Forest volume decompositions and Abel---Cayley---Hurwitz multinomial expansions

Authors:
Jim Pitman
Affiliations:
Department of Statistics, University of California, 367 Evans Hall #3860, Berkeley, California
Venue:
Journal of Combinatorial Theory Series A
Year:
2002

Citing 10
Cited 0

Enumerative combinatorics

Enumerative combinatorics
Counting forests

Discrete Mathematics
Voting blocks, reluctant functions, and a formula of Hurwitz

Discrete Mathematics
Identities of Rothe-Abel-Schla¨fli-Hurwitz-type

Discrete Mathematics - Special volume: algebraic combinatorics
A new proof of Cayley's formula for counting labeled trees

Journal of Combinatorial Theory Series A
Multinomial convolution polynomials

Discrete Mathematics
The art of computer programming, volume 1 (3rd ed.): fundamental algorithms

The art of computer programming, volume 1 (3rd ed.): fundamental algorithms
Coalscent random forests

Journal of Combinatorial Theory Series A
A family of random trees with random edge lengths

Random Structures & Algorithms
Asymptotics of the Hurwitz Binomial Distribution Related to Mixed Poisson Galton–Watson Trees

Combinatorics, Probability and Computing

Quantified Score

Hi-index	0.01

Visualization

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 ⊑ 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 (Σxr)(Σxs)|S|-|R|-1.