Regularity and Gorenstein property of the $L$-convex Polyominoes

We study the coordinate ring of an $L$-convex polyomino, determine its regularity in terms of the maximal number of rooks that can be placed in the polyomino. We also characterize the Gorenstein $L$-convex polyominoes and those which are Gorenstein on the punctured spectrum, and compute the Cohen--Macaulay type of any $L$-convex polyomino in terms of the maximal rectangles covering it.


Introduction
Commonly, a polyomino is a shape in the Cartesian plane N × N consisting of unit squares which are joined edge-to-edge. Classical examples of polyominoes are Ferrer diagrams, the stack and parallelogram polyominoes. They have a long history in combinatorics. Originally, polyominoes appeared in mathematical recreations, but it turned out that they have applications in various fields, for example, theoretical physics and bio-informatics. Among the most popular topics in combinatorics related to polyominoes one finds enumerating polyominoes of given size, including the asymptotic growth of the numbers of polyominoes, tilling problems, and reconstruction of polyominoes. A very nice introduction to the combinatorics of polyominoes and tilings is given in the monograph [23].
In the last decade, polyominoes have been related to the study of binomial ideals generated by collections of 2-minors of a generic matrix. Ideals of this kind where first introduced and studied by Qureshi [19]. For a polyomino P, in [19], the ideal I P generated by the inner minors of P was considered. For a field K, the K-algebra K[P] whose relations are given by I P is called the coordinate ring of P. Several authors studied the algebraic properties and invariants of K [P], relating them to the shape of P.
The study of the ideal of t-minors and related ideals of an m × n-matrix X = (x ij ) of indeterminates is a classical subject of commutative algebra and algebraic geometry; see for example the lecture notes [2] and its references to original articles. Several years after the appearance of the these lecture notes, a new aspect of the theory was introduced by considering Gröbner bases of determinantal ideals. These studies were initiated by the articles [18], [3] and [22]. More generally and motivated by geometric applications, ideals of t-minors of 2-sided ladders have been studied, see [7], [5], [6] and [11]. For the case of 2-minors, these classes of ideals may be considered as special cases of the ideal I P of inner 2-minors of a polyomino P.
One of the most challenging problems in the algebraic theory of polyminoes is the classification of the polyminoes P whose coordinate ring K[P] is a domain. It has been shown in [13] and [20] that this is the case if the polyomino is simply connected. In a more recent paper by Mascia, Rinaldo and Romeo [16], it is shown that if K[P] is a domain then P should not have a zig-zag walks, and they conjecture that this is also a sufficient condition for K[P] to be a domain. They verify this conjecture computationally for polyominoes of rank 14. It is expected that K[P] is always reduced.
Additional structural results on K[P] for special classes of polyominoes were already shown in Qureshi's article [19]. There she proved that K[P] is a Cohen-Macaulay normal domain when P is a convex polyomino, and characterized all stack polyominoes for which K [P] is Gorenstein by computing the class group of this algebra.
In the present paper we focus on so-called L-convex polyominoes. This is a particularly nice class of convex polyominoes which is distinguished by the property that any two cells of the polyomino can be connected by a path of cells with at most one change of directions. The combinatorics of this class of polyominoes is described in the paper [9] and [4] by Castiglione et al. In Section 1 we recall some of the remarkable properties of L-convex polyominoes referring to the above mentioned papers. In particular, if P is an L-convex polyomino, then there is a natural bipartite graph F P whose edges correspond to the cells of P. By using this correspondence, we show in Proposition 2.3 that there exists a polyomino P * which is a Ferrer diagram and such that the bipartite graphs F P and F P * are isomorphic. We call P * the Ferrer diagram projected by P. Similarly there exists a bipartite graph G P whose edges correspond to the coordinates of the vertices of P. By using the intimate relationship between F P and G P it can be shown that G P and G * P are isomorphic as well, see Corollary 2.7. The crucial observation which then follows from these considerations is the result (Theorem 3.1) that K[P] and K[P * ] are isomorphic as standard graded K-algebras. Therefore all algebraic invariants and properties of K[P] are shared by K[P * ]. For many arguments this allows us to assume that P itself is a Ferrer diagram. Since the coordinate ring of a Ferrer diagram can be identified with the edge ring of a Ferrer graph, results of Corso and Nagel [8] can be used to compute the Castelnuovo-Mumford regularity of K[P], denoted by reg(K[P]). It turns out that reg(K[P]) has a very nice combinatorial interpretation. Namely, for an L-convex polyomino, reg(K[P]) is equal to maximal number of non-attacking rooks that can be placed on P, as shown in Theorem 3.3. This is the main result of Section 2.
In Section 3 we study the Gorenstein property of L-convex polyominoes. We first observe that if we remove the rectangle of maximal width from P, then the result is again an L-convex polyomino. Repeating this process we obtain a finite sequence of L-convex polyominoes, which we call the derived sequence of P. In Theorem 4.3 we then shown that K [P] is Gorenstein if and only if the bounding boxes of the derived sequence of L-convex polyominoes of P are all squares. For the proof we use again that K[P] ∼ = K[P * ], and the characterization of Gorenstein stack polyominoes given by Qureshi in [19]. In addition, under the assumption K[P] is not Gorenstein, we show in Theorem 4.3 that K[P] is Gorenstein on the punctured spectrum if and only if P is a rectangle, but not a square. Here we use that the coordinate ring of a a Ferrer diagram may be viewed as a Hibi ring. Then we can apply a recent result of Herzog et al [14] which characterizes the Hibi rings which are Gorenstein on the punctured spectrum.
Finally, in Section 4 we compute the Cohen-Macaulay type of K[P] for an Lconvex polyomino P. Again we use the fact that K[P * ] may be viewed as a Hibi ring (of a suitable poset Q). The number of generators of the canonical module of K[P * ], which by definiton is the Cohen-Macaulay type, is described by Miyazaki [17] (based on results of Stanley [21] and Hibi [15]). It is the number of minimal strictly order reversing maps on Q. Then somewhat technical counting arguments provide us in Theorem 5.2 with the desired formula.

