# Cohen-Macaulay types of certain edge subrings of bipartite graphs and
(generalized) Fuss-Catalan numbers

Alin Ştefan

###### Abstract

We give an example of two non isomorphic coordinate rings of a special kind of
convex polyominoes whose Cohen-Macaulay types are generalized Fuss-Catalan
numbers. We further provide a determinantal formula for these numbers.

Keywords: Combinatorial identities, equations of a cone, canonical module,
toric rings
MSC 2010: 05A10, 05A19, 13A02, 13H10, 15A39

##  1 Introduction

Polyominoes are two-dimensional objects obtained by joining unitary squares
edge by edge. They appear in puzzles, designs, and tiling problems (see [5]
for more details).

Let $\mathcal{P}$ be a polyomino and $K$ be a field. We denote by
$I_{\mathcal{P}}$, the polyomino ideal attached to $\mathcal{P}$, in a
suitable polynomial ring over $K$. The associated quotient ring is denoted by
$K[\mathcal{P}]$. In [8], when $\mathcal{P}$ is convex, Qureshi proved that
$K[\mathcal{P}]$ is a normal domain. She proved that $I_{\mathcal{P}}$ defines
the edge subring $R=K[x_{i}y_{j}\ |\ (i,j)\in V(\mathcal{P})]$, where
$V(\mathcal{P})$ is the set of all corners belonging to cells of
$\mathcal{P}$. In this case, $R$ is Cohen-Macaulay by a celebrated theorem of
Hochster. Danilov and Stanley described the canonical module $\omega_{R}$ of
$R$ via the polyhedral cone generated by the set of exponents
$\\{\log(x_{i}y_{j})\ |\ (i,j)\in V(\mathcal{P})\\}.$ The Cohen-Macaulay type
of $R$ is defined by $r(R)=\mu(R)=\operatorname{dim}_{K}\ \omega_{R}\otimes
K.$ The canonical module is a fundamental object in commutative algebra that
contains information about the last syzygy module of $K[\mathcal{P}]$ (see
[3], pp. 136-146). In [13] the authors show how to compute a generating set
for the canonical module of the edge ring of bipartite graph using
combinatorial optimization tehniques in terms of the incidence matrix of graph
and the vertices of a certain blocking polyhedron, but an explicit formula for
the Cohen-Macaulay type is not given.

Let $\mathbb{N}$ denote the set of nonnegative integers, $[m]$ the set
$\\{1,2,\ldots,m\\}$ when $m\in\mathbb{N}$ and ${\mathbb{Z}}_{>}$ the set of
positive integers. For $p\geq 1$ let $u_{1},\ldots,u_{p}$,
$r_{1},\ldots,r_{p}\in{\mathbb{Z}}_{>}$ and for $k\in[p]$ we write

$A_{k}=1+\sum_{i=1}^{k}u_{i},\ B_{k}=1+\sum_{i=1}^{k}r_{i}.$

Let
$\mathcal{P}=\mathcal{P}\left({\begin{array}[]{cccc}u_{1}&u_{2}&...&u_{p}\\\
r_{1}&r_{2}&...&r_{p}\\\ \end{array}}\right)$ be the polyomino as shown in
Fig. 1. This has the first $r_{1}$ columns of height $A_{1}-1$, next $r_{2}$
columns of height $A_{2}-1$ and so on. The coordinate ring $K[\mathcal{P}]$ is
a normal Cohen-Macaulay domain of dimension $|V(\mathcal{P})|$ \-
$|\mathcal{P}|$=$A_{p}+B_{p}-1$. Also, $K[\mathcal{P}]$ is isomorphic to the
toric ring $R=K[x_{i}y_{j}\ |\ (i,j)\in V(\mathcal{P})]$. We present the
irreducible representation of the cone generated by

$B=\\{\log(x_{i}y_{j})\ |\ (i,j)\in
V(\mathcal{P})\\}\subset{\mathbb{N}}^{B_{p}+A_{p}},$

the set of the exponents of the generators of the toric ring $R$.

$M_{0}(1,1)$$M_{1}(1,A_{1})$$M_{2}(B_{1},A_{2})$$M_{p-2}(B_{p-3},A_{p-2})$$M_{p-1}(B_{p-2},A_{p-1})$$M_{p}(B_{p-1},A_{p})$$M_{p+1}(B_{p},A_{p})$Fig.
1 The polyomino
$\mathcal{P}\left({\begin{array}[]{cccc}u_{1}&u_{2}&...&u_{p}\\\
r_{1}&r_{2}&...&r_{p}\\\ \end{array}}\right)$

If
$\mathcal{P}_{1}=\mathcal{P}\left({\begin{array}[]{cccc}u_{1}&u_{2}&...&u_{p}\\\
r_{1}&r_{2}&...&r_{p}\\\ \end{array}}\right)$ and
$\mathcal{P}_{2}=\mathcal{P}\left({\begin{array}[]{cccc}u_{1}&u_{2}&...&u_{p}\\\
s_{1}&s_{2}&...&s_{p}\\\ \end{array}}\right)$ where $u_{1}=\ldots=u_{n}=n,\
r_{1}=\ldots=r_{n}=t$ and $s_{1}=\ldots=s_{n}=n-t$, we show that two non
isomorphic coordinate rings $K[\mathcal{P}_{1}]$ and $K[\mathcal{P}_{2}]$ have
the same Cohen-Macaulay types $r(K[\mathcal{P}_{1}])=\begin{bmatrix}n\\\
t\end{bmatrix}_{p}=\begin{bmatrix}n\\\
n-t\end{bmatrix}_{p}=r(K[\mathcal{P}_{2}])$ which are the generalized Fuss-
Catalan numbers.

##  2 Prelimininaries

###  2.1 Cones

In this section we fix the notation and recall some basic results. For details
we refer the reader to [1], [2], [3], [10], [12] and [15].

We denote ${\mathbb{Q}}_{+},{\mathbb{R}}_{+}$ the set of nonnegative rationals
and real numbers respectively.

