A new approach to $e$-positivity for Stanley's chromatic functions

TL;DR

This paper introduces a new combinatorial object to study the $e$-positivity of Stanley's chromatic functions, proving positivity for specific coefficients in the case of $(3+1)$-free posets.

Contribution

It presents a novel combinatorial model using correct sequences of unit interval orders to establish $e$-positivity results for certain chromatic coefficients.

Findings

Proves positivity of specific coefficients $c_{n-k,1^k}$, $c_{n-2,2}$, $c_{n-3,2,1}$, $c_{2^k,1^{n-2k}}$

Introduces correct sequences of unit interval orders as a combinatorial tool

Establishes positivity results for $(3+1)$-free posets

Abstract

In this paper, we study positivity phenomena for the $e$-coefficients of Stanley's chromatic function of a graph. We introduce a new combinatorial object: the {\em correct} sequences of unit interval orders, and using these, in certain cases, we succeed to construct combinatorial models of the coefficients appearing in Stanley's conjecture. Our main result is the proof of positivity of the coefficients $c_{n-k,1^k}$, $c_{n-2,2}$, $c_{n-3,2,1}$ and $c_{2^k,1^{n-2k}}$ of the expansion of the chromatic symmetric function in terms of the basis of the elementary symmetric polynomials for the case of $(3+1)$-free posets.

Equations

$$c:V\to \mathbb{N}$$

$$x^c=\prod _{v\in V}{x}_{c(v)},$$

$$X_G=\sum _{c\in \Pi (G)}{x}^c.$$

$$e_m=\sum _{i_1<i_2<...<i_m}{x}_{i_1}\cdot x_{i_2}\cdot ...\cdot x_{i_m},$$

$$\lambda =({\lambda }_1\ge {\lambda }_2\ge ...\ge {\lambda }_k),{\lambda }_i\in \mathbb{N},$$

$${\lambda }_1={\lambda }_2=...={\lambda }_k=1.$$

$$p_m=\sum _{i\in \mathbb{N}}{x}_i^m.$$

$$m_{\lambda }=\sum _{i_1<i_2<...<i_k}\sum _{{\lambda }^{\prime }\in S_k(\lambda )}{x}_{i_1}^{{\lambda }_1^{\prime }}\cdot x_{i_2}^{{\lambda }_1^{\prime }}\cdot ...\cdot x_{i_k}^{{\lambda }_k^{\prime }},$$

$$\text{for }u,w\in U:u\succ w\text{ iff }u\ge w+1.$$

$$X_G=\sum _{\lambda }{c}_{\lambda }{e}_{\lambda }.$$

$$\{X_{\mathrm{inc}}(P^{\prime })|P^{\prime }\mathrm{is}\mathrm{a}(3+1)-\mathrm{and}(2+2)-\mathrm{free}\mathrm{poset}\}.$$

$$X_G=\sum _{\lambda }{c}_{\lambda }{e}_{\lambda },$$

$$\text{sink}(G,j)=\sum _{l(\lambda )=j}{c}_{\lambda }.$$

$$P_{n,k}=\bigg{\{}\frac{i}{k+1}\bigg{\}}_{i=1}^{n}$$

$$e_i^G=\sum _{\begin{array}{c}\#S=i\\ S-\mathrm{stable}\end{array}}\prod _{v\in S}v,$$

$$e_{\lambda }^G=\prod _{i=1}^k{e}_i^G,$$

$$s_{\lambda }^G=\det (e_{{\lambda }_i^{\ast }+j-i}^G).$$

$$v^{\alpha }=\prod _{v\in V}{v}^{\alpha (v)},$$

$$T(x,v)=\sum _{\lambda }{m}_{\lambda }(x)e_{\lambda }^G(v),$$

$$[v^{\alpha }]T(x,v)\prod _{v\in V}\alpha (v)!=X_{G^{\alpha }}.$$

$$\sum _{\lambda }{s}_{\lambda }(x)s_{{\lambda }^{\ast }}(y)=\sum _{\lambda }{m}_{\lambda }(x)e_{\lambda }(y)=\sum _{\lambda }{e}_{\lambda }(x)m_{\lambda }(y)$$

$$T(x,v)=\sum _{\lambda }{m}_{\lambda }(x)e_{\lambda }^G(v)=\sum _{\lambda }{s}_{\lambda }(x)s_{{\lambda }^{\ast }}^G(v)=T(v,x)=\sum _{\lambda }{e}_{\lambda }(x)m_{\lambda }^G(v).$$

$$p_k^U=\sum _{\vec{w}\in P_k^U}{w}_1\cdot ...\cdot w_k\in N[U],$$

$$P_{\lambda }^U=\{\vec{w}=(w_1,\dots w_{{\lambda }_1}|w_{{\lambda }_1+1},\dots ,w_{{\lambda }_1+{\lambda }_2}|\dots |w_{|\lambda |-{\lambda }_k+1},\dots ,w_{|\lambda |})|\vec{w}\text{ is }\lambda \text{-correct }\}$$

$$p_{\lambda }^U=\prod _{i=1}^k{p}_{{\lambda }_i}^U=\sum _{\vec{w}\in P_{\lambda }^U}{w}_1\cdot ...\cdot w_{|\lambda |}.$$

$$M_{l,1}^U=\{(\vec{w}|z)\in P_{l.1}|z\succ \vec{w}\vee z\prec w_l\},$$

$$m_{l,1}^U=\sum _{(\vec{w};z)\in M_{l,1}^U}{w}_1\cdot ...\cdot w_l\cdot z.$$

$$m_{l,1}^U=p_l^U\cdot p_1^U-p_{l+1}^U,$$

$$P_l^U\times P_1^U\setminus P_{l+1}^U=M_{l,1}^U,$$

$$m_{l,1}^U=\sum _{(\vec{w};z)\in M_{l,1}^U}{w}_1\cdot ...\cdot w_l\cdot z.$$

Full text

1 Introduction

Let $G$ be a finite graph, $V(G)$ - the set of vertices of $G$, $E(G)$ - the set of edges of $G$.

