Séminaire Lotharingien de Combinatoire, B50c (2003), 13 pp.
Vic Reiner, Dennis Stanton and Volkmar Welker
The Charney-Davis Quantity For Certain Graded Posets
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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