Fix an integer $n>0$. If $0\neq a\in\mathbb{Q}^{n},$ then $H_{a}$ will denote
the rational hyperplane of $\mathbb{R}^{n}$ through the origin with normal
vector $a,$ that is,

$H_{a}=\\{x\in\mathbb{R}^{n}\ |\ \langle x,a\rangle=0\\},$

where $\langle\ ,\ \rangle$ is the scalar product in $\mathbb{R}^{n}.$ The two
closed rational linear halfspaces bounded by $H_{a}$ are:

$H^{+}_{a}=\\{x\in\mathbb{R}^{n}\ |\ \langle x,a\rangle\geq 0\\}\ {\rm{and}}\
H^{-}_{a}=H^{+}_{-a}=\\{x\in\mathbb{R}^{n}\ |\ \langle x,a\rangle\leq 0\\}.$

The two open rational linear halfspaces bounded by $H_{a}$ are:

$H^{>}_{a}=\\{x\in\mathbb{R}^{n}\ |\ \langle x,a\rangle>0\\}\ {\rm{and}}\
H^{<}_{a}=H^{>}_{-a}=\\{x\in\mathbb{R}^{n}\ |\ \langle x,a\rangle<0\\}.$

If $S\subset{\mathbb{Q}}^{n}$, then the set

$\mathbb{R}_{+}{S}=\\{\sum_{i=1}^{r}a_{i}v_{i}\ :\ a_{i}\in\mathbb{R}_{+},\
v_{i}\in S,\ r\in{\mathbb{N}}\\}$

is called the _rational cone_ generated by $S$. The _dimension_ of a cone is
the dimension of the smallest vector subspace of ${\mathbb{R}}^{n}$ which
contains it.

By the theorem of Minkowski-Weyl, see [2], [7], [15], finitely generated
rational cones can also be described as intersection of finitely many rational
closed subspaces (of the form $H^{+}_{a}$). We further restrict this
presentation to the class of finitely generated rational cones, which will be
simply called cones. If a cone $C$ is presented as

$C=H^{+}_{a_{1}}\cap\ldots\cap H^{+}_{a_{r}}$

such that no $H^{+}_{a_{i}}$ can be omitted, then we say that this is an
_irredundant representation_ of $C$. If $\operatorname{dim}(C)=n$, then the
halfspaces $H^{+}_{a_{1}},\ldots,H^{+}_{a_{r}}$ in an irredundant
representation of $C$ are uniquely determined and we set

$\operatorname{relint}(C)=H^{>}_{a_{1}}\cap\ldots\cap H^{>}_{a_{r}}$

the _relative interior_ of $C$. If $a_{i}=(a_{i1},\ldots,a_{in}),$ then we
call

$H_{a_{i}}(x):=a_{i1}x_{1}+\cdots+a_{in}x_{n}=0,$

the _equations of the cone_ $C.$

A hyperplane $H$ is called a supporting hyperplane of a cone $C$ if $C\cap
H\neq\emptyset$ and $C$ is contained in one of the closed halfspaces
determined by $H$. If $H$ is a supporting hyperplane of $C$, then $F=C\cap H$
is called a _proper face_ of $C$. It is convenient to consider also the empty
set and $C$ as faces, the _improper faces_. The faces of a cone are themselves
cones. A face $F$ with $\operatorname{dim}(F)=\operatorname{dim}(C)-1$ is
called a _facet_. If $\operatorname{dim}\ \mathbb{R}_{+}S=n$ and $F$ is a
facet defined by the supporting hyperplane $H$, then $H$ is generated as a
linear subspace by a linearly independent subset of $S$.

A cone $C$ is _pointed_ if $0$ is a face of $C$. This is equivalent to say
that $x\in C$ and $-x\in C$ imply $x=0$. The faces of dimension $1$ of a
pointed cone are called _extreme rays_.

Let $\mathcal{A}\subset\mathbb{Z}^{n}$ be a collection of lattice points in
$\mathbb{Z}^{n}$ and $R=K[\mathcal{A}]$ be the toric ring defined by
$\mathcal{A}$ over field $K.$ By a celebrated theorem of Hochster, when $R$ is
normal, it is Cohen-Macaulay. In this case, Danilov and Stanley
$(\cite[cite]{[\@@bibref{}{BH}{}{}]},$ Theorem 6.3.5 and
$\cite[cite]{[\@@bibref{}{D}{}{}]})$ described the canonical module
$\omega_{R}$ of $R$ via the polyhedral cone generated by $\mathcal{A},$ that
is $\omega_{R}=(\\{x^{a}|\
a\in{\mathbb{N}}\mathcal{A}\cap\operatorname{relint}({\mathbb{R_{+}}}\mathcal{A})\\}).$
The Cohen-Macaulay type of $R$ is defined by
$r(R)=\mu(\omega_{R})=\operatorname{dim}_{K}\ \omega_{R}\otimes K.$

###  2.2 Polyominoes

First, we briefly recall fundamental materials and basic terminologies on
polyominoes and their binomial ideals. For further information on algebra and
combinatorics on polyominoes we refer the reader to [8].

Given $a=(i,j)$ and $b=(k,l)$ belonging to $\mathbb{N}^{2}$ we write $a<b$ if
$i<k$ and $j<l.$ When $a<b$, we define an $interval$ $[a,b]$ of
$\mathbb{N}^{2}$ to be $[a,b]=\\{c\in\mathbb{N}^{2}:a\leq c\leq
b\\}\subset\mathbb{N}^{2}.$ For an interval $[a,b]$, the $diagonal$ corners of
$[a,b]$ are $a$ and $b$ and the $anti$-$diagonal$ corners of $[a,b]$ are
$c=(i,l)$ and $d=(k,j)$.

A $cell$ of $\mathbb{N}^{2}$ with the lower left corner $a\in\mathbb{N}^{2}$
is the interval $C=[a,a+(1,1)]$. Its $vertices$ are $a$, $a+(1,1)$, $a+(1,0)$
and $a+(0,1)$. Its $edges$ are $\\{a,a+(1,0)\\}$, $\\{a,a+(0,1)\\}$,
$\\{a+(1,0),a+(1,1)\\}$, $\\{a+(0,1),a+(1,1)\\}$. Let $V(C)$ denote the set of
vertices of $C$ and $E(C)$ the set of edges of $C$.

Let $\mathcal{P}$ be a finite collection of cells of $\mathbb{N}^{2}$. Its
$vertex\ set$ is $V(\mathcal{P})=\bigcup_{C\in\mathcal{P}}V(C)$ and its $edge\
set$ is $E(\mathcal{P})=\bigcup_{C\in\mathcal{P}}E(C)$. We say that two cells
$C$ and $D$ are $connected$ if there exists a sequence of cells

$\mathcal{C}:C=C_{1},\ldots,C_{n}=D$

of $\mathcal{P}$ such that $C_{i}\cup C_{i+1}$ is an edge of $C_{i}$ for
$i\in[n-1]$; if $C_{i}\neq C_{j}$ for all $i\neq j$, then $\mathcal{C}$ is
called a $path$. We say that $\mathcal{P}$ is a $polyomino$ if any two cells
of $\mathcal{P}$ are connected. A polyomino $\mathcal{Q}$ is a $subpolyomino$
of $\mathcal{P}$ and denoted $\mathcal{Q}\subset\mathcal{P}$ if each cell
belonging to $\mathcal{Q}$ belongs to $\mathcal{P}$.

