm{mch}}(\sigma)=n is $n(j-1)+1$. We call elements of $\sigma\in S_{n(j-1)+1}$ such that \text{\tau$-$\mathrm{mch}}(\sigma)=n maximum packings of $\tau$. We let $\mathcal{MP}_{n(j-1)+1}$ denote the set of maximum packings for $\tau$ in $S_{n(j-1)+1}$ and $mp_{n(j-1)+1,\tau}=|\mathcal{MP}_{n(j-1)+1}|$. It is easy to see that if $\tau\in S_{j}$ is minimal overlapping, then the only $\tau$-clusters are maximum packings for $\tau$ where the start of each $\tau$-match is marked with an $x$. For example, $\tau=132$ is a minimal overlapping permutation and it is easy to compute the number of maximum packings of size $2n+1$ for any $n\geq 1$. That is, if $\sigma=\sigma_{1}\ldots\sigma_{2n+1}$ is in $\mathcal{MP}_{2n+1,132}$, then there must be $132$-matches starting at positions $1,3,5,\ldots,2n-1$. It easily follows that for each $i=0,\ldots,n-1$, $\sigma_{2i+1}$ is smaller than $\sigma_{j}$ for all $j>2i+1$. Hence $\sigma_{1}=1$ and $\sigma_{3}=2$. We then have $2n-1$ choices for $\sigma_{3}$. Hence, it follows that

[TABLE]

Thus $C_{2n+1,132}(x)=x^{n}\prod_{i=0}^{n-1}(2i+1)$ for all $n\geq 1$ and $C_{2n,132}(x)=0$ for all $n\geq 1$.

On the other hand, suppose that $\tau=1234$. It is easy to see that for any $\tau$-cluster in $\mathcal{CM}S_{n,\tau}$ where $n\geq 4$, the underlying permutation must be the identity permutation. Moreover, if $\sigma=\sigma_{1}\ldots\sigma_{n}\in\mathcal{CM}S_{n,\tau}$, then $\sigma_{1}$ must be marked with an $x$ because $\sigma_{1}$ must be an element in a marked $\tau$-match and $\sigma_{n-3}$ must be marked since $\sigma_{n}$ must be an element of a marked $\tau$-match. Thus for $n=7$, we are forced to mark 1 and 4, $\displaystyle\overset{x}{1}23\overset{x}{4}567$. However we free to mark either 2 or 3 with an $x$. Hence $C_{7,1234}(x)=x^{2}(1+x)^{2}.$

Then for example, Elizalde and Noy [10] proved the following theorem which is a simple application of the Goulden-Jackson cluster method [13].

Theorem 2**.**

Let $\tau\in S_{j}$ where $j\geq 2$. Then

[TABLE]

It is easy to generalize this result to deal with elements of $\mathcal{P}^{0,0,k}_{nk}$. Suppose that we are given a set of fillings $\Gamma\subseteq\mathcal{P}^{0,0,k}_{tk}$ where $t\geq 2$. For any $n\geq 1$, we let $\mathcal{MP}^{0,0,k}_{kn,\Gamma}$ denote the set of all fillings $F\in\mathcal{P}^{0,0,k}_{nk}$ where we have marked some of the $\Gamma$-matches in $F$ by placing an $x$ on top of the column that starts a $\Gamma$-match in $F$. For example, suppose that suppose $\Gamma=\{P\}$ where P=\begin{array}[]{|c|c|c|}\hline\cr 4&6&5\\ \hline\cr 1&2&3\\ \hline\cr\end{array} and $F\in\mathcal{P}^{0,0,2}_{12}$ pictured in Figure 4. Then there are two $P$-matches in $F$, one starting at column 1 and one starting at column 4. Thus $F$ gives rise to four elements of $\mathcal{MP}^{0,0,2}_{12}$. Given a $F\in\mathcal{MP}^{0,0,k}_{kn,\Gamma}$, we let $m_{\Gamma}(F)$ be the number of marked $\Gamma$-matches in $F$.

We can also extend the reduction operation to $\Gamma$-marked fillings. That is, suppose $\Gamma\subseteq\mathcal{P}^{0,0,k}_{tk}$ where $t\geq 2$ and $F$ is a filling of the $k\times n$ array with integers which strictly increasing in columns, reading from bottom to top, where we have marked some of the $\Gamma$-matches by placing an $x$ at the top of the column that starts a marked $\Gamma$-match. Then by $\mathrm{red}(F)$, we mean the element of $\mathcal{MP}^{0,0,k}_{kn}$ that results by replacing the $i^{th}$ smallest element in $F$ by $i$ and marking a column in $\mathrm{red}(F)$ if and only if it is marked in $F$.

A $\Gamma$-cluster is a filling of $F\in\mathcal{MP}^{0,0,k}_{kn,P}$ such that

every column of $F$ is contained in a marked $\Gamma$-match of $F$ and 2. 2.

any two consecutive consecutive marked $\Gamma$-matches share at least one column.

We let $\mathcal{CM}^{0,0,k}_{kn,\Gamma}$ denote the set of all $\Gamma$-clusters in $\mathcal{MP}^{0,0,k}_{kn,\Gamma}$. For each $n\geq 2$, we define the cluster polynomial

[TABLE]

where $m_{\Gamma}(F)$ is the number of marked $\Gamma$-matches in $F$. By convention, we let $C^{0,0,k}_{k,\Gamma}(x)=1$.

Theorem 3**.**

Let $\Gamma\subseteq\mathcal{P}^{0,0,k}_{rk}$ where $r\geq 2$. Then

[TABLE]

Proof.

Replace $x$ by $x+1$ in (5). Then the left-hand side of (5) is the generating function of $m_{\Gamma}(F)$ over all $F\in\mathcal{MP}^{0,0,k}_{kn,\Gamma}$. That is, it easy to see that

[TABLE]

Thus we must show that

[TABLE]

Now

[TABLE]

Taking the coefficient of $\frac{t^{ks}}{(ks)!}$ on both sides of (7) where $n\geq 1$, we see that we must show that

[TABLE]

The right-hand side of (9) is now easy to interpret. First we pick an $m$ such that $1\leq m\leq s$. Then we pick $a_{1},\ldots,a_{m}\geq 1$ such that $a_{1}+a_{2}+\cdots+a_{m}=s$. Next the binomial coefficient $\binom{kn}{ka_{1},\ldots,ka_{m}}$ allows us to pick a sequence of sets $S_{1},\ldots,S_{m}$ which partition $\{1,\ldots,ks\}$ such that $|S_{i}|=ka_{i}$ for $i=1,\ldots,m$. Finally the product $\prod_{j=1}^{m}C^{0,0,k}_{ka_{j},\Gamma}(x)$ allows us to pick clusters $C_{i}\in\mathcal{CM}^{0,0,k}_{ka_{i},\Gamma}$ for $i=1,\ldots,m$ with weight $\prod_{j=1}^{m}x^{m_{\Gamma}(C_{i})}$. Note that in the cases where $a_{i}=1$, we will interpret $C_{i}$ as just a column of height $k$ filled with the numbers $1,\ldots,k$ which is increasing, reading from bottom to top.

For example, suppose that $k=2$ and $\Gamma=\{P\}$ where P=\begin{array}[]{|c|c|c|}\hline\cr 6&5&4\\ \hline\cr 1&2&3\\ \hline\cr\end{array}. Then in Figure 5, we have pictured $S_{1},S_{2},S_{3},S_{4},S_{5}$ which partition $\{1,\ldots,30\}$ and corresponding clusters $C_{1},\ldots,C_{5}$. Then for each $i=1,\ldots,m$, we create a cluster $D_{i}$ which results by replacing each $j$ in $C_{i}$ by the $j^{th}$ smallest element of $S_{i}$. If we concatenate $D_{1}\ldots D_{m}$ together, then we will obtain an element of $Q\in\mathcal{M}_{kn,\Gamma}^{0,0,k}$. It is easy to see that one can recover $D_{1},\ldots,D_{5}$ from $Q$. That is, given an element $F\in\mathcal{M}^{0,0,k}_{kn,\Gamma}$, we say that a marked subsequence $F[i,i+1,\ldots,j]$ is a maximal $\Gamma$-subcluster of $F$ if $\mathrm{red}(F[i,i+1,\ldots,j])$ is a $\Gamma$-cluster and $F[i,i+1,\ldots,j]$ is not properly contained in a marked subsequence $F[a,a+1,\ldots,b]$ such that $\mathrm{red}(F[a,a+1,\ldots,b])$ is a $\Gamma$-cluster. In the special case where $i=j$ and the column $F[i]$ is not marked, then we say $F[i]$ is maximal $\Gamma$-subcluster if $F[i]$ is not properly contained in a marked subsequence $F[a,a+1,\ldots,b]$ such that $\mathrm{red}(F[a,a+1,\ldots,b])$ is a $\Gamma$-cluster. Thus $D_{1},\ldots,D_{5}$ are the maximal $\Gamma$-subclusters of $Q$. Of course, once we have recovered $D_{1},\ldots,D_{5}$, we can recover the sets $S_{1},\ldots,S_{5}$ and the $\Gamma$-clusters $C_{1},\ldots,C_{5}$.

In this manner, we can see that the right-hand side of (9) just classifies the elements of $\mathcal{M}^{0,0,k}_{kn,\Gamma}$ by its maximal $\Gamma$-subclusters which proves our theorem.

∎

Next suppose that we are given a binary relation $\mathscr{R}$ between $k\times 1$ arrays of integers and a set of patterns $\Gamma\subseteq\mathcal{P}^{0,0,k}_{rk}$ where $r\geq 2$.

Definition 4**.**

*We say that $Q\in\mathcal{MP}^{0,0,k}_{kn,\Gamma}$ is a generalized $\Gamma,\mathscr{R}$-cluster if we can write

$Q=B_{1}B_{2}\cdots B_{m}$ where $B_{i}$ are blocks of consecutive columns in $Q$ such that*

either $B_{i}$ is a single column or $B_{i}$ consists of $r$-columns where $r\geq 2$, $\mathrm{red}(B_{i})$ is a $\Gamma$-cluster in $\mathcal{MP}_{kr,\Gamma}$, and any pair of consecutive columns in $B_{i}$ are in $\mathscr{R}$ and 2. 2.

for $1\leq i\leq m-1$, the pair $(\mathrm{last}(B_{i}),\mathrm{first}(B_{i+1})$ is not in $\mathscr{R}$ where for any $j$, $\mathrm{last}(B_{j})$ is the right-most column of $B_{j}$ and $\mathrm{first}(B_{j})$ is the left-most column of $B_{j}$.

Let $\mathcal{GC}^{0,0,k}_{kn,\Gamma,\mathscr{R}}$ denote the set of all generalized $\Gamma,\mathscr{R}$-clusters which have $n$ columns of height $k$. For example, suppose that $\mathscr{R}$ is the relation that holds for a pair of columns $(C,D)$ if and only if the top element of column $C$ is greater than the bottom element of column $D$ and $\Gamma=\{P\}$ where P=\begin{array}[]{|c|c|c|}\hline\cr 6&5&4\\ \hline\cr 1&2&3\\ \hline\cr\end{array}. Then in Figure 6, we have pictured a generalized $\Gamma,\mathscr{R}$-cluster with 5 blocks $B_{1},B_{2},B_{3},B_{4},B_{5}$.

Given $Q=B_{1}B_{2}\ldots B_{m}\in\mathcal{GC}^{0,0,k}_{kn,\Gamma,\mathscr{R}}$, we define the weight of $B_{i}$, $w_{\Gamma,\mathscr{R}}(B_{i})$, to be 1 if $B_{i}$ is a single column and $x^{m_{\Gamma}(\mathrm{red}(B_{i}))}$ if $B_{i}$ is order isomorphic to a $\Gamma$-cluster. Then we define the weight of $Q$, $w_{\Gamma,\mathscr{R}}(Q)$, to be $(-1)^{m-1}\prod_{i=1}^{m}w_{\Gamma,\mathscr{R}}(B_{i})$. We let

[TABLE]

Then we have the following theorem.

Theorem 5**.**

Let $\mathscr{R}$ be a binary relation on pairs of columns $(C,D)$ of height $k$ which are filled with integers which are increasing from bottom to top. Let $\Gamma\subseteq\mathcal{P}^{0,0,k}_{rk}$ where $r\geq 2$. Then

[TABLE]

Proof.

Replace $x$ by $x+1$ in (11). Then it easy to see that

[TABLE]

Thus we must show that

[TABLE]

Now

[TABLE]

Taking the coefficient of $\frac{t^{ks}}{(ks)!}$ on both sides of (7) where $n\geq 1$, we see that we must show that

[TABLE]

The right-hand side of (15) is now easy to interpret. First we pick an $m$ such that $1\leq m\leq s$. Then we pick $a_{1},\ldots,a_{m}\geq 1$ such that $a_{1}+a_{2}+\cdots+a_{m}=s$. Next the binomial coefficient $\binom{ks}{ka_{1},\ldots,ka_{m}}$ allows us to pick a sequence of sets $S_{1},\ldots,S_{m}$ which partition $\{1,\ldots,ks\}$ such that $|S_{i}|=ka_{i}$ for $i=1,\ldots,m$. Finally the product $\prod_{j=1}^{m}GC^{0,0,k}_{ka_{j},\Gamma,\mathscr{R}}(x)$ allows us to pick generalized $\Gamma,\mathscr{R}$-clusters $G_{i}\in\mathcal{GC}^{0,0,k}_{ka_{i},\Gamma,\mathscr{R}}$ for $i=1,\ldots,m$ with weight $\prod_{j=1}^{m}w_{\Gamma,\mathscr{R}}(G_{i})$. Note that in the cases where $a_{i}=1$, our definitions imply that $C_{i}$ is just a column of height $k$ filled with the numbers $1,\ldots,k$ which is increasing, reading from bottom to top.

