Properties of the coordinate ring of a convex polyomino

We classify all convex polyomino whose coordinate rings are Gorenstein. We also compute the Castelnuovo-Mumford regularity of the coordinate ring of any stack polyomino in terms of the smallest interval which contains its vertices. We give a recursive formula for computing the multiplicity of a stack polyomino.


Introduction
A polyomino P is a finite connected set of adjacent cells in the cartesian plane N 2 . A cell in N 2 is simply a unitary square. A polyomino P is said to be column convex (respectively row convex) if every column (respectively row) is connected. According to [2], P is a convex polyomino if for every two cells of P there is a monotone path between them, that is a path having only two directions, contained in P. Convex polyominoes include one-sided ladders, 2-sided ladders and stack polyominoes.
Let K be a field and consider the polynomial ring S = K[x ij |(i, j) vertex of P]. The polyomino ideal I P is the ideal of S generated by all 2-inner minors of P, where a 2-inner minor of P is a 2-minor of the matrix X = (x ij ) ij which involves only indeterminates of the vertices of P. The coordinate ring of P is defined as the quotient ring K[P] = S/I P . The ideal I P and the ring K[P] were first studied by Qureshi in [10]. There it was shown that if P is a convex polyomino, then K[P] is a normal Cohen-Macaulay domain. This was proved by viewing the ring K[P] as the edge ring of a suitable bipartite graph G P associated with P.
Understanding the graded free resolution of polyomino ideals is a difficult task. A first step in this direction was done in [5], where the convex polyomino ideals which are linearly related or have a linear resolution are classified.
In this paper, we continue the study of the algebraic properties of K[P].
In Section 1, we recall the basic terminology related to convex polyominoes and their associated bipartite graphs. The first main result of this paper appears in Section 2, where we classify all convex polyominoes whose coordinate rings are Gorenstein (Theorem 21). For this classification, we use a result due to Ohsugi and Hibi ([8]) who classified all 2-connected bipartite graphs whose edge rings are Gorenstein. In the case of stack polyominoes, we recover the classification of all Gorenstein stack polyominoes given in [10,Corollary 4.12]; see Section 3.
In Section 4, we give an upper bound for the Castelnuovo-Mumford regularity of the coordinate ring of any convex polyomino in terms of the smallest interval which contains its vertices (Proposition 37). The computation of the upper bound of the regularity uses as an important tool the formula of the a-invariant of the edge ring of a bipartite graph given in [11].
Finally, in Section 5 we give a recursive formula for computing the multiplicity of K[P] if P is a stack polyomino and we show some concrete cases when this formula may be applied.

Preliminaries
To begin with, we recall some concepts and introduce notation about collections of cells and polyominoes.
We consider on N 2 the natural partial order defined as follows: (i, j) (k, l) if and only if i k and j l. If a, b ∈ N 2 with a b, then the set  is called a cell interval. In the case that j = l (respectively i = k), the cell interval [A, B] is called a horizontal (respectively vertical) cell interval.
Let P be a finite collection of cells of N 2 . The vertex set of P and the edge set of P are V (P) = ∪ C∈P V (C) and E(P) = ∪ C∈P E(C), where C are the cells of P. Two cells A and B of P are connected, if there is a sequence of cells of P given by A = A 1 , A 2 , . . . , A n−1 , A n = B such that A i ∩ A i+1 is an edge of A i and A i+1 for each i ∈ {1, . . . , n − 1}. Such a sequence is called a path connecting the cells A and B. Definition 1. A collection of cells P is called a polyomino if any two cells of P are connected. Definition 2. A polyomino P is called row (respectively column) convex, if for any two cells A and B of P with left lower corners a = (i, j) and b = (k, j) (respectively a = (i, j) and b = (i, l)), the horizontal (respectively vertical) cell interval [A, B] is contained in P. If P is row and column convex, then P is called a convex polyomino.
