dmtcs:2927 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2011,
DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011)

https://doi.org/10.46298/dmtcs.2927
The short toric polynomial
Authors: Gábor Hetyei ^{1}
NULL
Gábor Hetyei
1 Department of Mathematics and Statistics
We introduce the short toric polynomial associated to a graded Eulerian poset. This polynomial contains the same information as Stanley's pair of toric polynomials, but allows different algebraic manipulations. Stanley's intertwined recurrence may be replaced by a single recurrence, in which the degree of the discarded terms is independent of the rank. A short toric variant of the formula by Bayer and Ehrenborg, expressing the toric hvector in terms of the cdindex, may be stated in a rankindependent form, and it may be shown using weighted lattice path enumeration and the reflection principle. We use our techniques to derive a formula expressing the toric hvector of a dual simplicial Eulerian poset in terms of its fvector. This formula implies Gessel's formula for the toric hvector of a cube, and may be used to prove that the nonnegativity of the toric hvector of a simple polytope is a consequence of the Generalized Lower Bound Theorem holding for simplicial polytopes.