This page is under construction. I will progressively add material related to forest and tree formulae for use in cluster and Mayer expansions in statistical mechanics and constructive field theory.
The original reference for the the Brydges-Kennedy tree formula is: D. Brydges and T. Kennedy. Mayer expansions and the Hamilton-Jacobi equation. J. Stat. Phys. 48 (1987), 19.
The original reference for the fundamental theorem of calculus forest version of this identity is: A. Abdesselam and V. Rivasseau. "Trees, forests and jungles: a botanical garden for cluster expansions". Constructive physics (Palaiseau, 1994), 7--36, Lecture Notes in Phys., 446, Springer, Berlin, 1995. preprint version .
The first proof I found for the partial fraction decomposition identity Eq. (II.6) in the previous article, uses inversion of Kronecker products of triangular matrices of Stirling numbers of the first and second kind (pdf scan).
I mentioned these partial fraction identities to Vincent Lafforgue and he also found a proof which uses minimal decompositions of permutation as products of transpositions (pdf scan).
A pedagogical presentation can be found here.