Black Cell Capacity in Catalan polyominoes

A Catalan word is a sequence $w_1{w}_2\cdots w_n$ of nonnegative integers such that $w_1=0$ and $w_i\le w_{i-1}+1$ for $2\le i\le n$. 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.