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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