Combinatorial results for certain semigroups of order-decreasing partial isometries of a finite chain

TL;DR

This paper explores the structure and size of specific subsemigroups of the symmetric inverse semigroup related to order-decreasing partial isometries on finite chains, providing combinatorial insights into their cardinalities.

Contribution

It introduces new combinatorial results for the cardinalities of certain semigroups of order-decreasing partial isometries and their order-preserving variants, advancing understanding of their algebraic structure.

Findings

Cardinalities of ${ m DDP}_n$ and ${ m ODDP}_n$ computed

Equivalence classes on these semigroups characterized

Semigroup orders explicitly determined

Abstract

Let ${\cal I}_n$ be the symmetric inverse semigroup on $X_n = \{1, 2, \ldots , n\}$ and let ${\cal DDP}_n$ and ${\cal ODDP}_n$ be its subsemigroups of order-decreasing partial isometries and of order-preserving and order-decreasing partial isometries of $X_n$, respectively. In this paper we investigate the cardinalities of some equivalences on ${\cal DDP}_n$ and ${\cal ODDP}_n$ which lead naturally to obtaining the order of the semigroups

Equations

$${\mathcal{DDP}}_n=\{\alpha \in {\mathcal{DP}}_n:(\forall x\in \mathrm{Dom}\alpha )x\alpha \le x\}.$$

$${\mathcal{O}\mathcal{DDP}}_n=\{\alpha \in {\mathcal{DDP}}_n:(\forall x,y\in \mathrm{Dom}\alpha )x\le y⟹x\alpha \le y\alpha \}$$

$$F(\alpha )=\{x\in X_n:x\alpha =x\}.$$

$$F(n;p_i)=\mid \{\alpha \in S:h(\alpha )=\mid \mathrm{Im}\alpha \mid =i\}\mid ,$$

$$F(n;m_i)=\mid \{\alpha \in S:f(\alpha )=i\}\mid$$

$$F(n;p)$$

$$\{\alpha \in {\mathcal{O}\mathcal{DDP}}_n:h(\alpha )=0\text{mbox}or1\}=\{\alpha \in {\mathcal{DDP}}_n^{\ast }:h(\alpha )=0\text{mbox}or1\}$$

$$\sum_{i\geq 0}{{n-1-2i\choose 2}}=\left\{\begin{array}[]{ll}\frac{(n+1)(n-1)(2n-3)}{24},&\,\mbox{if}\,\,$n$\,\,\mbox{is odd};\\ \frac{n(n-2)(2n+1)}{24},&\,\mbox{if}\,\,$n$\,\,\mbox{is even}.\end{array}\right.$$

$$F(n;p_{2})=\left\{\begin{array}[]{ll}\frac{(n+1)(n-1)(2n-3)}{24},&\,\mbox{if}\,\,$n$\,\,\mbox{is odd};\\ \frac{n(n-2)(2n+1)}{24},&\,\mbox{if}\,\,$n$\,\,\mbox{is even}.\end{array}\right.$$

$$F(n;p_2)$$

$$F(m+2;p)$$

$$2\sum _{i=1}^{a-1}{2}^{i-1}+2^{a-1}=2(2^{a-1}-1)+2^{a-1}=3.2^{a-1}-2$$

$$2\sum _{i=1}^a{2}^{i-1}=2(2^a-1)=2^{a+1}-2$$

$$a_n=\mid {\mathcal{DDP}}_n\mid =3a_{n-1}-2a_{n-2}-2^{\lfloor \frac{n}{2}\rfloor }+n+1,$$

$$\begin{array}[]{|c|c|c|c|c|c|c|c|c|c|}\hline\cr\,\,\,\,\,n{\backslash}p&0&1&2&3&4&5&6&7&\sum F(n;p)=\mid{\cal DDP}^{*}_{n}\mid\\ \hline\cr 0&1&&&&&&&&1\\ \hline\cr 1&1&1&&&&&&&2\\ \hline\cr 2&1&3&0&&&&&&4\\ \hline\cr 3&1&6&1&0&&&&&8\\ \hline\cr 4&1&10&3&0&0&&&&14\\ \hline\cr 5&1&15&7&1&0&0&&&24\\ \hline\cr 6&1&21&13&3&0&0&0&&38\\ \hline\cr 7&1&28&22&8&1&0&0&0&60\\ \hline\cr\end{array}$$

$$\begin{array}[]{|c|c|c|c|c|c|c|c|c|c|}\hline\cr\,\,\,\,\,n{\backslash}p&0&1&2&3&4&5&6&7&\sum F(n;p)=\mid{\cal DDP}_{n}\mid\\ \hline\cr 0&1&&&&&&&&1\\ \hline\cr 1&1&1&&&&&&&2\\ \hline\cr 2&1&3&1&&&&&&5\\ \hline\cr 3&1&6&5&1&&&&&13\\ \hline\cr 4&1&10&13&5&1&&&&30\\ \hline\cr 5&1&15&27&16&6&1&&&66\\ \hline\cr 6&1&21&48&38&21&7&1&&137\\ \hline\cr 7&1&28&78&78&57&28&8&1&279\\ \hline\cr\end{array}$$

