Counting Rectangles of Size r × s in Nondecreasing and Smirnov Words

The rectangle capacity, a word statistic that Mansour and the author recently introduced, counts, for two fixed positive integers r and s, the number of occurrences of a rectangle of size r × 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 rectangles of size r × s 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.