Hypergraphs and a functional equation of Bouwkamp and de Bruijn

摘要

Let Phi(u, v) = Sigma(m=0)(infinity) Sigma(n=0)(infinity) c(mn) u(m) v(m). Bouwkamp and de Bruijn found that there exists a power series Psi(u, v) satisfying the equation t Psi(tz, z) = log(Sigma(k=0)(infinity) t(k)/k! exp(k Phi(kz, z))). We show that this result can be interpreted combinatorially using hypergraphs. We also explain some facts about Phi(u, 0) and Psi(u, 0), shown by Bouwkamp and de Bruijn, by using hypertrees, and we use Lagrange inversion to count hypertrees by number of vertices and number of edges of a specified size. (c) 2005 Elsevier Inc. All rights reserved.