$$g(\mathrm{Im}\alpha )=(\mid a_2\alpha -a_1\alpha \mid ,\mid a_3\alpha -a_2\alpha \mid ,\dots ,\mid a_p\alpha -a_{p-1}\alpha \mid )$$

$$g^R(\mathrm{Im}\alpha )=(\mid a_p\alpha -a_{p-1}\alpha \mid ),\dots ,\mid a_3\alpha -a_2\alpha \mid ,\mid a_2\alpha -a_1\alpha \mid ),$$

$$g(\mathrm{Im}\alpha )=(d_1,d_2,\dots ,d_{p-1})\text{mbox}and{g}^R(\mathrm{Im}\alpha )=(d_{p-1},d_{p-2},\dots ,d_1).$$

$$\alpha=\pmatrix{1&2&4&7&8\cr 3&4&6&9&10},\beta=\pmatrix{2&4&7&8\cr 10&8&5&4}\in{\cal DP}_{10}$$

$$\sum _{m=p}^n\left(\genfrac{}{}{0ex}{}{m}{p}\right)=\left(\genfrac{}{}{0ex}{}{n+1}{p+1}\right).$$

$$\displaystyle g(\mathop{\rm Im}\nolimits\alpha)=\left\{\begin{array}[]{ll}g(\mathop{\rm Im}\nolimits\beta);\,\,\mbox{or}\\ g^{R}(\mathop{\rm Im}\nolimits\beta),\,\,\mbox{if}\,\,p\leq a_{p}-a_{1}\leq(n-1)/2.\end{array}\right.$$

$$(\alpha ,\beta )\in {\mathcal{D}}^{\ast }\text{mbox}ifandonlyifg(\mathrm{Im}\alpha )=g(\mathrm{Im}\beta ).$$

$$a_p-a_1=\sum _{i=1}^{p-1}(a_{i+1}-a_i)=\sum _{i=1}^{p-1}{d}_i,$$

$$(d_1,d_2,\dots ,d_p)=(d_1,d_2,\dots ,d_p{)}^R=(d_p,d_{p-1},\dots ,d_1).$$

$$g(n,p)=\frac{d(n-1,p-1)-d_s(n-1,p-1)}{2}.$$

$$B(n,p)$$

$$d_n$$

Full text

MSC2010: 20M18, 20M20, 05A10, 05A15.

1 Introduction and Preliminaries

Let $X_{n}=\{1,2,\ldots,n\}$ and ${\cal I}_{n}$ be the partial one-to-one transformation semigroup on $X_{n}$ under composition of mappings. Then ${\cal I}_{n}$ is an inverse semigroup (that is, for all $\alpha\in{\cal I}_{n}$ there exists a unique $\alpha^{\prime}\in{\cal I}_{n}$ such that $\alpha=\alpha\alpha^{\prime}\alpha$ and $\alpha^{\prime}=\alpha^{\prime}\alpha\alpha^{\prime}$). The importance of ${\cal I}_{n}$ (more commonly known as the symmetric inverse semigroup or monoid) to inverse semigroup theory may be likened to that of the symmetric group ${\cal S}_{n}$ to group theory. Every finite inverse semigroup $S$ is embeddable in ${\cal I}_{n}$, the analogue of Cayley’s theorem for finite groups. Thus, just as the study of symmetric, alternating and dihedral groups has made a significant contribution to group theory, so has the study of various subsemigroups of ${\cal I}_{n}$, see for example [3, 5, 6, 10, 14, 19, 20].

A transformation $\alpha\in{\cal I}_{n}$ is said to be order-preserving (order-reversing) if $(\forall x,y\in\ \mathop{\rm Dom}\nolimits\alpha)\ x\leq y\;\Longrightarrow\;x\alpha\leq y\alpha\ (x\alpha\geq y\alpha)$ and, an isometry (or distance-preserving) if ($\forall x,y\in\mathop{\rm Dom}\nolimits\alpha)\mid x-y\mid=\mid x\alpha-y\alpha\mid$. We shall denote by ${\cal DP}_{n}$ and ${\cal ODP}_{n}$, the semigroups of partial isometries and of order-preserving partial isometries of an $n-$chain, respectively. Eventhough semigroups of partial isometries on more restrictive but richer mathematical structures have been studied by Wallen [21], and Bracci and Picasso [4] the study of the corresponding semigroups on chains was only initiated recently by Al-Kharousi et al. [1, 2]. A little while later, Kehinde et al. [13] studied ${\cal DDP}_{n}$ and ${\cal ODDP}_{n},$ the order-decreasing analogues of ${\cal DP}_{n}$ and ${\cal ODP}_{n}$, respectively.

