  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.
