Black Cell Capacity in Catalan polyominoes

TL;DR

This work investigates the distribution of the black cell capacity on Catalan polyominoes derived from Catalan words, linking polyomino statistics to Dyck-path structure. It develops multiple generating-function frameworks: (i) a four-equation system for parity-split counts $F_{ab}$, (ii) a matrix continued-fraction (MCF) approach via an automaton for partial Dyck paths, and (iii) a bistatistic construction that yields closed forms for joint distributions of vertical black/white capacities. A further functional-equation treatment using a mixed statistic $\texttt{s}$ provides a closed expression in terms of $q$-Pochhammer factors and a determinant, with explicit series expansions and enumerations (e.g., polyominoes with $\texttt{verblack}=3$). The results deliver exact enumerations, structural insights, and new techniques for Catalan-polyomino statistics, while highlighting open questions on algebraicity, $D$-finiteness, and asymptotics. Together, these methods advance exact enumeration for polyomino statistics through continued fractions and $q$-series.

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.

Figures (9)

• Figure 1: The polyomino $P$ associated with the Catalan Word $w=0012012310110$. We have $\texttt{length}(P)=13$, $\texttt{last}(P)=1=w_{13}+1$. The black cell capacity $\texttt{black}(P)$ equals to 13, the vertical black cell capacity $\texttt{verblack}(P)$ equals to 12, and the vertical white cell capacity $\texttt{verwhite}(P)$ equals to 13.
• Figure 2: The image by $f$ of the Catalan polyomino $P=1232121$ is $f(P)=1121223$. The length of $P$ is odd and thus, we have $\texttt{black}(P)=\texttt{verblack}(f(P))=8$.
• Figure 3: The image by $f$ of the Catalan polyomino $P=12321212$ is $f(P)=12121223$. The length of $P$ is even and thus, we have $\texttt{black}(P)=\texttt{verwhite}(f(P))=9$.
• Figure 4: Illustration of Case 1: $\texttt{length}(P)$ and $\texttt{last}(P)$ are odd. The left part shows a polyomino $P$ where $Q$ satisfies $\texttt{last}(Q)=0\mod 2$, while the right part is for $\texttt{last}(Q)=1\mod 2$.
• Figure 5: The 14 Catalan polyominoes of length 4. There are 2 (resp 5, 4,2,1) polyominoes with black cell capacity 2 (resp. 3,4,5,6).

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 2.1
• proof
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Lemma 3.1
• proof
• Theorem 3.2
• proof