Forest Volume Decompositions and Abel - Cayley - Hurwitz Multinomial Expansions

This paper presents a systematic approach to the discovery, interpretation and veri cation of various extensions of Hurwitz's multinomial identities, involving polynomials de ned by sums over all subsets of a nite set. The identities are interpreted as decompositions of forest volumes de ned by the enumerator polynomials of sets of rooted labeled forests. These decompositions involve the following basic forest volume formula, which is a re nement of Cayley's 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 Pr2R xr Ps2S xsjSj,jRj,1: Supported in part by N.S.F. Grants DMS 97-03961 and DMS-0071448 1 

• Type: Journal
• Year: 2002
• Where: JCT
• Authors: Jim Pitman
• Keywords: Forest Volume, JCT 2002, Multinomial Identities, Polynomial