Polyominoes
In this subsection we recall definitions and notation about polyominoes. If a = (i, j), b = (k, ) ∈ N 2 , with i k and j , the set [a, b] = {(r, s) ∈ N 2 : i r k and j s } is called an interval of N 2 . If i < k and j < , [a, b] is called a proper interval, and the elements a, b, c, d are called corners of [a, b], where c = (i, ) and d = (k, j). In particular, a and b are called the diagonal corners whereas c and d are called the anti-diagonal corners of [a, b]. The corner a (resp. c) is also called the lower left (resp. upper) corner of [a, b], and d (resp. b) is the right lower (resp. upper) corner of [a, b]. A proper interval of the form C = [a, a + (1, 1)] is called a cell. Its vertices V (C) are a, a + (1, 0), a + (0, 1), a + (1, 1) and its edges E(C) are Let P be a finite collection of cells of N 2 , and let C and D be two cells of P. Then C and D are said to be connected, if there is a sequence of cells C = C 1 , . . . , C m = D from P such that C i and C i+1 have a common edge for all i = 1, . . . , m − 1. In addition, if C i = C j for all i = j, then C 1 , . . . , C m is called a path (connecting C and D). A collection of cells P is called a polyomino if any two cells of P are connected. We denote by V (P) = ∪ C∈P V (C) the vertex set of P. A polyomino P whose cells belong to P is called a subpolyomino of P.
A polyomino P is called row convex if for any two of its cells with lower left corners a = (i, j) and b = (k, j), with k > i, all cells with lower left corners (l, j) with i l k are cells of P. Similarly, P is called column convex if for any two of its cells with lower left corners a = (i, j) and b = (i, k), with k > j, all cells with lower left corners (i, l) with j l k are cells of P. If a polyomino P is simultaneously row and column convex then P is called convex.
Each proper interval [(i, j), (k, l)] in N 2 can be identified as a polyomino and it is referred to as rectangular polyomino, or simply as rectangle. A rectangular subpolyomino P of P is called maximal if there is no rectangular subpolyomino P of P that properly contains P . A rectangle has size m × n if it contains m columns and n rows of cells. Given a polyomino P, the rectangle that contains P and has the smallest size with this property is called bounding box of P. After a shift of coordinates, we may assume that the bounding box is [(0, 0), (m, n)] for some m, n ∈ N. In this case, the width of P, denoted by w(P) is m. Similarly, the height of P, denoted by h(P) is n.
Moreover, an interval [a, b] with a = (i, j) and b = (k, ) is called a horizontal edge interval of P if j = and the sets {(r, j), (r +1, j)} for r = i, . . . , k −1 are edges of cells of P. If a horizontal edge interval of P is not strictly contained in any other horizontal edge interval of P, then we call it a maximal horizontal edge interval. Similarly one defines vertical edge intervals and maximal vertical edge intervals of P.

