Decorated hypertrees

C. Jensen, J. McCammond and J. Meier have used weighted hypertrees to compute the Euler characteristic of a subgroup of the automorphism group of a free product. Weighted hypertrees also appear in the study of the homology of the hypertree poset. We link them to decorated hypertrees after a general study on decorated hypertrees, which we enumerate using box trees.---C. Jensen, J. McCammond et J. Meier ont utilis\'e des hyperarbres pond\'er\'es pour calculer la caract\'eristique d'Euler d'un sous-groupe du groupe des automorphismes d'un produit libre. Un autre type d'hyperarbres pond\'er\'es appara\^it aussi dans l'\'etude de l'homologie du poset des hyperarbres. Nous \'etudions les hyperarbres d\'ecor\'es puis les comptons \`a l'aide de la notion d'arbre en bo\^ite avant de les relier aux hyperarbres pond\'er\'es.Comment: nombre de pages : 3

Grassmann Integral Representation for Spanning Hyperforests

Given a hypergraph G, we introduce a Grassmann algebra over the vertex set, and show that a class of Grassmann integrals permits an expansion in terms of spanning hyperforests. Special cases provide the generating functions for rooted and unrooted spanning (hyper)forests and spanning (hyper)trees. All these results are generalizations of Kirchhoff's matrix-tree theorem. Furthermore, we show that the class of integrals describing unrooted spanning (hyper)forests is induced by a theory with an underlying OSP(1|2) supersymmetry.Comment: 50 pages, it uses some latex macros. Accepted for publication on J. Phys.