For example, suppose that $k=2$ and $\Gamma=\{P\}$ where P=\begin{array}[]{|c|c|c|}\hline\cr 6&5&4\\ \hline\cr 1&2&3\\ \hline\cr\end{array}. Suppose that $\mathscr{R}$ is relation where, for any two columns $C$ and $D$ which filled with integers and are strictly increasing in columns, $(C,D)\in\mathscr{R}$ if and only if the top element of $C$ is greater than the bottom element of $D$. Then in Figure 7, we have pictured $S_{1},S_{2},S_{3},S_{4},S_{5}$ which partition $\{1,\ldots,30\}$ and corresponding generalized $\Gamma,\mathscr{R}$-clusters $G_{1},\ldots,G_{5}$. For each $i$, we have indicated the separation between the blocks of $G_{i}$ by dark black lines. Then for each $i=1,\ldots,m$, we create a cluster $E_{i}$ which results by replacing each $j$ in $G_{i}$ by the $j^{th}$ smallest element of $S_{i}$. If we concatenate $E_{1}\ldots E_{5}$ together, then we will obtain an element of $Q\in\mathcal{M}_{kn,\Gamma}^{0,0,k}$. The weight of $Q$ equals $\prod_{j=1}^{5}w_{\Gamma,\mathscr{R}}(G_{i})$ where

[TABLE]

In Figure 7, we have indicated the boundaries between the $E_{i}$s by light lines.

We let $\mathcal{HGC}_{ks,\Gamma,\mathscr{R}}$ denote the set of all elements that can be constructed in this way. Thus $Q=E_{1}\ldots E_{m}$ is an element of $\mathcal{HGC}_{ks,\Gamma,\mathscr{R}}$ if and only if for each $i=1,\ldots,m$, $\mathrm{red}(E_{i})$ is a generalized $\Gamma,\mathscr{R}$-cluster. Next we define a sign reversing involution $\theta:\mathcal{HGC}_{ks,\Gamma,\mathscr{R}}\rightarrow\mathcal{HGC}_{ks,\Gamma,\mathscr{R}}$. Given $Q=E_{1}\ldots E_{m}\in\mathcal{HGC}_{ks,\Gamma,\mathscr{R}}$, look for the first $i$ such that either

the block structure of $\mathrm{red}(E_{i})=B^{(i)}_{1}\ldots B^{(i)}_{k_{i}}$ consists of more than one block or 2. 2.

$E_{i}$ consists of a single block $B^{(i)}_{1}$ and $(\mathrm{last}(B^{(i)}_{1}),\mathrm{first}(E_{i+1}))$ is not in $\mathscr{R}$.

In case (1), we let $\theta(E_{1}\ldots E_{m})$ be the result of replacing $E_{i}$ by two generalized $\Gamma,\mathscr{R}$-clusters, $E_{i}^{*}$ and $E_{i}^{**}$ where $E_{i}*$ consists just of $B^{(i)}_{1}$ and $E_{i}^{**}$ consists of $B^{(i)}_{2}\ldots B^{(i)}_{k_{i}}$. Note that in this case $w_{\Gamma,\mathscr{R}}(E_{i})=(-1)^{k_{i}-1}\prod_{j=1}^{k_{i}}w_{\Gamma,\mathscr{R}}(B^{(i)}_{j})$ while $w_{\Gamma,\mathscr{R}}(E_{i}^{*})w_{\Gamma,\mathscr{R}}(E_{i}^{**})=(-1)^{k_{i}-2}\prod_{j=1}^{k_{i}}w_{\Gamma,\mathscr{R}}(B^{(i)}_{j})$. In case (2), we let $\theta(E_{1}\ldots E_{m})$ be the result of replacing $E_{i}$ and $E_{i+1}$ by the single generalized $\Gamma,\mathscr{R}$-cluster $E=B^{(i)}_{1}B^{(i+1)}_{1}\ldots B^{(i+1)}_{k_{i+1}}$. Note that since $(\mathrm{last}(B^{(i)}_{1}),\mathrm{first}(E_{i+1}))$ is not in $\mathscr{R}$, $B^{(i)}_{1}B^{(i+1)}_{1}\ldots B^{(i+1)}_{k_{i+1}}$ reduces to a generalized $\Gamma,\mathscr{R}$-cluster. In this case, $w_{\Gamma,\mathscr{R}}(E_{i})w_{\Gamma,\mathscr{R}}(E_{i+1})=(-1)^{k_{i+1}-1}w_{\Gamma,\mathscr{R}}(B^{(i)}_{1})\prod_{j=1}^{k_{i+1}}w_{\Gamma,\mathscr{R}}(B^{(i+1)}_{j})$ and $w_{\Gamma,\mathscr{R}}(E)=(-1)^{k_{i+1}}w_{\Gamma,\mathscr{R}}(B^{(i)}_{1})\prod_{j=1}^{k_{i+1}}w_{\Gamma,\mathscr{R}}(B^{(i+1)}_{j})$. If neither case (1) or case (2) applies, then we let $\theta(E_{1}\ldots E_{m})=E_{1}\ldots E_{m}$. For example, suppose that $\mathscr{R}$ is the binary relation where, for any two columns $C$ and $D$, which filled with integers and are strictly increasing in columns, $(C,D)\in\mathscr{R}$ if and only if the top element of $C$ is greater than the bottom element of $D$ and $\Gamma=\{P\}$ where P=\begin{array}[]{|c|c|c|}\hline\cr 6&5&4\\ \hline\cr 1&2&3\\ \hline\cr\end{array}. Then if $Q=E_{1}\ldots E_{5}$ is the generalized $\Gamma,\mathscr{R}$-cluster pictured in Figure 7, then we are in case (1) with $i=1$ since $E_{1}$ consists of more than one block. Thus $\theta(Q)$ results by breaking that generalized $\Gamma,\mathscr{R}$-cluster into to two clusters $E_{1}^{*}$ of size 1 and $E^{**}_{1}$ of size 4. $\theta(Q)$ is pictured in Figure 8.

It is easy to see that $\theta$ is an involution. That is, if $Q=E_{1}\ldots E_{m}$ is in case (1) using $E_{i}$, then $\theta(Q)$ will be in case (2) using $E_{i}^{*}$ and $E_{i}^{**}$. Similarly if $Q=E_{1}\ldots E_{m}$ is in case (1) using $E_{i}$ and $E_{i+1}$, then $\theta(Q)$ will be in case (2) using $E=E_{i}E_{i+1}$. It follows that if $\theta(E_{1}\ldots E_{m})\neq E_{1}\ldots E_{m}$, then $w_{\Gamma,\mathscr{R}}(E_{1}\ldots E_{m})=-w_{\Gamma,\mathscr{R}}(\theta(E_{1}\ldots E_{m}))$ so that the right-hand side of (15) equals

[TABLE]

Thus we must examine the fixed points of $\theta$.

If $Q=E_{1}\ldots E_{m}\in\mathcal{HGC}_{ks,\Gamma,\mathscr{R}}$ and $\theta(Q)=Q$, then it must be the case that each $E_{i}$ consists of single column of weight 1 or it reduces to generalized $\Gamma,\mathscr{R}$-cluster $\overline{E}_{i}$ consisting of a single block $B^{(i)}_{1}$ whose weight is the weight of $\mathrm{red}(B^{(i)}_{1})$ as a $\Gamma$-cluster. Moreover, it must be the case that for all $i=1,\ldots m-1$, $(\mathrm{last}(E_{i}),\mathrm{first}(E_{i+1})$ is in $\mathscr{R}$. But this means for all $j=1,\ldots,n-1$, $(Q[j],Q[j+1])$ is in $\mathscr{R}$. That is, either $Q[j]$ equals $\mathrm{last}(E_{i})$ for some $i$ or column $j$ is contained in one of the $\Gamma$-clusters $E_{i}$ in which case $(Q[j],Q[j+1])$ is in $\mathscr{R}$ by our definition of generalized $\Gamma,\mathscr{R}$-clusters. Thus any fixed point $Q$ of $\theta$ is an element $\mathcal{MP}^{0,0,k}_{ks,\mathscr{R}}$. Then just like our proof Theorem 3, it follows that $E_{1},\ldots,E_{m}$ are just the maximal $\Gamma$-subclusters of an element in $\mathcal{P}^{0,0,k}_{ks,\mathscr{R}}$. Vice versa, if $T=F_{1}\ldots F_{r}$ is an element of $\mathcal{P}^{0,0,k}_{ks,\mathscr{R}}$ where $F_{1},\ldots,F_{r}$ are the maximal $\Gamma$-subclusters of $T$, then $T=F_{1}\ldots F_{r}$ is a fixed point of $\theta$. Thus we have proved that the right-hand side of (15) equals

[TABLE]

which is what we wanted to prove. ∎

We note that Theorem 3 is a special case of Theorem 5 in the case where $\mathscr{R}$ holds for any pair of columns $(C,D)$.

2.1 Examples

We end this section with two examples. Our goal is to study patterns in generalized Euler permutations $E^{0,0,k}_{kn}$. Thus the relation that we want to consider is that if $C$ and $D$ are fillings of column of height $k$ in which the numbers increase, reading from bottom to top, then $(C,D)\in\mathscr{R}$ if and only if the top element of column $C$ is greater than the bottom element of column $D$. Let $A_{k,3}$ be the element of $\mathcal{P}^{0,0,k}_{3}$ whose $2j+1^{\mathrm{st}}$ row consists of the elements $6j+1,6j+2,6j+3$, reading from left to right. and whose $2j+2^{\mathrm{th}}$ row consists of the $6j+4,6j+5,6j+6$, reading from right to left. For example, we have pictured $A_{3,3}$ and $A_{4,3}$ in Figure 9. There is a natural poset associated to $A_{3,3}$ and $A_{4,3}$. The Hasse diagrams of such posets are obtained by having a node for each integer and drawing a directed arrow from the node corresponding to $i$ to the node corresponding to $i+1$ for each $i\geq 1$. The Hasse diagrams of the posets corresponding to $A_{3,3}$ and $A_{4,3}$ are pictured in Figure 9.

It is easy to see that in a $A_{k,3}$-cluster, the fact that every column is part of marked cluster and the fact that any two consecutive marked clusters must share a column implies that in a $A_{k,3}$-cluster with $n$ columns, the elements in the first row are the $n$ smallest elements, the elements in the second row are the next $n$ smallest elements, etc. Moreover, elements in odd rows are increasing, reading from left to right, and the elements in even rows are increasing, reading from right to left. For example, we have pictured the Hasse diagrams corresponding $A_{3,3}$-clusters at the top of Figure 10 and the Hasse diagrams corresponding to $A_{4,3}$-clusters at the bottom of Figure 10. This means that in a $A_{k,3}$-cluster with $n$ columns, there are $A_{k,3}$-matches starting a positions $1,\ldots,n-2$. Thus to determine $C_{A_{k,3}}(x)$, we need only have to consider the possible ways to mark $A_{k,3}$-matches. Clearly we must mark the $A_{k,3}$-match starting at column 1 and the $A_{k,3}$-match starting at column $n-2$ since the first and last column must be part of marked $A_{k,3}$-matches. It follows that for all $k\geq 2$,

[TABLE]

For $C^{0,0,k}_{nk,A_{k,3}}(x)$, for $k\geq 5$, there are two cases depending on whether the $A_{k,3}$-match starting at column $n-3$ is marked or not. If the $A_{k,3}$-match starting at column $n-3$ is marked, then we can remove the last column, the elements $3n,3n-1,3n-2$, and the mark on column $n-2$ to obtain of $A_{k,3}$-cluster with $n-1$-columns after we reorder the elements. If the $A_{k,3}$-match starting at column $n-3$ is not marked, then we can remove the last two columns, the elements $3n,3n-1,3n-2,3n-3,3n-4,3n-5$, and the mark on column $n-2$ to obtain of $A_{k,3}$-cluster with $n-2$-columns after we reorder the elements. These two cases are pictured in Figure 10. It follows that for $n\geq 5$,

[TABLE]

Then one can easily compute that for all $k\geq 2$,

[TABLE]

Next we consider generalized $A_{k,3},\mathscr{R}$-clusters. There are two types depending on whether $k$ is even or odd. That is, when $k=2j+1$ is odd, then it is easy to see that for each $A_{2j+1,3}$-cluster $C$ with more than 2 columns, the smallest element in the cluster occupies the bottom left-hand position and the largest element in the cluster occupies the top right-hand position. In a generalized $A_{2j+1,3},\mathscr{R}$-cluster with blocks $B_{1}B_{2}\ldots B_{r}$, we must have that the top right-hand element of $B_{i}$ is less than the bottom left-most element of $B_{i+1}$ for $i=1,\ldots,r-1$. If we draw the Hasse diagram of such a generalized cluster, it is easy to see that these conditions force a total order on elements of the generalized $A_{2j+1},\mathscr{R}$-cluster. See for example, Figure 11 where we have pictured two different $A_{3,3},\mathscr{R}$-clusters.

This means that we can develop simple recursions for the generalized cluster polynomials $GC^{0,0,2j+1}_{(2j+1)n,A_{2j+1,3},\mathscr{R}}(x)$. That is, it easy to check that

[TABLE]

Then for $n\geq 3$, we can classify the generalized $A_{2j+1,3},\mathscr{R}$-clusters with $n$ columns $B_{1}\ldots B_{s}$ by the size of the first block $B_{1}$. It could be that $s=1$ so that $B_{1}$ is a $A_{2j+1,3}$-cluster. Such generalized clusters contribute $C^{0,0,2j+1}_{(2j+1)n,A_{2j+1,3}}(x)$ to $GC^{0,0,2j+1}_{(2j+1)n,A_{2j+1,3},\mathscr{R}}(x)$. Otherwise $B_{1}$ has $r$ columns where $1\leq r\leq n-1$. In that situation, our remarks imply that $B_{1}$ consists of the elements $1,\ldots,(2j+1)r$ and if we subtract $(2j+1)r$ from the elements of $B_{2}\ldots B_{s}$, we will end up with a generalized $A_{2j+1,3},\mathscr{R}$-clusters with $n-r$ columns. Thus if $B_{1}$ has $r$ columns, then such generalized clusters contribute $-C^{0,0,2j+1}_{(2j+1)r,A_{2j+1,3}}(x)GC^{0,0,2j+1}_{(2j+1)(n-r),A_{2j+1,3},\mathscr{R}}(x)$ to $GC^{0,0,2j+1}_{(2j+1)n,A_{2j+1,3},\mathscr{R}}(x)$. It follows that for $n\geq 3$,

[TABLE]

For example, we have used this recursion to compute some initial values of $GC^{0,0,3}_{3n,A_{3,3},\mathscr{R}}(x)$.