L-convex polyominoes
Let C : C 1 , C 2 , . . . , C m be a path of cells and (i k , j k ) be the lower left corner of C k for 1 k m. Then C has a change of direction at C k for some 2 k m − 1 if i k−1 = i k+1 and j k−1 = j k+1 .
A convex polyomino P is called k-convex if any two cells in P can be connected by a path of cells in P with at most k change of directions. The 1-convex polyominoes are simply called L-convex polyomino. A maximal rectangle R of size m × n is said to have unique occurrence in a polyomino P, if R is the only rectangular subpolyomino of P with size m × n. The next lemma gives information about the maximal rectangles of an L-convex polyomino. The maximal rectangles of the polyomino in Figure 1 are of sizes 7 × 2, 4 × 5, 3 × 6, 2 × 7 and 1 × 10.  As a consequence of Lemma 2.1 we have that, given an L-convex polyomino P, there is a unique maximal rectangle R w such that w(P) = w(R w ) and a unique maximal rectangle R h such that h(P) = h(R h ).

The bipartite graphs associated to polyominoes
Let P be a convex polyomino with bounding box [(0, 0), (m, n)]. In P there are n rows of cells, numbered increasingly from the top to the bottom, and m columns of cells, numbered increasingly from the left to the right. We attach a bipartite graph F P to the polyomino P in the following way.
if the i-th row of P intersects the j-th column of P non-trivially. The unique cell in the intersection of i-th row and j-th column is labelled as C ij . For all 1 i n, we define the i-th horizontal projection of P as the number of cells in the i-th row, and denote it by h i . Similarly, for all 1 j m, we define the j-th vertical projection of P as the number of cells in the j-th column and denote it by v j . The degree of a vertex v in a graph G, denoted by deg v, is the number of vertices adjacent to v in G. Note that h i = deg Y i and v j = deg X j in the graph F P . In the sequel, we will refer to the vector H P = (h 1 , h 2 , . . . , h n ) as the horizontal projections of P and V P = (v 1 , v 2 , . . . , v m ) as the vertical projection of P. For an L-convex polyomino one has Theorem 2.2 ([1,Lemma 1,2,3 ]). Let P be an L-convex polyomino, then: (a) P is uniquely determined by H P and V P ; for all 1 r i and for all 1 s j. Let G be a Ferrer graph and P be a polyomino such that Then P is called a Ferrer diagram. Note that a Ferrer diagram is a special type of stack polyomino (after a counterclockwise rotation by 90 degrees). Note that if [(0, 0), (m, n)] is the bounding box of a Ferrer diagram P, then (0, 0), (m, n) ∈ V (P).
We first prove that after a suitable relabelling of vertices of F P , it can be viewed as a Ferrer graph. Let T 1 , T 2 , . . . , T m and U 1 , U 2 , . . . , U n be the relabelling of the vertices of F P such that deg Then Hence F P is a Ferrer graph up to relabelling. Let P * be the unique polyomino with horizontal and vertical projections , then P * is a Ferrer diagram and F P ∼ = F P * . From the proof of the above proposition, one sees that given an L-convex polyomino P, the Ferrer diagram P * such that F P ∼ = F P * is uniquely determined. We refer to P * as the Ferrer diagram projected by P.
The Ferrer diagram P * projected by P Let r(P, k) be the number of ways of arranging k non-attacking rooks in cells of P. Recall that, for a graph G with n vertices, a k-matching of G is the set of k pairwise disjoint edges in G. Let p(G, k) be the number of k matchings of G. It is a fact, for example see [10, page 56], that r(P, k) = p(F P , k). Let r(P) denote the maximum number of rooks that can be arranged in P in non-attacking position, that is r(P) = max k r(P, k). We have the following Lemma 2.4. Let P be an L-convex polyomino and P * be the Ferrer diagram projected by P. Then r(P, k) = r(P * , k). In particular, r(P) = r(P * ).
Proof. From Proposition 2.3, we have F P ∼ = F P * then p(F P , k) = p(F P * , k). Then by using the theorem on [10, page 56], we see that r(P, k) = r(P * , k). As described in [19, Section 2], we can associate another bipartite graph G P to P in the following way. Let I = [(0, 0), (m, n)] be the bounding box of P. Since P is convex, in P there are m + 1 maximal vertical edge intervals and n + 1 maximal horizontal edge intervals, namely there are m + 1 columns and n + 1 rows of vertices. We number the rows in an increasing order from left to right and we number the columns in an increasing order from top to bottom. Let H 0 , . . . , H n denote the rows and V 0 , . . . , V m denote the columns of vertices of P.
To distinguish between G P and F P , we refer to them as follows: • The graph F P is the graph associated to the cells of P.
• The graph G P is the graph associated to the vertices of P. Figure 6: The two labellings on P of Figure 2 the electronic journal of combinatorics 28(1) (2021), #P1.50 The bipartite graph G P of the polyomino P in Figure 2.
The bipartite graph F P of the polyomino P in Figure 2. The relation between F P and G P is deducible from the following Let Similarly, by using unimodality of H P , we get for some 1 i m. Then h j = deg y j−1 − 1 for all 1 j < i and h j = deg y j − 1 for i j n.
(ii) As a consequence of (i), if P is a Ferrer diagram then Let G P and F P be the graphs associated to P as described above with V (G P ) =

