Forest Volume Decompositions and Abel–Cayley–Hurwitz Multinomial Expansions

This paper presents a systematic approach to the discovery, interpretation and verification of various extensions of Hurwitzʹs multinomial identities, involving polynomials defined by sums over all subsets of a finite set. The identities are interpreted as decompositions of forest volumes defined by the enumerator polynomials of sets of rooted labeled forests. These decompositions involve the following basic forest volume formula, which is a refinement of Cayleyʹs multinomial expansion: for R⊆S the polynomial enumerating out-degrees of vertices of rooted forests labeled by S whose set of roots is R, with edges directed away from the roots, is∑r∈R xr∑s∈S xs|S|−|R|−1.