[TABLE]

Then one can use these initial values of $GC^{0,0,3}_{3n,A_{3,3},\mathscr{R}}(x)$ and Theorem 5 to compute the following initial values of the polynomials $\sum_{\sigma\in\mathcal{E}^{0,0,3}_{3n}}x^{A_{3,3}\text{-}\mathrm{mch}(\sigma)}$.

[TABLE]

When $k=2j$ is even, then it is easy to see that for each $A_{2j,3}$-cluster $C$ with $n\geq 2$ columns, the smallest element in the cluster occupies the bottom left-most position and the element in the top right-most position equals $(2j-1)n+1$. In a generalized $A_{2j,3},\mathscr{R}$-cluster $G$ with blocks $B_{1}B_{2}\ldots B_{r}$, we must have that top right-most element of $B_{i}$ is less than the bottom left-most element of $B_{i+1}$ for $i=1,\ldots,r-1$. There are three cases to consider. First the generalized $A_{2j+1,3},\mathscr{R}$-clusters which consist of a single block $B_{1}$ contribute $C^{0,0,2j}_{(2j)n,A_{2j,3}}(x)$ to $GC^{0,0,2j}_{(2j)n,A_{2j,3},\mathscr{R}}(x)$. There are two additional cases if $G$ has more than one block which are pictured in Figure 12. It is easy to see that if $B_{1}$ is size one, which is pictured at the top of Figure 12, then the elements in $B_{1}$ must be the smallest $2j$ elements in $G$. Then $B_{2}\ldots B_{r}$ is order isomorphic to a $A_{2j+1,3},\mathscr{R}$-cluster with $n-1$ columns. Hence such generalized $A_{2j+1,3},\mathscr{R}$-clusters contribute $-GC^{0,0,2j}_{(2j)(n-1),A_{2j,3},\mathscr{R}}(x)$ to $GC^{0,0,2j}_{(2j)n,A_{2j,3},\mathscr{R}}(x)$. However if $B_{1}$ has $2\leq s\leq n-1$ columns, which is the situation pictured at the bottom of Figure 12, then it is easy to see that elements in the first $2j-1$ rows of $B_{1}$ plus right-most element of the top row of $B_{1}$, must be the elements $1,\ldots,s(2j-1)+1$. Then we are free to choose the remaining elements in the top row of $B_{1}$ in $\binom{2jn-((2j-1)s+1)}{s-1}$ ways and $B_{2}\ldots B_{r}$ will be order isomorphic to a $A_{2j+1,3},\mathscr{R}$-cluster with $n-s$ columns. Hence such generalized $A_{2j+1,3},\mathscr{R}$-clusters contribute $-\binom{2jn-((2j-1)s+1)}{s-1}C^{0,0,2j}_{2js}(x)GC^{0,0,2j}_{(2j)(n-s),A_{2j,3},\mathscr{R}}(x)$ to $GC^{0,0,2j}_{(2j)n,A_{2j,3},\mathscr{R}}(x)$.

It is easy to check that for $j\geq 1$,

[TABLE]

Then it follows from our arguments above that for $n\geq 3$,

[TABLE]

For example, we have used this recursion to compute some initial values of $GC^{0,0,2}_{2n,A_{2,3},\mathscr{R}}(x)$.

[TABLE]

One can then use these values of $GC^{0,0,2}_{2n,A_{2,3},\mathscr{R}}(x)$ and Theorem 5 to compute the following initial values of the polynomials $\sum_{\sigma\in\mathcal{E}^{0,0,3}_{3n}}x^{A_{2,3}\text{-}\mathrm{mch}(\sigma)}$.

[TABLE]

3 Generalized Clusters for fillings

of $D^{0,j,k}_{kn+j}$, $D^{i,0,k}_{i+kn}$, and $D^{i,j,k}_{ikn+j}$.

In this section, we shall extend the generalized cluster method to deal with various types of fillings of $D^{0,j,k}_{kn+j}$, $D^{i,0,k}_{i+kn}$, and $D^{i,j,k}_{ikn+j}$ in the case where $k\geq 2$ and $i$ and $j$ are positive integers which are not equal to $k$.

3.1 Generalized Clusters for fillings

of $D^{0,j,k}_{kn+j}$

We will start by considering fillings of $D^{0,j,k}_{kn+j}$. Fix a set of patterns $\Gamma\subseteq\mathcal{P}^{0,0,k}_{rk}$ where $r\geq 2$. We let $\mathcal{MP}^{0,j,k}_{kn+j,\Gamma}$ denote the set of elements that arise by starting with an element $F$ of $\mathcal{P}^{0,j,k}_{kn+j}$ and marking some of the $\Gamma$-matches in $F$ by placing on $x$ on the column which starts the $\Gamma$-match. Given an element $F\in\mathcal{MP}^{0,j,k}_{kn+j,\Gamma}$, we let $m_{\Gamma}(F)$ denote the number of marked $\Gamma$-matches in $F$.

To find an extension of Theorem 5 for these types of arrays, we need to define a special type of generalized cluster which we call a generalized end cluster. That is, suppose that we are given a binary relation $\mathscr{R}$ on columns of integers.

Definition 6**.**

We say that $Q\in\mathcal{MP}^{0,j,k}_{kn+j,\Gamma}$ is a generalized $\Gamma,\mathscr{R}$-end-cluster if we can write $Q=B_{1}B_{2}\cdots B_{m}$ where $B_{i}$ are blocks of consecutive columns in $Q$ such that

$B_{m}$* is a column of height $j$,* 2. 2.

for $i<m$, either $B_{i}$ is a single column or $B_{i}$ consists of $r$-columns where $r\geq 2$, $\mathrm{red}(B_{i})$ is a $\Gamma$-cluster in $\mathcal{MP}_{kr,\Gamma}$, and any pair of consecutive columns in $B_{i}$ are in $\mathscr{R}$ and 3. 3.

for $1\leq i\leq m-1$, the pair $(\mathrm{last}(B_{i}),\mathrm{first}(B_{i+1})$ is not in $\mathscr{R}$.

Let $\mathcal{GEC}^{0,j,k}_{kn+j,\Gamma,\mathscr{R}}$ denote the set of all generalized $\Gamma,\mathscr{R}$-end-clusters which have $n$ columns of height $k$ followed by a column of height $j$. Given $Q=B_{1}B_{2}\ldots B_{m}\in\mathcal{GEC}^{0,j,k}_{kn+j,\Gamma,\mathscr{R}}$, we define the weight of $B_{i}$, $w_{\Gamma,\mathscr{R}}(B_{i})$, to be 1 if $B_{i}$ is a single column and $x^{m_{\Gamma}(\mathrm{red}(B_{i}))}$ if $B_{i}$ is order isomorphic to a $\Gamma$-cluster. Then we define the weight of $Q$, $w_{\Gamma,\mathscr{R}}(Q)$, to be $(-1)^{m-1}\prod_{i=1}^{m}w_{\Gamma,\mathscr{R}}(B_{i})$. We let

[TABLE]

Let $\mathcal{P}^{0,j,k}_{kn+j,\mathscr{R}}$ denote the set of all elements $F\in\mathcal{P}^{0,j,k}_{kn+j}$ such that the relation $\mathscr{R}$ holds for any pair of consecutive columns in $F$. We let $\mathcal{MP}^{0,j,k}_{kn+j,\Gamma,\mathscr{R}}$ denote the set of elements that arise by starting with an element $F$ of $\mathcal{P}^{0,j,k}_{kn+j,\mathscr{R}}$ and marking some of the $\Gamma$-matches in $F$ by placing on $x$ on the column which starts the $\Gamma$-match.

Then we have the following theorem.

Theorem 7**.**

Let $\mathscr{R}$ be a binary relation on pairs of columns $(C,D)$ which are filled with integers which are increasing from bottom to top. Let $\Gamma\subseteq\mathcal{P}^{0,0,k}_{tk}$ where $t\geq 2$. Then

[TABLE]

Proof.

Replace $x$ by $x+1$ in (20). Then it easy to see that

[TABLE]

Thus we must show that

[TABLE]

Now

[TABLE]

Taking the coefficient of $\frac{t^{ks+j}}{(ks+j)!}$ on both sides of (22) where $s\geq 0$, we see that we must show that

[TABLE]

The right-hand side of (24) is now easy to interpret. First we pick non-negative integers $a$ and $b$ such that $a+b=s$. Then we pick an $m$ such that $1\leq m\leq a$. Next we pick $a_{1},\ldots,a_{m}\geq 1$ such that $a_{1}+a_{2}+\cdots+a_{m}=a$. Next the binomial coefficient $\binom{ks+j}{ka_{1},\ldots,ka_{m},kb+j}$ allows us to pick a sequence of sets $S_{1},\ldots,S_{m},S_{m+1}$ which partition $\{1,\ldots,ks+j\}$ such that $|S_{i}|=ka_{i}$ for $i=1,\ldots,m$ and $|S_{m+1}|=kb+j$. The factor $GEC^{0,j,k}_{kb+j,\Gamma,\mathscr{R}}(x)$ allows us to pick a generalized $\Gamma,\mathscr{R}$-end-cluster $G_{m+1}$ of size $kb+j$ with weight $w_{\Gamma,\mathscr{R}}(G_{m+1})$. Note that in the cases where $b=0$, our definitions imply that $G_{m+1}$ is just a column of height $j$ filled with the numbers $1,\ldots,j$ which is increasing, reading from bottom to top. Finally the product $\prod_{j=1}^{m}GC^{0,0,k}_{ka_{j},\Gamma,\mathscr{R}}(x)$ allows us to pick generalized $\Gamma,\mathscr{R}$-clusters $G_{i}\in\mathcal{GC}^{0,0,k}_{ka_{i},\Gamma,\mathscr{R}}$ for $i=1,\ldots,m$ with weight $\prod_{i=1}^{m}w_{\Gamma,\mathscr{R}}(G_{i})$. Note that in the cases where $a_{i}=1$, our definitions imply that $G_{i}$ is just a column of height $k$ filled with the numbers $1,\ldots,k$ which is increasing, reading from bottom to top.

For example, suppose that $k=2$ and $j=1$ and $\Gamma=\{P\}$ where P=\begin{array}[]{|c|c|c|}\hline\cr 6&5&4\\ \hline\cr 1&2&3\\ \hline\cr\end{array}. Suppose that $\mathscr{R}$ is the relation such that for any two increasing columns of integers $C$ and $D$, $(C,D)\in\mathscr{R}$ if and only if the top element of $C$ is greater than the bottom element of $D$. Then in Figure 13, we have pictured $S_{1},S_{2},S_{3},S_{4},S_{5},S_{6}$ which partition $\{1,\ldots,39\}$ and corresponding generalized $\Gamma,\mathscr{R}$-clusters $G_{1},\ldots,G_{5}$ and a general $\Gamma,\mathscr{R}$-end-cluster $G_{6}$. For each $i$, we have indicated the separation between the blocks of $G_{i}$ by dark black lines. Then for each $i=1,\ldots,m+1$, we create a cluster $E_{i}$ which results by replacing each $j$ in $G_{i}$ by the $j^{th}$ smallest element of $S_{i}$. If we concatenate $E_{1}\ldots E_{6}$ together, then we will obtain an element of $Q\in\mathcal{M}_{ks+j,\Gamma}^{0,j,k}$. The weight of $Q$ equals $\prod_{j=1}^{6}w_{\Gamma,\mathscr{R}}(G_{i})$ where

[TABLE]

In Figure 13, we have indicated the boundaries between the is by light lines.

We let $\mathcal{HGEC}_{ks+j,\Gamma,\mathscr{R}}$ denote the set of all elements that can be constructed in this way. Thus $Q=E_{1}\ldots E_{m}E_{m+1}$ is an element of $\mathcal{HGEC}_{ks+j,\Gamma,\mathscr{R}}$ if and only if for each $i=1,\ldots,m$, $\mathrm{red}(E_{i})$ is a generalized $\Gamma,\mathscr{R}$-cluster and $\mathrm{red}(E_{m+1})$ is a generalized $\Gamma,\mathscr{R}$-end-cluster. Next we define a sign reversing involution $\theta:\mathcal{HGEC}_{ks+j,\Gamma,\mathscr{R}}\rightarrow\mathcal{HGEC}_{ks+j,\Gamma,\mathscr{R}}$. Given $Q=E_{1}\ldots E_{m}E_{m+1}\in\mathcal{HGEC}_{ks+j,\Gamma,\mathscr{R}}$, look for the first $i$ such that either

the block structure of $\mathrm{red}(E_{i})=B^{(i)}_{1}\ldots B^{(i)}_{k_{i}}$ consists of more than one block or 2. 2.

$E_{i}$ consists of a single block $B^{(i)}_{1}$ and $(\mathrm{last}(B^{(i)}_{1}),\mathrm{first}(E_{i+1}))$ is not in $\mathscr{R}$.

