Hyperforests on the Complete Hypergraph by Grassmann Integral Representation

TL;DR

This paper introduces a novel Grassmann integral method to analyze hyperforests in complete hypergraphs, providing new insights and applications in counting and asymptotic analysis.

Contribution

It develops a Grassmann integral representation for hyperforest generating functions, enabling new counting formulas and asymptotic insights for hypergraphs.

Findings

New Grassmann integral representation for hyperforests

Explicit counting formulas for hyperforests in k-uniform hypergraphs

Asymptotic analysis of hyperforest components

Abstract

We study the generating function of rooted and unrooted hyperforests in a general complete hypergraph with n vertices by using a novel Grassmann representation of their generating functions. We show that this new approach encodes the known results about the exponential generating functions for the different number of vertices. We consider also some applications as counting hyperforests in the k-uniform complete hypergraph and the one complete in hyperedges of all dimensions. Some general feature of the asymptotic regimes for large number of connected components is discussed.