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

Abstract We present a conjecture about the reduced Hilbert series of coordinate ring simple polyomino in terms particular arrangements non-attacking rooks that can be placed on polyomino. By using computational approach, we prove above holds for all polyominoes up to rank 11. In addition, true class parallelogram polyominoes, by looking at those as planar distributive lattices. Finally, give co...

problems about polyominoes have been discussed. \Ve introduce new one: Determine the smallest size fen) of cut-set C(n) of n-ominoes, such that every sufficiently polyomino contains at one n-omino C(n) a partial polyomino. This question may be of interest in biology or pharmacy, for ex:arrlpl IC, if every infinite cell of polyominoes can be avoided the control of all nominoes for fixed n then i...

We consider the smallest superpolyomino problem: given a set of colored polyominoes, find the smallest polyomino containing each input polyomino as a subshape. This problem is shown to be NP-hard, even when restricted to a set of polyominoes using a single common color. Moreover, for sets of polyominoes using two or more colors, the problem is shown to be NP-hard to approximate within a O(n1/3−...

In the plane Z × Z a cell is a unit square, and a polyomino is a finite connected union of cells having no cut point. Polyominoes are defined up to translations. A column (row) of a polyomino is the intersection between the polyomino and an infinite strip of cells whose centers lie on a vertical (horizontal) line. The enumeration problem for general polyominoes is difficult to solve and still o...

In this work, we present a new method for generating a threshold structure. This kind of structure can be advantageously used in various halftoning algorithms such as clustered-dot or dispersed-dot dithering, error diffusion with threshold modulation, etc. The proposed method is based on rectifiable polyominoes -a non-periodic hierarchical structure, which tiles the Euclidean plane with no gaps...

A permutomino of size n is a polyomino determined by particular pairs (π1, π2) of permutations of size n, such that π1(i) 6= π2(i), for 1 ≤ i ≤ n. Here we determine the combinatorial properties and, in particular, the characterization for the permutations defining convex permutominoes. Using such a characterization, these permutations can be uniquely represented in terms of the so called square...

We consider the tilings by translation of a single polyomino or tile on the square grid Z. It is well-known that there are two regular tilings of the plane, namely, parallelogram and hexagonal tilings. Although there exist tiles admitting an arbitrary number of distinct hexagon tilings, it has been conjectured that no polyomino admits more than two distinct parallogram tilings. In this paper, w...

A Polyomino is an edge–connected union of cells in the planar square lattice. Here we consider generalized linear polyominoes; that is, the polyominoes supported by a n × 2 lattice. In this paper, we obtain the Green function and the Kirchhoff index of a generalized linear polyomino as a perturbation of a 2n–path by adding weighted edges between opposite vertices. This approach deeply links gen...