A formula for enumerating permutations with a fixed pinnacle set

Billey, 2013, Permutations with given peak set, J. Integer Seq., 16, 18 Brändén, 2008, Actions on permutations and unimodality of descent polynomials, Eur. J. Combin., 29, 514, 10.1016/j.ejc.2006.12.010 Davis, 2017, The pinnacle set of a permutation, Discrete Math., 341, 3249, 10.1016/j.disc.2018.08.011 Foata, 1974, Rearrangements of the symmetric group and enumerative properties of the tangent and secant numbers, Math. Z., 137, 257, 10.1007/BF01237393 Gaetz, 2019 Postnikov, 2008, Faces of generalized permutohedra, Doc. Math., 13, 207, 10.4171/dm/248 Rusu, 2020 Rusu, 2020 Stanley, 2015