Pinnacle sets revisited

In 2017, Davis, Nelson, Petersen, and Tenner [Discrete Math. 341 (2018),3249--3270] initiated the combinatorics of pinnacles in permutations. We provide a simple and efficient recursion to compute $p_n(S)$, the number of permutations of $S_n$ with pinnacle set $S$, and a conjectural closed formula for the related numbers $q_n(S)$. We determine the lexicographically minimal elements of the orbits of the modified Foata-Strehl action, prove that these elements form a lower ideal of the left weak order and characterize and count the maximal elements of this ideal.