In case (1), if $i\leq m$, we let $\theta(E_{1}\ldots E_{m+1})$ be the result of replacing $E_{i}$ by two generalized $\Gamma,\mathscr{R}$-clusters, $E_{i}^{*}$ and $E_{i}^{**}$, where $E_{i}*$ consists just of $B^{(i)}_{1}$ and $E_{i}^{**}$ consists of $B^{(i)}_{2}\ldots B^{(i)}_{k_{i}}$. If $i=m+1$, we let $\theta(E_{1}\ldots E_{m+1})$ be the result of replacing $E_{m+1}$ by a generalized $\Gamma,\mathscr{R}$-cluster $E_{m+1}^{*}$ followed by a generalized $\Gamma,\mathscr{R}$-end-cluster $E_{m+1}^{**}$ where $E_{m+1}*$ consists just of $B^{(m+1)}_{1}$ and $E_{m+1}^{**}$ consists of $B^{(m+1)}_{2}\ldots B^{(m+1)}_{k_{m+1}}$. Note that in either case, it is easy to check that $w_{\Gamma,\mathscr{R}}(E_{i})=-w_{\Gamma,\mathscr{R}}(E_{i}^{*})w_{\Gamma,\mathscr{R}}(E_{i})^{**}$. In case (2), if $i<m$, we let $\theta(E_{1}\ldots E_{m+1})$ be the result of replacing $E_{i}$ and $E_{i+1}$ by the single generalized $\Gamma,\mathscr{R}$-cluster $E=B^{(i)}_{1}B^{(i+1)}_{1}\ldots B^{(i+1)}_{k_{i+1}}$. If $i=m$, we let $\theta(E_{1}\ldots E_{m+1})$ be the result of replacing $E_{m}$ and $E_{m+1}$ by the single generalized $\Gamma,\mathscr{R}$-end-cluster $E=B^{(m)}_{1}B^{(m+1)}_{1}\ldots B^{(m+1)}_{k_{m+1}}$. In either case, it is easy to check that $w_{\Gamma,\mathscr{R}}(E_{i})w_{\Gamma,\mathscr{R}}(E_{i+1})=-w_{\Gamma,\mathscr{R}}(E)$. If neither case (1) or case (2) applies, then we let $\theta(E_{1}\ldots E_{m+1})=E_{1}\ldots E_{m+1}$. For example, suppose that $\mathscr{R}$ is the binary relation where for any two increasing columns of integers $C$ and $D$, $(C,D)\in\mathscr{R}$ if and only if the top element of $C$ is greater than the bottom element of $D$ and $\Gamma=\{P\}$ where P=\begin{array}[]{|c|c|c|}\hline\cr 6&5&4\\ \hline\cr 1&2&3\\ \hline\cr\end{array}. Then if $Q=E_{1}\ldots E_{6}$ is the generalized $\Gamma,\mathscr{R}$-cluster pictured in Figure 13, then we are in case (1) with $i=1$ since $E_{1}$ consists of more than one block. Thus $\theta(Q)$ results by breaking that generalized $\Gamma,\mathscr{R}$-cluster into two clusters $E_{1}^{*}$ of size 1 and $E^{**}_{1}$ of size 4. $\theta(Q)$ is pictured in Figure 14.

It is easy to see that $\theta$ is an involution. That is, if $Q=E_{1}\ldots E_{m+1}$ is in case (1) using $E_{i}$, then $\theta(Q)$ will be in case (2) using $E_{i}^{*}$ and $E_{i}^{**}$. Similarly if $Q=E_{1}\ldots E_{m+1}$ is in case (1) using $E_{i}$ and $E_{i+1}$, then $\theta(Q)$ will be in case (2) using $E=E_{i}E_{i+1}$. It follows that if $\theta(E_{1}\ldots E_{m+1})\neq E_{1}\ldots E_{m+1}$, then $w_{\Gamma,\mathscr{R}}(E_{1}\ldots E_{m+1})=-w_{\Gamma,\mathscr{R}}(\theta(E_{1}\ldots E_{m+1}))$ so that the right-hand side of (15) equals

[TABLE]

Thus we must examine the fixed points of $\theta$.

If $Q=E_{1}\ldots E_{m+1}\in\mathcal{HGEC}_{ks+j,\Gamma,\mathscr{R}}$ and $\theta(Q)=Q$, then it must be the case that for each $i\leq m$, $E_{i}$ consists of single column of weight 1 or it reduces to generalized $\Gamma,\mathscr{R}$-cluster $\overline{E}_{i}$ consisting of a single block $B^{(i)}_{1}$ whose weight is the weight of $\mathrm{red}(B^{(i)}_{1})$ as a $\Gamma$-cluster. Moreover, it must be the case that for all $i=1,\ldots m-1$, $(\mathrm{last}(E_{i}),\mathrm{first}(E_{i+1})$ is in $\mathscr{R}$. Similarly, $E_{m+1}$ must consists of a single column of height $j$ and $(\mathrm{last}(E_{m}),E_{m+1})$ must be in $\mathscr{R}$. But this means for all $j=1,\ldots,s-1$, $(Q[j],Q[j+1])$ is in $\mathscr{R}$. That is, either $Q[j]$ equals $\mathrm{last}(E_{i})$ for some $i$ or column $j$ is contained in one of the $\Gamma$-clusters $E_{i}$ in which case $(Q[j],Q[j+1])$ is in $\mathscr{R}$ by our definition of generalized $\Gamma,\mathscr{R}$-clusters. Thus any fixed point $Q$ of $\theta$ is an element $\mathcal{MP}^{0,j,k}_{ks+j,\Gamma,\mathscr{R}}$. Then just like our proof Theorem 3, it follows that $E_{1},\ldots,E_{m}$ are just the maximal $\Gamma$-subclusters of an element in $\mathcal{P}^{0,j,k}_{kn,\mathscr{R}}$. Vice versa, if $T=F_{1}\ldots F_{r}F_{r+1}$ is an element of $\mathcal{P}^{0,0,k}_{ks+j,\mathscr{R}}$ where $F_{1},\ldots,F_{r}$ are the maximal $\Gamma$-subclusters of $T$ and $F_{r+1}$ is the last column of height $j$, then $T=F_{1}\ldots F_{r}$ is a fixed point of $\theta$. Thus we have proved that the right-hand side of (24) equals

[TABLE]

which is what we wanted to prove. ∎

3.2 Generalized Clusters for fillings

of $D^{i,0,k}_{i+kn}$.

In this subsection, we shall extend the generalized cluster method to deal with various types of fillings of $D^{i,0,k}_{i+kn}$. Fix a set of patterns $\Gamma\subseteq\mathcal{P}^{0,0,k}_{tk}$ where $t\geq 2$. We let $\mathcal{MP}^{i,0,k}_{i+kn,\Gamma}$ denote the set of elements that arise by starting with an element $F$ of $\mathcal{P}^{i,0,k}_{i+kn}$ and marking some of the $\Gamma$-matches in $F$ by placing on $x$ on the column which starts the $\Gamma$-match. Given an element $F\in\mathcal{MP}^{i,0,k}_{i+kn,\Gamma}$, we let $m_{\Gamma}(F)$ denote the number of marked $\Gamma$-matches in $F$.

To find an extension of Theorem 5 for these types of arrays, we need to define a special type of generalized cluster which we call a generalized start cluster. That is, suppose that we are given a binary relation $\mathscr{R}$ on columns of integers.

Definition 8**.**

We say that $Q\in\mathcal{MP}^{i,0,k}_{i+kn,\Gamma}$ is a generalized $\Gamma,\mathscr{R}$-start-cluster if we can write $Q=B_{1}B_{2}\cdots B_{m}$ where $B_{i}$ are blocks of consecutive columns in $Q$ such that

$B_{1}$* is a single column of height $i$,* 2. 2.

for $2\leq a\leq m$, either $B_{a}$ is a single column or $B_{a}$ consists of $r$-columns where $r\geq 2$, $\mathrm{red}(B_{a})$ is a $\Gamma$-cluster in $\mathcal{MP}_{kr,\Gamma}$, and any pair of consecutive columns in $B_{i}$ are in $\mathscr{R}$ and 3. 3.

for $1\leq i\leq m-1$, the pair $(\mathrm{last}(B_{i}),\mathrm{first}(B_{i+1})$ is not in $\mathscr{R}$.

Let $\mathcal{GSC}^{i,0,k}_{i+kn,\Gamma,\mathscr{R}}$ denote the set of all generalized $\Gamma,\mathscr{R}$-start-clusters which start with a column of height $i$ and which is followed by $n$ columns of height $k$. Given $Q=B_{1}B_{2}\ldots B_{m}\in\mathcal{GEC}^{i,0,k}_{i+kn,\Gamma,\mathscr{R}}$, we define the weight of $B_{i}$, $w_{\Gamma,\mathscr{R}}(B_{i})$, to be 1 if $B_{i}$ is a single column and $x^{m_{\Gamma}(\mathrm{red}(B_{i}))}$ if $B_{i}$ is order isomorphic to a $\Gamma$-cluster. Then we define the weight of $Q$, $w_{\Gamma,\mathscr{R}}(Q)$, to be $(-1)^{m-1}\prod_{i=1}^{m}w_{\Gamma,\mathscr{R}}(B_{i})$. We let

[TABLE]

Let $\mathcal{P}^{i,0,k}_{i+kn,\mathscr{R}}$ denote the set of all elements $F\in\mathcal{P}^{i,0,k}_{i+kn}$ such that the relation $\mathscr{R}$ holds for any pair of consecutive columns in $F$. We let $\mathcal{MP}^{i,0,k}_{i+kn,\Gamma,\mathscr{R}}$ denote the set of elements that arise by starting with an element $F$ of $\mathcal{P}^{i,0,k}_{i+kn,\mathscr{R}}$ and marking some of the $\Gamma$-matches in $F$ by placing on $x$ on the column which starts the $\Gamma$-match.

Then we have the following theorem.

Theorem 9**.**

Let $\mathscr{R}$ be a binary relation on pairs of columns $(C,D)$ which are filled with integers which are increasing from bottom to top. Let $\Gamma\subseteq\mathcal{P}^{0,0,k}_{tk}$ where $t\geq 2$. Then

[TABLE]

The proof of Theorem 9 is very similar to the proof of Theorem 7 so that we shall not give the details.

3.3 Generalized Clusters for fillings of $D^{i,j,k}_{i+kn+j}$.

In this subsection, we shall combine the results of the previous two subsections to extend the generalized cluster method to deal with various types of fillings of $D^{i,j,k}_{i+kn+j}$. Fix a set of patterns $\Gamma\subseteq\mathcal{P}^{0,0,k}_{tk}$ where $t\geq 2$. We let $\mathcal{MP}^{i,j,k}_{i+kn+j,\Gamma}$ denote the set of elements that arise by starting with an element $F$ of $\mathcal{P}^{i,j,k}_{i+kn+j}$ and marking some of the $\Gamma$-matches in $F$ by placing on $x$ on the column which starts the $\Gamma$-match. Given an element $F\in\mathcal{MP}^{i,j,k}_{i+kn+j,\Gamma}$, we let $m_{\Gamma}(F)$ denote the number of marked $\Gamma$-matches in $F$. Clearly, neither the first or the last column of $F$ is contained in a $\Gamma$-match.

To find an extension of Theorem 5 for fillings of $D_{kn+j}^{0,j,k}$, we defined generalized end clusters and for fillings of $D_{kn+i}^{i,0,k}$, we defined generalized start clusters. However, they are not enough to extend Theorem 5 for fillings of $D_{i+kn+j}^{i,j,k}$. Next we define a special type of generalized cluster which we call a generalized start-end cluster. That is, suppose that we are given a binary relation $\mathscr{R}$ on columns of integers.

Definition 10**.**

We say that $Q\in\mathcal{MP}^{i,j,k}_{i+kn+j,\Gamma}$ is a generalized $\Gamma,\mathscr{R}$-start-end-cluster if we can write $Q=B_{1}B_{2}\cdots B_{m}$ where $B_{i}$ are blocks of consecutive columns in $Q$ such that

$m\geq 2$, 2. 2.

$B_{1}$* is a column of height $i$,* 3. 3.

$B_{m}$* is a column of height $j$,* 4. 4.

for $1<i<m$, either $B_{i}$ is a single column or $B_{i}$ consists of $r$-columns where $r\geq 2$, $\mathrm{red}(B_{i})$ is a $\Gamma$-cluster in $\mathcal{MP}_{kr,\Gamma}$, and any pair of consecutive columns in $B_{i}$ are in $\mathscr{R}$ and 5. 5.

for $1\leq i\leq m-1$, the pair $(\mathrm{last}(B_{i}),\mathrm{first}(B_{i+1}))$ is not in $\mathscr{R}$.

Let $\mathcal{GSEC}^{i,j,k}_{i+kn+j,\Gamma,\mathscr{R}}$ denote the set of all generalized $\Gamma,\mathscr{R}$-start-end-clusters which have $n$ columns of height $k$ between a column of height $i$ and a column of height $j$. Given $Q=B_{1}B_{2}\ldots B_{m}\in\mathcal{GSEC}^{i,j,k}_{i+kn+j,\Gamma,\mathscr{R}}$, we define the weight of $B_{i}$, $\omega_{\Gamma,\mathscr{R}}(B_{i})$, to be 1 if $B_{i}$ is a single column and $x^{m_{\Gamma}(\mathrm{red}(B_{i}))}$ if $B_{i}$ is order isomorphic to a $\Gamma$-cluster. Then we define the weight of $Q$, $\omega_{\Gamma,\mathscr{R}}(Q)$, to be $(-1)^{m-1}\prod_{i=1}^{m}\omega_{\Gamma,\mathscr{R}}(B_{i})$. We let

[TABLE]

Let $\mathcal{P}^{i,j,k}_{i+kn+j,\mathscr{R}}$ denote the set of all elements $F\in\mathcal{P}^{i,j,k}_{i+kn+j}$ such that the relation $\mathscr{R}$ holds for any pair of consecutive columns in $F$. We let $\mathcal{MP}^{i,j,k}_{i+kn+j,\Gamma,\mathscr{R}}$ denote the set of elements that arise by starting with an element $F$ of $\mathcal{P}^{i,j,k}_{i+kn+j,\mathscr{R}}$ and marking some of the $\Gamma$-matches in $F$ by placing an $x$ on the column which starts the $\Gamma$-match.

Then we have the following theorem.

Theorem 11**.**

Let $\mathscr{R}$ be a binary relation on pairs of columns $(C,D)$ which are filled with integers which are increasing from bottom to top. Let $\Gamma\subseteq\mathcal{P}^{0,0,k}_{tk}$ where $t\geq 2$. Then

[TABLE]

Proof.

Replace $x$ by $x+1$ in (28). Then it easy to see that

[TABLE]

Thus we must show that

[TABLE]

Now we rewrite the right-hand side of (30) as

[TABLE]

Taking the coefficient of $\frac{t^{ks+j}}{(ks+j)!}$ on both sides of (31) where $s\geq 0$, we see that we must show that

[TABLE]

Now we interpret the right-hand side of (32). Our situation is a bit different from our previous proofs in that (32) allows two ways to construct an array in $\mathcal{MP}_{i+ks+j,\Gamma,\mathscr{R}}^{i,j,k}$.