In Figure 1 we have two examples of polyominoes: the one on the right is a convex polyomino, while the other one is row convex but not column convex, hence it is not convex.
Let P be a convex polyomino and [a, b] ⊂ N 2 be the smallest interval which contains V (P). After a shift of coordinates, we may assume that a = (1, 1) and b = (m, n) and thus, we say that P is a convex polyomino on [m] × [n], where for a positive integer a, [a] denotes the set {1, . . . , a}. For example, the right side polyomino in Figure 1 is a convex polyomino on [4] × [4].
y j y j+1 Figure 2: The bipartite graph attached to a cell in N 2 Fix a field K and a polynomial ring . We consider the ideal I P ⊂ S generated by all binomials . , x m , y 1 , . . . , y n ] can be viewed as the edge ring of the bipartite graph G P with vertex set V (G P ) = X ∪ Y , where X = {x 1 , . . . , x m } and Y = {y 1 , . . . , y n } and edge set E(G P ) = {{x i , y j } | (i, j) ∈ V (P)}. In Figure 2, we displayed the bipartite graph attached to a cell in N 2 . According to [10], K[P] can be identified with K[G P ].

Gorenstein convex polyominoes
Let P be a convex polyomino on [m]×[n]. We set X = {x 1 , . . . , x m } and Y = {y 1 , . . . , y n } and, if needed, we identify the point (x i , y j ) in the plane with the vertex (i, j) ∈ V (P).
Let A and B be two cells in P. Recall that A and B are connected by a path if there is a sequence of cells in P, A = A 1 , A 2 , . . . , A r−1 , A r = B, with the property that A i ∩ A i+1 is an edge of A i and A i+1 , for each i ∈ [r − 1]. We denote by (x j i , y k i ) the lower left corner of A i , for all i ∈ [r]. Every path in P may go in at most four directions which are given below: In the next proposition we show that the bipartite graph G P associated with P is 2-connected. Let us first recall the definition of 2-connectivity. Definition 4. If G is a finite connected graph on the vertex set V , then given a subset ∅ = W ⊂ V , G W denotes the induced subgraph of G on W . We say that G is 2-connected if G together with G V \{v} are connected for all v ∈ V .
Proposition 5. Let P be a convex polyomino on [m] × [n]. Then the bipartite graph G P is 2-connected.
Proof. Firstly, we prove that the bipartite graph G P is connected. For that it is sufficient to choose x, x ∈ {x 1 , . . . , x m } and to find a path between them in G P . Let x, x , y, y ∈ V (G P ) such that (x, y), (x , y ) ∈ V (P). Without loss the generality, we may suppose that (x, y − 1), (x , y − 1) / ∈ V (P). Since P is a convex polyomino, there exists a monotone path Γ from a cell containing (x, y) to a cell containing (x , y ), both as outside corners of Γ. Let us consider the sequence γ of vertices of P belonging to the cells of Γ. Now, let is a lower outside or inside corner of the path Γ}.
We claim that A γ is a path in G P containing x and x . Clearly, {x, y} and {x , y } belong to A γ by definition of Γ. Since Γ is a monotone path, for every {x i k , y j k } ∈ A γ \ {{x, y}, {x y }}, there exist exactly two other edges of the form {x i k , y r } and {x s , y j k } in A γ , with r = j k and s = i k .
In order to complete the proof, we show that for any k ∈ [m], the graph . In a similar way as in the first part of the proof, we consider Γ to be a monotone path in P from a cell containing (x, y) to a cell containing (x , y ). Let A = {{x i l , y j l } | (x i l , y j l ) is a lower outside or inside corner of the path Γ}.
If for any {x i l , y j l } ∈ A, we have x i l = x k , then A is a path in G containing x and x by the argument used above.
If there is {x i l , y j l } ∈ A such that i l = k, then we have exactly two edges If (x k−1 , y j l 1 ), (x k−1 , y j l 2 ) ∈ V (P), then A ∪ {{x k−1 , y j l 1 }, {x k−1 , y j l 2 }} is a path in G containing x and x else A ∪ {{x k+1 , y j l 1 }, {x k+1 , y j l 2 }} is a path in G containing x and x by the argument used in the first part of the proof.
