Journal of Integer Sequences, Vol. 27 (2024), Article 24.8.4

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


Sela Fried
Department of Computer Science
Israel Academic College in Ramat Gan
Pinhas Rutenberg 87
Ramat Gan 52275
Israel

Abstract:

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.


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(Concerned with sequences A027480 A033487 A045943 A050534 A059845 A077414 A105938 A115067 A140090 A140091 A140672 A140673 A140674 A140675 A151542 A240440 A241204 A253945 A266732 A266733 A271040 A281200 A370238.)


Received June 27 2024; revised version received October 1 2024. Published in Journal of Integer Sequences, November 26 2024.


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