Abstract
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 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. Nous introduisons le polynôme torique court associé à un ensemble ordonné Eulérien. Ce polynôme contient la même information que le couple de polynômes toriques de Stanley, mais il permet des manipulations algébriques différentes. La récurrence entrecroisée de Stanley peut être remplacée par une seule récurrence dans laquelle le degré des termes écartés est indépendant du rang. La variante torique courte de la formule de Bayer et Ehrenborg, qui exprime le vecteur torique d'un ensemble ordonné Eulérien en termes de son cd-index, est énoncée sous une forme qui ne dépend pas du rang et qui peut être démontrée en utilisant une énumération des chemins pondérés et le principe de réflexion. Nous utilisons nos techniques pour dériver une formule exprimant le vecteur h-torique d'un ensemble ordonné Eulérien dont le dual est simplicial, en termes de son f-vecteur. Cette formule implique la formule de Gessel pour le vecteur h-torique d'un cube, et elle peut être utilisée pour démontrer que la positivité du vecteur h-torique d'un polytope simple est une conséquence du Théorème de la Borne Inférieure Généralisé appliqué aux polytopes simpliciaux.
Highlights
We often look for a “magic” simplification that makes known results easier to state, and provides the language to state new results
There is no change when we want to extract the coefficients of the individual polynomials only, but when we consider a sequence {pn(x)}n≥0 of multiplicatively symmetric polynomials, given by some rule, switching to the additively symmetric variant {qn(x)}n≥0 greatly changes the appearance of the rules, making them sometimes easier to manipulate
The short toric polynomial t(P, x), associated to a graded Eulerian poset P is defined in Section 3 as the multiplicatively symmetric variant of f (P, x)
Summary
We often look for a “magic” simplification that makes known results easier to state, and provides the language to state new results. The short toric polynomial t(P, x), associated to a graded Eulerian poset P is defined in Section 3 as the multiplicatively symmetric variant of f (P, x). An application showing the usefulness of our invariant may be found, where we express the toric h-vector of an Eulerian dual simplicial poset in terms of its f -vector. This question was raised by Kalai, see [19]. An equivalent form of our formula implies that the nonnegativity of the toric h-vector of simple polytope is an elementary consequence of the Generalized Lower Bound Theorem (GLBT) holding for simplicial polytopes [20]. The word elementary has to be stressed since Karu [13] has shown that the GLBT holds for all polytopes
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