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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
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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