Let $A$ and $B$ be cells of $\mathbb{N}^{2}$ for which $(i,j)$ is the lower
left corner of $A$ and $(k,l)$ is the lower left corner of $B$. If $i\leq k$
and $j\leq l$, then the $cell\ interval$ of $A$ and $B$ is the set $[A,B]$
which consists of those cells $E$ of $\mathbb{N}^{2}$ whose lower left corner
$(r,s)$ satisfies $i\leq r\leq k$ and $j\leq s\leq l$.

Let $\mathcal{P}$ be a finite collection of cells of $\mathbb{N}^{2}$. We call
$\mathcal{P}$ $row\ convex$ if the horizontal cell interval $[A,B]$ is
contained in $\mathcal{P}$ for any cells $A$ and $B$ of $\mathcal{P}$ whose
lower left corners are in horizontal position. Similary one can define
$column\ convex$. We call $\mathcal{P}$ $convex$ if it is row convex and
column convex.

Let $\mathcal{P}$ be a finite collection of cells of $\mathbb{N}^{2}$ with
$V(\mathcal{P})$ its vertex set. Let $S$ denote the polynomial ring over a
field $K$ whose the variables are those $x_{a}$ with $a\in V(\mathcal{P})$. We
say that an interval $[a,b]$ is $an\ interval\ of\mathcal{P}$ if
$\mathcal{P}_{[a,b]}\subset\mathcal{P}$, where $\mathcal{P}_{[a,b]}$ is the
interval $[a,b]$ regarded as a polyomino. For each interval $[a,b]$ of
$\mathcal{P}$, we introduce the binomial $f_{a,b}=x_{a}x_{b}-x_{c}x_{d}$,
where $c$ and $d$ are the anti-diagonals of $[a,b]$. Such a binomial $f_{a,b}$
is said to be an $inner$ 2-$minor$ of $\mathcal{P}$. Write $I_{\mathcal{P}}$
for the ideal generated by all inner 2-minors of $\mathcal{P}$. If
$\mathcal{P}$ happens to be a polyomino, then $I_{\mathcal{P}}$ will be called
a $polyomino$ $ideal$. The $K$-algebra $S/I_{\mathcal{P}}$ is denoted
$K[\mathcal{P}]$ and is called the coordinate ring of $\mathcal{P}$.

The following result has been shown by Qureshi in [8].

######  Theorem 1.

Let $\mathcal{P}$ be a convex polyomino. Then $K[\mathcal{P}]$ is a normal
Cohen-Macaulay domain of dimension $|V(\mathcal{P})|$ \- $|\mathcal{P}|$.

We may assume that $V(\mathcal{P})\subset[(1,1),(m,n)]$. To $\mathcal{P}$ we
attach the toric ring

$R=K[x_{i}y_{j}\ |\ (i,j)\in V(\mathcal{P})]\subset
K[x_{1},\ldots,x_{m},y_{1},\ldots,y_{n}].$

For the sake of convenience, we denote for $a=(i,j)\in V(\mathcal{P})$ the
variable $x_{a}$ in $S$ by $x_{ij}$. The polyomino ideal $I_{\mathcal{P}}$ may
be viewed as follows as a toric ideals. Consider the $K$-algebra homomorphism
$\varphi:S\rightarrow R$ with $\varphi(x_{ij})=x_{i}y_{j}$ for all $(i,j)\in
V(\mathcal{P})$. It follows that $I_{\mathcal{P}}=Ker(\varphi)$ and
$K[\mathcal{P}]$ may be indentified with the toric ring $R$. It follows that
$R$ is normal and we recall that by a well known result of Danilov and Stanley
$(\cite[cite]{[\@@bibref{}{BG}{}{}]},\cite[cite]{[\@@bibref{}{BH}{}{}]},\cite[cite]{[\@@bibref{}{MS}{}{}]},\cite[cite]{[\@@bibref{}{V}{}{}]})$
the canonical module $\omega_{R}$ of $R,$ with respect to standard grading,
can be expressed as an ideal of $R$ generated by monomials, that is
$\omega_{R}=(\\{x^{a}|\
a\in{\mathbb{N}}B\cap\operatorname{relint}({\mathbb{R_{+}}}B)\\}),$ where

$B:=\\{\log(x_{i}y_{j})\ |\ (i,j)\in
V(\mathcal{P})\\}\subset{\mathbb{N}}^{m+n}$

is the set of the exponents of the generators of the toric ring $R$.

###  2.3 Simple paths with general bounderies

We briefly recall the most general problem to count paths in a region that is
bounded by nonliniar upper and lower boundaries (see [9] for more details).

Let $a_{1}\leq a_{2}\leq\cdots\leq a_{n}$ and $b_{1}\leq b_{2}\leq\cdots\leq
b_{n}$ be integers with $a_{i}\geq b_{i}.$ We abbreviate
$\bf{a}$=$(a_{1},a_{2},\ldots,a_{n})$ and
$\bf{b}$=$(b_{1},b_{2},\ldots,b_{n}).$ By $L((0,b_{1})\rightarrow(n,a_{n})\ |\
\bf{a}\geq\bf{y}\geq\bf{b})$ we denote the set of all lattice paths from
$(0,b_{1})$ to $(n,a_{n})$ that satisfy the property that for all
$i=1,2,\dots,n$ the height $y_{i}$ of the $i-$th horizontal step is in the
interval $[b_{i},a_{i}].$

######  Theorem 2.

$(\cite[cite]{[\@@bibref{}{K}{}{}]},$ Theorem 10.7.1.$)$ Let
$\bf{a}$=$(a_{1},a_{2},\ldots,a_{n})$ and
$\bf{b}$=$(b_{1},b_{2},\ldots,b_{n})$ be integer sequences with $a_{1}\leq
a_{2}\leq\cdots\leq a_{n},$ $b_{1}\leq b_{2}\leq\cdots\leq b_{n}$ and
$a_{i}\geq b_{i},$ $i=1,2,\dots,n.$ The number of all paths from $(0,b_{1})$
to $(n,a_{n})$ satisfying the property that for all $i=1,2,\dots,n$ the height
$y_{i}$ of the $i-$th horizontal step is between $b_{i}$ and $a_{i}$ is given
by

$|\ L((0,b_{1})\rightarrow(n,a_{n})\ |\ {\bf{a}\geq\bf{y}\geq\bf{b})}\
|=\det_{1\leq i,j\leq n}\left(\binom{a_{i}-b_{j}+1}{j-i+1}\right).$

##  3 The Main result

######  Definition 3.

For $n,t,p\in\mathbb{N}$, $1\leq t<n$, $p\geq 1$ we define
$\begin{bmatrix}n\\\ t\end{bmatrix}_{p}$ to be the number of elements of
$A(n,t,p)$, where

$A(n,t,p):=\\{\alpha\in{\mathbb{Z}}_{>}^{pt+1}\ |\
\sum_{i=1}^{pt+1}\alpha_{i}=p(n-t),\sum_{i=1}^{kt}\alpha_{i}\leq k(n-t)\
\textrm{for}\ \textrm{all}\ k\in[p-1]\\}.$

