Further enumeration results concerning a recent equivalence of restricted inversion sequences

Let asc and desc denote respectively the statistics recording the number of ascents or descents in a sequence having non-negative integer entries. In a recent paper by Andrews and Chern, it was shown that the distribution of asc on the inversion sequence avoidance class $I_n(\geq,\neq,>)$ is the...

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Published in:Discrete Mathematics and Theoretical Computer Science Vol. 24, no. 1; no. Combinatorics; pp. 1 - 28
Main Authors: Mansour, Toufik, Shattuck, Mark
Format: Journal Article
Language:English
Published: Nancy DMTCS 01.01.2022
Discrete Mathematics & Theoretical Computer Science
Subjects:
05a05
2021. 2010 mathematics subject classification. 05a15
65q20
65q30 pattern avoidance
[math.math-co]mathematics [math]/combinatorics [math.co]
[math.math-mp]mathematics [math]/mathematical physics [math-ph]
Algorithms
combinatorial statistic
Combinatorics
Enumeration
Equivalence
Functional equations
inversion sequence
Inversions (Geometry)
july 22
kernel method
Mathematical Physics
Mathematical research
Mathematics
Sequences
Sequences (Mathematics)
Israel
2021. 2010 Mathematics Subject Classification. 05A15
65Q30 pattern avoidance
65Q20
July 22
kernel method
05A05
inversion sequence
combinatorial statistic
ISSN:1365-8050, 1462-7264, 1365-8050
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Summary:Let asc and desc denote respectively the statistics recording the number of ascents or descents in a sequence having non-negative integer entries. In a recent paper by Andrews and Chern, it was shown that the distribution of asc on the inversion sequence avoidance class $I_n(\geq,\neq,>)$ is the same as that of $n-1-\text{asc}$ on the class $I_n(>,\neq,\geq)$, which confirmed an earlier conjecture of Lin. In this paper, we consider some further enumerative aspects related to this equivalence and, as a consequence, provide an alternative proof of the conjecture. In particular, we find recurrence relations for the joint distribution on $I_n(\geq,\neq,>)$ of asc and desc along with two other parameters, and do the same for $n-1-\text{asc}$ and desc on $I_n(>,\neq,\geq)$. By employing a functional equation approach together with the kernel method, we are able to compute explicitly the generating function for both of the aforementioned joint distributions, which extends (and provides a new proof of) the recent result $|I_n(\geq,\neq,>)|=|I_n(>,\neq,\geq)|$. In both cases, an algorithm is formulated for computing the generating function of the asc distribution on members of each respective class having a fixed number of descents.
Bibliography:SourceType-Scholarly Journals-1
ObjectType-Feature-1
content type line 14
ISSN:1365-8050
1462-7264
1365-8050
DOI:10.46298/dmtcs.8330