One way is via $\mathcal{GSC}$, $\mathcal{GEC}$ and $\mathcal{GC}$. First we pick non-negative integers $a$, $b$ and $c$ such that $a+b+c=s$. Then we pick an $m$ such that $1\leq m\leq b$. Next we pick $b_{1},\ldots,b_{m}\geq 1$ such that $b_{1}+b_{2}+\cdots+b_{m}=b$. Next the multinomial coefficient $\binom{i+ks+j}{i+ka,kb_{1},\ldots,kb_{m},kc+j}$ allows us to pick a sequence of sets $S_{1},\ldots,S_{m},S_{m+1},S_{m+2}$ which partition $\{1,2,\ldots,i+ks+j\}$ such that $|S_{1}|=i+ka$, $|S_{h+1}|=kb_{h}$ for $h=1,\ldots,m$ and $|S_{m+2}|=kc+j$. The factor $GSC^{i,0,k}_{i+ka,\Gamma,\mathscr{R}}(x)$ allows us to pick a generalized $\Gamma,\mathscr{R}$-start-cluster $G_{1}$ of size $i+ka$ with weight $\omega_{\Gamma,\mathscr{R}}(G_{1})$. Note that in the cases where $a=0$, our definitions imply that $G_{1}$ is just a column of height $i$ filled with the numbers $1,\ldots,i$ which is increasing, reading from bottom to top. The factor $GEC^{0,j,k}_{kc+j,\Gamma,\mathscr{R}}(x)$ allows us to pick a generalized $\Gamma,\mathscr{R}$-end-cluster $G_{m+2}$ of size $kc+j$ with weight $\omega_{\Gamma,\mathscr{R}}(G_{m+2})$. Note that in the cases where $c=0$, our definitions imply that $G_{m+2}$ is just a column of height $j$ filled with the numbers $1,\ldots,j$ which is increasing, reading from bottom to top. Finally, the product $\prod_{j=1}^{m}GC^{0,0,k}_{kb_{h},\Gamma,\mathscr{R}}(x)$ allows us to pick generalized $\Gamma,\mathscr{R}$-clusters $G_{h}\in\mathcal{GC}^{0,0,k}_{kb_{h},\Gamma,\mathscr{R}}$ for $h=2,\ldots,m+1$ with weight $\prod_{h=1}^{m}\omega_{\Gamma,\mathscr{R}}(G_{h})$. Note that in the cases where $b_{h}=1$, our definitions imply that $G_{h}$ is just a column of height $k$ filled with the numbers $1,\ldots,k$ which is increasing, reading from bottom to top.

Another way is via $\mathcal{GSEC}$. The term $GSEC^{i,j,k}_{i+ks+j,\Gamma,\mathscr{R}}(x)$ allows us to have a generalized $\Gamma,\mathscr{R}$-start-end-cluster $G_{*}$ of size $i+ks+j$. Note that in the case where $s=0$, our definitions implies that $G_{*}$ has two blocks and the first block is a single column of height $i$ and the second block is a single column of height $j$.

For example, suppose that $i=3$, $k=2$ and $j=1$ and $\Gamma=\{P\}$ where P=\begin{array}[]{|c|c|c|}\hline\cr 6&5&4\\ \hline\cr 1&2&3\\ \hline\cr\end{array}. Suppose that $\mathscr{R}$ is relation where for any two increasing columns $C$ and $D$ of integers, $(C,D)\in\mathscr{R}$ if and only if the top element of $C$ is greater than the bottom element of $D$. Then in Figure 15, we have pictured $S_{1},S_{2},S_{3},S_{4},S_{5},S_{6}$ which partition $\{1,\ldots,42\}$ and corresponding a generalized $\Gamma,\mathscr{R}$- start-cluster $G_{1}$, generalized $\Gamma,\mathscr{R}$-clusters $G_{2},\ldots,G_{5}$ and a generalized $\Gamma,\mathscr{R}$-end-cluster $G_{6}$. This is a construction for the right-hand side of (32) via $\mathcal{GSC}$, $\mathcal{GEC}$ and $\mathcal{GC}$. For each $h$, we have indicated the separation between the blocks of $G_{i}$ by dark black lines. Then for each $h=1,\ldots,m+1$, we let $E_{i}$ be the result of replacing each $j$ in $G_{i}$ by the $j^{th}$ smallest element of $S_{i}$. If we concatenate $E_{1}\ldots E_{6}$ together, then we will obtain an element of $Q\in\mathcal{MP}_{i+ks+j,\Gamma}^{i,j,k}$. The weight of $Q$ equals $\prod_{h=1}^{6}\omega_{\Gamma,\mathscr{R}}(G_{h})$ where

[TABLE]

In Figure 15, we have indicated the boundaries between the $E_{h}$’s by light lines.

In Figure 16, we have pictured $S_{*}=\{1,2,\ldots,18\}$ and corresponding a generalized $\Gamma,\mathscr{R}$-start-end-cluster $G_{*}$. This is a construction for the right-hand side of (32) via $\mathcal{GSEC}$. Clearly, $G_{*}\in\mathcal{MP}_{i+ks+j,\Gamma}^{i,j,k}$. The weight of $G_{*}$ is

[TABLE]

We let $\mathcal{HGC}_{i+ks+j,\Gamma,\mathscr{R}}$ denote the set of all elements that can be constructed via $\mathcal{GSC}$, $\mathcal{GEC}$ and $\mathcal{GC}$, and let $\mathcal{HGSEC}_{i+ks+j,\Gamma,\mathscr{R}}$ denote the set of all elements that can be constructed via $\mathcal{GSEC}$. We let

[TABLE]

Then $Q=E_{1}\ldots E_{m+1}E_{m+2}$ is an element of $\mathcal{HGC}_{i+ks+j,\Gamma,\mathscr{R}}$ if and only if $m\geq 0$, $\mathrm{red}(E_{1})$ is a generalized $\Gamma,\mathscr{R}$-start-cluster, for each $h=1,\ldots,m$, $\mathrm{red}(E_{h+1})$ is a generalized $\Gamma,\mathscr{R}$-cluster and $\mathrm{red}(E_{m+2})$ is a generalized $\Gamma,\mathscr{R}$-end-cluster. On the other hand, $Q=E_{1}$ is an element of $\mathcal{HGSEC}_{i+ks+j},\Gamma,\mathscr{R}$ if and only if $\mathrm{red}(E_{1})$ is a generalized $\Gamma,\mathscr{R}$-start-end-cluster.

Next we define a sign reversing involution $\theta:\mathcal{H}_{i+ks+j}\rightarrow\mathcal{HGEC}_{i+ks+j}$. We would like to use the same involutions as in our previous proofs. However, there is one potential problem. That is, suppose that $Q=E_{1}E_{2}$ where $E_{1}$ consists of a single column of height $i$, $E_{2}=B_{1}\ldots B_{m+1}$ is a generalized $\Gamma,\mathscr{R}$-end-cluster, and $(E_{1},\mathrm{first}(B_{1})$ is not in $\mathscr{R}$. Then if we followed our previous involutions, we would like to replace $E_{1}E_{2}$ by a single cluster $E$. If we did not have generalized $\Gamma,\mathscr{R}$-start-end-clusters, then we could not define $\theta(E_{1}E_{2})=E$. This is the reason, we added a term corresponding to generalized $\Gamma,\mathscr{R}$-start-end-clusters. Formally, $\theta$ is defined as follows. Let $Q\in\mathcal{HGC}_{i+ks+j,\Gamma,\mathscr{R}}$.

 

Case 1. $Q=B_{1}B_{2}\ldots B_{m}B_{m+1}$ is a generalized $\Gamma,\mathscr{R}$-start-end-cluster, then we let $\theta(Q)=E_{1}E_{2}$ where $E_{1}=B_{1}$ is a column of height $i$ and $E_{2}=B_{2}\ldots B_{m}B_{m+1}$ is a generalized $\Gamma,\mathscr{R}$-end-cluster. Note by our definitions, $(E_{1},\mathrm{first}(B_{2}))$ is not in $\mathscr{R}$.

 

Case 2. $Q=E_{1}E_{2}$ where $E_{1}=B_{1}$ is a column of height $i$, $E_{2}=B_{2}\ldots B_{m}B_{m+1}$ is a generalized $\Gamma,\mathscr{R}$-end-cluster, and $(E_{1},\mathrm{first}(B_{2}))$ is not in $\mathscr{R}$, then we let $\theta(E_{1}E_{2})=E$ where $E=B_{1}B_{2}\ldots B_{m}B_{m+1}$ is a generalized $\Gamma,\mathscr{R}$-start-end-cluster.

 

Otherwise assume we are not in Case 1 or Case 2 and that $Q=E_{1}\ldots E_{m+1}E_{m+2}$ where $E_{j}=B^{(j)}_{1}\ldots B^{(j)}_{k_{j}}$ for $j=1,\ldots,m+2$. Then we look for the first $h$ such that either

 

Case 3 the block structure of $\mathrm{red}(E_{h})=B^{(h)}_{1}B^{(h)}_{2}\ldots B^{(h)}_{k_{h}}$ consists of more than one block or

 

Case 4 $E_{h}$ consists of a single block $B^{(h)}_{1}$ and $(\mathrm{last}(B^{(h)}_{1}),\mathrm{first}(B^{(h+1)}_{1}))$ is not in $\mathscr{R}$.

If $Q$ is in Case 3, then we let $\theta(Q)$ be the result of replacing $E_{h}$ by $E_{h}^{*}=B_{1}^{(h)}$ and $E^{**}=B_{2}^{(h)}\ldots B^{(h)}_{k_{h}}$ in $Q$. If $Q$ is in Case 4, we let $\theta(Q)$ be the result of replacing $E^{(h)}E^{(h+1)}$ by $E=B_{1}^{(h)}B_{1}^{(h+1)}\ldots B^{(h+1)}_{k_{h+1}}$ in $Q$.

If neither Case (1), Case (2), Case (3), or Case (4) applies, then we let $\theta(Q)=Q$. It is straightforward to check that $\theta$ is an involution and that if $\theta(Q)\neq Q$, then $w_{\Gamma,\mathscr{R}}(Q)=-w_{\Gamma,\mathscr{R}}(\theta(Q))$. The fixed points of $\theta$ are those $Q\in\mathcal{HGC}_{i+ks+j,\Gamma,\mathscr{R}}$ of the form $E_{1}\ldots E_{m+1}$ where each $E_{i}$ consists of a single block and $E_{1}$ consists a single column of height $i$, $E_{m+1}$ consists of a single column of height $j$, and for each $2\leq r\leq m$, $E_{r}$ is either a single column of height $k$ or $E_{r}$ reduces to a $\Gamma$ cluster such any two consecutive columns in $E_{r}$ satisfy $\mathscr{R}$. In addition, it must be the case that $(\mathrm{last}(E_{i}),\mathrm{first}(E_{i+1})\in\mathscr{R}$ for $i=1,\ldots,m$. As in our previous proofs, it follows that any two consecutive columns in $Q$ must be in $\mathscr{R}$ so that $Q\in\mathcal{MP}_{i+ks+j,\Gamma,\mathscr{R}}^{i,j,k}$. In addition, $E_{1},\ldots,E_{m+1}$ are just the maximal $\Gamma$-subclusters in $R$. Hence the theorem follows as in the previous proofs. ∎

3.4 Examples

In this subsection, we will illustrate our generalized cluster method with two examples. First we shall compute the generating function for the number of matches of 162534 in down-up permutations and then compute the number of 124356 matches in 3-generalized Euler permutations.

3.4.1 Up-down pattern 162534 in down-up permutations

We say a permutation $\sigma=\sigma_{1}\sigma_{2}\cdots\sigma_{n}\in\mathcal{S}_{n}$ is an up-down permutation if $\sigma_{1}<\sigma_{2}>\sigma_{3}<\sigma_{4}>\sigma_{5}<\cdots$. More precisely, $\sigma\in\mathcal{S}_{n}$ is up-down if and only if

[TABLE]

We let $\mathcal{UD}_{n}$ denote the set of all the up-down permutations in $\mathcal{S}_{n}$. We say a permutation $\sigma=\sigma_{1}\sigma_{2}\cdots\sigma_{n}\in\mathcal{S}_{n}$ is an down-up permutation if $\sigma_{1}>\sigma_{2}<\sigma_{3}>\sigma_{4}<\sigma_{5}>\cdots$. Similarly, a permutation $\sigma\in\mathcal{S}_{n}$ is down-up if and only if

[TABLE]

We let $\mathcal{DU}_{n}$ denote the set of all the up-down permutations in $\mathcal{S}_{n}$.

We note that down-up permutations can be represented by arrays. Consider a binary relation $\mathscr{R}$ such that holds for a pair of columns $(C,D)$ of height $\leq 2$ if and only if the top element of column $C$ is greater than the bottom element of column $D$. Then $\{w(F):F\in\mathcal{P}_{2n+2,\mathscr{R}}^{1,1,2}\}=\mathcal{DU}_{2n+2}$ and $\{w(F):F\in\mathcal{P}_{2n+2,\mathscr{R}}^{1,0,2}\}=\mathcal{DU}_{2n+1}$.

Let $\tau=1~{}6~{}2~{}5~{}3~{}4\in\mathcal{UD}_{6}$. Then the generating function

[TABLE]

where

[TABLE]

and P=\begin{array}[]{|c|c|c|}\hline\cr 6&5&4\\ \hline\cr 1&2&3\\ \hline\cr\end{array}\in\mathcal{P}_{0,0,2j}^{0,0,2}. Note that (33) and (34) are equivalent to generating functions for the distribution of $\tau$-matches in down-up permutations of even length and odd length, respectively.

By Theorem 9,

[TABLE]

and, by Theorem 11,

[TABLE]

Thus we need to compute $GC^{0,0,2}_{2n,P,\mathscr{R}}(x)$, $GSC^{0,0,2}_{2n,P,\mathscr{R}}(x)$, $GEC^{0,0,2}_{2n,P,\mathscr{R}}(x)$, and $GSEC^{0,0,2}_{2n,P,\mathscr{R}}(x)$.

We shall start by discussing structures of $P$-clusters. Note that $P$ is one of the patterns that we discussed in Section 2. In this case, the Hasse diagram of a $P$-clusters is of the form pictured in Figure 17.

