{"title":"Parity statistics on restricted permutations and the Catalan–Schett polynomials","authors":"Zhicong Lin , Jing Liu , Sherry H.F. Yan","doi":"10.1016/j.jcta.2025.106049","DOIUrl":null,"url":null,"abstract":"<div><div>Motivated by Kitaev and Zhang's recent work on non-overlapping ascents in stack-sortable permutations and Dumont's permutation interpretation of the Jacobi elliptic functions, we investigate some parity statistics on restricted permutations. Some new related bijections are constructed and two refinements of the generating function for descents over 321-avoiding permutations due to Barnabei, Bonetti and Silimbanian are obtained. In particular, an open problem of Kitaev and Zhang about non-overlapping ascents on 321-avoiding permutations is solved and several combinatorial interpretations for the Catalan–Schett polynomials are found. The stack-sortable permutations are at the heart of our approaches.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106049"},"PeriodicalIF":0.9000,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Theory Series A","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0097316525000445","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Motivated by Kitaev and Zhang's recent work on non-overlapping ascents in stack-sortable permutations and Dumont's permutation interpretation of the Jacobi elliptic functions, we investigate some parity statistics on restricted permutations. Some new related bijections are constructed and two refinements of the generating function for descents over 321-avoiding permutations due to Barnabei, Bonetti and Silimbanian are obtained. In particular, an open problem of Kitaev and Zhang about non-overlapping ascents on 321-avoiding permutations is solved and several combinatorial interpretations for the Catalan–Schett polynomials are found. The stack-sortable permutations are at the heart of our approaches.
期刊介绍:
The Journal of Combinatorial Theory publishes original mathematical research concerned with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series A is concerned primarily with structures, designs, and applications of combinatorics and is a valuable tool for mathematicians and computer scientists.