Pinnacle sets revisited

Abstract

In 2017, Davis, Nelson, Petersen, and Tenner (Davis et al. (2018) [1]) initiated the combinatorics of pinnacles in permutations. We provide a simple and efficient recursion to compute pn(S), the number of permutations in Sn with pinnacle set S, and a conjectural closed formula for the related numbers qn(S). This conjecture was proved between the first draft of this paper and its final version by Fang (2022) [4] and later by Minnich [7]. We also 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.