2012/03/14 by Matthieu Josuat-Vergès, Matthieu Josuat Vergès
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Commutative Algebra and Its Applications #Cumulant #Enumeration #Gaussian #Hypergeometric distribution #Hypergeometric function #Interpretation (philosophy) #Polynomial #Polynomial and algebraic computation #Tutte polynomial #math.CO
paper · pdf · doi:10.4153/cjm-2012-042-9
published as Can. J. Math.-J. Can. Math. 65 (2013) 863-878 · 14 pages
arxiv created 2012/03/14 · openalex publication_date 2012/08/28 · arxiv updated 2019/08/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Abstract The q-semicircular distribution is a probability law that interpolates between the Gaussian law and the semicircular law. There is a combinatorial interpretation of itsmoments in terms ofmatchings, where q follows the number of crossings, whereas for the free cumulants one has to restrict the enumeration to connected matchings. The purpose of this article is to describe combinatorial properties of the classical cumulants. We show that like the free cumulants, they are obtained by an enumeration of connected matchings, the weight being now an evaluation of the Tutte polynomial of a so-called crossing graph. The case q = 0 of these cumulants was studied by Lassalle using symmetric functions and hypergeometric series. We show that the underlying combinatorics is explained through the theory of heaps, which is Viennot's geometric interpretation of the Cartier–Foata monoid. This method also gives a general formula for the cumulants in terms of free cumulants.