Forest Volume Decompositions and Abel–Cayley–Hurwitz Multinomial Expansions

Abstract

This paper presents a systematic approach to the discovery, interpretation and verification of various extensions of Hurwitz's 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 Cayley's multinomial expansion: for R ı 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

The Pfaff/Cauchy derivative identities and Hurwitz type extensions

“…The general case (again up to rescaling) is due to Raney [36]. Among other sources for Hurwitz type identities we mention only the very interesting recent work by Pitman [33,34], where one can find further references. If every w i = w then these become Rothe type identities.…”

A remark on the enumeration of rooted labeled trees

“…(Note, however, that in applying this identity, we must first fix N and M and then specialize valid for n ≥ 1 and k, ℓ ≥ 0 with k + ℓ ≤ n. We remark, finally, that many Abel identities, including (A.9), can be proven combinatorially: see e.g. [8,17,21].…”

A combinatorial identity for rooted labeled forests

“…It should be noted that variations of (1) are well-known and can be found in, e.g., [6,Theorem 5.3.4]. Equations of this type are related to Hurwitz multinomial identities, see [3], [4] and [5]. In particular, in [4] Pitman presents a systematic approach to interpreting identities of this type as decompositions of forest volumes, i.e., polynomials that enumerate special classes of rooted forests.…”

“…We choose to prove a variation of ( 1 ) that belongs to the same family of identities as discussed in [ 4 , Section 4 and Corollary 8 in particular].…”