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

Abstract

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. Though the main results are of algebraic nature, all proofs are combinatorial.

Hilbert series of Parallelogram Polyominoes

“…A parallelogram polyomino that is also L-convex is known as a Ferrer diagram. In [7] the authors prove that the coordinate ring of any L-convex polyomino is isomorphic to the coordinate ring of a Ferrer diagram.…”

“…Although the Gorenstein distributive lattices are completely characterized in [15], we plan to give a combinatorial interpretation of the Gorenstein parallelogram polyominoes in the language of polyominoes. Our aim is to compare the conditions on a parallelogram polyomino to be Gorenstein with the conditions found in [7] for L-convex polyominoes and in [22] for simple thin polyominoes.…”

“…This identification allows us to use the existing knowledge on Hibi rings arising from simple planar distributive lattices and translate it in terms of their structure as coordinate rings of parallelogram polyominoes. Moreover, from [7,Theorem 3.1] it is known that the polyomino ideals of L-convex polyominoes can be interpreted as polyomino ideals of certain Ferrer diagrams. The Ferrer diagrams are a special subclass of parallelogram polyominoes.…”

“…A rook polynomial n i=1 r i x i is a polynomial whose coefficients r i represents the number of distinct ways of arranging i rooks on squares of P in nonattacking positions. In [7] and [22], the authors linked the Castelnuovo-Mumford regularity of K[P] to the maximum number of non-attacking rooks that can be placed on the polyomino, for the classes of L-convex polyominoes and simple thin polyominoes. For the latter class, in [22], the authors proved that the polynomial h(t) of the reduced Hilbert series h(t)/(1−t) d of K[P] coincides with the rook polynomial of P. This result in [22] motivated us to study the relation between the Hilbert series and the rook polynomial for simple non-thin polyominoes.…”

“…The Gorensteinness of the coordinate ring of some classes of polyominoes has been studied by many authors, for example see [19], [7] and [22]. Even though the Gorenstein ladder determinantal rings and the Gorenstein Hibi rings are completely characterized, in Section 4 we give a combinatorial characterization of Gorenstein parallelogram polyominoes that is analogous to the characterizations given in [7] and [22] for L-convex and simple thin polyominoes, respectively. Such characterization involves the intersections of the maximal rectangles of the parallelogram polyominoes.…”