Regularity of L-convex polyominoes
Let K be a field. We denote by S the polynomial ring over K with variables  Proof. Since P is convex, it is known that K[P] is isomorphic to the edge ring K[G P ] of the bipartite graph G P (see [19,Section 2]). By Corollary 2.7, G P is isomorphic to G P * . Hence the assertion follows. Recall G P * is the bipartite graph associated to the vertices of P * . We may assume that V (G * P ) = {x 0 , . . . , x m } {y 0 , . . . , y n }. Then deg y 0 = m + 1 2 and deg x 0 = n + 1. Hence, [8,Proposition 5.7] gives We want to rewrite the formula above in terms of the horizontal projection of P * . According to Remark 2.5.(2), for any 1 j n we have h j = deg y j − 1. Hence and the assertion follows.
Let P be a Ferrer diagram with horizontal projections (h 1 , . . . , h n ). Then, by using a combinatorial argument, it is easy to see that for any r n the number of ways of placing r rooks in non-attacking position in P is given by By using this fact we obtain Theorem 3.3. Let P be an L-convex polyomino. Then

reg(K[P]) = r(P).
Proof. From Lemma 2.4 we know that r(P) = r(P * ) where P * is the Ferrer diagram projected by P. By Theorem 3.1, it is enough to show that where (h 1 , . . . , h n ) are the horizontal projections of P * . It follows from Equation (1) that r(P * ) is the greatest integer r n such that each factor of is positive. Hence, for any i ∈ {1, . . . , r} we must have h r−i+1 − (i − 1) > 0. Fix an integer i ∈ {1, . . . , r}. Then we see that Hence we conclude that r h r−i+1 + (r − i). Therefore, r(P * ) = max{r | r n and r min{h r−i+1 + (r − i) | 1 i r}} We observe that, by exchanging the role of rows and columns in P * , we obtain r(P * ) = min{m, v j + j − 1 | 1 j m} which is similar to (2).

On the Gorenstein property of L-convex polyominoes
Let P be a L-convex polyomino with width m. Assume that the unique maximal rectangle of P with width m, has height d. Then for some positive integer s, . . . , h s , m . . . , m, h s+d+1 , . . . , h n ) with h s , h s+d+1 < m. Let P 1 be the collection of cells with n − d rows satisfying the following property: C ij is a cell of P if and only if C ij is a cell of P 1 for 1 i s, and for s + d + 1 i n, C i−d,j is a cell of P 1 . Proof. P 1 could be seen as the polyomino P from which we remove the maximal rectangle R having width m. Hence, each cell in P 1 corresponds uniquely to a cell in P. Let C, D ∈ P 1 . Then we consider the corresponding cells C , D ∈ P. We observe that neither C nor D is a cell of R. Since P is L-convex, there exists a path of cells C in P connecting C and D with at most one change of direction. If no cell of C belongs to R, then C determines a path of cells C of P 1 with at most one change of direction connecting C and D.
Otherwise, since neither C nor D are cells of R, the path C crosses R and the induced path C ∩ R has no change of direction. Therefore, the path C in P 1 , obtained by cutting off the induced path C ∩ R from C , is a path of cells with at most one change of direction connecting C and D.
If P 1 = ∅, we may again remove the unique rectangle of maximal width from P 1 to obtain P 2 in a similar way. After a finite number of such steps, say t steps, we arrive at P t which is a rectangle. Then P t+1 = ∅. We set P 0 = P, and call the sequence P 0 , P 1 , . . . , P t the derived sequence of L-convex polyominoes of P.