If we let $\mathcal{C}_{2n,P}$ denote the set of $P$-clusters consisting of $n$ columns, then we proved that

[TABLE]

In section 2, we also gave a recursion to compute $GC_{2n,P,\mathscr{R}}^{0,0,2}(x)$. However, in this section, we shall give an alternative way to compute $GC_{2n,P,\mathscr{R}}^{0,0,2}(x)$. Suppose $Q\in\mathcal{GC}_{2n,P,\mathscr{R}}^{0,0,2}$ has $m$ blocks, i.e., $Q=B_{1}B_{2}\ldots B_{m}$. For an array $F$, we let $\text{Col}(F)$ denote the number of columns in $\text{Col}(F)$. Each $B_{i}$ is either order isomorphic to a $P$-cluster or a single column. We let $\mathcal{GC}_{\text{Col}=(b_{1},b_{2},\ldots,b_{m})}$ to denote the set of generalized clusters such that $\text{Col}(B_{i})=b_{i}$. Given $\text{Col}(B_{i})$ for each $1\leq i\leq m$, the set of words of generalized clusters $\mathcal{GC}_{\text{Col}=(b_{1},b_{2},\ldots,b_{m})}$ is equal to the set of linear extensions of the poset whose Hasse diagram corresponds to $(b_{1},b_{2},\ldots,b_{m})$. That is, we are assuming that for any block $(\mathrm{last}(B_{i}),\mathrm{first}(B_{i+1})$ is not in $\mathscr{R}$ so that there must be an arrow directed from the top of the last column of $B_{i}$ to the bottom of the first column of $B_{i+1}$. We let $\Gamma(b_{1},b_{2},\ldots,b_{m})$ denote the Hasse diagram corresponding to $\mathcal{GC}_{\text{Col}=(b_{1},b_{2},\ldots,b_{m})}$. For example, $\Gamma(3,1,1,5,1)$ is pictured in Figure 18.

It is clear that

[TABLE]

where $\text{LE}(\Gamma(b_{1},b_{2},\ldots,b_{m}))$ is the number of linear extensions of $\Gamma(b_{1},b_{2},\ldots,b_{m})$ and by convention, we let $C_{2}(x)=1$. Therefore, to compute (42), we only need to count linear extensions of $\Gamma(b_{1},b_{2},\ldots,b_{m})$. Continue using $\Gamma(3,1,1,5,1)$ as example, the Hasse diagram in Figure 18 is actually a tree-like diagram, as drawn in Figure 19. Then we can easily compute that

[TABLE]

That is, it is easy to see that the first four elements in the bottom row must be labeled with $1,\ldots,4$, reading from left to right, since there is a directed path from these elements to any other elements in the poset. Once we remove these four elements, the Hasse diagram becomes disconnected so that $\binom{18}{2}$ ways to pick the labels $a<b$ for the two elements above the element labeled with 4 and only one way to assign $a$ and $b$ to those two elements. Once we have picked those two elements $a$ and $b$, the next 10 elements must be the smallest elements of $\{1,\ldots,22\}-\{1,2,3,4,,a,b\}$. Once we remove those 10 elements, the Hasse diagram again becomes disconnected so that there are $\binom{6}{4}$ ways to pick the labels of the vertical segment and only one way to order them.

In general, it is not difficult to see that

[TABLE]

and then hence

[TABLE]

Since a generalized $P$-cluster can not have a block of 2 columns,

[TABLE]

Using a computer program, we computed that

[TABLE]

Next we compute $GSC_{2n+1,P,\mathscr{R}}^{1,0,2}(x)$. Suppose $Q\in\mathcal{GSC}_{2n+1,P,\mathscr{R}}^{1,0,2}$ has $m$ blocks, i.e., $Q=B_{1}B_{2}\ldots B_{m}$. $B_{1}$ has to be a block consisting of one element, and for $2\leq i\leq m$, each $B_{i}$ is either order isomorphic to a $P$-cluster or a single column. We let $\mathcal{GSC}_{\text{Col}=(1,b_{2},\ldots,b_{m})}$ to denote the set of generalized start-clusters such that $\text{Col}(B_{i})=b_{i}$. Given $\text{Col}(B_{i})$ for each $2\leq i\leq m$, the set of words of generalized start-clusters $\mathcal{GSC}_{\text{Col}=(b_{1},b_{2},\ldots,b_{m})}$ is equal to the set of linear extensions of the poset whose Hasse diagram corresponds is equal to $\Gamma_{S}(1,b_{2},\ldots,b_{m})$ where $\Gamma_{S}(1,b_{2},\ldots,b_{m})$ is the Hasse diagram obtained from $\Gamma(b_{2},\ldots,b_{m})$ by adding an arrow from the element in $B_{1}$ to the left-most element in the bottom row of $\Gamma(b_{2},\ldots,b_{m})$. For example, $\Gamma_{S}(1,3,1,1,5,1)$ is pictured in Figure 20.

It is clear that

[TABLE]

By comparing Figures 18 and 20, it is easy to see that

[TABLE]

Therefore, for $n\geq 1$,

[TABLE]

and $GSC_{1,P,\mathscr{R}}^{1,0,2}(x)=1$. Then by (3.4.1), we have

[TABLE]

Using these initial values of $GC_{2n,P,\mathscr{R}}^{0,0,2}(x-1)$ and $GSC_{2n+1,P,\mathscr{R}}^{1,0,2}(x-1)$ in (33), we computed the following initial terms of $A_{P}(x,t)$:

[TABLE]

Next we compute $GEC_{2n+1,P,\mathscr{R}}^{0,1,2}(x)$. Suppose $Q\in\mathcal{GEC}_{2n+1,P,\mathscr{R}}^{0,1,2}$ has $m$ blocks, i.e., $Q=B_{1}B_{2}\ldots B_{m}$. $B_{m}$ has to be a block consisting of one element, and for $1\leq i\leq m-1$, each $B_{i}$ is either order isomorphic to a $P$-cluster or a single column. We let $\mathcal{GEC}_{\text{Col}=(b_{1},\ldots,b_{m-1},1)}$ to denote the set of generalized end-clusters such that $\text{Col}(B_{i})=b_{i}$. Given $\text{Col}(B_{i})$ for each $1\leq i\leq m-1$, we can represent the set of words of such generalized end-clusters uniquely by as the set of linear extensions of the poset whose Hasse diagram is $\Gamma_{E}(b_{1},\ldots,b_{m-1},1)$ where $\Gamma_{E}(b_{1},\ldots,b_{m-1},1)$ is the Hasse diagram obtained from $\Gamma(b_{1},\ldots,b_{m-1})$ by adding an arrow from the right-most element in the top row of $\Gamma(b_{1},\ldots,b_{m-1})$ to the element in $B_{m}$. For example, $\Gamma_{E}(3,1,1,5,1,1)$ is pictured in Figure 21.

In this case, one can show that

[TABLE]

Hence

[TABLE]

Using a computer program, we computed that

[TABLE]

Finally we shall compute $GSEC_{2n+2,P,\mathscr{R}}^{1,1,2}(x)$. Suppose $Q\in\mathcal{GSEC}_{2n+2,P,\mathscr{R}}^{1,1,2}$ has $m$ blocks, i.e., $Q=B_{1}B_{2}\ldots B_{m}$. Because of the definition of generalized start-end clusters, $m\geq 2$. $B_{1}$ as well as $B_{m}$ has to be a block consisting of one element, and for $2\leq i\leq m-1$, each $B_{i}$ is either order isomorphic to a $P$-cluster or a single column. We let $\mathcal{GEC}_{\text{Col}=(1,b_{2},\ldots,b_{m-1},1)}$ to denote the set of generalized start-end-clusters such that $\text{Col}(B_{i})=b_{i}$. Given $\text{Col}(B_{i})$ for each $2\leq i\leq m-1$, we can represent the set of words of such generalized start-end-clusters as the linear extensions of the poset whose Hasse diagram $\Gamma_{SE}(1,b_{2},\ldots,b_{m-1},1)$ is obtained from the Hasse diagram $\Gamma(b_{2},\ldots,b_{m-1})$ by adding an arrow connecting the point corresponding to $B_{1}$ to the left-most element in the bottom row of $\Gamma(b_{2},\ldots,b_{m-1})$ and add an arrow from the right-most element in the top row of $\Gamma(b_{2},\ldots,b_{m-1})$ to a point corresponding to $B_{m}$. For example, $\Gamma_{SE}(1,3,1,1,5,1,1)$ is pictured in Figure 22.

Comparing Figure 21 and 22, it is easy to see that

[TABLE]

and then hence

[TABLE]

Based on (3.4.1),

[TABLE]

Then substituting the initial values of $GC_{2n,P,\mathscr{R}}^{0,0,2}(x-1)$, $GSC_{2n+1,P,\mathscr{R}}^{1,0,2}(x-1)$, $GEC_{2n+1,P,\mathscr{R}}^{0,1,2}(x-1)$ and $GSEC_{2n+2,P,\mathscr{R}}^{1,1,2}(x-1)$ into (33) by expressions obtained in (3.4.1), (3.4.1), (3.4.1) and (3.4.1), we have computed that

[TABLE]

Adding $A_{P}(x,t)$ and $B_{P}(x,t)$, we obtain the following initial terms of the generating function for the distribution of the number of $123654$-matches in down-up permutations. The initial terms of this generating functions are

[TABLE]

3.4.2 Pattern 124356 in 3-generalized Euler permutations

In this section, we shall study the generating function of $\tau$-matches where $\tau=124356$ in generalized Euler permutations $\mathcal{E}_{3n+j}^{0,j,3}$ for $j=0,1,2$. We have pictured $\tau$ as a $3\times 2$ array $\Gamma_{\tau}$ at the top-left of Figure 23 and its corresponding Hasse diagram at the top-right of Figure 23. We picked $\tau$ because it has two key properties. The first key property is that there is unique $\Gamma_{\tau}$-cluster for in $\mathcal{CM}_{3n,\Gamma_{\tau}}^{0,0,3}$ for every $n\geq 1$. That is, a $\Gamma_{\tau}$ cluster with $n$ columns must be a linear extension of the Hasse diagram which is obtained from the Hasse diagram $D$ of $\Gamma_{\tau}$ by adding the arrows of the Hasse diagram for $\Gamma_{\tau}$ in any pair of consecutive columns. This process is pictured at the bottom left of Figure 23 in the case where $n=6$. We claim that vertical arrows in each column other than the first and last column are redundant. That is, for each vertical arrow in columns $2,\ldots,n=1$, there is another directed path to its end points that does not use vertical arrows. If we eliminate such vertical arrows as pictured at the bottom-right of Figure 23, it then easy to see that remaining arrows linear order the points with the smallest element at the left-most cell in the bottom row of the diagram and the largest element in the top row of the diagram. Moreover, in the cluster, we must mark column except the last one with an $x$. It follows that $C_{3n,\Gamma_{\tau}}^{0,0,3}(x)=x^{n-1}$ for all $n\geq 2$.

This fact is actually a special case of a more general theorem proved by Harmse and Remmel [15]. That is, Harmse and Remmel completely classified those elements $P\in\mathcal{P}_{2k}^{0,0,k}$ such that there is a unique $P$-cluster with $n$ columns for every $n$ greater than or equal to 2 as those $P$ such that in every pair of consecutive rows $j$ and $j+1$, either left column contains a pair of consecutive elements or the right column contains a pair of consecutive elements. For example, in $\Gamma_{\tau}$, 1 and 2 are consecutive elements that appear in the left column of rows 1 and 2 and 5 and 6 are consecutive elements that appear in the left column of rows 2 and 3.

Let $\mathscr{R}$ be the relation that holds between any pair of increasing columns of integers $(C,D)$ if and only if the top element of $C$ is greater than the bottom element of $D$. Then for all $n\geq 0$,

[TABLE]

It follows from Theorems 5 and 7 that

[TABLE]

The second key property of $\Gamma_{\tau}$ is that in generalized clusters and generalized end-clusters $(B_{1},\ldots,B_{m})$, the order of the elements is completely determined. That is, the arrows between consecutive blocks $B_{i},B_{i+1}$ are directed from the largest element in block $B_{i}$ to the smallest element in block $B_{i+1}$. This is illustrated in Figure 24 where we have pictured a generalized $\Gamma_{\tau}$-cluster at the top and a generalized $\Gamma_{\tau}$-end-cluster at the bottom.

Next we shall compute $GC^{0,0,3}_{3n,\Gamma_{\tau},\mathscr{R}}(x)$, $GEC^{0,0,3}_{3n+1,\Gamma_{\tau},\mathscr{R}}(x)$, and $GEC^{0,0,3}_{3n+2,\Gamma_{\tau},\mathscr{R}}(x)$. We will start with computing $GC^{0,0,3}_{3n,\Gamma_{\tau},\mathscr{R}}(x)$. It is easy check that $GC^{0,0,3}_{3,\Gamma_{\tau},\mathscr{R}}(x)=1$ and $GC^{0,0,3}_{6,\Gamma_{\tau},\mathscr{R}}(x)=x-1$. We claim that for $n\geq 2$,

[TABLE]

That is, if $C=(B_{1},\ldots,B_{m})$ is a generalized $\Gamma_{\tau},\mathscr{R}$-cluster, then either $B_{1}$ consists of a single column in which case

[TABLE]

or $B_{1}$ has $k\geq 2$ columns and weight $x^{k-1}$ in which case we can replace $B_{1}$ by the unique cluster $B_{1}^{\prime}$ with $k-1$ columns and

[TABLE]

We let

[TABLE]

Based on the recursion above, we have

[TABLE]

with boundary conditions

[TABLE]

We solved this differential equation using Mathematica and obtained that

[TABLE]

Next we compute $GEC^{0,0,3}_{3n+1,\Gamma_{\tau},\mathscr{R}}(x)$. One can check that $GEC^{0,0,3}_{1,\Gamma_{\tau},\mathscr{R}}(x)=1$ and $GEC^{0,0,3}_{4,\Gamma_{\tau},\mathscr{R}}(x)=-1$. Then we can argue as we did in the computation of $GEC^{0,0,3}_{3n+1,\Gamma_{\tau},\mathscr{R}}(x)$ that for $n\geq 2$,

[TABLE]

We let

[TABLE]

Based on the recursion above, we have

[TABLE]

with boundary conditions

[TABLE]

Again, we used Mathematica to compute that

[TABLE]

where $z=\sqrt[3]{x-1}$.