Definition 1.1**.**

A proper coloring $c$ of $G$ is a map

[TABLE]

such that no two adjacent vertices are colored in the same color.

For each coloring $c$ we define a monomial

[TABLE]

where $x_{1},x_{2},...,x_{n},...$ are commuting variables. We denote by $\Pi(G)$ the set of all proper colorings of $G$, and by $\Lambda$ the ring of symmetric functions in the infinite set of variables $\{x_{1},x_{2},...\}.$

In [2], Stanley defined the chromatic symmetric function of a graph.

Definition 1.2**.**

The chromatic symmetric function $X_{G}\in\Lambda$ of a graph $G$ is the sum of the monomials $x^{c}$ over all proper colorings of $G$:

[TABLE]

Definition 1.3**.**

Denote by $e_{m}$ the $m$-th elementary symmetric function:

[TABLE]

where $i_{1},..,i_{k}\in\mathbb{N}$. Given a non-increasing sequence of positive integers (we will call these partitions)

[TABLE]

we define the elementary symmetric function $e_{\lambda}=\prod\limits_{i=1}^{k}e_{\lambda_{i}}.$ These functions form a basis of $\Lambda.$

For a natural number $k$, we denote by $1^{k}$ the partition $\lambda$ of length $k$, where

[TABLE]

Definition 1.4**.**

A symmetric function $X\in\Lambda$ is $e$*-positive *if it has non-negative coefficients in the basis of the elementary symmetric functions.

Definition 1.5**.**

Denote by $p_{m}$ the $m$-th power sum symmetric function:

[TABLE]

Given a partition $\lambda=(\lambda_{1}\geq\lambda_{2}\geq...\geq\lambda_{k})$, we define the power sum symmetric function $p_{\lambda}=\prod\limits_{i=1}^{k}p_{\lambda_{i}}.$ These functions also form a basis of $\Lambda.$

Definition 1.6**.**

Given a partition $\lambda=(\lambda_{1}\geq\lambda_{2}\geq...\geq\lambda_{k})$, we define the monomial symmetric function

[TABLE]

where the inner sum is taken over the set of all permutations of the sequence $\lambda$, denoted by $S_{k}(\lambda)$.

Example 1.7**.**

The chromatic symmetric function of $K_{n}$, the complete graph on $n$ vertices, is $e$-positive: $X_{K_{n}}=n!\,e_{n}$.

Definition 1.8**.**

For a poset $P$, the incomparability graph, $\textnormal{inc}(P)$, is the graph with elements of $P$ as vertices, where two vertices are connected if and only if they are not comparable in $P$.

Definition 1.9**.**

Given a pair of natural numbers $a,b\in\mathbb{N}^{2}$, we say that a poset $P$ is *(a+b)-free *if it does not contain a length-$a$ and a length-$b$ chain, whose elements are incomparable.

Definition 1.10**.**

A unit interval order (UIO) is a partially ordered set which is isomorphic to a finite subset of $U\subset\mathbb{R}$ with the following poset structure:

[TABLE]

Thus $u$ and $w$ are incomparable precisely when $|u-w|<1$ and we will use the notation $u\sim w$ in this case.

Theorem 1.11** (Scott-Suppes [1]).**

A finite poset $P$ is a UIO if and only if it is $(2+2)$- and $(3+1)$-free.

Stanley [2] initiated the study of incomparability graphs of $(3+1)$-free partially ordered sets. Analyzing the chromatic symmetric functions of these incomparability graphs, Stanley [2] stated the following positivity conjecture.

Conjecture 1.12** (Stanley).**

If $P$ is a $(3+1)$-free poset, then $X_{\textnormal{inc}(P)}$ is $e$-positive.

For a graph $G$ let us denote by ${c_{\lambda}}(G)$ the coefficients of $X_{G}$ with respect to the $e$-basis. We omit the index $G$ whenever this causes no confusion:

[TABLE]

Conjecture 1.12 has been verified with the help of computers for up to 20-element posets [6]. In 2013, Guay-Paquet [6] showed that to prove this conjecture, it would be sufficient to verify it for the case of $(3+1)$- and $(2+2)$-free posets, i.e. for unit interval orders (see Theorem 1.11). More precisely:

Theorem 1.13** (Guay-Paquet).**

Let $P$ be a $(3+1)$-free poset. Then, $X_{\mathrm{inc}}(P)$ is a convex combination of the chromatic symmetric functions

[TABLE]

The strongest general result in this direction is that of Gasharov [3].

Definition 1.14**.**

For a partition $\lambda=(\lambda_{1}\geq\lambda_{2}\geq...\geq\lambda_{k})$, define the Schur functions $s_{\lambda}=\mathrm{det}(e_{\lambda_{i}^{*}+j-i})_{i,j}$*, *where $\lambda^{*}$ is the conjugate partition to $\lambda$. The functions $\{s_{\lambda}\}$ form a basis of $\Lambda$.

Definition 1.15**.**

A symmetric polynomial $X$ is $s$*-positive *if it has non-negative coefficients in the basis of Schur functions.

Obviously, a product of $e$-positive functions is $e$-positive. This also holds for $s$-positive functions. Thus, the equality $e_{n}=s_{1^{n}}$ implies that $e$-positive functions are $s$-positive, and thus $s$-positivity is weaker than $e$-positivity.

Theorem 1.16** (Gasharov).**

If $P$ is a $(3+1)$-free poset, then $X_{\textnormal{inc}(P)}$ is $s$-positive.

