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Burstein’s permutation conjecture, Hong and Li’s inversion sequence conjecture and restricted Eulerian distributions

Part of: Combinatorics

Published online by Cambridge University Press:  23 October 2023

Shane Chern Open the ORCID record for Shane Chern [Opens in a new window] ,
Shishuo Fu Open the ORCID record for Shishuo Fu [Opens in a new window]  and
Zhicong Lin
Shane Chern
Affiliation:
Department of Mathematics and Statistics, Dalhousie University, Halifax, NS, Canada ([email protected]; [email protected])
Shishuo Fu
Affiliation:
College of Mathematics and Statistics, Chongqing University, Huxi Campus, Chongqing, P.R. China ([email protected])
Zhicong Lin
Affiliation:
Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, Qingdao, Shandong, P.R. China ([email protected])
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Abstract

Recently, Hong and Li launched a systematic study of length-four pattern avoidance in inversion sequences, and in particular, they conjectured that the number of 0021-avoiding inversion sequences can be enumerated by the OEIS entry A218225. Meanwhile, Burstein suggested that the same sequence might also count three sets of pattern-restricted permutations. The objective of this paper is not only a confirmation of Hong and Li’s conjecture and Burstein’s first conjecture but also two more delicate generating function identities with the $\mathsf{ides}$ statistic concerned in the restricted permutation case and the $\mathsf{asc}$ statistic concerned in the restricted inversion sequence case, which yield a new equidistribution result.

Keywords

permutationsinversion sequencespattern avoidanceEulerian statisticsgenerating functionsequidistributions

MSC classification

Primary: 05A05: Permutations, words, matrices 05A15: Exact enumeration problems, generating functions
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 66 , Issue 4 , November 2023 , pp. 1179 - 1201
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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