The computation of $GEC^{0,0,3}_{3n+2,\Gamma_{\tau},\mathscr{R}}(x)$ is similar to the computation of $GEC^{0,0,3}_{3n+1,\Gamma_{\tau},\mathscr{R}}(x)$. That is, we have

[TABLE]

We let

[TABLE]

Based on the recursion above, we have the following differential equation

[TABLE]

with boundary conditions

[TABLE]

Then we used Mathematica to compute that

[TABLE]

where $z=\sqrt[3]{x-1}$.

Finally we have

[TABLE]

A few initial terms of expansion of 1+\sum{j=0}^{2}\sum_{n\geq 1}\frac{t^{3n+j}}{(3n+j)!}\sum_{\sigma\in\mathcal{E}^{0,j,3}_{3n+j}}x^{\text{\tau$-$\mathrm{mch}}(\sigma)} are

[TABLE]

The key properties that made this example work is that $\Gamma_{\tau}$ had unique $\Gamma_{\tau}$-clusters, i.e. Harmse and Remmel’s characterization applies, and that the bottom-left element of $\Gamma_{\tau}$ is equal to 1 and the top-right element of $\Gamma_{\tau}$ is the largest element. There are many such examples $k$-Euler permutations where $k>3$. For example, when $k=4$, $\alpha=12354678$ is such a permutation. Thus we can carry out a similar analysis for the distribution of $\alpha$-matches in 4-generalized Euler permutations.

4 Joint Clusters and Generalized Joint Clusters

In this section, we extend the notions $\Gamma$-clusters and generalized $\Gamma$-clusters for a single set of patterns $\Gamma$ to joint clusters and generalized joint clusters for a sequence of sets of patterns $\Gamma_{1},\ldots,\Gamma_{s}$ such that $\Gamma_{i}\subseteq\mathcal{P}^{0,0,k}_{r_{i}k}$ where $r_{i}\geq 2$ for $i=1,\ldots,s$. Our goal here is to compute generating functions of the form

[TABLE]

Similarly, if we are given some binary relation $\mathscr{R}$ on a pairs of columns, then we want to the compute multivariate generating function

[TABLE]

For each the theorems in the previous sections, we will state multivariate analogues of that theorem. In each case, the proofs of the multivariate analogues are essentially the same of the corresponding proofs in the previous sections so that we shall not supply the details.

4.1 Joint clusters

Let $\Gamma_{i}\subseteq\mathcal{P}^{0,0,k}_{r_{i}k}$ where $r_{i}\geq 2$ for $i=1,\ldots,s$. For any $i,j\geq 0$, $k\geq 2$, and $n\geq 1$, we let $\mathcal{MP}^{i,j,k}_{kn,\Gamma_{1},\ldots,\Gamma_{s}}$ denote the set of all fillings $F\in\mathcal{P}^{i,j,k}_{i+nk+j}$ where we have marked some of the $\Gamma_{i}$-matches in $F$ by placing an $x_{i}$ on top of the column that starts a $\Gamma_{i}$-match in $F$ for $i=1,\ldots,s$. We let $m_{\Gamma_{i}}(F)$ equal the number of columns in $F$ which are marked with $x_{i}$. Note that our definitions allow a given column to be marked with some subset of $\{x_{1},\ldots,x_{s}\}$. For example, suppose that $\Gamma_{1}=\{P\}$ and $\Gamma_{2}=\{Q\}$ where Q=\begin{array}[]{|c|c|}\hline\cr 3&4\\ \hline\cr 1&2\\ \hline\cr\end{array} and P=\begin{array}[]{|c|c|c|}\hline\cr 4&5&6\\ \hline\cr 1&2&3\\ \hline\cr\end{array}. Then Figure 25 pictures an element of $\mathcal{MP}^{0,0,k}_{12,\Gamma_{1},\Gamma_{2}}$ where there are $\Gamma_{1}$-matches starting at columns 2 and 3 and $\Gamma_{2}$-matches starting at columns 2, 3, and 4. Thus we can have columns which have more than one label since a columns 2 and 3 starts a $\Gamma_{1}$-match and a $\Gamma_{2}$-match. For the $F$ pictured in Figure 25, $m_{\Gamma_{1}}(F)=2=m_{\Gamma_{2}}(F)$.

A joint $\Gamma_{1},\ldots,\Gamma_{s}$-cluster is a filling of $F\in\mathcal{MP}^{0,0,k}_{kn,\Gamma_{1},\ldots,\Gamma_{s}}$ such that

every column of $F$ is contained in a marked $\Gamma_{i}$-match of $F$ for some $i=1,\ldots,s$ and 2. 2.

for all $1\leq a<b\leq n$, if columns $a$ and $b$ are marked and there is no column $c$ such that $a<c<b$ and column $c$ is marked, then there is an $i$ such that there is a $\Gamma_{i}$-match starting at column $a$ which is marked with $x_{i}$ and that $\Gamma_{i}$-match contains column $b$.

We let $\mathcal{CM}^{0,0,k}_{kn,\Gamma_{1},\ldots,\Gamma_{s}}$ denote the set of all joint $\Gamma_{1},\ldots,\Gamma_{s}$-clusters in $\mathcal{MP}^{0,0,k}_{kn,\Gamma_{1},\ldots,\Gamma_{s}}$. For each $n\geq 2$, we define the cluster polynomial

[TABLE]

By convention, we let $C^{0,0,k}_{k,\Gamma}(x_{1},\ldots,x_{s})=1$.

Then we have the following multivariate version of Theorem 3.

Theorem 12**.**

Let $\Gamma_{i}\subseteq\mathcal{P}^{0,0,k}_{r_{i}k}$ where $r_{i}\geq 2$ for $i=1,\ldots,s$. Then

[TABLE]

4.1.1 Example

For any fixed $k>0$, we let $P_{k,m}$ be the element of $\mathcal{P}^{0,0,k}_{km}$ whose word is the identity permutation. Thus the $i^{\mathrm{th}}$ column of $P_{k,m}$ contains the numbers $k(i-1)+1,\ldots,ki$, reading from bottom to top. For example, $P_{3,4}$ and its corresponding Hasse diagram is pictured in Figure 26.

Our goal in this subsection is to compute the generating function

[TABLE]

for any $m\geq 2$ and $k\geq 1$.

By Theorem 3, we can compute (50) if we can compute the polynomials $G^{0,0,k}_{kn,P_{k,m},P_{k,m+1}}(x,y)$. Our definitions ensures that if $C$ is a $P_{k,m},P_{k,(m+1)}$ cluster, then the underlying array of $C$ is just $P_{k,t}$ for some $t$.

It is easy to check that

[TABLE]

For $n\geq m+1$, we can classify the $P_{k,m},P_{k,(m+1)}$ clusters $C$ with $n$ columns by the label of the first column. We have to cases.

 

Case 1. $y$ is a label on the first column $C$. In this case, we let $j\geq 1$ be the least number such that column $j+1$ is also marked. Clearly, we must have that $1\leq j\leq m$ because if columns $2,\ldots,m$ are not marked, then column $m+1$ must be marked. In this case, we can remove, the first $j$ columns of $C$ and we will be left with a $P_{k,m},P_{k,(m+1)}$ cluster with $n-j$ columns. Note that that the first columns can be marked with $x$ or not. It follows that such clusters contribute

[TABLE]

to $C^{0,0,k}_{kn,P_{k,m},P_{k,m+1}}(x,y)$.

 

Case 2. $y$ is not a label on the first column $C$. In this case, the first column of $C$ is marked with an $x$. Again, we let $j\geq 1$ be the least number such that column $j+1$ is also marked. In this case, we must have that $1\leq j\leq m-1$ because if columns $2,\ldots,m-1$ are not marked, then column $m$ must be marked. In this case, we can remove, the first $j$ columns of $C$ and we will be left with a $P_{k,m},P_{k,(m+1)}$ cluster with $n-j$ columns. Note that that the first columns can be marked with $x$ or not. It follows that such clusters contribute

[TABLE]

to $C^{0,0,k}_{kn,P_{k,m},P_{k,m+1}}(x,y)$.

It follows that for $n\geq m+1$,

[TABLE]

Let

[TABLE]

Then it follows that

[TABLE]

Taking the partial derivative $\frac{\partial^{mk}}{\partial t^{mk}}$ of both sides of this equation leads to the following partial differential equation:

[TABLE]

In the case where $m=2$ and $k=1$ so that $P_{m,k}=12$ and $P_{k,m+1}=123$, then we obtain the differential equation

[TABLE]

with boundary conditions

[TABLE]

Solving the differential equation above, we get

[TABLE]

Then

[TABLE]

where $z(x,y)=\sqrt{x(y(xy+2)-4)+1}$.

A few initial terms of the expansion of $1+\sum_{n\geq 1}\frac{t^{n}}{n!}\sum_{\sigma\in S_{n}}x^{12\mathrm{-mch}(\sigma)}y^{123\mathrm{-mch}(\sigma)}$ are

[TABLE]

In the case where $k=m=2$, the we obtain the differential equation

[TABLE]

with boundary conditions

[TABLE]

We used Mathematica to solve this differential equation and obtained that

[TABLE]

where

[TABLE]

Thus

[TABLE]

The first few terms of this series are

[TABLE]

4.2 Generalized joint clusters

Next suppose that we are given a binary relation $\mathscr{R}$ between $k\times 1$ arrays of integers and sequences of sets of patterns where $\Gamma_{i}\subseteq\mathcal{P}^{0,0,k}_{r_{i}k}$ where $r_{i}\geq 2$ for , $1\leq i\leq m$. Given our definition of joint $\Gamma_{1},\ldots,\Gamma_{s}$-clusters, we can easily modify the definition of generalized $\Gamma$-clusters given in Definition 4 to generalized joint $\Gamma_{1},\ldots,\Gamma_{s}$-clusters.

Definition 13**.**

We say that $Q\in\mathcal{MP}^{0,0,k}_{kn,\Gamma_{1},\ldots,\Gamma_{s}}$ is a ** generalized joint $\Gamma_{1},\ldots,\Gamma_{s},\mathscr{R}$-cluster* if we can write $Q=B_{1}B_{2}\cdots B_{h}$ where $B_{i}$ are blocks of consecutive columns in $Q$ such that*

either $B_{i}$ is a single column or $B_{i}$ consists of $r$-columns where $r\geq 2$, $\mathrm{red}(B_{i})$ is a joint $\Gamma_{1},\ldots,\Gamma_{s}$-cluster in $\mathcal{MP}^{0,0,k}_{kn,\Gamma_{1},\ldots,\Gamma_{s}}$, and any pair of consecutive columns in $B_{i}$ are in $\mathscr{R}$ and 2. 2.

for $1\leq i\leq h-1$, the pair $(\mathrm{last}(B_{i}),\mathrm{first}(B_{i+1}))$ is not in $\mathscr{R}$.

Let $\mathcal{GC}^{0,0,k}_{kn,\Gamma_{1},\ldots,\Gamma_{s},\mathscr{R}}$ denote the set of all generalized joint $\Gamma_{1},\ldots,\Gamma_{s},\mathscr{R}$-clusters which have $n$ columns of height $k$. For example, suppose that $\mathscr{R}$ is the relation that holds for a pair of columns $(C,D)$ if and only if the top element of column $C$ is greater than the bottom element of column $D$ and $\Gamma_{1}=\{P_{1}\}$ and $\Gamma_{2}=\{P_{2}\}$ where P_{1}=\begin{array}[]{|c|c|c|}\hline\cr 6&5&4\\ \hline\cr 1&2&3\\ \hline\cr\end{array} and P_{2}=\begin{array}[]{|c|c|c|c|}\hline\cr 8&7&6&5\\ \hline\cr 1&2&3&4\\ \hline\cr\end{array}. Then in Figure 27, we have pictured a generalized joint $\Gamma_{1},\Gamma_{2}$-cluster with 5 blocks $B_{1},B_{2},B_{3},B_{4},B_{5}$.

Given $Q=B_{1}B_{2}\ldots B_{h}\in\mathcal{GC}^{0,0,k}_{kn,\Gamma_{1},\ldots,\Gamma_{s},\mathscr{R}}$, we define the weight of $B_{i}$,

[TABLE]

where $\text{Col}(B_{i})$ is the number of columns in $B_{i}$. We define the weight of $Q$, $\omega_{\Gamma_{1},\ldots,\Gamma_{s},\mathscr{R}}(Q)$, to be $(-1)^{h-1}\prod_{i=1}^{h}\omega_{\Gamma_{1},\ldots,\Gamma_{s},\mathscr{R}}(B_{i})$. We then define the generalized joint $\Gamma_{1},\ldots,\Gamma_{s},\mathscr{R}$-cluster polynomial to be

[TABLE]

Then we have the following theorem which is the multivariate version of Theorem 5.

Theorem 14**.**

Let $\mathscr{R}$ be a binary relation on pairs of columns $(C,D)$ of height $k$ which are filled with integers which are increasing from bottom to top. Let $\Gamma_{1},\ldots,\Gamma_{s}$ be a sequence of sets of patterns such that $\Gamma_{i}\subseteq\mathcal{P}^{0,0,k}_{r_{i}k}$ where $r_{i}\geq$ for $1\leq i\leq s$. Then

[TABLE]

There are obvious analogues of start, end, and start-end general clusters in the joint cluster setting. Throughout the rest of this section, we shall assume that we are given a binary relation $\mathscr{R}$ on columns of integers and a sequence $\Gamma_{1},\ldots,\Gamma_{s}$ of a sets of patterns such that $\Gamma_{i}\subseteq\mathcal{P}^{0,0,k}_{r_{i}k}$ where $r_{i}\geq 2$ for $i=1,\ldots,s$.

