Polyominoes determined by permutations

Abstract

International audience In this paper we consider the class of $\textit{permutominoes}$, i.e. a special class of polyominoes which are determined by a pair of permutations having the same size. We give a characterization of the permutations associated with convex permutominoes, and then we enumerate various classes of convex permutominoes, including parallelogram, directed-convex, and stack ones.

A Closed Formula for the Number of Convex Permutominoes

“…In fact a stack permutomino can be uniquely represented by a permutomino in class B having the same size. Hence the generating function of stack permutominoes is given by B(1, t), and then the number of stack permutominoes of size n, as already stated in [15], is equal to 2 n .…”

“…The two permutations associated with P 1 and P 2 are indicated by π 1 and π 2 , respectively (see Figure 2). While it is clear that any permutomino of size n uniquely individuates two point-by-point distinct permutations π 1 and π 2 of [n+1], not all the couples of permutations π 1 and π 2 of n such that π 1 (i) = π 2 (i), 1 ≤ i ≤ n + 1 define a permutomino, as it was partially investigated in [15] (see Figure 3). Permutominoes were introduced by F. Incitti in [17] while studying the problem of determining the R-polynomials (related with the Kazhdan-Lusztig R-polynomials) associated with a pair (x, y) of permutations.…”

“…In [15], using bijective techniques, it was proved that the number of parallelogram permutominoes of size n is equal to the nth Catalan number,…”

“…and then the number of directed convex permutominoes of size n, as already stated in [15], is equal to…”

Permutations defining convex permutominoes

“…Let us recall the main enumerative results concerning convex permutominoes. In [14], using bijective techniques, it was proved that the number of parallelogram permutominoes of size n + 1 is equal to c n and that the number of directed-convex permutominoes of size n + 1 is equal to 1 2 b n , where, throughout all the paper, c n and b n will denote, respectively, the Catalan numbers and the central binomial coefficients. Finally, in [13] it was proved, using the ECO method, that the number of convex permutominoes of size n + 1 is:…”