Formal Power Series and Algebraic Combinatorics pp 743-753 | Cite as
An Application of Dumont’s Statistic
Abstract
In 1974, Dumont [5] gave an interpretation of the Eulerian numbers which extends to a number of statistics on permutations [7] and on arbitrary words [8]. We apply one such statistic to a special case of a result of Stanley concerning the flag h-vectors of Cohen-Macaulay complexes [9]. Specifically, we give a bijective proof that for each distributive lattice J(P) which is a product of chains, there is a poset Q such that the f-vector of Q is the h-vector of P. We conjecture that the result holds for all finite distributive lattices.
Résumé
En 1974, Dumont [5] a donné une interprétation des nombres Eulériens qui s’étend à diverses statistiques sur les permutations [7] et sur les mots arbitraires [8]. Nous appliquons une telle statistique à un cas particulier d’un résultat de Stanley en ce qui concerne les vecteurs h des complexes de Cohen-Macaulay [9]. Plus spécifiquement, nous montrons par une bijection que pour chaque treillis distributif J(P) qui est un produit de chaînes, il y a un ensemble partiellement ordonné Q tel que le vecteur f de Q est le vecteur h de P. Nous conjecturons que ce résultat demeure vrai pour tous les treillis distributifs finis.
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