Geodesic
The Hodge structure on a filtered Boolean algebra
Journal of Algebraic Combinatorics, Tome 20 (2004) no. 1, pp. 61-70.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: Let $Delta( B _{n})$ be the order complex of the Boolean algebra and let $B( n, k)$ be the part of $Delta( B _{n})$ where all chains have a gap at most $k$ between each set. We give an action of the symmetric group $S _{l}$ on the $l$-chains that gives $B( n, k)$ a Hodge structure and decomposes the homology under the action of the Eulerian idempontents. The $S _{n}$ action on the chains induces an action on the Hodge pieces and we derive a generating function for the cycle indicator of the Hodge pieces. The Euler characteristic is given as a corollary. We then exploit the connection between chains and tabloids to give various special cases of the homology. Also an upper bound is obtained using spectral sequence methods.
Keywords: Hodge structure, Boolean algebra, Euler characteristics 
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Kravitz, Scott. The Hodge structure on a filtered Boolean algebra. Journal of Algebraic Combinatorics, Tome 20 (2004) no. 1, pp. 61-70. http://geodesic.mathdoc.fr/item/JAC_2004__20_1_a2/