Lemma 4.2.
Let P be an L-convex polyomino and P 0 , P 1 , . . . , P t be the derived sequence of L-convex polyominoes of P. Let P * be the Ferrer diagram projected by P and let (P * ) 0 , (P * ) 1 , . . . , (P * ) t be the derived sequence of L-convex polyominoes of P * . Then t = t and for any 0 k t the polyomino (P * ) k is the Ferrer diagram projected by P k . In other words, for all k (P * ) k = (P k ) * .
From Proposition 2.3 it follows that P * has a maximal rectangle R * of width m and height d and with v * 1 · · · v * l > d. Hence the L-convex polyomino (P * ) 1 is uniquely determined by the projections On the other hand, P 1 is the L-convex polyomino uniquely determined by the projections H P 1 = (h 1 , . . . , h s , h s+d+1 , . . . , h n ) and, By reordering the two vectors in a decreasing order, we obtain the Ferrer diagram projected by P 1 which coincides with (P * ) 1 . This proves the assertion for k = 1. By inductively applying the above argument, the assertion follows for all k. (a) P is Gorenstein.   (v 1 , v 2 , . . . , v m ). We set and let = 1, . . . , r, and b Then the following conditions are equivalent: (a) P is Gorenstein. (b) P is not a square, and K[P] has an isolated singularity.
(c) P is rectangle, but not a square.
Before we start the proof of the theorem, we note that if P is a Ferrer diagram, then K[P] can be viewed as a Hibi ring. Recall for a given finite poset Q = {v 1 , . . . , v n } and a field K, the Hibi ring over the field K associated to Q, which we denote by K[Q] ⊂ K[y, x 1 , . . . , x n ], is defined as follows. The K-algebra K[Q] is generated by the monomials yx I := y v i ∈I x i for every I ∈ I(Q), that is

The algebra K[Q]
is standard graded if we set deg(yx I ) = 1 for all I ∈ I(Q). Here I(Q) is the set of poset ideals of Q. The poset ideals of Q are just the subset I ⊂ Q with the property that if p ∈ Q and q p, then q ∈ Q.
Let P be a Ferrer diagram with maximal horizontal edge intervals {H 0 , . . . , H n }, numbered increasingly from the bottom to the top, and maximal vertical edge intervals {V 0 , . . . , V m }, numbered increasingly from the left to the right. We let Q be the poset on the set {H 1 , . . . , H n , V 1 , . . . , V m } consisting of two chains H 1 < . . . < H n and V 1 < . . . < V m and the covering relations H i < V j , if H i intersects V j in a way such that there is no 0 i < i for which H i intersects V j , and j is the smallest integer with this property. Proof. We may assume that the interval [(0, 0), (m, n)] is the bounding box of the Ferrer diagram P. It follows from the definition of Ferrer diagrams that (0, 0) and (m, n) belong to V (P). For any two vertices a = (i, j) and b = (k, l) of P we define the meet a ∧ b = (min{i, k}, min{j, l}) and the join a ∨ b = (max{i, k}, max{j, l}). With this operations of meet and join, P is a distributive lattice. An element c of this lattice is called join-irreducible, if c = (0, 0) and whenever a ∧ b = c, then a = c or b = c. By Birkhoff's fundamental structure theorem [1], any finite distributive lattice is the ideal lattice of the poset of its join irreducible elements. The join irreducible elements of P can be described as follows: 1. Every a j = (0, j) with 1 j n is a join irreducible element in P and a 1 < a 2 < . . . < a n .
. It shows that in each vertical edge interval V 1 , . . . , V m of P, there is exactly one join irreducible element. We denote by b i , the unique join irreducible element in V i with 1 i m.
In the poset of join irreducible elements of P, we have two chains a 1 < a 2 < . . . < a n and b 1 < b 2 < . . . < b m , and the covering relations a j = (0, j) < b i = (i, k) if j = k and b i is the minimal element with this property. Then, it follows that the poset of join irreducible elements of P is exactly the poset Q described above. Thus the elements a ∈ P are in bijection with the poset ideals of Q. In fact, the poset ideal I a ∈ I(Q) corresponding ot a is the set of join irreducible elements q ∈ Q with q a. Thus we have a surjective K-algebra homomorphism As shown by Hibi [15] (see also [12,Theorem 10.1.3]), Ker(ϕ) is generated by the relations x a x b − x a∧b x a∨b . This shows that Ker(ϕ) = I P , as desired.

