Black Cell Capacity in Catalan polyominoes

TL;DR

This paper investigates the distribution of black cell capacity in Catalan polyominoes, deriving generating functions to encode this combinatorial statistic, thus advancing understanding of their structural properties.

Contribution

It introduces a new statistic, black cell capacity, on Catalan polyominoes and provides generating functions to analyze its distribution.

Findings

Derived explicit generating functions for black cell capacity

Established combinatorial properties of Catalan polyominoes related to black cells

Connected black cell capacity to Catalan word structures

Abstract

A Catalan word is a sequence $w_1w_2\cdots w_n$ of nonnegative integers such that $w_1=0$ and $w_{i}\leq w_{i-1}+1$ for $2\leq i\leq n$. Given a Catalan word, we construct a column-convex polyomino (or \emph{bargraph}) by placing, at position $i$, a column of height $w_i + 1$, with all columns aligned along their bottom edges. On these Catalan polyominoes we define the black cell capacity by coloring the cells in a chessboard pattern and we count the number of black cells in the polyomino. We study the distribution of the black cell capacity over Catalan polyominoes and derive generating functions that encode this statistic.