A refinement of Cayley's formula for trees

TL;DR

This paper generalizes Cayley's formula for counting labeled rooted trees by introducing polynomials that encode the number of proper vertices, providing new proofs and hook length formulas, and connecting to parking functions.

Contribution

It offers a new proof of existing hook length formulas, introduces a polynomial framework for counting trees by proper vertices, and links these counts to parking function interpretations.

Findings

Derived polynomials P_n(a,b,c) generalize tree counts

Provided new hook length formulas for trees

Connected tree enumeration to parking functions

Abstract

A proper vertex of a rooted tree with totally ordered vertices is a vertex that is less than all its proper descendants. We count several kinds of labeled rooted trees and forests by the number of proper vertices. Our results are all expressed in terms of the polynomials P_n(a,b,c)= c(a+(n-1)b+c)(2a+(n-2)b+c)...((n-1)a+b+c) which reduce to (n+1)^{n-1} for a=b=c=1. Our study of proper vertices was motivated by A. Postnikov's hook length formula for binary trees (arXiv:math.CO/0507163), which was also proved by W. Y. C. Chen and L. L. M. Yang (arXiv:math.CO/0507163) and generalized by R. R. X. Du and F. Liu (arXiv:math.CO/0501147). Our approach gives a new proof of Du and Liu's results and gives new hook length formulas. We also find an interpretation of the polynomials P_n(a,b,c) in terms of parking functions: we count parking functions according to the number of cars that park in their preferred parking spaces.