Increasing Trees and the Degree-Chromatic Polynomial

TL;DR

This paper derives a general formula for the number of m-ary increasing trees using degree-chromatic polynomials and Bell polynomials, connecting combinatorial structures with algebraic and permutation enumeration methods.

Contribution

It introduces a unified formula for m-ary increasing trees and links it to degree-chromatic polynomials, Bell polynomials, and permutation enumeration.

Findings

Derived a general formula for m-ary increasing trees.

Connected degree-chromatic polynomial evaluations to special permutation counts.

Applied combinatorial and algebraic methods like Lagrange inversion.

Abstract

This paper studies increasing trees on $n$ labeled vertices, in which labels increase from the root to the leaves. It is known that the number of binary increasing trees coincides with the number of alternating permutations (Euler numbers). Riordan obtained explicit formulas for the numbers of ternary and quaternary trees. This article derives a general formula for the number of $m\text{-ary}$ increasing trees for any $m$. The main result is expressed in terms of the degree-chromatic polynomial of the complete graph and Bell polynomials. It is shown how the corresponding generating function is related to the inversion problem and how combinatorial methods, including the lemma on coefficients of the multiplicative inverse function and the Lagrange inversion formula, can be used to compute the coefficients. A connection is also established between the values of the degree-chromatic polynomial at $\lambda=-1$ and the numbers of special permutations studied by Gessel.