Analogous to Al-Kharousi et al. [2], this paper investigates the combinatorial properties of ${\cal DDP}_{n}$ and ${\cal ODDP}_{n}$, thereby complementing the results in Kehinde et al. [13] which dealt mainly with the algebraic and rank properties of these semigroups. In this section we introduce basic definitions and terminology as well as quote some elementary results from Section 1 of Al-Kharousi et al. [1] and Kehinde et al. [13] that will be needed in this paper. In Section 2 we obtain the cardinalities of two equivalences defined on ${\cal ODDP}_{n}$ and ${\cal DDP}_{n}$. These equivalences lead to formulae for the orders of ${\cal ODDP}_{n}$ and ${\cal DDP}_{n}$ as well as new triangles of numbers that were as a result of this work recently recorded in [18].

For standard concepts in semigroup and symmetric inverse semigroup theory, see for example [12, 16]. In particular $E(S)$ denotes the set of idempotents of $S$. Let

[TABLE]

be the subsemigroup of ${\cal I}_{n}$ consisting of all order-decreasing partial isometries of $X_{n}$. Also let

[TABLE]

be the subsemigroup of ${\cal DDP}_{n}$ consisting of all order-preserving and order-decreasing partial isometries of $X_{n}$. Then we have the following result.

Lemma 1.1

${\cal DDP}_{n}$* and ${\cal ODDP}_{n}$ are subsemigroups of ${\cal I}_{n}$.*

Remark 1.2

${\cal DDP}_{n}={\cal DP}_{n}\cap{\cal I}_{n}^{-}$* and ${\cal ODDP}_{n}={\cal ODP}_{n}\cap{\cal I}_{n}^{-}$, where ${\cal I}_{n}^{-}$ is the semigroup of partial one-to-one order-decreasing transformations of $X_{n}$ [19].*

Next, let $\alpha$ be an arbitrary element in ${\cal I}_{n}$. The height or rank of $\alpha$ is $h(\alpha)=\mid\mathop{\rm Im}\nolimits\alpha\mid$, the right [left] waist of $\alpha$ is $w^{+}(\alpha)=max(\mathop{\rm Im}\nolimits\alpha)\,[w^{-}(\alpha)=min(\mathop{\rm Im}\nolimits\alpha)]$, the right [left] shoulder of $\alpha$ is $\varpi^{+}(\alpha)=max(\mathop{\rm Dom}\nolimits\alpha)$  [$\varpi(\alpha)=min(\mathop{\rm Dom}\nolimits\alpha)]$, and fix of $\alpha$ is denoted by $f(\alpha)$, and defined by $f(\alpha)=|F(\alpha)|$, where

[TABLE]

Next we quote some parts of [1, Lemma 1.2] that will be needed as well as state some additional observations that will help us understand more the cycle structure of order-decreasing partial isometries.

Lemma 1.3

Let $\alpha\in{\cal DP}_{n}$. Then we have the following:

• (a)

The map $\alpha$ is either order-preserving or order-reversing. Equivalently, $\alpha$ is either a translation or a reflection.

• (b)

If $f(\alpha)=p>1$ then $f(\alpha)=h(\alpha)$. Equivalently, if $f(\alpha)>1$ then $\alpha$ is a partial identity.

• (c)

If $\alpha$ is order-preserving and $f(\alpha)\geq 1$ then $\alpha$ is a partial identity.

• (d)

If $\alpha$ is order-preserving then it is either strictly order-decreasing

($x\alpha<x$* for all $x$ in $\mathop{\rm Dom}\nolimits\alpha$) or strictly order-increasing ($x\alpha>x$ for all $x$ in $\mathop{\rm Dom}\nolimits\alpha$) or a partial identity.*

• (e)

If $F(\alpha)=\{i\}$ (for $1\leq i\leq n$) then for all $x\in\mathop{\rm Dom}\nolimits\alpha$ we have that $x+x\alpha=2i$.

• (f)

If $\alpha$ is order-decreasing and $i\in F(\alpha)$ ($1\leq i\leq n)$* then for all $x\in\mathop{\rm Dom}\nolimits\alpha$ such that $x<i$ we have $x\alpha=x$.*

• (g)

If $\alpha$ is order-decreasing and $F(\alpha)=\{i\}$ then $\mathop{\rm Dom}\nolimits\alpha\subseteq\{i,i+1,\ldots,n\}$.

2 Combinatorial results

For a nice survey article concerning combinatorial problems in the symmetric inverse semigroup and some of its subsemigroups we refer the reader to Umar [20].

As in Umar [20], for natural numbers $n\geq p\geq m\geq 0$ and $n\geq i\geq 0$ we define

[TABLE]

[TABLE]

where $S$ is any subsemigroup of ${\cal I}_{n}$. From [2, Proposition 2.4] we have

Theorem 2.1

Let $S={\cal ODP}_{n}.$ Then $F(n;p)=\frac{(2n-p+1)}{p+1}{n\choose p}$, where $n\geq p\geq 1$.

We now have

Proposition 2.2

Let $S={\cal ODDP}_{n}$. Then $F(n;p)=\pmatrix{n+1\cr p+1}$, where $n\geq p\geq 1$.

Proof.   By virtue of Lemma 1.3[d] and Theorem 2.1 we see that

[TABLE]

$\Box$

The proof of the next lemma is routine using Proposition 2.2

Lemma 2.3

Let $S={\cal ODDP}_{n}$. Then $F(n;p)=F(n-1;p-1)+F(n-1;p)$, for all $n\geq p\geq 2$.

