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

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