Combinatorial Statistics on Alternating Permutations

We consider two combinatorial statistics on permutations. One is the genus. The other, , is defined for alternating permutations, as the sum of the number of descents in the subwords formed by the peaks and the valleys. We investigate the distribution of on genus zero permutations and Baxter permutations. Our q-enumerative results relate the statistic to lattice path enumeration, the rank generating function and characteristic polynomial of noncrossing partition lattices, and polytopes obtained as face-figures of the associahedron.