Gasharov proved $s$-positivity by constructing so-called $P$-tableau and finding a one-to-one correspondence between these tableau and $s$-coefficients [3]. However, $e$-positivity conjecture 1.4 is still open. The strongest known result on the $e$-coefficients was obtained by Stanley in [2]. He showed that sums of $e$-coefficients over the partitions of fixed length are non-negative:

Theorem 1.17** (Stanley).**

For a finite graph $G$ and $j\in\mathbb{N}$, suppose

[TABLE]

and let $\text{sink}(G,j)$ be the number of acyclic orientation of $G$ with $j$ sinks. Then

[TABLE]

Remark 1.18**.**

By taking $j=1$, it follows from the theorem that $c_{n}$ is non-negative.

Stanley in [2] showed that for $n\in\mathbb{N}$ and the unit interval order $P_{n}=\{\frac{i}{2}\}_{i=1}^{n}$, the corresponding $X_{\text{inc}(P_{n})}$ is $e$-positive, while $e$-positivity for the UIOs

[TABLE]

with $k>1$ has not yet been proven. It was checked for small $n$ and some $k$ (see [2]).

Next, we introduce correct sequences (abbreviated as corrects), defined below. These play a major role in the article.

Definition 1.19**.**

Let U be a UIO. We will call a sequence $\vec{w}=(w_{1},\dots,w_{k})$ of elements of $U$ correct if

• •

$w_{i}\not\succ w_{i+1}$ for $i=1,2,\dots,k-1$

• •

and for each $j=2,\dots,k$, there exists $i<j$ such that $w_{i}\not\prec w_{j}$.

Every sequence of length 1 is correct, and sequence $(w_{1},w_{2})$ is correct precisely when $w_{1}\sim w_{2}$. The second condition (supposing that the first one holds) may be reformulated as follows: for each $j=1,\dots k$, the subset $\{w_{1},\dots,w_{j}\}\subset U$ is connected with respect to the graph structure ${(U,\sim)}$. Using this notation, we prove the following theorems.

Theorem 1.20**.**

Let $X_{\text{inc}(U)}=\sum\limits_{\lambda}c_{\lambda}e_{\lambda}$ be a chromatic symmetric function of the $n$-element unit interval order $U$. Then $c_{n}$ is equal to the number of corrects of length $n$, in which every element of $U$ is used exactly once.

Corollary 1.20.1**.**

Let $X_{\text{inc}(P)}=\sum\limits_{\lambda}c_{\lambda}e_{\lambda}$ be a chromatic symmetric function of $n$-element $(3+1)$-free poset $P$, then $c_{n}$ is a nonnegative integer.

Indeed, positivity for the general case follows from Theorem 1.13, which presents the chromatic symmetric function of a $(3+1)$-free poset as a convex combination of the chromatic symmetric functions of unit interval orders.

Stanley [7] and Chow [5] showed the positivity of $c_{n}$ for $(3+1)$-free posets using combinatorial techniques, and linked $e$-coefficients with the acyclic orientations of the incomparability graphs. The construction of corrects not only serves this purpose for UIOs (see [10]), but also creates a new approach, which allows us to obtain the following new result:

Theorem 1.21**.**

Let $X_{\text{inc}(P)}=\sum\limits_{\lambda}c_{\lambda}e_{\lambda}$ be a chromatic symmetric function of the $(3+1)$-free poset $P$, and $k\in\mathbb{N}$. Then $c_{n-k,1^{k}}$, $c_{n-2,2}$, $c_{n-3,2,1}$ and $c_{2^{k},1^{n-2k}}$ are non-negative integers.

The proofs of Theorem 1.20 and Theorem 1.16, and positivity of correspondent $G$-power sum symmetric functions and Schur $G$-symmetric functions can be found in [9] and  [10]. The article is structured as follows: in Section 2, we describe the $G$-homomorphism introduced by Stanley in [7], which is essential for our approach. Positivity of $c_{n-k,1^{k}}$, $c_{n-2,2}$, $c_{n-3,2,1}$ and $c_{2^{k},1^{n-2k}}$ (Theorem 1.21) is proven in Section 3.

Acknowledgements. We are grateful to Emanuele Delucchi and Bart Vandereycken for their help and useful discussions.

2 Stanley’s $G$-homomorphism

For a graph $G$, Stanley [7, p. 6] defined $G$-analogues of the standard families of symmetric functions. Let $G$ be a finite graph with vertex set $V(G)=\{v_{1},...,v_{n}\}$ and edge set $E(G)$. We will think of the elements of $V(G)$ as commuting variables.

Definition 2.1**.**

For a positive integer $i$, $1\leq i,\leq n$, we define the $G$*-analogues *of the elementary symmetric polynomials, or the elementary $G$-symmetric polynomials, as follows

[TABLE]

where the sum is taken over all $i$-element subsets $S$ of $V$, in which no two vertices form an edge, i.e. stable subsets. We set $e_{0}^{G}=1$, and $e_{i}^{G}=0$ for $i<0$.

Note that these polynomials are not necessarily symmetric.

Let $\Lambda_{G}\subset\mathbb{R}[v_{1},...,v_{n}]$ be the subring generated by $\{e_{i}^{G}\}_{i=1}^{n}$. The map $e_{i}\mapsto e_{i}^{G}$ extends to a ring homomorphism $\phi_{G}:\Lambda\rightarrow\Lambda_{G}$, called the $G$-homomorphism. For $f\in\Lambda$, we will use the notation $f^{G}$ for $\phi_{G}(f)$.

Example 2.2**.**

Given a partition $\lambda=\lambda_{1}\geq\lambda_{2}\geq...\geq\lambda_{k},\ k\in\mathbb{N},$ we have

[TABLE]

[TABLE]

For an integer function $\alpha:V\rightarrow\mathbb{N}$ and $f^{G}\in\Lambda_{G}$, let

[TABLE]

and $[v^{\alpha}]f^{G}$ stands for the coefficient of $v^{\alpha}$ in the polynomial $f^{G}\in\Lambda_{G}$.

