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Further enumeration results concerning a recent equivalence of restricted inversion sequences

Abstract : 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.
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Submitted on : Monday, January 31, 2022 - 10:43:20 AM
Last modification on : Saturday, June 25, 2022 - 8:10:33 PM


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Toufik Mansour, Mark Shattuck. Further enumeration results concerning a recent equivalence of restricted inversion sequences. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2022, vol. 24, no. 1, ⟨10.46298/dmtcs.8330⟩. ⟨hal-03295362v3⟩



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