######  Proposition 4.

For any $n,t,p\in\mathbb{N}$, $1\leq t<n$, $p\geq 1$ we have:
$1)$ $\begin{bmatrix}n\\\ t\end{bmatrix}_{p}=\begin{bmatrix}n\\\
n-t\end{bmatrix}_{p}.$
$2)$ $\begin{bmatrix}n\\\ 1\end{bmatrix}_{p}=\begin{bmatrix}n\\\
n-1\end{bmatrix}_{p}=C_{p+1}(n),\textrm{where}\
C_{p}(n)=\frac{1}{(n-1)p+1}\binom{np}{p}\ \textrm{are}\ \textrm{the}\
\textrm{Fuss-Catalan}\ \textrm{numbers}.$

###### Proof.

Let $n,t,p\in\mathbb{N}$, $1\leq t<n$, $p\geq 1$ be fixed. If we examine the
elements of $A(n,t,p)$ then we see that we can get rid of $\alpha_{pt+1}$ at
the expense of getting another inequality: so the number $\begin{bmatrix}n\\\
t\end{bmatrix}_{p}$ is the same as the number of all vectors

$(\alpha_{1},\ldots,\alpha_{pt})\in{\mathbb{Z}}_{>}^{pt}\ \textrm{with}\
\sum_{i=1}^{kt}\alpha_{i}\leq k(n-t)\ \textrm{for}\ \textrm{all}\ k\in[p].\ \
\ \ \ \ \ \ \ \ \ \ (1)$

These vectors can be better represented by lattice paths from $(0,0)$ to
$(pt,p(n-t))$ consisting of unit horizontal and vertical steps with the
constraint that is imposed by the inequalities $(1).$ More precisely, between
the vertical lines $x=(k-1)t$ and $x=kt$ the path must stay weakly below
height $k(n-t)$ for any $k\in[p].$ If we denote by
$\bf{a}$=$(a_{1},a_{2},\ldots,a_{pt})$ and
$\bf{b}$=$(b_{1},b_{2},\ldots,b_{pt})$ where $a_{1}=\cdots=a_{t}=n-t,$
$a_{t+1}=\cdots=a_{2t}=2(n-t),$ …, $a_{(p-1)t+1}=\cdots=a_{pt}=p(n-t)$ and
$b_{1}=\cdots=b_{pt}=0$ using $(\cite[cite]{[\@@bibref{}{K}{}{}]},$ Theorem
10.7.1.$)$ we have

$\begin{bmatrix}n\\\ t\end{bmatrix}_{p}=|\ L((0,0)\rightarrow(pt,a_{pt})\ |\
{\bf{a}\geq\bf{y}\geq\bf{b})}\ |=\det_{1\leq i,j\leq
pt}\left(\binom{a_{i}+1}{j-i+1}\right).$

By using similar argument, if we denote by
$\bf{a^{\prime}}$=$(a_{1}^{\prime},a_{2}^{\prime},\ldots,a_{p(n-t)}^{\prime})$
and
$\bf{b^{\prime}}$=$(b_{1}^{\prime},b_{2}^{\prime},\ldots,b_{p(n-t)}^{\prime})$
where $a_{1}^{\prime}=\cdots=a_{n-t}^{\prime}=t,$
$a_{n-t+1}^{\prime}=\cdots=a_{2(n-t)}^{\prime}=2t,$ …,
$a_{(p-1)(n-t)+1}^{\prime}=\cdots=a_{p(n-t)}^{\prime}=pt$ and
$b_{1}^{\prime}=\cdots=b_{p(n-t)}^{\prime}=0$ we have

$\begin{bmatrix}n\\\ n-t\end{bmatrix}_{p}=|\
L((0,0)\rightarrow(p(n-t),a_{pt}^{\prime})\ |\
{\bf{a^{\prime}}\geq\bf{y}\geq\bf{b^{\prime}})}\ |=\det_{1\leq i,j\leq
p(n-t)}\left(\binom{a_{i}^{\prime}+1}{j-i+1}\right).$

Since any path of $L((0,0)\rightarrow(p(n-t),a_{pt}^{\prime})$ is a reflection
of a path from $L((0,0)\rightarrow(pt,a_{pt})$ in a line parallel to the
antidiagonal $x+y=0$ we get $\begin{bmatrix}n\\\
t\end{bmatrix}_{p}=\begin{bmatrix}n\\\ n-t\end{bmatrix}_{p}.$
For $t=1$ one obtains $\begin{bmatrix}n\\\ 1\end{bmatrix}_{p}=C_{p+1}(n),$
Fuss-Catalan numbers since the above lattice path model reduces in that case
to the standard lattice path model for Fuss-Catalan numbers. ∎

######  Corollary 5.

For any $n,t,p\in\mathbb{N}$, $1\leq t<n$, $p\geq 1$ we have:
$1)$ For $a_{1}=\cdots=a_{t}=n-t,$ $a_{t+1}=\cdots=a_{2t}=2(n-t),$ …,
$a_{(p-1)t+1}=\cdots=a_{pt}=p(n-t)$ we have

$\begin{bmatrix}n\\\ t\end{bmatrix}_{p}=\det_{1\leq i,j\leq
pt}\left(\binom{a_{i}+1}{j-i+1}\right).$

$2)$ For $a_{1}^{\prime}=\cdots=a_{n-t}^{\prime}=t,$
$a_{n-t+1}^{\prime}=\cdots=a_{2(n-t)}^{\prime}=2t,$ …,
$a_{(p-1)(n-t)+1}^{\prime}=\cdots=a_{p(n-t)}^{\prime}=pt$ we have

$\begin{bmatrix}n\\\ n-t\end{bmatrix}_{p}=\det_{1\leq i,j\leq
p(n-t)}\left(\binom{a_{i}^{\prime}+1}{j-i+1}\right).$

###### Proof.

See the proof above. ∎

######  Example 6.

For $n=3$, $t=1$ and $p=3$ we have:
$i)$ $\bf{a}$=$(2,4,6)$,

$\begin{bmatrix}3\\\ 1\end{bmatrix}_{3}=\det_{1\leq i,j\leq
3}\left(\binom{a_{i}+1}{j-i+1}\right)=\begin{vmatrix}3&3&1\\\ 1&5&10\\\
0&1&7\end{vmatrix}=55=C_{4}(3).$

$ii)$ $\bf{a}^{\prime}$=$(1,1,2,2,3,3)$,

$\begin{bmatrix}3\\\ 2\end{bmatrix}_{3}=\det_{1\leq i,j\leq
6}\left(\binom{a_{i}^{\prime}+1}{j-i+1}\right)=\begin{vmatrix}2&1&0&0&0&0\\\
1&2&1&0&0&0\\\ 0&1&3&3&1&0\\\ 0&0&1&3&3&1\\\ 0&0&0&1&4&6\\\
0&0&0&0&1&4\end{vmatrix}=55.$