For the characterisation of Gorenstein convex polyominoes we need the following theorem due to Ohsugi and Hibi ([8] be the polynomial ring. The edge ring of G is the toric ring for some x ∈ T } represents the set of the neighbors of the subset T . Theorem 6. [8, Theorem 2.1] Let G be a bipartite graph on X ∪ Y and suppose that G is 2-connected. Then the edge ring of G is Gorenstein if and only if x 1 · · · x m y 1 · · · y n ∈ K[G] and one has |N (T )| = |T | + 1 for every subset T ⊂ X such that G T ∪N (T ) is connected and that G (X∪Y )\(T ∪N (T )) is a connected graph with at least one edge.
Note that x 1 · · · x m y 1 · · · y n ∈ K[G] if and only if G possesses a perfect matching (i.e. there is a set of edges E ⊂ E(G) with the property that no two of them have a common vertex and ∪ {x,y}∈E {x, y} = V (G)). A characterization of the bipartite graph which possesses a perfect matching is given by Villarreal. Recall that a subset of vertices of G is called independent if no two of them are adjacent.
From now on, whenever we consider a convex polyomino P, we consider it endowed with its associated bipartite graph G P on the vertex set V (G P ) = X ∪ Y . . Then x 1 · · · x m y 1 · · · y n ∈ K[G P ] if and only if |N (T )| |T | for every T ⊂ X or T ⊂ Y .
Proof. If x 1 · · · x m y 1 · · · y n ∈ K[G P ], then by Theorem 7, we obtain |N (T )| |T |, for every independent subset of vertices T ⊂ X ∪ Y . Notice that all subsets T ⊂ X and U ⊂ Y are independent.
Conversely, we suppose |N (T )| |T | for every T ⊂ X or T ⊂ Y . Let be an independent set of vertices with r, s 1. Then, by assumption, Thus, |T | |N (T )| and according to Theorem 7, x 1 · · · x m y 1 · · · y n ∈ K[G P ].
Remark 11. If the subset T = {x} ⊂ X has only one element, then N Y (T ) is a neighbor vertical interval. Indeed, let y i 1 , y i 2 ∈ N Y (x) with i 1 < i 2 . By Proposition 3, there exists a monotone path between the cells containing (x, y i 1 ) and (x, y i 2 ) as corners. We display some of the monotone paths between two cells in Figure 3.
Example 12. In the polyomino of Figure 4, let Lemma 13. Let P be a convex polyomino. Then for each ∅ = T X, the following conditions are equivalent: Proof. For (1)⇒(2), it is sufficient to choose x, z ∈ T and to find a path d between them in G T ∪N (T ) . Without loss of generality, we may choose y s ∈ N Y (x) and y t ∈ N Y (z) with s < t. Then by hypothesis, {y s , y s+1 , . . . , So the path between x and z in G P is is an edge in P. In particular, it also follows that i j+1 = i j + 1 for each j ∈ [s − 1], which will end the proof. Let Since there is no y l ∈ N Y (T ) between y i j and y i j+1 , the only cases that can occur are: the electronic journal of combinatorics 28(1) (2021), #P1.45 is an edge in P. We proceed in a similar way in the case that for all a ∈ [r − 2] we have Since P is a convex polyomino, by Proposition 3, there is a monotone path between the cells containing (x i 1 , y) and (x i 2 , y) as corners. We display some of the monotone paths between two cells in Figure 5. Then we have [(x i 1 , y), (x i 2 , y)] ⊂ P.
Lemma 17. If P is a convex polyomino, then for each ∅ = T ⊂ X, Proof. First assume that for every : Example 18. In Figure 6 Lemma 19. Let P be a convex polyomino. Then for each ∅ = T X, the following conditions are equivalent: is a connected graph with at least one edge.
