André permutations, lexicographic shellability and the 𝑐𝑑-index of a convex polytope

Purtill, Mark

doi:10.1090/S0002-9947-1993-1094560-9

André permutations, lexicographic shellability and the $cd$-index of a convex polytope
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by Mark Purtill

Trans. Amer. Math. Soc. 338 (1993), 77-104

DOI: https://doi.org/10.1090/S0002-9947-1993-1094560-9

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Abstract:

The $cd$-index of a polytope was introduced by Fine; it is an integer valued noncommutative polynomial obtained from the flag-vector. A result of Bayer and Fine states that for any integer "flag-vector," the existence of the $cd$-index is equivalent to the holding of the generalized Dehn-Sommerville equations of Bayer and Billera for the flag-vector. The coefficients of the $cd$-index are conjectured to be nonnegative. We show a connection between the $cd$-index of a polytope $\mathcal {P}$ and any $CL$-shelling of the lattice of faces of $\mathcal {P}$ ; this enables us to prove that each André polynomial of Foata and Schützenberger is the $cd$-index of a simplex. The combinatorial interpretation of this $cd$-index can be extended to cubes, simplicial polytopes, and some other classes (which implies that the $cd$-index has nonnegative coefficients for these polytopes). In particular, we show that any polytope of dimension five or less has a positive $cd$-index.

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