Let
$\mathcal{P}=\mathcal{P}\left({\begin{array}[]{cccc}u_{1}&u_{2}&...&u_{p}\\\
r_{1}&r_{2}&...&r_{p}\\\ \end{array}}\right)$ be the polyomino of Figure 1.
The coordinate ring $K[\mathcal{P}]$ is a normal Cohen-Macaulay domain and it
may be identified with the toric ring $R=K[x_{i}y_{j}\ |\ (i,j)\in
V(\mathcal{P})]$. We will describe the irreducible representation of the cone
generated by

$B=\\{\log(x_{i}y_{j})\ |\ (i,j)\in
V(\mathcal{P})\\}\subset{\mathbb{N}}^{B_{p}+A_{p}},$

the set of the exponents of the generators of the toric ring $R$.
Let

$\nu_{i}=-\sum_{k\in[B_{i}-1]}e_{k}+\sum_{k\in[A_{i}]}e_{B_{p}+k},\ \
\nu=\sum_{k\in[B_{p}]}e_{k}-\sum_{k\in[A_{p}]}e_{B_{p}+k},$

for $i\in[p-1]$, where $\\{e_{k}\\}_{1\leq k\leq A_{p}+B_{p}}$ is the
canonical base of ${\mathbb{R}}^{A_{p}+B_{p}}$.

######  Lemma 7.

The cone generated by $B=\\{\log(x_{i}y_{j})\ |\ (i,j)\in
V(\mathcal{P})\\}\subset{\mathbb{N}}^{B_{p}+A_{p}}$, the set of the exponents
of the generators of the toric ring $R$, has irreducible representation

${\mathbb{R}}_{+}B=(\bigcap_{a\in N}H_{a}^{+})\cap H_{\nu},$

where $N=\\{\nu_{i},\ e_{k}\ |\ i\in[p-1],\ k\in[A_{p}+B_{p}]\\}$.

###### Proof.

The set of the exponents of the generators of the toric ring $R$ consists of
the vectors $g_{i,j}^{1}=e_{i}+e_{B_{p}+j}$ for $i\in[B_{1}-1],\ j\in[A_{1}],$
and $g_{i,j}^{t}=e_{i}+e_{B_{p}+j}$ for $2\leq t\leq p,\ B_{t-1}\leq i\leq
B_{t}-1,\ j\in[A_{t}]$.
Now we will prove that ${\mathbb{R}}_{+}B\subseteq H_{a}^{+}\cap H_{\nu}$ for
any $a\in N$. It is clear that ${\mathbb{R}}_{+}B\subseteq H_{\nu}$ and
${\mathbb{R}}_{+}B\subseteq H_{e_{k}}^{+}$ for all $k\in[A_{p}+B_{p}]$. If
$i\in[B_{1}-1]$ and $j\in[A_{1}]$ we have $g_{i,j}^{1}\in H_{\nu_{s}}$ for any
$s\in[p-1]$. If $2\leq t\leq p$, $B_{t-1}\leq i\leq B_{t}-1$, $j\in[A_{t}]$
then $g_{i,j}^{t}\in H_{\nu_{s}}^{+}$ for any $1\leq s\leq t-1$ and
$g_{i,j}^{t}\in H_{\nu_{s}}$ for any $t\leq s\leq p$. Thus
${\mathbb{R}}_{+}B\subseteq(\bigcap_{a\in N}H_{a}^{+})\cap H_{\nu}$.
Now we will prove the converse inclusion
${\mathbb{R}}_{+}B\supseteq(\bigcap_{a\in N}H^{+}_{a})\cap H_{\nu}.$
It is enough to prove that the extremal rays of the cone $\bigcap_{a\in
N}H^{+}_{a}$ are in ${{\mathbb{R}}_{+}}B.$ Any extremal ray of the cone
$\bigcap_{a\in N}H^{+}_{a}$ can be written as the intersection of
$A_{p}+B_{p}-2$ hyperplanes $H_{a}$, with $a\in N.$ We will prove that all the
exponents of the generators of the toric ring $R$ are the extremal rays of the
cone $\bigcap_{a\in N}H^{+}_{a}$. There are the following posibilities to
obtain extremal rays by intersection of $A_{p}+B_{p}-2$ hyperplanes.

$1.$ For $i\in[B_{1}-1]$ and $j\in[A_{1}]$, let
$k_{1}<\ldots<k_{A_{p}+B_{p}-3}$ be the sequence of integers such that
$\\{k_{1},\ldots,k_{A_{p}+B_{p}-3}\\}\subset[A_{p}+B_{p}]\setminus\\{i,B_{p}+j\\}$.
The system of equations :
$\ (*)\ \begin{cases}z_{k_{1}}=0,\\\ z_{k_{2}}=0,\\\ \vdots\\\
z_{k_{A_{p}+B_{p}-3}}=0,\\\ H_{\nu_{1}}(z)=0.\end{cases}$
admits the solution $g_{i,j}^{1}.$

$2.$ For $B_{t-1}\leq i\leq B_{t}-1$ and $j\in[A_{t}]$ with $2\leq t\leq p-1$,
let $k_{1}<\ldots<k_{A_{p}+B_{p}-3}$ be the sequence of integers such that
$\\{k_{1},\ldots,k_{A_{p}+B_{p}-3}\\}\subset[A_{p}+B_{p}]\setminus\\{i,B_{p}+j\\}$.
The system of equations :
$\ (**)\ \begin{cases}z_{k_{1}}=0,\\\ z_{k_{2}}=0,\\\ \vdots\\\
z_{k_{A_{p}+B_{p}-3}}=0,\\\ H_{\nu_{t}}(z)=0.\end{cases}$
admits the solution $g_{i,j}^{t}$.

$3.$ For $B_{p-1}\leq i\leq B_{p}$ and $j\in[A_{p}]$, let
$k_{1}<\ldots<k_{A_{p}+B_{p}-2}$ be the sequence of integers such that
$\\{k_{1},\ldots,k_{A_{p}+B_{p}-2}\\}=[A_{p}+B_{p}]\setminus\\{i,B_{p}+j\\}$.
The system of equations :
$\ (***)\ \begin{cases}z_{k_{1}}=0,\\\ z_{k_{2}}=0,\\\ \vdots\\\
z_{k_{A_{p}+B_{p}-2}}=0.\end{cases}$
admits the solution $g_{i,j}^{p}$.
Thus ${\mathbb{R}}_{+}B\supseteq(\bigcap_{a\in N}H_{a}^{+})\cap H_{\nu}$. ∎

######  Remark 8.

If $G=(V(G),E(G))$ is a finite connected graph on the vertex set
$V(G)=[n],n\geq 2$ and $e=\\{i,j\\}\in E(G)$ is an edge then we define
$\rho(e)\in{\mathbb{R}}^{n}$ by $\rho(e)=e_{i}+e_{j}$ where $e_{i}$ is the
$i$th unit coordinate vector in ${\mathbb{R}}^{n}$. We write
$\mathcal{P}_{G}\subset{\mathbb{R}}^{n}$ for the convex hull of the finite set
$\\{\rho(e)\ |\ e\in E(G)\\}\subset{\mathbb{R}}^{n}$ and we call
$\mathcal{P}_{G}$ the _edge polytope_ of $G.$ In $($[11], Theorem 1.7$)$
Ohsugi and Hibi described the set of facets of the edge polytope
$\mathcal{P}_{G}$. Lemma 7 can be derived from this result and using
$(\cite[cite]{[\@@bibref{}{Z}{}{}]},$ Theorems 1.1, 1.2, 1.3.$)$.