Theorem 2.4

$\mid{\cal ODDP}_{n}\mid=2^{n+1}-(n+1).$**

Proof.   It is enough to observe that $\mid{\cal ODDP}_{n}\mid=\sum_{p=0}^{n}F(n;p)$.

Lemma 2.5

Let $S={\cal ODDP}_{n}$. Then $F(n;m)={n\choose m}$, for all $n\geq m\geq 1$.

Proof.   It follows directly from Lemma 1.3[b,c] and the fact that all idempotents are necessarily order-decreasing. $\Box$

Proposition 2.6

Let  $U_{n}=\{\alpha\in{\cal ODDP}_{n}:f(\alpha)=0\}$. Then $\mid{U_{n}}\mid=\\ \mid{\cal ODDP}_{n-1}\mid$.

Proof.   The proof is similar to that of [19, Theorem 4.3]. $\Box$

Remark 2.7

The triangles of numbers $F(n;p)$ and $F(n;m)$, have as a result of this work appeared in Sloane [18] as [A184049] and [A184050], respectively.

Now we turn our attention to counting order-reversing partial isometries. First recall from [13, Section3.2(c)] that order-decreasing and order-reversing partial isometries exist only for heights less than or equal to $n/2$. We now have

Lemma 2.8

Let $S={\cal DDP}^{*}_{n}$ be the set of order-reversing partial isometries of $X_{n}$. Then $F(n;p_{0})=1$ and $F(n;p_{1})=\pmatrix{n+1\cr 2}$, for all $n\geq 1$.

Proof.   These follow from the simple observation that

[TABLE]

and Proposition 2.2. $\Box$

Lemma 2.9

*Let $\alpha\in{\cal DDP}^{*}_{n}$. Then for all $p\geq 1$ we have

$F(2p+1,p+1)=1$ and $F(2p,p)=3$.*

Proof.   (i) By Lemma 1.3[f,g] we see that for $i\in\{0,1,\ldots,p\}$, ${p+1+i\choose p+1-i}$ is the unique order-reversing isometry of height $p+1;$ and (ii) for $i\in\{0,1,\ldots,p-1\}$, ${p+i\choose p-i}$, ${p+1+i\choose p-i}$ and ${p+1+i\choose p+1-i}$ are the only order-reversing isometries of height $p$. $\Box$

The following technical lemma will be useful later.

Lemma 2.10

Let $\alpha\in{\cal DDP}^{*}_{n}$. Suppose $\varpi^{+}(\alpha)-r\in\mathop{\rm Dom}\nolimits\alpha$ and $\varpi^{+}(\alpha)-s\notin\mathop{\rm Dom}\nolimits\alpha$ for all $1\leq s<r$. Then $\varpi(\alpha)>r$.

Proof.   By order-reversing we see that $(\varpi^{+}(\alpha))\alpha=w^{-}(\alpha)$ and $(\varpi(\alpha))\alpha=w^{+}(\alpha)$. Thus $\varpi^{+}(\alpha)-r\geq\varpi(\alpha)\;\Longrightarrow\;\varpi^{+}(\alpha)-\varpi(\alpha)\geq r.$ So by isometry we have $w^{+}(\alpha)-w^{-}(\alpha)=\varpi^{+}(\alpha)-\varpi(\alpha)\geq r\;\Longrightarrow\;w^{+}(\alpha)\geq w^{-}(\alpha)+r\;\Longrightarrow\;w^{+}(\alpha)>r\;\Longrightarrow\;\varpi(\alpha)>r,$ as required. $\Box$

Lemma 2.11

Let $S={\cal DDP}^{*}_{n}$. Then $F(n;p)=F(n-2;p-1)+F(n-2;p)$, for all $n\geq p\geq 2$.

Proof.   Let $\alpha\in{\cal DDP}^{*}_{n}$ and $h(\alpha)=p$. Define $A=\{\alpha\in{\cal DDP}^{*}_{n-2}:h(\alpha)=p\}$ and $B=\{\alpha\in{\cal DDP}^{*}_{n-2}:h(\alpha)=p-1\}$. Clearly, $A\cap B=\emptyset$. Define a map $\theta:\{\alpha\in{\cal DDP}^{*}_{n}:h(\alpha)=p\}\rightarrow A\cup B$ by $(\alpha)\theta=\alpha^{\prime}$ where

(i) $x\alpha^{\prime}=x\alpha\,(x\in\mathop{\rm Dom}\nolimits\alpha),$ if $\alpha\in A$. It is clear that $\alpha^{\prime}$ is an order-decreasing isometry and $h(\alpha)=p$;

(ii) if $\{n-1,n\}\subseteq\mathop{\rm Dom}\nolimits\alpha\}$ and $\alpha\in B$, let $\mathop{\rm Dom}\nolimits\alpha^{\prime}=\{x-1:x\in\mathop{\rm Dom}\nolimits\alpha\,\,\mbox{and}\,\,x<n\}$ and $(x-1)\alpha^{\prime}=x\alpha-1\leq x-1$ and so $\alpha^{\prime}$ is order-decreasing and $h(\alpha)=p-1$;

