Counting $r\times s$ rectangles in (Catalan) words

TL;DR

This paper introduces the rectangle capacity statistic to count specific rectangles in words and Catalan words, providing generating functions for their distribution and total counts, advancing combinatorial enumeration methods.

Contribution

It defines a new rectangle capacity statistic and derives generating functions for its distribution and total counts in words and Catalan words.

Findings

Derived bivariate generating functions for rectangle counts.

Established total count generating functions over all words.

Extended results specifically to Catalan words.

Abstract

Generalizing previous results, we introduce and study a new statistic on words, that we call rectangle capacity. For two fixed positive integers $r$ and $s$, this statistic counts the number of occurrences of a rectangle of size $r\times s$ in the bargraph representation of a word. We find the bivariate generating function for the distribution on words of the number of $r\times s$ rectangles and the generating function for their total number over all words. We also obtain the analog results for Catalan words.