######  Example 9.

We present two polyominoes $\mathcal{P}_{1},\mathcal{P}_{2}$ whose Cohen-
Macaulay types of coordinate rings are
$r(K[\mathcal{P}_{1}])=\begin{bmatrix}3\\\ 1\end{bmatrix}_{3}$ and
$r(K[\mathcal{P}_{2}])=\begin{bmatrix}3\\\ 2\end{bmatrix}_{3}$.

$i)$ Consider the polyomino
$\mathcal{P}_{1}=\mathcal{P}\left({\begin{array}[]{cccc}3&3&3\\\ 1&1&1\\\
\end{array}}\right)$. We have $A_{1}=4,\ A_{2}=7,\ A_{3}=10$ and $B_{1}=2,\
B_{2}=3,\ B_{3}=4$. The quotient ring by the polyomino ideal
$I_{\mathcal{P}_{1}}$ is isomorphic to the edge subring

$R_{1}=K[x_{1}y_{1},\ldots,x_{1}y_{4},x_{2}y_{1},\ldots,x_{2}y_{7},x_{3}y_{1},\ldots,x_{3}y_{10},x_{4}y_{1},\ldots,x_{4}y_{10}].$

Computation with NORMALIZ, we get the Hilbert series of $R_{1}$

$H_{R_{1}}(t)=\frac{1+18t+66t^{2}+55t^{3}}{(1-t)^{13}}$

and the generators of the canonical module of $R_{1}$

$\omega_{R_{1}}$=($x_{1}x_{2}x_{3}x_{4}^{7}y$, |  $x_{1}x_{2}x_{3}^{2}x_{4}^{6}y$, |  $x_{1}x_{2}x_{3}^{3}x_{4}^{5}y$, |  $x_{1}x_{2}x_{3}^{4}x_{4}^{4}y$, |  $x_{1}x_{2}x_{3}^{5}x_{4}^{3}y$, |  $x_{1}x_{2}x_{3}^{6}x_{4}^{2}y$,
---|---|---|---|---|---
$x_{1}x_{2}x_{3}^{7}x_{4}y$, |  $x_{1}x_{2}^{2}x_{3}x_{4}^{6}y$, |  $x_{1}x_{2}^{2}x_{3}^{2}x_{4}^{5}y$, |  $x_{1}x_{2}^{2}x_{3}^{3}x_{4}^{4}y$, |  $x_{1}x_{2}^{2}x_{3}^{4}x_{4}^{3}y$, |  $x_{1}x_{2}^{2}x_{3}^{5}x_{4}^{2}y$,
$x_{1}x_{2}^{2}x_{3}^{6}x_{4}y$, |  $x_{1}x_{2}^{3}x_{3}x_{4}^{5}y$, |  $x_{1}x_{2}^{3}x_{3}^{2}x_{4}^{4}y$, |  $x_{1}x_{2}^{3}x_{3}^{3}x_{4}^{3}y$, |  $x_{1}x_{2}^{3}x_{3}^{4}x_{4}^{2}y$, |  $x_{1}x_{2}^{3}x_{3}^{5}x_{4}y$,
$x_{1}x_{2}^{4}x_{3}x_{4}^{4}y$, |  $x_{1}x_{2}^{4}x_{3}^{2}x_{4}^{3}y$, |  $x_{1}x_{2}^{4}x_{3}^{3}x_{4}^{2}y$, |  $x_{1}x_{2}^{4}x_{3}^{4}x_{4}y$, |  $x_{1}x_{2}^{5}x_{3}x_{4}^{3}y$, |  $x_{1}x_{2}^{5}x_{3}^{2}x_{4}^{2}y$,
$x_{1}x_{2}^{5}x_{3}^{3}x_{4}y$, |  $x_{1}^{2}x_{2}x_{3}x_{4}^{6}y$, |  $x_{1}^{2}x_{2}x_{3}^{2}x_{4}^{5}y$, |  $x_{1}^{2}x_{2}x_{3}^{3}x_{4}^{4}y$, |  $x_{1}^{2}x_{2}x_{3}^{4}x_{4}^{3}y$, |  $x_{1}^{2}x_{2}x_{3}^{5}x_{4}^{2}y$,
$x_{1}^{2}x_{2}x_{3}^{6}x_{4}y$, |  $x_{1}^{2}x_{2}^{2}x_{3}x_{4}^{5}y$, |  $x_{1}^{2}x_{2}^{2}x_{3}^{2}x_{4}^{4}y$, |  $x_{1}^{2}x_{2}^{2}x_{3}^{3}x_{4}^{3}y$, |  $x_{1}^{2}x_{2}^{2}x_{3}^{4}x_{4}^{2}y$, |  $x_{1}^{2}x_{2}^{2}x_{3}^{5}x_{4}y$,
$x_{1}^{2}x_{2}^{3}x_{3}x_{4}^{4}y$, |  $x_{1}^{2}x_{2}^{3}x_{3}^{2}x_{4}^{3}y$, |  $x_{1}^{2}x_{2}^{3}x_{3}^{3}x_{4}^{2}y$, |  $x_{1}^{2}x_{2}^{3}x_{3}^{4}x_{4}y$, |  $x_{1}^{2}x_{2}^{4}x_{3}x_{4}^{3}y$, |  $x_{1}^{2}x_{2}^{4}x_{3}^{2}x_{4}^{2}y$,
$x_{1}^{2}x_{2}^{4}x_{3}^{3}x_{4}y$, |  $x_{1}^{3}x_{2}x_{3}x_{4}^{5}y$, |  $x_{1}^{3}x_{2}x_{3}^{2}x_{4}^{4}y$, |  $x_{1}^{3}x_{2}x_{3}^{3}x_{4}^{3}y$, |  $x_{1}^{3}x_{2}x_{3}^{4}x_{4}^{2}y$, |  $x_{1}^{3}x_{2}x_{3}^{5}x_{4}y$,
$x_{1}^{3}x_{2}^{2}x_{3}x_{4}^{4}y$, |  $x_{1}^{3}x_{2}^{2}x_{3}^{2}x_{4}^{3}y$, |  $x_{1}^{3}x_{2}^{2}x_{3}^{3}x_{4}^{2}y$, |  $x_{1}^{3}x_{2}^{2}x_{3}^{4}x_{4}y$, |  $x_{1}^{3}x_{2}^{3}x_{3}x_{4}^{3}y$, |  $x_{1}^{3}x_{2}^{3}x_{3}^{2}x_{4}^{2}y$,
$x_{1}^{3}x_{2}^{3}x_{3}^{3}x_{4}y)R_{1},$ |  |  |  |  |

where $y=y_{1}\cdot\ldots\cdot y_{10}.$
Hence, we have the Cohen-Macaulay type of $R_{1}$ is 55$=\begin{bmatrix}3\\\
1\end{bmatrix}_{3}$.