Let $G^{\alpha}$ denote the graph, obtained by replacing every vertex $v$ of $G$ by the complete subgraph of size $\alpha(v)$: $K_{\alpha(v)}^{v}$. Given vertices $u$ and $v$ of $G$, a vertex of $K_{\alpha(v)}^{v}$ is connected to a vertex of $K_{\alpha(u)}^{u}$ if and only if $u$ and $v$ form an edge in $G$.

Considering the Cauchy product [8, ch. 4.2], Stanley [7, p. 6] found a connection between the $G$-analogues of symmetric functions and $X_{G}$. Following Stanley [7], we set

[TABLE]

where the sum is taken over all partitions. Then

[TABLE]

Using the Cauchy identity

[TABLE]

and applying the $G$-homomorphism, one obtains:

[TABLE]

An immediate consequence of the formulas (1) and (2) is the following result of Stanley:

Theorem 2.3** (Stanley).**

For every finite graph G

$X_{G^{\alpha}}$* is s-positive for every $\alpha:V(G)\rightarrow\mathbb{N}$ if and only if $s_{\lambda}^{G}\in\mathbb{N}[V(G)]$ for every partition $\lambda$.* 2. 2.

$X_{G^{\alpha}}$* is e-positive for every $\alpha:V(G)\rightarrow\mathbb{N}$ if and only if $m_{\lambda}^{G}\in\mathbb{N}[V(G)]$ for every partition $\lambda$.*

Remark 2.4**.**

If $X_{G^{\alpha}}=\sum\limits_{\lambda}c^{\alpha}_{\lambda}e_{\lambda},$ then $c_{\lambda}^{\alpha}=[v^{\alpha}]m^{G}_{\lambda}.$ Hence, monomial positivity of $m^{G}_{\lambda}$ is equivalent to the positivity of $c_{\lambda}^{\alpha}$ for every $\alpha$.

The proofs of positivity of $G$-power sum symmetric functions and Schur $G$-symmetric functions for the case of unit interval orders can be found in [9].

3 Proofs of the theorems

It follows from Theorem 2.3 that to prove that the graph $G$ is $e$-positive, it is enough to show the monomial positivity of its monomial $G$-symmetric functions. On the other hand, Guay-Paquet in Theorem 1.13 showed that it is sufficient to check $e$-positivity for unit interval orders, in order to prove it for the general case of $(3+1)$-free posets. Therefore, in the following section 3 we analyze the functions $m_{\lambda}^{G}$ for the case $G=\text{inc}(U),$ where $U$ is UIO.

Let us repeat the definition of a central notion for our work, that of correct sequences of elements of a unit interval order.

Definition 3.1**.**

Let $(U,\prec)$ be a unit interval order, and $G=\text{inc}(U)$. We will call a sequence $\vec{w}=(w_{1},\dots,w_{k})$ of elements of $U$ correct if

• •

$w_{i}\not\succ w_{i+1}$ for $i=1,2,\dots,k-1$

• •

and for each $j=2,\dots,k$, there exists $i<j$ such that $w_{i}\not\prec w_{j}$.

We denote by $P^{U}_{k}$ the set of all correct sequences (abbreviated as corrects) of length $k$. Since $G$ is uniquely defined by $U$, and we are working only with UIO, here and below we use the $U$-index instead of $G$. The $U$-analogues of symmetric functions will be analyzed.

Theorem 3.2**.**

Let $U$ be a unit interval order and $p_{k}^{U}$ the Stanley power-sum function of the corresponding incomparability graph. Then, for every natural $k$, we have

[TABLE]

where the sum is taken over all corrects of length $k$.

The proof of this theorem can be found in [9].

Below, we prove positivity of $m^{U}_{l,1^{k}}$, $m^{U}_{l,2}$, $m^{U}_{l,2,1},$ and $m^{U}_{2^{l},1^{k}}$. We need the following mild technical generalization of correct sequences: let $\lambda=(\lambda_{1}\geq\dots\geq\lambda_{k})$ be a partition of $|\lambda|=\sum\limits_{i=1}^{k}\lambda_{i}$. Then, we will call sequence $(w_{1},\dots,w_{|\lambda|})$ $\lambda$-correct if each of the subsequences $(w_{1},\dots w_{\lambda_{1}})$, $(w_{\lambda_{1}+1},\dots,w_{\lambda_{1}+\lambda_{2}})$,$\dots$ $(w_{|\lambda|-\lambda_{k}+1},\dots,w_{|\lambda|})$ are correct. Introduce the set

[TABLE]

of $\lambda$-correct sequences of length-$|\lambda|$. In particular, $P_{l}^{U}$ is the set of $l$-corrects of $U$. This definition is consistent with Theorem 3.2, and we have:

[TABLE]

For $\vec{w}=(w_{1},\dots w_{l})\in P^{U}_{l}$ and $z\in U$ we write $z\succ\vec{w}$, if $z\succ w_{i}$ for every $1\leq i\leq l.$

Theorem 3.3**.**

Let

[TABLE]

then

[TABLE]

Remark 3.4**.**

According to Remark 2.4, this implies $c_{n-1,1}(U)\geq 0$.

Proof.

Since $P^{U}_{l+1}\subset P^{U}_{l}\times P^{U}_{1}$, using the following relation

[TABLE]

we have

[TABLE]

and, as a consequence,

[TABLE]

∎

Next, we introduce the set

[TABLE]

Theorem 3.5**.**

For natural numbers $l$ and $k$, let

[TABLE]

Then,

[TABLE]

Remark 3.6**.**

According to Remark 2.4, this implies $c_{n-k,1^{k}}(U)\geq 0$.

Proof.

We prove this by induction on $k$. Note that for $k=1$, the definition of $M^{U}_{l,1^{k}}$ coincides with $M^{U}_{l,1}$ from Theorem 3.3. Thus, the case $k=1$ follows from Theorem 3.3.

Assume the statement is true for $k$, and consider the standard equation

[TABLE]

Below, we construct a pair of inverse maps, $\phi_{l,1^{k+1}}$ and $\psi_{l,1^{k+1}}$ from the left part to the right part of the latter equation and vice versa respectively. Every case is followed by a visual illustration.

