Séminaire Lotharingien de Combinatoire, B50c (2003), 13 pp.

Vic Reiner, Dennis Stanton and Volkmar Welker

The Charney-Davis Quantity For Certain Graded Posets

Abstract. Given a naturally labelled graded poset P with r ranks, the alternating sum

$\displaystyle W(P,-1):=\sum_{w \in \JH(P)} (-1)^{\des(w)}

is related to a quantity occurring in the Charney-Davis Conjecture on flag simplicial spheres. When |P|-r is odd it vanishes. When |P|-r is even and P satisfies the Neggers-Stanley Conjecture, it has sign (-1)(|P|-r)/2. We interpret this quantity combinatorially for several classes of graded posets P, including certain disjoint unions of chains and products of chains. These interpretations involve alternating multiset permutations, Baxter permutations, Catalan numbers, and Franel numbers.

Received: June 6, 2003. Accepted: October 19, 2003. Final Version: December 12, 2003.

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