{"docId":9032,"paperId":8330,"url":"https:\/\/dmtcs.episciences.org\/8330","doi":"10.46298\/dmtcs.8330","journalName":"Discrete Mathematics & Theoretical Computer Science","issn":"","eissn":"1365-8050","volume":[{"vid":608,"name":"vol. 24, no. 1"}],"section":[{"sid":6,"title":"Combinatorics","description":[]}],"repositoryName":"Hal","repositoryIdentifier":"hal-03295362","repositoryVersion":3,"repositoryLink":"https:\/\/hal.archives-ouvertes.fr\/hal-03295362v3","dateSubmitted":"2021-08-03 07:33:36","dateAccepted":"2022-01-19 20:52:59","datePublished":"2022-02-07 11:56:51","titles":{"en":"Further enumeration results concerning a recent equivalence of restricted inversion sequences"},"authors":["Mansour, Toufik","Shattuck, Mark"],"abstracts":{"en":"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."},"keywords":[{"en":"65Q30 pattern avoidance"},{"en":"combinatorial statistic"},{"en":"July 22"},{"en":"2021. 2010 Mathematics Subject Classification. 05A15"},{"en":"05A05"},{"en":"65Q20"},{"en":"kernel method"},{"en":"inversion sequence"},"[MATH.MATH-CO]Mathematics [math]\/Combinatorics [math.CO]","[MATH.MATH-MP]Mathematics [math]\/Mathematical Physics [math-ph]"]}