\Romannum

1. We define

[TABLE]

as follows:

Let $(\vec{w}\ |\vec{\varepsilon}\ )\in P^{U}_{l}\times E_{k+1}^{U}$

If $\varepsilon_{i}\prec w_{l}\ \vee\ \varepsilon_{i}\succ\vec{w}$ for $1\leq i\leq k+1$, then

[TABLE]

\Romannum

2. The inverse of map $\psi_{l,1^{k+1}}$:

$\displaystyle\psi^{1}_{l,1^{k+1}}:M^{U}_{l,1^{k+1}}\to P^{U}_{l}\times E_{k+1}^{U};$

$\displaystyle\psi^{2}_{l,1^{k+1}}:M^{U}_{l+1,1^{k}}\to P^{U}_{l}\times E_{k+1}^{U}.$

Let $(\vec{w}\ ;\vec{\varepsilon}\ )\in M^{U}_{l,1^{k+1}}$ and $(\vec{w}\ ,z;\vec{\nu}\ )\in M^{U}_{l+1,1^{k}}$.

For $(\vec{w}\ ;\vec{\varepsilon}\ )\in M^{U}_{l,1^{k+1}}$, we have:

\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\psi^{1}_{l,1^{k+1}}(\vec{w}\ ;\vec{\varepsilon}\ )=(\vec{w}\ |\vec{\varepsilon}\ )\in P^{U}_{l}\times E_{k+1}^{U}.

If $\exists\ i$, s.t. $\varepsilon_{i}\nprec w_{l}\ \wedge\ \varepsilon_{i}\nsucc\vec{w}$, then define

[TABLE]

then we have

[TABLE]

For $(\vec{w}\ ,z;\vec{\nu}\ )\in M^{U}_{l+1,1^{k}}$, we define

$j=\min(i\ |1\leq i\leq k,z\prec\nu_{i}),$

then we have

\displaystyle\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\psi^{2}_{l,1^{k+1}}(\vec{w}\ ,z;\vec{\nu}\ )=

\displaystyle\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}=(\vec{w}\ |\nu_{1},...,\nu_{j-1},z,\nu_{j},..,\nu_{k})\in P^{U}_{l}\times E_{k+1}^{U}.

This completes the proof. ∎

Given a correct $\vec{w}\in P^{U}_{l}$, let

[TABLE]

Theorem 3.7**.**

For natural $l\geq 2$, let

[TABLE]

Then,

[TABLE]

Remark 3.8**.**

According to Remark 2.4, this implies $c_{n-2,2}(U)\geq 0$.

Remark 3.9**.**

There is a slightly more elegant version of $M^{U}_{l,2},$ which we will use in the future:

[TABLE]

Here is an illustration of an element of $M^{U}_{l,2}$:

Proof.

We can write

[TABLE]

The conditions in (3) have the form $A\wedge B$. We begin with a few remarks.

Observe that the conditions of $A$ and $B$ are mutually exclusive, so we can consider the two statements independently. 2. 2.

Define $\tau=\max\{i\leq l|\;q_{1}\nprec w_{i}\vee w_{i}\sim w_{i+1}\}$. Note that it could happen that $q_{1}\prec\theta$, but clearly $\tau\geq\theta$. 3. 3.

For $\vec{u}\in P^{U}_{l+2}$, let $U_{l-1}=(u_{1},...,u_{l-1})$ and $\breve{\theta}=\max(\{i<l|\;u_{i}\sim u_{i+1}\}).$

To prove the theorem, we consider the following formula

[TABLE]

and construct two injective maps.

{Parallel}

[v]0.480.48 \ParallelLText \Romannum1. We define

[TABLE]

as follows:

Let $(\vec{w},q_{0},q_{1})\in P^{U}_{l,2}\setminus M^{U}_{l,2}.$

If $\vec{w}\prec q_{0}$, then we define

[TABLE]

which is in $P^{U}_{l+2},$ since

[TABLE]

\ParallelRText\Romannum

2.

[TABLE]

the inverse of map $\psi$:

Let $(u_{1},...,u_{l+2})\in P^{U}_{l+2}.$

If $u_{l+2}\succ U_{l-1}$ and $u_{l+2}\succ u_{l+1}$, then

[TABLE]

\ParallelPar

{Parallel}

[v]0.480.48 \ParallelLText

$q_{0}\prec w_{\tau}$ implies $A$, and implies $\tau>\theta$; then we define

[TABLE]

Here and below, $\tau+1$ might be equal to $l+1$, in which case we simply omit $w_{\tau+1}$.

\ParallelRText

if $u_{\breve{\theta}+1}\prec u_{\breve{\theta}-1}$, then

[TABLE]

Here and below, $\breve{\theta}+1$ might be equal to $l+2$, in which case we simply omit $u_{\breve{\theta}+2}$.

\ParallelPar

{Parallel}

[v]0.480.48

\ParallelLText

Finally, if $\vec{w}\nprec q_{0}$ and $q_{0}\nprec w_{\tau}$, then

[TABLE]

Here and below, $\tau+1$ might be equal to $l+1$, in which case we simply omit $w_{\tau+1}$.

\ParallelRText

if $u_{\breve{\theta}+1}\nprec u_{\breve{\theta}-1}$, then

[TABLE]

Here and below, $\breve{\theta}+1$ might be equal to $l+2$, in which case we simply omit $u_{\breve{\theta}+2}$.

\ParallelPar

∎

To find a combinatorial interpretation of $m^{U}_{l,2,1}$, we construct a bijection between the right and left hand sides of the following equality:

[TABLE]

This formula and its proof are similar to the previous one.

Theorem 3.10**.**

For natural $l$, let

[TABLE]

Then,

[TABLE]

Remark 3.11**.**

According to Remark 2.4, this implies $c_{n-3,2,1}(U)\geq 0$.