(iii) if $\{n-2,n-1\}\subseteq\mathop{\rm Dom}\nolimits\alpha\}$ and $\alpha\in B$, let $\mathop{\rm Dom}\nolimits\alpha^{\prime}=\{x-1:x\in\mathop{\rm Dom}\nolimits\alpha\,\,\mbox{and}\,\,x<n-1\}$ and $(x-1)\alpha^{\prime}=x\alpha-1\leq x-1$ and so $\alpha^{\prime}$ is order-decreasing and $h(\alpha)=p-1$;

(iv) otherwise, if $\alpha\in B$, let $\mathop{\rm Dom}\nolimits\alpha^{\prime}=\{x-r:x\in\mathop{\rm Dom}\nolimits\alpha\,\,\mbox{and}\,\,x<\varpi^{+}(\alpha)\}$, where $r$ is such that $\varpi^{+}(\alpha)-r\in\mathop{\rm Dom}\nolimits\alpha$ and $\varpi^{+}(\alpha)-s\notin\mathop{\rm Dom}\nolimits\alpha$ for all $1\leq s<r$. Define $(x-r)\alpha^{\prime}=x\alpha-r\leq x-r$ and so $\alpha^{\prime}$ is order-decreasing and Lemma 2.10 ensures that $h(\alpha)=p-1$.

Moreover, in (ii) and (iii), we have $\mid(x-1)\alpha^{\prime}-(y-1)\alpha^{\prime}\mid=\mid(x\alpha-1)-(y\alpha-1)\mid=\mid x\alpha-y\alpha\mid=\mid x-y\mid=\mid(x-1)-(y-1)\mid,$ and in (iv), we have $\mid(x-r)\alpha^{\prime}-(y-r)\alpha^{\prime}\mid=\mid(x\alpha-r)-(y\alpha-r)\mid=\mid x\alpha-y\alpha\mid=\mid x-y\mid=\\ \mid(x-r)-(y-r)\mid.$ Hence $\alpha^{\prime}$ is an isometry.

Also observe that in (ii), we have $\varpi^{+}(\alpha^{\prime})=n-2$; in (iii) we have $\varpi^{+}(\alpha^{\prime})=n-3$; and in (iv) we have $\varpi^{+}(\alpha^{\prime})<n-3$. These observations coupled with the definitions of $\alpha^{\prime}$ ensures that $\theta$ is a bijection.

To show that $\theta$ is onto it is enough to note that we can in a symmetric manner define $\theta^{-1}$ from $A\cup B\rightarrow\{\alpha\in{\cal DDP}^{*}_{n}:h(\alpha)=p\}$. This establishes the statement of the lemma. $\Box$

The next lemma which can be proved by induction, is necessary.

Lemma 2.12

Let $S={\cal DDP}^{*}_{n}$. Then we have the following:

[TABLE]

Lemma 2.13

Let $S={\cal DDP}^{*}_{n}$. Then we have the following:

[TABLE]

Proof.   By applying Lemmas 2.8 and 2.11 sucessively we get

[TABLE]

By iteration the result follows from Lemma 2.12 and the facts that $F(2;p_{2})=0$ and $F(3;p_{2})=1={2\choose 2}$. $\Box$

Proposition 2.14

Let $S={\cal DDP}^{*}_{n}$. Then for all $\lfloor(n+1)/2\rfloor\geq p\geq 1$, we have F(n;p)=\left\{\begin{array}[]{ll}\frac{(n+1)(n-1)(n-3)\cdots(n-2p+3)(2n-3p+3)}{2^{p}(p+1)!},&\,\mbox{if}\,\,n\,\,\mbox{is odd};\\ \frac{n(n-2)(n-4)\cdots(n-2p+2)(2n-p+3)}{2^{p}(p+1)!},&\,\mbox{if}\,\,n\,\,\mbox{is even}.\end{array}\right..

Proof.   (By Induction).

Basis Step: $F(n;p_{1})=\pmatrix{n+1\cr 1+1}=\pmatrix{n+1\cr 2}$ is true by Lemma 2.8 and the observation made in its proof, while the formula for $F(n;p_{2})$ is true by Lemma 2.13.

Inductive Step: Suppose $F(m;p)$ is true for all $\lfloor(n+1)/2\rfloor>m\geq p\geq 1$.

Case 1. If $m$ is odd, consider (using the induction hypothesis)

[TABLE]

which is the formula for $F(m+2;p)$ when $m$ is odd.

Case 2. If $m$ is even, consider (using the induction hypothesis)

[TABLE]

which is the formula for $F(m+2;p)$ when $m$ is even. $\Box$

Proposition 2.15

Let $S={\cal DDP}^{*}_{n}$ and let $b_{n}=\sum_{p\geq 0}{F(n;p)}$. Then for $n\geq 0$, we have

$b_{2n+1}=5\cdot 2^{n+1}-4n-8$; 2. 2.

$b_{2n}=7\cdot 2^{n}-4n-6$.