$ii)$ Consider the polyomino
$\mathcal{P}_{2}=\mathcal{P}\left({\begin{array}[]{cccc}3&3&3\\\ 2&2&2\\\
\end{array}}\right).$ We have $A_{1}=4,\ A_{2}=7,\ A_{3}=10$ and $B_{1}=3,\
B_{2}=5,\ B_{3}=7$. The quotient ring by the polyomino ideal
$I_{\mathcal{P}_{2}}$ is isomorphic to the edge subring

$R_{2}=K[x_{1}y_{1},\ldots,x_{1}y_{4},x_{2}y_{1},\ldots,x_{2}y_{4},x_{3}y_{1},\ldots,x_{3}y_{7},x_{4}y_{1},\ldots,x_{4}y_{7},$
$x_{5}y_{1},\ldots,x_{5}y_{10},x_{6}y_{1},\ldots,x_{6}y_{10},x_{7}y_{1},\ldots,x_{7}y_{10}].\
\ \ \ $

Computation with NORMALIZ, we get the Hilbert series of $R_{2}$

$H_{R_{2}}(t)=\frac{1+36t+318t^{2}+960t^{3}+1071t^{4}+444t^{5}+55t^{6}}{(1-t)^{16}}$

and the generators of the canonical module of $R_{2}$

$\omega_{R_{2}}$=($x_{1}x_{2}x_{3}x_{4}x_{5}x_{6}x_{7}^{4}y$, |  $x_{1}x_{2}x_{3}x_{4}x_{5}x_{6}^{2}x_{7}^{3}y$, |  $x_{1}x_{2}x_{3}x_{4}x_{5}x_{6}^{3}x_{7}^{2}y$, |  $x_{1}x_{2}x_{3}x_{4}x_{5}x_{6}^{4}x_{7}y$,
---|---|---|---
$x_{1}x_{2}x_{3}x_{4}x_{5}^{2}x_{6}x_{7}^{3}y$, |  $x_{1}x_{2}x_{3}x_{4}x_{5}^{2}x_{6}^{2}x_{7}^{2}y$, |  $x_{1}x_{2}x_{3}x_{4}x_{5}^{2}x_{6}^{3}x_{7}y$, |  $x_{1}x_{2}x_{3}x_{4}x_{5}^{3}x_{6}x_{7}^{2}y$,
$x_{1}x_{2}x_{3}x_{4}x_{5}^{3}x_{6}^{2}x_{7}y$, |  $x_{1}x_{2}x_{3}x_{4}x_{5}^{4}x_{6}x_{7}y$, |  $x_{1}x_{2}x_{3}x_{4}^{2}x_{5}x_{6}x_{7}^{3}y$, |  $x_{1}x_{2}x_{3}x_{4}^{2}x_{5}x_{6}^{2}x_{7}^{2}y$,
$x_{1}x_{2}x_{3}x_{4}^{2}x_{5}x_{6}^{3}x_{7}y$, |  $x_{1}x_{2}x_{3}x_{4}^{2}x_{5}^{2}x_{6}x_{7}^{2}y$, |  $x_{1}x_{2}x_{3}x_{4}^{2}x_{5}^{2}x_{6}^{2}x_{7}y$, |  $x_{1}x_{2}x_{3}x_{4}^{2}x_{5}^{3}x_{6}x_{7}y$,
$x_{1}x_{2}x_{3}x_{4}^{3}x_{5}x_{6}x_{7}^{2}y$, |  $x_{1}x_{2}x_{3}x_{4}^{3}x_{5}x_{6}^{2}x_{7}y$, |  $x_{1}x_{2}x_{3}x_{4}^{3}x_{5}^{2}x_{6}x_{7}y$, |  $x_{1}x_{2}x_{3}^{2}x_{4}x_{5}x_{6}x_{7}^{3}y$,
$x_{1}x_{2}x_{3}^{2}x_{4}x_{5}x_{6}^{2}x_{7}^{2}y$, |  $x_{1}x_{2}x_{3}^{2}x_{4}x_{5}x_{6}^{3}x_{7}y$, |  $x_{1}x_{2}x_{3}^{2}x_{4}x_{5}^{2}x_{6}x_{7}^{2}y$, |  $x_{1}x_{2}x_{3}^{2}x_{4}x_{5}^{2}x_{6}^{2}x_{7}y$,
$x_{1}x_{2}x_{3}^{2}x_{4}x_{5}^{3}x_{6}x_{7}y$, |  $x_{1}x_{2}x_{3}^{2}x_{4}^{2}x_{5}x_{6}x_{7}^{2}y$, |  $x_{1}x_{2}x_{3}^{2}x_{4}^{2}x_{5}x_{6}^{2}x_{7}y$, |  $x_{1}x_{2}x_{3}^{2}x_{4}^{2}x_{5}^{2}x_{6}x_{7}y$,
$x_{1}x_{2}x_{3}^{3}x_{4}x_{5}x_{6}x_{7}^{2}y$, |  $x_{1}x_{2}x_{3}^{3}x_{4}x_{5}x_{6}^{2}x_{7}y$, |  $x_{1}x_{2}x_{3}^{3}x_{4}x_{5}^{2}x_{6}x_{7}y$, |  $x_{1}x_{2}^{2}x_{3}x_{4}x_{5}x_{6}x_{7}^{3}y$,
$x_{1}x_{2}^{2}x_{3}x_{4}x_{5}x_{6}^{2}x_{7}^{2}y$, |  $x_{1}x_{2}^{2}x_{3}x_{4}x_{5}x_{6}^{3}x_{7}y$, |  $x_{1}x_{2}^{2}x_{3}x_{4}x_{5}^{2}x_{6}x_{7}^{2}y$, |  $x_{1}x_{2}^{2}x_{3}x_{4}x_{5}^{2}x_{6}^{2}x_{7}y$,
$x_{1}x_{2}^{2}x_{3}x_{4}x_{5}^{3}x_{6}x_{7}y$, |  $x_{1}x_{2}^{2}x_{3}x_{4}^{2}x_{5}x_{6}x_{7}^{2}y$, |  $x_{1}x_{2}^{2}x_{3}x_{4}^{2}x_{5}x_{6}^{2}x_{7}y$, |  $x_{1}x_{2}^{2}x_{3}x_{4}^{2}x_{5}^{2}x_{6}x_{7}y$,
$x_{1}x_{2}^{2}x_{3}^{2}x_{4}x_{5}x_{6}x_{7}^{2}y$, |  $x_{1}x_{2}^{2}x_{3}^{2}x_{4}x_{5}x_{6}^{2}x_{7}y$, |  $x_{1}x_{2}^{2}x_{3}^{2}x_{4}x_{5}^{2}x_{6}x_{7}y$, |  $x_{1}^{2}x_{2}x_{3}x_{4}x_{5}x_{6}x_{7}^{3}y$,
$x_{1}^{2}x_{2}x_{3}x_{4}x_{5}x_{6}^{2}x_{7}^{2}y$, |  $x_{1}^{2}x_{2}x_{3}x_{4}x_{5}x_{6}^{3}x_{7}y$, |  $x_{1}^{2}x_{2}x_{3}x_{4}x_{5}^{2}x_{6}x_{7}^{2}y$, |  $x_{1}^{2}x_{2}x_{3}x_{4}x_{5}^{2}x_{6}^{2}x_{7}y$,
$x_{1}^{2}x_{2}x_{3}x_{4}x_{5}^{3}x_{6}x_{7}y$, |  $x_{1}^{2}x_{2}x_{3}x_{4}^{2}x_{5}x_{6}x_{7}^{2}y$, |  $x_{1}^{2}x_{2}x_{3}x_{4}^{2}x_{5}x_{6}^{2}x_{7}y$, |  $x_{1}^{2}x_{2}x_{3}x_{4}^{2}x_{5}^{2}x_{6}x_{7}y$,
$x_{1}^{2}x_{2}x_{3}^{2}x_{4}x_{5}x_{6}x_{7}^{2}y$, |  $x_{1}^{2}x_{2}x_{3}^{2}x_{4}x_{5}x_{6}^{2}x_{7}y$, |  $x_{1}^{2}x_{2}x_{3}^{2}x_{4}x_{5}^{2}x_{6}x_{7}y)R_{2},$ |

