episciences.org_8330_1660008578
1660008578
episciences.org
raphael.tournoy+crossrefapi@ccsd.cnrs.fr
episciences.org
Discrete Mathematics & Theoretical Computer Science
13658050
02
07
2022
vol. 24, no. 1
Combinatorics
Further enumeration results concerning a recent equivalence of restricted inversion sequences
Toufik
Mansour
Mark
Shattuck
Let asc and desc denote respectively the statistics recording the number of ascents or descents in a sequence having nonnegative 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 $n1\text{asc}$ 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 $n1\text{asc}$ 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.
02
07
2022
8330
https://hal.archivesouvertes.fr/hal03295362v3
https://hal.archivesouvertes.fr/hal03295362v2
https://hal.archivesouvertes.fr/hal03295362v1
10.46298/dmtcs.8330
https://dmtcs.episciences.org/8330

https://dmtcs.episciences.org/9032/pdf