Wenjie Fang - Efficient recurrence for the enumeration of permutations with fixed pinnacle set

dmtcs:8321 - Discrete Mathematics & Theoretical Computer Science, March 11, 2022, vol. 24, no. 1 - https://doi.org/10.46298/dmtcs.8321
Efficient recurrence for the enumeration of permutations with fixed pinnacle setArticle

Authors: Wenjie Fang ORCID

    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 |Sn(P)| of permutations of size n with a given pinnacle set P, with arithmetic complexity O(k4+klogn) for P of size k. A symbolic expression can also be computed in this way for pinnacle sets of fixed size. A weighted sum qn(P) of |Sn(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 qn(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.


    Volume: vol. 24, no. 1
    Section: Combinatorics
    Published on: March 11, 2022
    Accepted on: February 15, 2022
    Submitted on: July 30, 2021
    Keywords: Mathematics - Combinatorics

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