Geodesic
Combinatorial statistics on alternating permutations
Journal of Algebraic Combinatorics, Tome 8 (1998) no. 2, pp. 169-191.

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Summary: We consider two combinatorial statistics on permutations. One is the genus. The other, [^$(des)$] widehattextdes , 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 [^$(des)$] widehattextdes on genus zero permutations and Baxter permutations. Our $q$-enumerative results relate the [^$(des)$] widehattextdes 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.
Keywords: lattice path, permutation, associahedron, Catalan, schröder 
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     author = {Dulucq, Serge and Simion, Rodica},
     title = {Combinatorial statistics on alternating permutations},
     journal = {Journal of Algebraic Combinatorics},
     pages = {169--191},
     publisher = {mathdoc},
     volume = {8},
     number = {2},
     year = {1998},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_1998__8_2_a2/}
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Dulucq, Serge; Simion, Rodica. Combinatorial statistics on alternating permutations. Journal of Algebraic Combinatorics, Tome 8 (1998) no. 2, pp. 169-191. http://geodesic.mathdoc.fr/item/JAC_1998__8_2_a2/