Let us explain the meaning of $M_{l,2,1}^{U}$. This theorem states that in addition to combinations of pairwise comparable corrects of lengths $l$, 2 and 1 we have two more cases:

Proof.

To prove Theorem 3.10 using Formula 4, we construct the maps $\varphi_{l|2,1}$ and $\psi_{l|2,1}$.

\Romannum

1. We construct the map from the left hand side to the right hand side

[TABLE]

Let us take

[TABLE]

Let

[TABLE]

We will use this $\theta$ for $w$ on the right hand side as well.

If $z\succ\vec{q}$.

If $z\succ\vec{w}$.

If $(\vec{w}\ ;\vec{q}\ )\in M^{U}_{l,2}$, then

[TABLE]

\Romannum

2. We construct

$\psi^{1}_{l|2,1}:M^{U}_{l+2,1}\to P^{U}_{l}\times M^{U}_{2,1},$

$\psi^{2}_{l|2,1}:M^{U}_{l+1,2}\to P^{U}_{l}\times M^{U}_{2,1},$

$\psi^{3}_{l|2,1}:M^{U}_{l,2,1}\to P^{U}_{l}\times M^{U}_{2,1}.$

We take

$(\vec{u}\ ;\xi)\in M^{U}_{l+2,1};$

$(\vec{w},\xi\ ;\ q_{0},q_{1})\in M^{U}_{l+1,2};$

$(\vec{w}\ ;q_{0},q_{1};z)\in M^{U}_{l,2,1}.$

Let $\breve{\theta}=\max(\{i<l+2|\ u_{i}\sim u_{i+1}\})$.

If $(\vec{w}\ ;q_{0},q_{1};z)\in M^{U}_{l,2,1}$ and $z\succ\vec{q}$ and $z\succ\vec{w}$, then

\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\psi^{3}_{l|2,1}(\vec{w}\ ;q_{0},q_{1};z)=(\vec{w}\ |\ q_{0},q_{1}\ ;z).

In this case we have 2 illustrations:

If $(\vec{w}\ |\vec{q}\ )\notin M^{U}_{l,2}$, then using

[TABLE]

we have

[TABLE]

If $(\vec{u}\ ;\xi)\in M^{U}_{l+2,1}$ and $\xi\succ\vec{u}$, then using

$\psi_{l|2}(\vec{u}\ )\in M^{U}_{l,2}$

we have

\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\psi^{1}_{l|2,1}(\vec{u}\ ;z)=(\psi_{l|2}(\vec{u}\ );\ \xi).

Here, we provide illustrations for the right hand side, see Theorem 3.7, where $\phi_{l|2}$ and $\psi_{l|2}$ are defined, for more details.

Note that the picture above illustrates most of the cases, except the special one, when $u_{l+2}\leavevmode\nobreak\ \succ\leavevmode\nobreak\ u_{l+1}$, $u_{l+2}\succ J_{l-1}$ and $u_{l+2}\sim u_{l}$. In this case map $\psi_{l|2}$ takes out $u_{l+2}$ and $u_{l}$:

If $z\nprec w_{l}$ and $z\nsucc\vec{w}$, then denote

[TABLE]

If $\hat{w}_{\theta}\succ q_{0}$ and $\hat{w}_{\theta+1}\succ q_{1}$, then

[TABLE]

For $(\vec{w},\xi\ ;q_{0},q_{1})\in M^{U}_{l+1,2}$, such that

($\xi\succ w_{l},$ $w_{\theta}\succ q_{0},$ $w_{\theta+1}\succ q_{1},$ and $\xi\succ q_{0}$),

or

($\xi\sim w_{l},$ $w_{l}\succ q_{0},$ $\xi\succ q_{1},$ and $\xi\succ q_{0}$),

we have:

\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\psi^{2}_{l|2,1}(\vec{w},\xi\ ;q_{0},q_{1})=(\vec{w}\ |q_{0},q_{1};\xi).

If $\hat{w}_{\theta+1}\nsucc q_{1}$ or $\hat{w}_{\theta}\nsucc q_{0}$, we take

[TABLE]

(note that $\tau\geq\theta$), and insert $q_{0}$ or $q_{1}$ after it:

If $q_{0}\prec w_{\tau}$, then

[TABLE]

4

For $(\vec{u}\ ;\xi)\in M^{U}_{l+2,1},$ such that $\xi\prec u_{l+2}$, define

$\eta=\max(0,\{i\geq\breve{\theta}|u_{i}\sim\xi\}).$

If $\eta>0$, then we take out $u_{\eta},\ u_{l+2}$ and $\xi$:

If $\eta=\breve{\theta}+1$ and $u_{\breve{\theta}}\succ\xi$, then

\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\psi^{1}_{l|2,1}(\vec{u}\ ;\xi)=(u_{1},...,u_{\eta-1},u_{\eta+1},...,u_{l+1}\ |\xi,u_{\eta};u_{l+2}).

If $q_{0}\nprec s_{\tau}$, then

[TABLE]

If ($\eta=\theta$ and $\xi\prec u_{\theta+1}$) or ($\eta=\theta+1$ and ($u_{\theta}\sim u_{\breve{\theta}+2}$ or $u_{\breve{\theta}}\nsucc\xi$ ) or $\eta>\breve{\theta}+1$, then

\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\psi^{1}_{l|2,1}(\vec{u}\ ;\xi)=(u_{1},...,u_{\eta-1},u_{\eta+1},...,u_{l+1}\ |u_{\eta},\xi\ ;u_{l+2}).

Here we have a special case when $\tau=\theta$. Note that it is possible that $w_{\tau}\succ q_{1}$:

If $z\prec w_{l}$.

