The short toric polynomial

TL;DR

This paper introduces the short toric polynomial for graded Eulerian posets, simplifying algebraic manipulations and providing new formulas for toric invariants, with implications for polytope theory.

Contribution

It defines a new short toric polynomial that simplifies existing recurrence relations and expresses toric invariants in a rank-independent manner, advancing combinatorial algebra.

Findings

Derived a rank-independent formula for the toric h-vector in terms of the f-vector.

Proved the nonnegativity of the toric h-vector for simple polytopes via the Generalized Lower Bound Theorem.

Unified the algebraic framework for toric polynomials with new combinatorial interpretations.

Abstract

We introduce the short toric polynomial associated to a graded Eulerian poset. This polynomial contains the same information as the two toric polynomials introduced by Stanley, but allows different algebraic manipulations. The intertwined recurrence defining Stanley's toric polynomials 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 $h$-vector in terms of the $cd$-index, may be stated in a rank-independent 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 $h$-vector of a dual simplicial Eulerian poset in terms of its $f$-vector. This formula implies Gessel's formula for the toric $h$-vector of a cube, and may be used to prove that the nonnegativity of the toric $h$-vector of a simple polytope is a consequence of the Generalized Lower Bound Theorem holding for simplicial polytopes.