Proof. Let T be a subset in X which satisfies the conditions given in (1). By Lemma 17 and the fact that T X, there is For the connectivity of the graph G (X∪Y )\(T ∪N Y (T )) , it is sufficient to choose y, z ∈ Y \ N Y (T ) and to find a path between them in G (X∪Y )\(T ∪N Y (T )) . Without loss of generality, we consider x s ∈ N X (y) and the electronic journal of combinatorics 28(1) (2021), #P1.45 In other words, the path between y and z is Conversely, we suppose that The claim follows using the same argument of the proof of Lemma 13 (swapping the x i 's with the y i 's and replacing T with Y \ N Y (T )).
Example 20. In Figure 6, let Figure 7). The graph G (X∪Y )\(T 2 ∪N Y (T 2 )) is represented by the two isolated vertices x 4 and x 5 .
Let P be a convex polyomino. Since the coordinate ring of P can be viewed as an edge ring of a bipartite graph, by applying Theorem 6, Corollary 8, Lemma 13 and Lemma 19, we get the following result.
Then K[P] is Gorenstein if and only if the following conditions are fulfilled:

For
4. For T = {x 6 }, we have property (a), but N X (Y \ N Y (T )) = X \ T is not a neighbor horizontal interval.
The polyomino P 2 of Figure 9 is Gorenstein, because x 1 y 1 · x 2 y 2 · x 3 y 4 · x 4 y 3 ∈ K[P 2 ] and for each T which satisfies the properties (a), (b), one has |N Y (T )| = |T | + 1. In this case, we need to check the conditions of the Theorem 21 only for two sets: Definition 23. Let P be a convex polyomino. A vertex a ∈ V (P) is called an interior vertex of P, if a is a vertex of four distinct cells of P. We denote by int(P) the set of all interior vertices of P. The set ∂P = V (P) \ int(P) is called the boundary of P. We say that the vertex a ∈ ∂P is an inside (outside) corner of P if it belongs to exactly three (one) different cells of P. (Figure 10) Let P be a convex polyomino on [m]×[n]. Then P is called two-sided ladder ( Figure 11) if for every (i, j), (k, l) ∈ V (P) with i k, j l, we have (i, l), (k, j) ∈ V (P).
As a consequence of Theorem 21, we may recover the characterisation of Gorenstein two-sided ladder polyominoes obtained by Conca in [4, Theorem 5.2]. Proof. Let P be a two-sided ladder polyomino on [m] × [n] such that K[P] is Gorenstein. By the first condition of Theorem 21 and Remark 9, we obtain m = n.
Let (x r , y t ) ∈ V (P) be an inside corner of P. If (x r , y t ) is a lower inside corner, then we consider T = {x r+1 , x r+2 , . . . , x n }. Since P is a two-sided ladder polyomino, N Y (T ) = {y 1 , y 2 , . . . , y t } and 1 < r, t < n, T satisfies the second condition of Theorem 21. Thus, we obtain r + t = n − |T | + |N Y (T )| = n + 1 and (x r , y t ) ∈ {(x i , y j ) ∈ V (P)|i + j = n + 1}. In the case that (x r , y t ) ∈ V (P) is an upper inside corner of P, we proceed in a similar way.
