A local injective proof of log-concavity for increasing spanning forests

TL;DR

This paper provides a local combinatorial proof of log-concavity for the generating polynomial of increasing spanning forests, offering new insights into the structure of these combinatorial objects and related Stirling numbers.

Contribution

It introduces a local injection-based proof of log-concavity for increasing spanning forests, differing from previous global approaches.

Findings

Proves weighted strong log-concavity for increasing spanning forests.

Provides a new combinatorial proof of log-concavity of Stirling numbers of the first kind.

Demonstrates a local edge-moving injection technique.

Abstract

We give an explicit combinatorial proof of a weighted version of strong log-concavity for the generating polynomial of increasing spanning forests of a finite simple graph equipped with a total ordering of the vertices. In contrast to similar proofs in the literature, our injection is local in the sense that it proceeds by moving a single edge from one forest to the other. In the particular case of the complete graph, this gives a new combinatorial proof of log-concavity of unsigned Stirling numbers of the first kind where a pair of permutations is transformed into a new pair by breaking a single cycle in the first permutation and gluing two cycles in the second permutation, while all the other spectator cycles are left untouched.