Efficient recurrence for the enumeration of permutations with fixed pinnacle set - Base de données, Automate, Analyse d'agorithmes et Modèles
Article Dans Une Revue Discrete Mathematics and Theoretical Computer Science Année : 2022

Efficient recurrence for the enumeration of permutations with fixed pinnacle set

Wenjie Fang

Résumé

Initiated by Davis, Nelson, Petersen and Tenner (2018), the enumerative study of pinnacle sets of permutations has attracted a fair amount of attention recently. In this article, we provide a recurrence that can be used to compute efficiently the number $|\mathfrak{S}_n(P)|$ of permutations of size $n$ with a given pinnacle set $P$, with arithmetic complexity $O(k^4 + k\log n)$ for $P$ of size $k$. A symbolic expression can also be computed in this way for pinnacle sets of fixed size. A weighted sum $q_n(P)$ of $|\mathfrak{S}_n(P)|$ proposed in Davis, Nelson, Petersen and Tenner (2018) seems to have a simple form, and a conjectural form is given recently by Flaque, Novelli and Thibon (2021+). We settle the problem by providing and proving an alternative form of $q_n(P)$, which has a strong combinatorial flavor. We also study admissible orderings of a given pinnacle set, first considered by Rusu (2020) and characterized by Rusu and Tenner (2021), and we give an efficient algorithm for their counting.
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Dates et versions

hal-03667552 , version 1 (08-09-2024)

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Wenjie Fang. Efficient recurrence for the enumeration of permutations with fixed pinnacle set. Discrete Mathematics and Theoretical Computer Science, 2022, 24 (1), pp.8. ⟨10.46298/dmtcs.8321⟩. ⟨hal-03667552⟩
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