Proof.   Apply induction and use the fact that $\mid{\cal DDP}^{*}_{n}\mid=\sum_{p=0}^{n}F(n;p)$.

Proposition 2.16

Let $S={\cal DDP}_{n}$. Then

• (1)

*if $n$ is odd and $(n+1)/2\geq p\geq 2$ *

$F(n;p)=\frac{(n+1)(n-1)(n-3)\cdots(n-2p+3)(2n-3p+3)}{2^{p}(p+1)!}+{n+1\choose p+1}$;

• (2)

*if $n$ is even and $n/2\geq p\geq 2$ *

$F(n;p)=\frac{n(n-2)(n-4)\cdots(n-2p+2)(2n-p+3)}{2^{p}(p+1)!}+{n+1\choose p+1}$;

• (3)

if $\lfloor(n+1)/2\rfloor<p$, $F(n;p)={n+1\choose p+1}$.

Proof.   It follows from Propositions 2.2 & 2.14 and Lemmas 1.3[c] & 2.8. $\Box$

Combining Theorem 2.4, Lemmas 1.3[a,c] & 2.9, Proposition 2.15 and the observation made in the proof of Lemma 2.8 we get the order of ${\cal DDP}_{n}$ which we record as a theorem below.

Theorem 2.17

Let ${\cal DDP}_{n}$. Then for all $n\geq 0$ we have

• (1)

$\mid{\cal DDP}_{2n+1}\mid=2^{2n+2}+5\cdot 2^{n+1}-(2n^{2}+9n+12)$;

• (2)

$\mid{\cal DDP}_{2n}\mid=2^{2n+1}+7\cdot 2^{n}-(2n^{2}+7n+8)$.

Lemma 2.18

Let $S={\cal DDP}_{n}$. Then $F(n;m)={n\choose m}$, for all $n\geq m\geq 2$.

Proof.   It follows directly from [13, Lemma 3.18] and the fact that all idempotents are necessarily order-decreasing. $\Box$

Proposition 2.19

Let $S={\cal DDP}_{n}$. Then $F(2n;m_{1})=2^{n+1}-2$ and $F(2n-1;m_{1})=3\cdot 2^{n-1}-2$, for all $n\geq 1$.

Proof.   Let $F(\alpha)=\{i\}$. Then by Lemma 1.3[e], for any $x\in\mathop{\rm Dom}\nolimits\alpha$ we have $x+x\alpha=2i$. Thus, by Lemma 1.3[g], there $2i-2$ possible elements for $\mathop{\rm Dom}\nolimits\alpha:(x,x\alpha)\in\{(i,i),(i+1,i-1),(i+2,i-2),\ldots,(2i-1,1)\}$. However, (excluding $(i,i)$) we see that there are $\sum_{j=0}{i-1\choose j}=2^{i-1}$, possible partial isometries with $F(\alpha)=\{i\}$, where $2i-1\leq n\Longleftrightarrow i\leq(n+1)/2$. Moreover, by symmetry we see that $F(\alpha)=\{i\}$ and $F(\alpha)=\{n-i+1\}$ give rise to equal number of decreasing partial isometries. Note that if $n$ is odd (even) the equation $i=n-i+1$ has one (no) solution. Hence, if $n=2a-1$ we have

[TABLE]

decreasing partial isometries with exactly one fixed point; if $n=2a$ we have

[TABLE]

decreasing partial isometries with exactly one fixed point. $\Box$

Theorem 2.20

Let ${\cal DDP}_{n}$. Then

[TABLE]

with $a_{0}=1$ and $a_{-1}=0$.

Proof.   It follows from Propositions 2.6 & 2.19, Lemma 2.18 and the fact that $\mid{\cal DDP}_{n}\mid=\sum_{m=0}^{n}F(n;m)$. $\Box$

Remark 2.21

The triangle of numbers $F(n;m)$ and sequence $\mid{\cal DDP}_{n}\mid$ have as a result of this work appeared in Sloane [18] as [A184051] and [A184052], respectively. However, the triangles of numbers $F(n;p)$ for ${\cal DDP}_{n}$ and ${\cal DDP^{*}}_{n}$ and the sequence $\mid{\cal DDP}^{*}_{n}\mid$ are as at the time of submitting this paper not in Sloane [18]. For some computed values of $F(n;p)$, see Tables 3.1 and 3.2.

[TABLE]

Table 3.1

[TABLE]

Table 3.2

3 Number of ${\cal D}^{*}$-classes

For the definitions of the Green’s relations (${\cal L},{\cal R}$ and ${\cal D}$) and their starred analogues (${\cal L}^{*},{\cal R}^{*}$ and ${\cal D}^{*}$), we refer the reader to Howie [12] and Fountain [8], (respectively) or Ganyushkin and Mazorchuk [9].

