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

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Sela Fried

Combinatorics · Submitted 27 Jun 2024

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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.

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arXiv ID2406.18923 (version 1)
Pith IDYXPNDJ43, Pith's permanent record of this paper and its review
CitationSela Fried. "Counting $r\times s$ rectangles in nondecreasing and Smirnov words." arXiv:2406.18923 (2024). https://pith.science/paper/YXPNDJ43
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,683 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.15 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.53 · compares with prior results 0.77 · mentions code or data availability 0.02.

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arXiv listing abstract v1. Results and the main claim are underlined; .

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.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 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. background 1.00 0.26 0.04 asserted 1.00
s1 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. contribution 0.55 0.97 0.67 asserted 1.00
s2 We also obtain the analog results for Smirnov words, which are words that have no consecutive equal letters. contribution 0.56 0.94 0.12 asserted 1.00
s3 This complements our recent results concerned with general words (i.e., not restricted) and Catalan words. background 0.80 0.73 0.05 asserted 0.99

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Original 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.

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