A closed formula for the number of convex permutominoes

The conference GASCom brings together researchers in combinatorics, algorithms, probabilities, and more generally mathematical computer science, around the theme of random and exhaustive generation of combinatorial structures, mostly considered from a theoretical point of view. In connection with this main theme, the conference is also interested in contributions in enumerative or analytic combinatorics, and interactions with other areas of mathematics, computer science, phys...

In this paper, we obtain the counting formulaes of convex pentagons and convex hexagons, respectively, in an $n$-triangular net by solving the corresponding recursive formulaes.

The aim of this work is the study of the class of periodic parallelogram polyominoes, and two of its variantes. These objets are related to 321-avoiding affine permutations. We first provide a bijection with the set of triangles under Dyck paths. We then prove the ultimate periodicity of the generating series of our objects, and introduced a notion of primitive polyominoes, which we enumerate. We conclude by an asymptotic analysis.

We describe a class of fixed polyominoes called $k$-omino towers that are created by stacking rectangular blocks of size $k\times 1$ on a convex base composed of these same $k$-omino blocks. By applying a partition to the set of $k$-omino towers of fixed area $kn$, we give a recurrence on the $k$-omino towers therefore showing the set of $k$-omino towers is enumerated by a Gauss hypergeometric function. The proof in this case implies a more general hypergeometric identity wit...

Parallelogram polyominoes are a subclass of convex polyominoes in the square lattice that has been studied extensively in the literature. Recently congruence classes of convex polyominoes with respect to rotations and reflections have been enumerated by counting orbits under the action of the dihedral group D4, of symmetries of the square, on (translation-type) convex polyominoes. Asymmetric convex polyominoes were also enumerated using Moebius inversion in the lattice of sub...

A periodic parallelogram polyomino is a parallelogram polyomino such that we glue the first and the last column. In this work we extend a bijection between ordered trees and parallelogram polyominoes in order to compute the generating function of periodic parallelogram polyominoes with respect to the height, the width and the intrinsic thickness, a new statistic unrelated to the existing statistics on parallelogram polyominoes. Moreover we define a rotation over periodic para...

We study a new class of polyominoes, called $p$-Fibonacci polyominoes, defined using $p$-Fibonacci words. We enumerate these polyominoes by applying generating functions to capture geometric parameters such as area, semi-perimeter, and the number of inner points. Additionally, we establish bijections between Fibonacci polyominoes, binary Fibonacci words, and integer compositions with certain restrictions.

Recent work of Brlek \textit{et al.} gives a characterization of digitally convex polyominoes using combinatorics on words. From this work, we derive a combinatorial symbolic description of digitally convex polyominoes and use it to analyze their limit properties and build a uniform sampler. Experimentally, our sampler shows a limit shape for large digitally convex polyominoes.

For positive integers $m$ and $n$, the partial permutohedron $\mathcal{P}(m,n)$ is a certain integral polytope in $\mathbb{R}^m$, which can be defined as the convex hull of the vectors from $\{0,1,\ldots,n\}^m$ whose nonzero entries are distinct. For $n=m-1$, $\mathcal{P}(m,m-1)$ is (after translation by $(1,\ldots,1)$) the polytope $P_m$ of parking functions of length $m$, and for $n\ge m$, $\mathcal{P}(m,n)$ is combinatorially equivalent to an $m$-stellohedron. The main res...

We introduce some new classes of words and permutations characterized by the second difference condition $\pi(i-1) + \pi(i+1) - 2\pi(i) \leq k$, which we call the $k$-convexity condition. We demonstrate that for any sized alphabet and convexity parameter $k$, we may find a generating function which counts $k$-convex words of length $n$. We also determine a formula for the number of 0-convex words on any fixed-size alphabet for sufficiently large $n$ by exhibiting a connection...