If $w_{\theta+1}\succ q_{1}$ and $w_{\theta}\succ q_{0}$, then

[TABLE]

If $(\vec{w}\ ;q_{0},q_{1};z)\in M^{U}_{l,2,1}$ and $z\succ\vec{q},$ and $z\prec w_{l},$ and $w_{\theta+1}\succ q_{1}$ and $w_{\theta}\succ q_{0}$, then

\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\psi^{3}_{l|2,1}(\vec{w}\ ;q_{0},q_{1};z)=(\vec{w}\ |q_{0},q_{1};z).

If $w_{\theta+1}\nsucc q_{1}$ or $w_{\theta}\nsucc q_{0}$, then Let

[TABLE]

if $\gamma>\theta$ and $\gamma>\tau$ (i.e. $\exists\ w_{\gamma}\sim z$), then

[TABLE]

This is the first type exceptional element, shown before on the Figure 7 and on the picture below:

If $(\vec{w}\ ;q_{0},q_{1};z)\in M^{U}_{l,2,1}$ is the first type exceptional element , then

\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\psi^{3}_{l|2,1}(\vec{w}\ ;q_{0},q_{1};z)=(\vec{w}\ |q_{0},q_{1};z).

Otherwise (i.e. if $\gamma=0$ or $\gamma<\tau$), we have:

[TABLE]

For $(\vec{u}\ ;\xi)\in M^{U}_{l+2,1}$, such that

$\xi\prec u_{l+2}\text{ and }\eta=0,$

what implies

$(\xi\succ u_{\theta+1}\text{ and }\xi\succ u_{\theta}),$

we have

\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\psi^{2}_{l|2,1}(\vec{u}\ ;\xi)=(\psi_{l|2}(\vec{u}\ );\xi).

If $q_{1}\succ z$ and $q_{0}\sim z$.

If $q_{0}\succ\vec{w}$

If $z\succ\vec{w}$, then

[TABLE]

If For $(\vec{w}\ ;q_{0},q_{1};z)\in M^{U}_{l,2,1}$, such that

$q_{1}\succ z\text{ and }q_{0}\sim z\text{ and }z\succ\vec{w},$

we have

\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\psi^{3}_{l|2,1}(\vec{w}\ ;q_{0},q_{1};z)=(\vec{w}\ |q_{0},q_{1};z).

If $z\nsucc\vec{w}$ and $z\nprec w_{l}$, then

[TABLE]

For $(\vec{w},\xi\ ;q_{0},q_{1})\in M^{U}_{l+1,2}$, such that $q_{0}\succ\vec{w},$ $q_{0}\sim\xi$ and $q_{1}\succ\xi$, we have

\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\psi^{2}_{l|2,1}(\vec{w},\xi\ ;q_{0},q_{1})=(\vec{w}\ |q_{0},q_{1};\xi).

If $q_{0}\nsucc\vec{w}$ and $q_{0}\nprec w_{l}$, then

[TABLE]

For $(\vec{u}\ ;\xi)\in M^{U}_{l+2,1}$, such that

$u_{l+1}\sim\xi$, $u_{l+1}\sim u_{l+2}$, and $u_{l+2}\succ\xi$, we have

\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\psi^{2}_{l|2,1}(\vec{u}\ ;\xi)=(u_{1},...,u_{l}|u_{l+1},u_{l+2}\;\xi).

If $q_{0}\prec w_{l}$.

If $q_{1}\sim w_{l}$, then

[TABLE]

For $(\vec{w},\xi\ ;q_{0},q_{1})\in M^{U}_{l+1,2}$, such that

$\xi\sim w_{l}\text{ and }\xi\sim q_{0}\text{ and }q_{0}\prec w_{l}\text{ and }q_{1}\prec\xi,$

we have

\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\psi^{2}_{l|2,1}(\vec{w},\xi\ ;q_{0},q_{1})=(\vec{w}\ |q_{0},\xi\ ;q_{1}).

If $q_{1}\prec w_{l}$ and $w_{\theta}\succ q_{0}$ and $w_{\theta+1}\succ q_{1}$, then

[TABLE]

For $(\vec{w}\ ;q_{0},q_{1};z)\in M^{U}_{l,2,1}$, such that

$w_{\theta}\succ q_{0}\text{ and }w_{\theta+1}\succ q_{1}\text{ and }q_{0}\sim z\text{ and }q_{1}\succ z$, we have

\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\psi^{3}_{l|2,1}(\vec{w}\ ;q_{0},q_{1};z)=(\vec{w}\ |q_{0},q_{1};z).

If $q_{1}\prec w_{l}$ and ($w_{\theta}\nsucc q_{0}$ or $w_{\theta+1}\nsucc q_{1}$), let

[TABLE]

If $q_{0}\prec w_{\tau}$ ($\Rightarrow\ q_{1}\sim w_{\tau}$), then

[TABLE]

This case is isomorphic to the second exceptional element type, shown below:

If $(\vec{w}\ ;q_{0},q_{1};z)\in M^{U}_{l,2,1}$ has the second

exceptional element type, then

\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\psi^{3}_{l|2,1}(\vec{w}\ ;q_{0},q_{1};z)=(\vec{w}\ |q_{0},q_{1};z).

If $q_{0}\nprec w_{\tau}$, then

[TABLE]

The following 3 pictures illustrate this case:

For $(\vec{u}\ ;\xi)\in M^{U}_{l+2,1}$, such that $\xi\prec u_{l+2}$, $\eta=\theta$ and $\xi\succ u_{\theta+1}$, we have

\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\psi^{1}_{l|2,1}(\vec{u}\ ;\xi)=(u_{1},...,u_{\theta-1},u_{\theta+2},...,u_{l+2}\ |u_{\theta},\xi\ ;u_{\theta+1}).

As a reminder,

$\displaystyle\breve{\theta}=\max(\{i<l+2|\ u_{i}\sim u_{i+1}\}),$

$\displaystyle\eta=\max(0,\{i\geq\theta|u_{i}\sim\xi\}).$

If $q_{1}\succ z$ and $q_{0}\succ z$.

This is the easiest case, since we insert $\vec{q}$ and $z$ independently.

