Combinatorial statistics on alternating permutations
Following the definition of the genus of a hypermap given in [\textit{R. Cori}, ``Un code pur les graphes planaires et ses applications,'' Astérisque 27, 1-169 (1975; Zbl 0313.05115)], specialized to hypermaps of one vertex, the paper under review defines a permutation \(\alpha\) of \(\{1, 2,\dots,N\}\) to be of genus 0 if the number of cycles of \(\alpha\) plus the number of cycles of \(\alpha^{-1} (1,2, \dots,N)\) is equal to \(N+1\). It is shown that genus-0 alternating permutations are counted by Schröder numbers. This result is generalized to count these permutations according to the number of alternating descents (\(i\) is an alternating descent of \(\alpha\) if \(\alpha (i+2) >\alpha(i))\) in terms of the \(q\)-analogue of Schröder numbers. Alternating genus-0 permutations are shown to be Baxter permutations, and alternating Baxter permutations are counted in terms of the \(q\)-analogue of Catalan numbers. These results are used to relate the alternating descent 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 [\textit{C. W. Lee}, ``The associahedron and triangulations of the \(n\)-gon,'' Eur. J. Comb. 10, No. 6, 551-560 (1989; Zbl 0682.52004)].
scientific article; zbMATH DE number 1228568
scientific article