The Cohen-Macaulay type of L-convex polyominoes.
In this section, we give a general formula for the Cohen-Macaulay type of the coordinate ring of an L-convex polyomino. To illustrate our result, we first consider the special case of an L-convex polyomino with just two maximal rectangles. Proposition 5.1. Let P be an L-convex polyomino whose maximal rectangles are R 1 having size m × s and R 2 having size t × n with s < n and t < m. Let r = max{n, m, n + m − (s + t)}. Then Proof. Let P * be the Ferrer diagram projected by P and let Q be the poset of the join-irreducible elements associated to P * . It consists of the two chains V 1 < · · · < V m and H 1 < · · · < H n , and the cover relation H n−s < V t+1 . We have |Q| = m + n, and r = rank Q + 1. We compute the number of minimal generators of the canonical module ω K[P * ] . For this purpose, let Q be the poset obtained from Q by adding the elements −∞ and ∞ with ∞ > p and −∞ < p for all p ∈ Q, and let T ( Q) be the set of integer valued functions ν : Q → Z 0 with ν(∞) = 0 and ν(p) < ν(q) for all p > q. By using a result of Stanley [21], Hibi shows in [15, (3.3)] that the monomials of the form for ν ∈ T ( Q) form a K-basis for ω K[P * ] . By using [17,Corollary 2.4], we have that the number of generators of ω K[P * ] is the number of minimal maps ν ∈ T ( Q) with respect to the order given in [17,Page 5]. In fact, ν µ for ν, µ ∈ T ( Q) if µ − ν is decreasing. We observe that the minimal maps ν necessarily assign the numbers 1, . . . , r to the vertices of a maximal chain of Q in reversed order, hence we have to find the possible values for the remaining |Q| − r = m + n − r elements, depending on r. We distinguish three cases: In the case (a), the maximal chain is V 1 < · · · < V m . Hence we must take ν(V m−i+1 ) = i for i ∈ {1, . . . , m}. We have to determine how many vectors (a 1 , . . . , a n ) with integers entries 0 < a 1 < · · · < a n satisfy m − t < a s+1 < r −(n−s) = m−(n−s)+2, where the left inequality follows from the cover relation, while the right inequality follows from the fact that a s+2 < · · · < a n < m + 1 are determined. Therefore, for fixed i = In the case (b), we assign to each element of the chain H 1 , . . . , H n a number in {1, . . . , n} in strictly decreasing order. We have to determine how many vectors where the left inequality follows from the fact that 0 < b 1 < · · · < b m−t−1 , while the rightmost inequality follows from the cover relation. Therefore, for i = b m−t , there are  Figure 10: We count the number of minimal maps assigning 1 < · · · < m−t+n−s to V m > · · · > V t+1 > H n−s > · · · > H 1 .
In the case (a), we assign to each element of the chain V 1 , · · · , V m a number in {1, . . . , m} in decreasing order. We have to determine how many vectors (a 1 , . . . , a n ) with integers entries 0 < a 1 < · · · < a n < m + 1 satisfy We observe that Theorem 4.3 can also deduced from Theorem 5.2. The following example demonstrates Theorem 5.2. Example 5.3. Let P be the Ferrer diagram in Figure 11.

Figure 11
We have t = 4 maximal rectangles whose sizes are There are 4 maximal chains in the poset Q corresponding to P containing 5 vertices. For example, the chain V 1 , . . . , V 5 and the chain H 1 , H 2 , V 3 , V 4 , V 5 , that correspond to the cases r = m and r = (n − d 3 ) + (m − c 2 ), hence h = 2. We are going to compute A and A 2 B 2 as in Theorem 5.2. We have In conclusion we want to point out that for L-convex polyominoes, important algebraic invariants, like the Castelnuovo-Mumford regularity, the Cohen-Macaulay type, and algebraic properties, like being Gorenstein, are now completely understood and have a nice combinatorial interpretation. It is still a challenge to prove similar results when the polyomino is k-convex for k > 1, rather than just L-convex.