Conversely, we suppose that m = n and the inside corners of P belong to the set {(x i , y j ) ∈ V (P)|i + j = n + 1}. According to Corollary 8, for the proof of the first condition of Theorem 21, it is sufficient to show that ( , we obtain x 1 · · · x n y 1 · · · y n ∈ K[P]. Let i ∈ [n] and set r = max{j ∈ [n]|y j ∈ N Y (x i )} and s = min{j ∈ [n]|y j ∈ N Y (x i )}. If 1 < s < r < n, then (x j , y r ) and (x l , y s ) are inside corners of P for some j ∈ {1, 2, . . . , i− 1, i} and some l ∈ {i, i + 1, i + 2, . . . , n}. By hypothesis, r = n + 1 − j n + 1 − i and s = n+1−l n+1−i. In other words, 1 < s n+1−i r < n and (x i , y n+1−i ) ∈ V (P). If s = 1, then (x k , y r ) is either an inside corner of P or a top left corner of P (i.e., (x 1 , y n )) for some k ∈ {1, 2, . . . , i − 1, i}. Thus, r = n + 1 − k n + 1 − i 1 = s and (x i , y n+1−i ) ∈ V (P). In the case that r = n, (x k , y s ) is an inside corner of P or a bottom right corner of P (i.e., (x n , y 1 )) for some k ∈ {i, i + 1, . . . , n}. Hence, s = n + 1 − k n + 1 − i n = r and (x i , y n+1−i ) ∈ V (P).

Gorenstein stack polyominoes
In this section we simplify the characterization of Theorem 21 for the subclass of stack polyominoes, recovering a result of Qureshi [10,Corollary 4.12]. Stack polyominoes have the nice property that N Y (T ) is a neighbor vertical interval for all ∅ = T ⊂ X. We consider P to be a polyomino and we may assume that [(1, 1), (m, n)] is the smallest interval containing V (P). Then P is called a stack polyomino (Figure 12), if it is a convex polyomino and for i ∈ [m − 1], the cell [(i, 1), (i + 1, 2)] belongs to P.
Remark 25. If P is a stack polyomino, then for every x ∈ X we have {y 1 , Let T = ∅ be a subset of X and y j ∈ N Y (T ) \ {y 1 , y 2 }. Hence, there exists x k ∈ T such that y j ∈ N Y (x k ). Since N Y (x k ) is a neighbor vertical interval, In other words, N Y (T ) = {y 1 , y 2 , . . . , y s } is a neighbor vertical interval for all ∅ = T X, where s = max{i ∈ [n] | (x k , y i ) ∈ V (P) for some x k ∈ T }.
In the case that I ⊂ X and |I| > |N Y (I)|, we consider We check conditions (1) and (2) for the set T . Since P is a stack polyomino, N Y (T ) = {y 1 , y 2 , . . . , y s } for some s n.
Thus, there is l > s such that y l ∈ N Y (x) \ N Y (T ) and condition (2) holds.
If there exists J ⊂ Y with |J| > |N X (J)|, then we set We check conditions (1) and (2) for the set T . For the proof of the first condition, it is sufficient to show that J ⊂ Y \ N Y (T ). Let y ∈ J. If y ∈ N Y (T ), then there is x ∈ T ∩N X (y). Since y ∈ J, we get x ∈ N X (y) ⊂ N X (J) = X\T . Thus, Hence, we proved X \T = N X (Y \N Y (T )). By Lemma 17 and the previous remark for any x ∈ X \ T , N Y (x) N (T ) and we have the second condition.
As a consequence of Theorem 21, we may recover the characterisation of Gorenstein stack polyominoes obtained by Qureshi in [ 2. m = n and for every T ⊂ X with the properties that Y \ N Y (T ) = ∅ and for every ) and this is a neighbor horizontal interval by Remark 15. By using Theorem 21 and Corollary 8, |N Y (T )| = |T | + 1 and x 1 · · · x m y 1 · · · y n ∈ K[P]. Thus, we also obtain m = n by Remark 9.
Let ∅ = T X such that N Y (T ) is a neighbor vertical interval and N X (Y \ N Y (T )) = X \ T is a neighbor horizontal interval. Since T X, there exists x ∈ X \ T with N Y (x) N Y (T ), by Lemma 17. It follows that we find We may reformulate Corollary 27 as follows.  Proof. Let K[P] be Gorenstein and (x r , y t ) be an inside corner of P.
For x ∈ X \ T we have that max{j ∈ [n] | y j ∈ N Y (x)} > t. By Corollary 27, it follows that |N Y (T )| = |T | + 1. Thus, |T | = t − 1. In other words, n − t + 1 = n − |T | and the minimal rectangle we are interested in is a square.