where $y=y_{1}\cdot\ldots\cdot y_{10}.$
Hence, we have the Cohen-Macaulay type of $R_{2}$ is 55$=\begin{bmatrix}3\\\
2\end{bmatrix}_{3}$.

In general it is almost impossible to describe the generators of the canonical
module of the coordinate ring for a polyomino. Our final result gives a
description in a particular case.

For $n,t,p\in\mathbb{N}$, $1\leq t<n$, $p\geq 1$, let $u_{1}=\cdots=u_{p}=n$
and $r_{1}=\cdots=r_{p}=t$ and we consider
$\mathcal{P}=\mathcal{P}\left({\begin{array}[]{cccc}u_{1}&u_{2}&...&u_{p}\\\
r_{1}&r_{2}&...&r_{p}\\\ \end{array}}\right)$.

######  Theorem 10.

Let $S=K[x_{1},\ldots,x_{pt+1},y_{1},\ldots,y_{pn+1}]$ be a standard graded
polynomial ring over a field $K$ and $R=K[x_{i}y_{j}\ |\ (i,j)\in
V(\mathcal{P})].$ Then the Cohen-Macaulay type of $R$ is
$r(R)=\begin{bmatrix}n\\\ t\end{bmatrix}_{p}$.

###### Proof.

We know that $K[\mathcal{P}]$ is a normal Cohen-Macaulay domain of dimension
$|V(\mathcal{P})|$ \- $|\mathcal{P}|=pt+pn+1$. It follows that $R$ is normal
and we recall that by a well known formula of Danilov-Stanley the canonical
module $\omega_{R}$ of $R,$ with respect to standard grading, can be expressed
as an ideal of $R$ generated by monomials, that is $\omega_{R}=(\\{x^{a}|\
a\in{\mathbb{N}}B\cap\operatorname{relint}({\mathbb{R_{+}}}B)\\}),$ where

$B:=\\{\log(x_{i}y_{j})\ |\ (i,j)\in
V(\mathcal{P})\\}\subset{\mathbb{N}}^{pt+pn+2}$

is the set of the exponents of the generators of the toric ring $R$. Thus the
canonical module is the ideal

$\omega_{R}=(\\{x^{\alpha}y\ |\ \alpha\in{\mathbb{Z}}_{>}^{pt+1},\
\sum_{i=1}^{pt+1}\alpha_{i}=pn+1,\ \sum_{i=1}^{kt}\alpha_{i}<kn+1\ for\ all\
k\in[p-1]\\})R,$

where $x^{\alpha}=x_{1}^{\alpha_{1}}\cdot\ldots\cdot x_{pt+1}^{\alpha_{pt+1}}$
and $y=y_{1}\cdot\ldots\cdot y_{pn+1}$.
Then

$\omega_{R}=(\\{x^{\alpha}y\ |\ \alpha-{\bf{1}}\in A(n,t,p)\\})R,$

where ${\bf{1}}=(1,1,\ \ldots,1)\in{\mathbb{Z}}_{>}^{pt+1}$ and so the Cohen-
Macaulay type of $R$ is $r(R)=\begin{bmatrix}n\\\ t\end{bmatrix}_{p}$. ∎

For $n,t,p\in\mathbb{N}$, $1\leq t<n$, $p\geq 1$, let $u_{1}=\cdots=u_{p}=n$
and $r_{1}=\cdots=r_{p}=n-t$ and we consider
$\mathcal{P}=\mathcal{P}\left({\begin{array}[]{cccc}u_{1}&u_{2}&...&u_{p}\\\
r_{1}&r_{2}&...&r_{p}\\\ \end{array}}\right)$.

######  Corollary 11.

Let $S=K[x_{1},\ldots,x_{p(n-t)+1},y_{1},\ldots,y_{pn+1}]$ be a standard
graded polynomial ring over a field $K$ and $R=K[x_{i}y_{j}\ |\ (i,j)\in
V(\mathcal{P})].$ Then the Cohen-Macaulay type of $R$ is
$r(R)=\begin{bmatrix}n\\\ t\end{bmatrix}_{p}$.

$\bf{Acknowledgement.}$ The author would like to thank the anonymous referee
for her/his valuable suggestions.

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Alin Ştefan
Department of Computer Science, Information Technology, Mathematics and
Physics
Petroleum and Gas University of Ploieşti
B-dul. Bucureşti, No. 39, Ploieşti 100680, Romania
<EMAIL_ADDRESS><EMAIL_ADDRESS>