Counting $r\times s$ rectangles in nondecreasing and Smirnov words

TL;DR

This paper derives generating functions for counting specific rectangle patterns in nondecreasing and Smirnov words, extending previous work on general and Catalan words with new combinatorial formulas.

Contribution

It provides the first explicit generating functions for the distribution and total count of $r\times s$ rectangles in nondecreasing and Smirnov words.

Findings

Derived bivariate generating functions for nondecreasing words.

Obtained generating functions for total number of rectangles in these words.

Extended previous results to new classes of words.

Abstract

The rectangle capacity, a word statistic that was recently introduced by the author and Mansour, counts, for two fixed positive integers $r$ and $s$, the number of occurrences of a rectangle of size $r\times s$ in the bargraph representation of a word. In this work we find the bivariate generating function for the distribution on nondecreasing words of the number of $r\times s$ rectangles and the generating function for their total number over all nondecreasing words. We also obtain the analog results for Smirnov words, which are words that have no consecutive equal letters. This complements our recent results concerned with general words (i.e., not restricted) and Catalan words.