Definition 15**.**

We say that $Q\in\mathcal{MP}^{0,j,k}_{kn+j,\Gamma_{1},\ldots,\Gamma_{s}}$ is a generalized joint $\Gamma_{1},\ldots,\Gamma_{s},\mathscr{R}$-end-cluster if we can write $Q=B_{1}B_{2}\cdots B_{m}$ where $B_{i}$ are blocks of consecutive columns in $Q$ such that

$B_{m}$* is a column of height $j$,* 2. 2.

for $i<m$, either $B_{i}$ is a single column or $B_{i}$ consists of $r$-columns where $r\geq 2$, $\mathrm{red}(B_{i})$ is a joint $\Gamma_{1},\ldots,\Gamma_{s}$-cluster in $\mathcal{MP}_{kr,\Gamma_{1},\ldots,\Gamma_{s}}$, and any pair of consecutive columns in $B_{i}$ are in $\mathscr{R}$ and 3. 3.

for $1\leq i\leq m-1$, the pair $(\mathrm{last}(B_{i}),\mathrm{first}(B_{i+1})$ is not in $\mathscr{R}$.

Let $\mathcal{GEC}^{0,j,k}_{kn+j,\Gamma_{1},\ldots,\Gamma_{s},\mathscr{R}}$ denote the set of all generalized $\Gamma_{1},\ldots,\Gamma_{s},\mathscr{R}$-end-clusters which have $n$ columns of height $k$ followed by a column of height $j$. Given $Q=B_{1}B_{2}\ldots B_{m}\in\mathcal{GEC}^{0,j,k}_{kn+j,\Gamma_{1},\ldots,\Gamma_{s},\mathscr{R}}$, we define the weight of $B_{i}$, $w_{\Gamma_{1},\ldots,\Gamma_{s},\mathscr{R}}(B_{i})$, to be 1 if $B_{i}$ is a single column and $\prod_{i=1}^{s}x_{i}^{m_{\Gamma_{i}}(\mathrm{red}(B_{i}))}$ if $B_{i}$ is order isomorphic to a $\Gamma_{1},\ldots,\Gamma_{s}$-cluster. We define the weight of $Q$, $w_{\Gamma_{1},\ldots,\Gamma_{s},\mathscr{R}}(Q)$, to be $(-1)^{m-1}\prod_{i=1}^{s}w_{\Gamma_{1},\ldots,\Gamma_{s},\mathscr{R}}(B_{i})$. We then let

[TABLE]

Then we have the following multivariate analogue of Theorem 7.

Theorem 16**.**

Let $\mathscr{R}$ be a binary relation on pairs of columns $(C,D)$ which are filled with integers which are increasing from bottom to top. Let $\Gamma_{1},\ldots,\Gamma_{s}$ be a sequence of a sets of patterns such that $\Gamma_{i}\subseteq\mathcal{P}^{0,0,k}_{r_{i}k}$ where $r_{i}\geq 2$ for $i=1,\ldots,s$. Then

[TABLE]

Definition 17**.**

We say that $Q\in\mathcal{MP}^{i,0,k}_{i+kn,\Gamma}$ is a generalized joint $\Gamma_{1},\ldots,\Gamma_{s},\mathscr{R}$-start-cluster if we can write $Q=B_{1}B_{2}\cdots B_{m}$ where $B_{i}$ are blocks of consecutive columns in $Q$ such that

$B_{1}$* is a single column of height $i$,* 2. 2.

for $2<a\leq m$, either $B_{a}$ is a single column or $B_{a}$ consists of $r$-columns where $r\geq 2$, $\mathrm{red}(B_{a})$ is a joint $\Gamma_{1},\ldots,\Gamma_{s}$-cluster in $\mathcal{MP}_{kr,\Gamma_{1},\ldots,\Gamma_{s}}$, and any pair of consecutive columns in $B_{i}$ are in $\mathscr{R}$ and 3. 3.

for $1\leq i\leq m-1$, the pair $(\mathrm{last}(B_{i}),\mathrm{first}(B_{i+1})$ is not in $\mathscr{R}$.

Let $\mathcal{GSC}^{i,0,k}_{i+kn,\Gamma,\mathscr{R}}$ denote the set of all generalized $\Gamma,\mathscr{R}$-start-clusters which start with a column of height $i$ and which is followed by $n$ columns of height $k$. Given $Q=B_{1}B_{2}\ldots B_{m}\in\mathcal{GEC}^{i,0,k}_{i+kn,\Gamma,\mathscr{R}}$, we define the weight of $B_{i}$, $w_{\Gamma_{1},\ldots,\Gamma_{s},\mathscr{R}}(B_{i})$, to be 1 if $B_{i}$ is a single column and $\prod_{j=1}^{s}x_{i}^{m_{\Gamma_{j}}(\mathrm{red}(B_{i}))}$ if $B_{i}$ is order isomorphic to a joint $\Gamma_{1},\ldots,\Gamma_{s}$-cluster. We define the weight of $Q$, $w_{\Gamma_{1},\ldots,\Gamma_{s},\mathscr{R}}(Q)$, to be $(-1)^{m-1}\prod_{i=1}^{m}w_{\Gamma_{1},\ldots,\Gamma_{s},\mathscr{R}}(B_{i})$. We then let

[TABLE]

Let $\mathcal{P}^{i,0,k}_{i+kn,\mathscr{R}}$ denote the set of all elements $F\in\mathcal{P}^{i,0,k}_{i+kn}$ such that the relation $\mathscr{R}$ holds for any pair of consecutive columns in $F$. Then we have the following theorem.

Theorem 18**.**

Let $\mathscr{R}$ be a binary relation on pairs of columns $(C,D)$ which are filled with integers which are increasing from bottom to top. Let $\Gamma_{1},\ldots,\Gamma_{s}$ be a sequence of a sets of patterns such that $\Gamma_{i}\subseteq\mathcal{P}^{0,0,k}_{r_{i}k}$ where $r_{i}\geq 2$ for $i=1,\ldots,s$. Then

[TABLE]

Definition 19**.**

We say that $Q\in\mathcal{MP}^{i,j,k}_{i+kn+j,\Gamma}$ is a generalized joint $\Gamma_{1},\ldots,\Gamma_{s},\mathscr{R}$-start-end-cluster if we can write $Q=B_{1}B_{2}\cdots B_{m}$ where $B_{i}$ are blocks of consecutive columns in $Q$ such that

$m\geq 2$, 2. 2.

$B_{1}$* is a column of height $i$,* 3. 3.

$B_{m}$* is a column of height $j$,* 4. 4.

for $1<i<m$, either $B_{i}$ is a single column or $B_{i}$ consists of $r$-columns where $r\geq 2$, $\mathrm{red}(B_{i})$ is a joint $\Gamma_{1},\ldots,\Gamma_{s}$-cluster in $\mathcal{MP}_{kr,\Gamma_{1},\ldots,\Gamma_{s}}$, and any pair of consecutive columns in $B_{i}$ are in $\mathscr{R}$ and 5. 5.

for $1\leq i\leq m-1$, the pair $(\mathrm{last}(B_{i}),\mathrm{first}(B_{i+1}))$ is not in $\mathscr{R}$.

Let $\mathcal{GSEC}^{i,j,k}_{i+kn+j,\Gamma_{1},\ldots,\Gamma_{s},\mathscr{R}}$ denote the set of all generalized $\Gamma_{1},\ldots,\Gamma_{s},\mathscr{R}$-start-end-clusters which have $n$ columns of height $k$ between a column of height $i$ and a column of height $j$. Given $Q=B_{1}B_{2}\ldots B_{m}\in\mathcal{GSEC}^{i,j,k}_{i+kn+j,\Gamma_{1},\ldots,\Gamma_{s},\mathscr{R}}$, we define the weight of $B_{i}$, $\omega_{\Gamma_{1},\ldots,\Gamma_{s},\mathscr{R}}(B_{i})$, to be 1 if $B_{i}$ is a single column and $\prod_{j=1}^{s}x^{m_{\Gamma_{j}}(\mathrm{red}(B_{i}))}$ if $B_{i}$ is order isomorphic to a joint $\Gamma_{1},\ldots,Gamma_{s}$-cluster. Then we define the weight of $Q$, $\omega_{\Gamma_{1},\ldots,\Gamma_{s},\mathscr{R}}(Q)$, to be $(-1)^{m-1}\prod_{i=1}^{m}\omega_{\Gamma_{1},\ldots,\Gamma_{s},\mathscr{R}}(B_{i})$. We let

[TABLE]

Theorem 20**.**

Let $\mathscr{R}$ be a binary relation on pairs of columns $(C,D)$ which are filled with integers which are increasing from bottom to top. Let $\Gamma_{1},\ldots,\Gamma_{s}$ be a sequence of a sets of patterns such that $\Gamma_{i}\subseteq\mathcal{P}^{0,0,k}_{r_{i}k}$ where $r_{i}\geq 2$ for $i=1,\ldots,s$. Then

[TABLE]

where

[TABLE]

and

[TABLE]

4.2.1 Example

In this section, we shall consider joint distribution of $\tau=1423$ and $\pi=162534$ in up-down permutations. That is, we shall compute the generating function

[TABLE]

As in the previous sections, we can think of up-down permutations as arrays as in Figure 28.

We let $\mathscr{R}$ be the binary relation such that $\mathscr{R}$ holds for a pair of columns $(C,D)$ if and only if the top element of column $C$ is greater than the bottom element of column $D$. Then

[TABLE]

In particular, $\tau=1423=w(P)$ and $\pi=162534=w(Q)$ where P=\begin{array}[]{|c|c|}\hline\cr 4&3\\ \hline\cr 1&2\\ \hline\cr\end{array} and Q=\begin{array}[]{|c|c|c|}\hline\cr 6&5&4\\ \hline\cr 1&2&3\\ \hline\cr\end{array}.

Thus the generating function $F_{\tau,\pi}(x,y,t)$ in (63) is equivalent to

[TABLE]

Thus we must consider the following two generating functions

[TABLE]

and

[TABLE]

because $F_{P,Q}(x,y,t)=A_{P,Q}(x,y,t)+B_{P,Q}(x,y,t)$.

We shall compute $A_{P,Q}(x,y,t)$ first. By Theorem 14, we have

[TABLE]

We shall start by discussing structures of joint $(P,Q)$-clusters. It is easy to see that the Hasse diagram of a joint $(P,Q)$-clusters is of the form pictured in Figure 29.

Let $\mathcal{CM}_{2n,P,Q}$ be the set of joint $(P,Q)$-clusters consisting of two rows and $n$ columns. Let $C_{2n,P,Q}(x,y)$ denote the joint cluster polynomial. We can use the exact same reasoning that we used to compute $C^{0,0,2}_{2n,P_{2,2},P_{2,3}}(x,y)$ to prove that

[TABLE]

Next we shall compute $GC^{0,0,2}_{2n,P,Q,\mathscr{R}}(x,y)$. In fact our computation is very similar to the computation of $GC^{0,0,2}_{2n,P}(x)$ in Section 3.4.1.

Suppose $J\in\mathcal{GC}_{2n,P,Q,\mathscr{R}}^{0,0,2}$ has $m$ blocks, i.e., $J=B_{1}B_{2}\ldots B_{m}$. For an array $F$, we let $\text{Col}(F)$ to denote the number of columns in $\text{Col}(F)$. Each $B_{i}$ is either order isomorphic to a joint $(P,Q)$-cluster or a single column. We let $\mathcal{GC}_{\text{Col}=(b_{1},b_{2},\ldots,b_{m})}$ to denote the set of generalized clusters such that $\text{Col}(B_{i})=b_{i}$. Given $\text{Col}(B_{i})$ for each $1\leq i\leq m$, we can represent the fillings for the set of such generalized clusters as the set of linear extensions of a poset whose Hasse diagram corresponds $(b_{1},\ldots,b_{m})$. That is, for any block $(\mathrm{last}(B_{i}),\mathrm{first}(B_{i+1})$ is not in $\mathscr{R}$ so that there must be an arrow directed from the top of the last column of $B_{i}$ to the bottom of the first column of $B_{i+1}$. We let $\Gamma(b_{1},b_{2},\ldots,b_{m})$ denote the Hasse diagram corresponding to $\mathcal{GC}_{\text{Col}=(b_{1},b_{2},\ldots,b_{m})}$. For example, $\Gamma(3,1,1,5,1)$ is pictured in Figure 30.

These are the same type of Hasse diagrams that we saw in Section 3.4.1. Then using the same reasoning that we used in Section 3.4.1, it follows that

[TABLE]

where $\text{LE}(\Gamma(b_{1},b_{2},\ldots,b_{m}))$ is the number of linear extensions of $\Gamma(b_{1},b_{2},\ldots,b_{m})$ and by convention, we let $C_{2}(x)=1$. Therefore, to compute (42), we only need to count linear extensions of $\Gamma(b_{1},b_{2},\ldots,b_{m})$. Continue using $\Gamma(3,1,1,5,1)$ as example, the Hasse diagram in Figure 30 is actually a tree-like diagram, as drawn in Figure 19. Hence

[TABLE]

The only difference between the generalized clusters considered in Section 3.4.1 and our current situation is that the blocks with more than one column are weighted as joint $P,Q$-clusters.

Using a computer program, we computed that

[TABLE]

Using these initial terms, we can use our formula for $GC_{2n,P,Q,\mathscr{R}}(x,y,t)$ given in (67) to compute that

[TABLE]

Next we shall compute $B_{P,Q}(x,y,t)$. By Theorem 16,

[TABLE]

Again, the computation of joint generalized end-cluster polynomials $GEC_{2n+1,P,Q,\mathscr{R}}^{0,1,2}(x,y,t)$ is the same as the computation of generalized end-cluster polynomials in Section 3.4.1. That is, the only difference between the generalized clusters considered in Section 3.4.1 and our current situation is that the blocks with more than one column are weighted as joint $P,Q$-clusters.

Using a computer program, we computed that

[TABLE]

Using these initial terms, we can use our formula for $GEC^{0,1,2}_{2n+1,P,Q,\mathscr{R}}(x,y,t)$ given in (70) to compute that

[TABLE]

Finally, taking sum of $A_{P,Q}(x,y,t)$ and $B_{P,Q}(x,y,t)$, we get the generating function for joint distribution of patterns $1423$ and $162534$ in up-down permutations,

[TABLE]