Efficient recurrence for the enumeration of permutations with fixed pinnacle set

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 + 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 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.