Geodesic
The Charney-Davis Quantity For Certain Graded Posets
Séminaire lotharingien de combinatoire, Tome 50 (2003-2005) 
Vic Reiner; Dennis Stanton; Volkmar Welker. The Charney-Davis Quantity For Certain Graded Posets. Séminaire lotharingien de combinatoire, Tome 50 (2003-2005). http://geodesic.mathdoc.fr/item/SLC_2003-2005_50_a2/
@article{SLC_2003-2005_50_a2,
     author = {Vic Reiner and Dennis Stanton and Volkmar Welker},
     title = {The {Charney-Davis} {Quantity} {For} {Certain} {Graded} {Posets}},
     journal = {S\'eminaire lotharingien de combinatoire},
     year = {2003-2005},
     volume = {50},
     url = {http://geodesic.mathdoc.fr/item/SLC_2003-2005_50_a2/}
}
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Voir la notice de l'acte provenant de la source Séminaire Lotharingien de Combinatoire website

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.