First, notice that from [1, Lemma 2.1] we deduce that number of ${\cal L}$-classes in $K(n,p)=\{\alpha\in{\cal DP}_{n}:h(\alpha)=p\}$ (as well as the number of ${\cal R}$-classes there) is ${n\choose p}$. To describe the ${\cal D}$-classes in ${\cal DP}_{n}$ and ${\cal ODP}_{n}$, first we recall (from [1]) that the gap and reverse gap of the image set of $\alpha$ (with $h(\alpha)=p$) are ordered $(p-1)$-tuples defined as follows:

[TABLE]

and

[TABLE]

where \alpha=\pmatrix{a_{1}&a_{2}&\cdots&a_{p}\cr a_{1}\alpha&a_{2}\alpha&\cdots&a_{p}\alpha} with $1\leq a_{1}<a_{2}<\cdots<a_{p}\leq n.$ Further, let $d_{i}=\mid a_{i+1}\alpha-a_{i}\alpha\mid$ for $i=1,2,\ldots,p-1$. Then

[TABLE]

For example, if

[TABLE]

then $g(\mathop{\rm Im}\nolimits\alpha)=(1,2,3,1),\,g(\mathop{\rm Im}\nolimits\beta)=(2,3,1),\,g^{R}(\mathop{\rm Im}\nolimits\alpha)=(1,3,2,1)$ and $g^{R}(\mathop{\rm Im}\nolimits\beta)=(1,3,2).$ Next, let $d(n,p)$ be the number of distinct ordered $p$-tuples: $(d_{1},d_{2},\ldots,d_{p})$ with $\sum_{i=1}^{p}d_{i}=n$. This is clearly the number of compositions of $n$ into $p$ parts. Thus, we have

Lemma 3.1

[17, p.151]** $d(n,p)={n-1\choose p-1}.$

We shall henceforth use the following well-known binomial identity when needed:

[TABLE]

We take this opportunity to state and prove a result which was omitted in [2].

Theorem 3.2

Let $S={\cal ODP}_{n}$. Then

• (1)

the number of ${\cal D}$-classes in $K(n,p)\,(p\geq 1)$ is ${n-1\choose p-1}$;

• (2)

the number of ${\cal D}$-classes in $S$ is $1+2^{n-1}$.

Proof.

• (1)

It follows from [1, Theorem 2.5]: $(\alpha,\beta)\in{\cal D}$ if and only if $g(\mathop{\rm Im}\nolimits\alpha)=g(\mathop{\rm Im}\nolimits\beta)$; [1, Lemma 3.3]: $p-1\leq\sum_{i=1}^{p-1}d_{i}\leq n-1$; Lemma 3.1; and so the number of ${\cal D}$-classes is $\sum_{i=p-1}^{n-1}d(i,p-1)=\sum_{i=p-1}^{n-1}{i-1\choose p-2}={n-1\choose p-1}.$

• (2)

The number of ${\cal D}$-classes in $S$ is $1+\sum_{p=1}^{n}{n-1\choose p-1}=1+2^{n-1}$.

$\Box$

The following results from [13] will be needed:

Lemma 3.3

[13, Lemma 2.3]** Let $\alpha,\beta\in{\cal DDP}_{n}$ or ${\cal ODDP}_{n}.$  Then

• (1)

$\alpha\leq_{{\cal R^{*}}}\beta$*   if and only if  $\mathop{\rm Dom}\nolimits\alpha\subseteq\mathop{\rm Dom}\nolimits\beta$;*

• (2)

$\alpha\leq_{{\cal L^{*}}}\beta$*  if and only if $\mathop{\rm Im}\nolimits\alpha\subseteq\mathop{\rm Im}\nolimits\beta$;*

• (3)

$\alpha\leq_{{\cal H^{*}}}\beta$*   if and only if  $\mathop{\rm Dom}\nolimits\alpha\subseteq\mathop{\rm Dom}\nolimits\beta$ and $\mathop{\rm Im}\nolimits\alpha\subseteq\mathop{\rm Im}\nolimits\beta$.*

From [13, (3)], for $\alpha,\beta\in{\cal DDP}_{n}$, we have $(\alpha,\beta)\in{\cal D^{*}}$ if and only if

[TABLE]

Similarly, from [13, (4)], for $\alpha,\beta\in{\cal ODDP}_{n}$, we have

[TABLE]

Now a corollary of Theorem 3.2 follows:

Corollary 3.4

Let $S={\cal ODDP}_{n}$. Then

• (1)

the number of ${\cal D}^{*}$-classes in $K(n,p)\,(p\geq 1)$ is ${n-1\choose p-1}$;

• (2)

the number of ${\cal D}^{*}$-classes in $S$ is $1+2^{n-1}$.

Observe that for all $\alpha\in{\cal DP}_{n}$ with $h(\alpha)=p$,

[TABLE]

where $g(\mathop{\rm Dom}\nolimits\alpha)=(d_{1},d_{2},\ldots,d_{p-1}).$ Moreover, an ordered $p$-tuple: $(d_{1},d_{2},\ldots,d_{p})$ is said to be symmetric if

[TABLE]

Now, let $d_{s}(n,p)$ be the number of distinct symmetric ordered $p$-tuples:

$(d_{1},d_{2},\ldots,d_{p})$ with $\sum_{i=1}^{p}d_{i}=n$. Then we have

Lemma 3.5

[2, Lemma 3.5]** d_{s}(n;p)=\left\{\begin{array}[]{ll}0,&\,\,\mbox{if}\,\,n$\,\,\mbox{is odd and}\,\,$p\,\,\mbox{is even};\\ {\lfloor{\frac{n-1}{2}}\rfloor\choose\lfloor{\frac{p-1}{2}}\rfloor},&\,\,\mbox{otherwise}.\end{array}\right.

