Every polyomino is a winner in the weak achievement game on a rectangular infinite board with large enough dimension. In particular, Snaky is a winner on the 41-dimensional board. An infinite polyomino is a winner on an infinite dimensional board but a loser on a finite dimensional board.

We compute an asymptotic estimate of a lower bound of the number of k-convex polyominoes of semiperimeter p. This approximation can be written as μ(k)p4p where μ(k) is a rational fraction of k which up to μ(k) is the asymptotics of convex polyominoes. A polyomino is a connected set of unit square cells drawn in the plane Z × Z [7]. The size of a polyomino is the number of its cells. A central p...

Abstract A remarkable family of discrete sets which has recently attracted the attention geometry community is convex polyominoes, that are counterpart Euclidean sets, and combine constraints convexity connectedness. In this paper we study problem their reconstruction from orthogonal projections, relying on approach defined by Barcucci et al. (Theor Comput Sci 155(2):321–347, 1996). particular,...

In this work, we provide a necessary and sufficient condition on polyomino ideal for having the set of inner 2-minors as graded reverse lexicographic Gröbner basis, due to combinatorial properties itself. Moreover, prove that when latter holds coincides with lattice associated polyomino, is prime. As an application, describe two new infinite families prime polyominoes.

We associate with a moon polyominoes a charge function C :M→ {±1} such that every cell of the polyomino is either positive or negative. The charge function induces naturally a sign on the northeast and southeast chains of length 2 in 01-fillings of the moon polyomino. We prove that the joint distribution of the numbers of positive chains and negative chains is symmetric and independent of the c...

Let $$\mathcal {P}$$ be a simple thin polyomino, namely polyomino that has no holes and does not contain square tetromino as subpolyomino. In this paper, we determine the reduced Hilbert–Poincaré series $$h(t)/(1-t)^d$$ of $$K[\mathcal {P}]$$ by proving h(t) is rook polynomial . As an application, characterize Gorenstein polyominoes.

Finding an efficient optimal partial tiling algorithm is still an open problem. We have worked on a special case, the tiling of Manhattan polyominoes with dominoes, for which we give an algorithm linear in the number of columns. Some techniques are borrowed from traditional graph optimisation problems. For our purpose, a polyomino is the (non necessarily connected) union of unit squares (for wh...

We define the Helly number of a polyomino P as the smallest number h such that the h-Helly property holds for the family of symmetric and translated copies of P on the integer grid. We prove the following: (i) the only polyominoes with Helly number 2 are the rectangles, (ii) there does not exist any polyomino with Helly number 3, (iii) there exist polyminoes of Helly number k for any k 6= 1, 3.