If $z\succ\vec{w}$, then

[TABLE]

For $(\vec{w}\ ;q_{0},q_{1};z)\in M^{U}_{l,2,1}$, such that

$z\succ\vec{w}$, $q_{0}\succ z$ and $q_{1}\succ z$,

we have

\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\psi^{3}_{l|2,1}(\vec{w}\ ;q_{0},q_{1};z)=(\vec{w}\ |q_{0},q_{1};z).

If $z\nprec w_{l}$ and $z\nsucc\vec{w}$ ($\Rightarrow q_{0}\nprec w_{l}$)

If $q_{0}\succ\vec{w}$, then

[TABLE]

If $(\vec{w},\xi\ ;q_{0},q_{1})\in M^{U}_{l+1,2}$,

and $q_{0}\succ\vec{w}$, and $q_{1}\succ\xi$, then

\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\psi^{2}_{l|2,1}(\vec{w},\xi\ ;q_{0},q_{1})=(\vec{w}\ |q_{0},q_{1};\xi).

If $q_{0}\nsucc\vec{w}$, then

[TABLE]

For $(\vec{u}\ ;\xi)\in M^{U}_{l+2,1}$, such that

$\xi\prec u_{l+2}\text{ and }\xi\prec u_{l+2},$

what implies

$(\xi\prec u_{\theta+1}\text{ and }\xi\prec u_{\theta}),$

we have

\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\psi^{2}_{l|2,1}(\vec{u}\ ;\xi)=(\psi_{l|2}(\vec{u}\ );\xi)=(u_{1},...,u_{l}|u_{l+1},u_{l+2};\xi).

If $z\prec w_{l}$

If $(\vec{w}\ ;\vec{q}\ )\in M_{l,2}^{U}$, then

[TABLE]

If $(\vec{w}\ ;q_{0},q_{1};z)\in M^{U}_{l,2,1}$, and

$q_{0}\succ z,\ q_{1}\succ z\text{ and }w_{l}\succ z,\text{ then}$

\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\psi^{3}_{l|2,1}(\vec{w}\ ;\vec{q}\ ;z)=(\vec{w}\ |\vec{q}\ ;z).

If $(\vec{w}\ |\vec{q}\ )\notin M_{l,2}^{U}$, then

[TABLE]

If $(\vec{u}\ |\xi)\in M^{U}_{l+2,1}$, $\xi\prec u_{l+2}$ and $\eta=0$, what implies

$(\xi\prec u_{\theta+1}\text{ and }\xi\prec u_{\theta}),$

then we have

\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\psi^{2}_{l|2,1}(\vec{u}\ ;\xi)=(\psi_{l|2}(\vec{u}\ );\xi).

It is easy to see that by construction the left hand side fully describes the set $P^{U}_{l}\times M^{U}_{2,1}$. Now, we check that the right hand side coincide with $M_{l+2,1}^{U}\sqcup M_{l+1,2}^{U}\sqcup M^{U}_{l,2,1}$:

• •

Let $(\vec{u}\ ;\xi)\in M^{U}_{l+2,1}$. Then the following cases from the right hand side clearly describe $M^{U}_{l,2,1}$:

$\xi\succ\vec{u};$ 2. 4.

$\xi\prec u_{l+2}$ and $\eta>0$; 3. 7.

$\xi\prec u_{l+2}$, $\eta=0$ and $(\xi\succ u_{\theta+1}\text{ and }\xi\succ u_{\theta})$; 4. 19.

$\xi\prec u_{l+2}$, $\eta=0$ and $(\xi\prec u_{\theta+1}\text{ and }\xi\prec u_{\theta})$.

• •

Let $(\vec{w}\ ,\xi;q_{0},q_{1})\in M^{U}_{l+1,2}$. Then the following cases from the right hand side clearly describe $M^{U}_{l,2,1}$.:

($\xi\succ w_{l},$ $w_{\theta}\succ q_{0},$ $w_{\theta+1}\succ q_{1},$ and $\xi\succ q_{0}$), or ($\xi\sim w_{l},$ $w_{l}\succ q_{0},$ $\xi\succ q_{1},$ and $\xi\succ q_{0}$); 2. 11.

$(\xi\sim w_{l},$ $w_{l}\succ q_{0},$ $\xi\succ q_{1},$ and $\xi\sim q_{0});$ 3. 9.

$q_{0}\succ\vec{w}$, $q_{1}\succ\xi$ and $q_{0}\sim\xi;$ 4. 16.

$q_{0}\succ\vec{w}$, $q_{1}\succ\xi$ and $q_{0}\succ\xi.$

• •

Let $(\vec{w}\ ;q_{0},q_{1};z)\in M^{U}_{l,2,1}$. Then the following cases from the right hand side clearly describe $M^{U}_{l,2,1}$.:

$z\succ\vec{q}$ and $z\succ{w}$; 2. 5.

$z\succ\vec{q}$ and $z\prec{w_{l}}$; 3. 8.

$q_{0}\sim z$, $q_{1}\succ z$ and $z\succ\vec{w}$; 4. 12.

$q_{0}\sim z$, $q_{1}\succ z$ and $(w_{\theta}\succ q_{0}\text{ and }w_{\theta+1}\succ q_{1})$; 5. 15.

$q_{0}\succ z$, $q_{1}\succ z$ and $z\prec w_{l}$; 6. 18.

$q_{0}\succ z$, $q_{1}\succ z$ and $z\prec w_{l}$; 7. 6.

First exceptional type element; 8. 13.

Second exceptional type element.

Since every number from 1 to 19 was used exactly once, this completes the proof.

∎

Theorem 3.12**.**

For natural numbers $l$ and $k$, let

[TABLE]

Then

[TABLE]

Remark 3.13**.**

According to Remark 2.4, this implies $c_{2^{k},1^{n-2k}}(U)\geq 0$.

The proof is omitted and can be found in [9].