Conversely, we suppose that T ⊂ X is a set with the properties that Y \ N Y (T ) = ∅ and for every x ∈ X \ T , Since P is a column convex polyomino, y r is the y-coordinate of an inside corner. Then by assumption, |X \ T | = n − r + 1. Hence, n − |T | = n − r + 1 and |T | + 1 = r = |N Y (T )|. By Corollary 27, K[P] is Gorenstein.
Notice that Corollaries 27 and 28 extend the classification of Gorenstein one-sided ladder polyominoes given in [3,Theorem 4.9(c)].
Examples 29. By Corollary 28, the first polyomino of Figure 13 is Gorenstein, while the second is not.

The regularity of K[P]
Let P be a convex polyomino on [m]×[n]. Recall that the coordinate ring of P is a finitely generated module over the polynomial ring S = K[x ij | (i, j) ∈ V (P)]. The Castelnuovo-Mumford regularity of K[P], denoted reg(K[P]), is defined to be the largest integer r such that, for every i, the i th syzygy of K[P] is generated in degree at most r + i.
We consider H K[P] (t) to be the Hilbert series of K[P]. Then Let G P be the bipartite graph attached to P on the vertex set X ∪ Y . In this section, we consider G P as a digraph with all its arrows leaving the vertex set Y . Hence, we denote the directed edges by (z, w), where z ∈ Y and w ∈ X. Following [11], we introduce the following notion.
is the set of edges leaving the vertex set T and is the set of edges entering the vertex set T .
x 1 x 2 x 3 y 1 y 2 y 3 y 4 Figure 14: A convex polyomino and its associated digraph Proof. Let T = ∅ be a proper subset in X ∪ Y . Then T = A ∪ B with A ⊂ X and B ⊂ Y . By Definition 30 and Remark 32, Then there exist x ∈ A and y ∈ Y \B such that (x, y) ∈ V (P). In other words, (x, y) ∈ δ − (T ) = ∅. Conversely, suppose that δ − (T ) = ∅. Then we find x ∈ A and y ∈ Y \ B such that (x, y) ∈ V (P). This is equivalent to saying that In [11], Valencia and Villarreal show that for any connected bipartite graph G, the a-invariant, a(K[G]) can be interpreted in combinatorial terms as follows.
are disjoint directed cuts and also, For every i ∈ [m], we define the height of i as Following the proof of [9, Theorem], we give a total order on the variables x ij , with (i, j) ∈ V (P), as follows: x ij > x kl if and only if (2) (height(i) > height(k)) or (height(i) = height(k) and i > k) or (i = k and j > l). m n Figure 16: Let < be the reverse lexicographical order induced by this order of variables. As we have already seen in the previous sections, the ideal I P can be viewed as the toric ideal of the edge ring K[G P ], where G P is the bipartite graph associated to P. As it follows from the proof of [9,Theorem], the reduced Gröbner basis of I P with respect to < consists of all 2-inner minors of P. In what follows, whenever we consider the Gröbner basis of I P , we assume that the variables x ij , with (i, j) ∈ V (P) are totally ordered as in (2). We notice that in < (I P ) is a squarefree monomial ideal. Thus, we may view in < (I P ) as the Stanley-Reisner ideal of a simplicial complex ∆ P on the vertex set V (P). It is known that ∆ P is a pure shellable simplicial complex by [ Let x ij be the smallest variable in S with respect to < and fix v = (i, height(i)) ∈ V (P). If i = 1, then we denote by P 1 the polyomino obtained from P by deleting the only cell which contains the vertex v. Otherwise, P 1 is given by deleting the only cell which contains the vertex (m, height(m)); see Figure 18. Notice that in both cases dim(∆ P 1 ) = d − 1 = m + n − 2.