Now by virtue of $(5)$ and [1, Theorem 2.5], it is not difficult to see that the number of ${\cal D}^{*}$-classes in ${\cal DDP}_{n}$ is the same as the number of ${\cal D}$-classes in ${\cal ODP}_{n}$ less those pairs that are merged into single ${\cal D}^{*}$-classes in ${\cal DDP}_{n}$. Thus, we have

Lemma 3.6

*Let $g(m,p)$ be the number of ${\cal D}$-classes in ${\cal ODP}_{n}$ (consisting of maps of height $p$ and $\sum d_{i}=m$) that are merged into single ${\cal D}^{*}$-classes in ${\cal DDP}_{n}$. Then $m\leq(n-1)/2$, and

g(m,p)=\left\{\begin{array}[]{ll}\frac{1}{2}{m-1\choose p-2},&\,\,\mbox{if}\,\,n$\,\,\mbox{is odd and}\,\,$p\,\,\mbox{is odd};\\ \frac{1}{2}[{m-1\choose p-2}-{\lfloor{\frac{m-1}{2}}\rfloor\choose\lfloor{\frac{p-2}{2}}\rfloor}],&\,\,\mbox{otherwise}.\end{array}\right.*

Proof.   The result follows from $(5)$, Lemmas 3.1 & 3.5 and the observation that

[TABLE]

$\Box$

Now have the main result of this section.

Theorem 3.7

*Let $B(n,p)$ be the number of ${\cal D}$-classes in ${\cal ODP}_{n}$ (consisting of maps of height $p$) that are merged into single ${\cal D}^{*}$-classes in ${\cal DDP}_{n}$. Then for $n\geq p\geq 1$, we have

B(n,p)=\left\{\begin{array}[]{ll}\frac{1}{2}[{\lfloor{\frac{n-1}{2}}\rfloor\choose p-1}-{\lfloor{\frac{n-1}{4}}\rfloor\choose\frac{p-1}{2}}],&\,\,\mbox{if}\,\,p\,\,\mbox{is odd};\\ \frac{1}{2}[{\lfloor{\frac{n-1}{2}}\rfloor\choose p-1}-2{\lfloor{\frac{n-1}{4}}\rfloor\choose\frac{p}{2}}],&\,\,\mbox{if}\,\,n\equiv 1,2\,(mod\,4),\,\&\,p\,\,\mbox{is even};\\ \frac{1}{2}[{\lfloor{\frac{n-1}{2}}\rfloor\choose p-1}-2{\lfloor{\frac{n-3}{4}}\rfloor\choose\frac{p}{2}}-{\lfloor{\frac{n-3}{4}}\rfloor\choose\frac{p-2}{2}}],&\,\,\mbox{if}\,\,n\equiv-1,0\,(mod\,4),\,\&\,p\,\,\mbox{is even}.\end{array}\right.*

Proof.   The result follows from $(5)$, $(7)$ and Lemma 3.6. To see this, let $n\equiv 0\,(mod\,4)$ and $p$ be even. Then $n=4k$ for some integer $k$, and

[TABLE]

All the other cases are handled similarly. $\Box$

Now have the main result of this section.

Corollary 3.8

The number of ${\cal D}^{*}$-classes in ${\cal DDP}_{n}$ (consisting of maps of height $p\geq 1$) is ${n-1\choose p-1}-B(n,p)$.

Proof.   The result follows from Theorem 3.7 and the remarks preceding Lemma 3.6. $\Box$

Corollary 3.9

*The number of ${\cal D}^{*}$-classes in ${\cal DDP}_{n}$ denoted by $d_{n}$ is

d_{n}=\left\{\begin{array}[]{ll}2^{n-1}-2^{\lfloor{\frac{n-3}{2}}\rfloor}+\cdot 2^{\lfloor{\frac{n+1}{4}}\rfloor},&\,\,\mbox{if}\,\,n\equiv-1,0\,(mod\,4);\\ 2^{n-1}-2^{\lfloor{\frac{n-3}{2}}\rfloor}+3\cdot 2^{\lfloor{\frac{n-3}{4}}\rfloor},&\,\,\mbox{if}\,\,n\equiv 1,2\,(mod\,4).\end{array}\right.*

Proof.   The result follows from Theorem 3.7 and Corollary 3.8. To see this, let $n\equiv 1,2\,(mod\,4)$. Then $n=4k+1,4k+2$ for some integer $k$, and

[TABLE]

The case $n\equiv-1,0\,(mod\,4)$ is handled similarly. $\Box$

Acknowledgements. The second named author would like to thank Bowen University, Iwo and Sultan Qaboos University for their financial support and hospitality, respectively.