Remark 41. Since x i1 is the smallest variable with respect to <, we have (i, 1) ∈ F for every F ∈ ∆ P . Indeed, x i1 is regular on S/ in < (I P ), thus it does not belong to any of the minimal primes of in < (I P ) which implies that x i1 belongs to all the facets of ∆ P .
In what follows we will sometimes confuse the point (i, j) of P with the vertex x ij of ∆ P .
Proof. Let x ij be the smallest variable in S with respect to < and set v = (i, height(i)) ∈ V (P).
We depict the points that are in F by black dots, the points removed from F by crosses and the points added to F by empty dots. Now, we observe that even if the order of the variables for P 1 is not induced by the order of the variables of P, the generators of in < (I P 1 ) are also generators of in < (I P ), since the 2-inner minors of P 1 are also 2-inner minors of P. Therefore, we may conclude that F ∈ ∆ P 1 and so F ∈ F(∆ P 1 ).
Therefore, we have shown that every facet F of del(v) determines uniquely a facet F of ∆ P 1 , if height(i) 3. for k = 1 to h do 14: if (j, k) ∈ F then 15: end for 24: end if 25: return F Conversely, let F be a facet of ∆ P 1 . Following the steps of Algorithm 1 in reverse order, we obtain a facet F of del(v). Algorithm 2 gives explicitly all the steps to get F from F . We thus get |F(∆ P 1 )| = |F(del(v))| if height(i) 3. Moreover, we have equality between the sets F(∆ P 1 ) and F(del(v)) if and only if i = 1 and height(i) 3.
In order to complete the proof, let us point out that the same two algorithms work for height(i) = 2. In fact, for i > 1 (respectively i = 1), F is a facet of del(v) if and only if F is a facet of the cone (m, 1) * ∆ P 1 (respectively (1, 1) * ∆ P 1 ).
For example, if we consider the polyomino P of Figure 20, v = (3, 2) ∈ V (P) and if we choose for j = m − 2 to i do 11: for k = 1 to h do 12: if (j, k) ∈ F then 13: if (m, k) ∈ F then end for 26: end if 27: return F Let P 2 be the polyomino obtained from P by deleting all the cells of P which lie below the horizontal edge interval containing the vertex v.
Proof. Let F be a facet of lk(v). Then F ∪ {v} ∈ F(∆ P ). We set j = height(i).
We now prove the main result of this section.
Theorem 44. Let P be a stack polyomino on [m] × [n] and v = (i, j) ∈ V (P) such that x i1 is the smallest variable in S and j = height(i). Then where P 1 and P 2 are the polyominoes defined before.
e(K[P(m − l + 1, n − j 1 − · · · − j l−1 , k 1 − j 1 − · · · − j l−1 )]) = k l −1 (m − l + 1) + (n − j 1 − · · · − j l−1 ) − 2 (m − l + 1) − 1 − (m − l + 1) + (n − j 1 − · · · − j l−1 ) − (k 1 − j 1 − · · · − j l−1 ) − 2 (m − l + 1) − 1 . One may, of course, approach the computation of the multiplicity in a recursive way for arbitrary convex polyominoes. Finding the appropriate order of the variables in concordance to the one described in [9] is not difficult as we will see in the example below. What is difficult in the general case is to identify the link of a suitable chosen vertex as a simplicial complex of another polyomino related to the original one. We illustrate part of these difficulties in the following example.
Example 48. Let P be the convex polyomino of Figure 24. According to the proof of [9], the generators of I P form the reduced Gröbner basis of I P with respect to the reverse lexicographical order induced by the following order of variables: x 32 > x 33 > x 34 > x 31 > x 22 > x 23 > x 24 > x 12 > x 13 > x 14 > x 42 > x 43 > x 41 > x 52 > x 53 . We consider the vertex v = (5, 3). The link of v in ∆ P is the cone of the vertex (5, 2) with the simplicial complex which we may associate to the collection of cells Q displayed in Figure 24 in a similar way to the one we used for stack polyominoes. The problem is that the collection Q is no longer a convex polyomino.