Increasing Trees and the Degree-Chromatic Polynomial
Medet Jumadildayev
TL;DR
Problem: enumerate $m$-ary increasing trees on $n$ labeled vertices. Approach: express $T_n(m)$ via the degree-chromatic polynomials $\chi_m(K_s,-1)$ of complete graphs and Bell polynomials using coefficient-extraction and Lagrange inversion techniques. Main contribution: a closed-form formula for $T_n(m)$ valid for any $m$, namely $T_n(m)=\sum_{k=0}^n (-1)^k B_{n+k-1,k}(\chi_m(K_0,-1),\ldots,\chi_m(K_{n-1},-1))$, with $\chi_m(K_n,\lambda)=\sum_{k=1}^n B_{n,k}(1_m)(\lambda)_k$ and $\chi_m(K_n,-1)=\sum_{k=1}^n (-1)^k k! B_{n,k}(1_m)$. Connections to Gessel's $a_n(m)$ and the Euler numbers for $m=2$ are established. Significance: unifies combinatorial enumeration of increasing trees with graph-coloring polynomials, enabling explicit counts for arbitrary $m$ and revealing rich algebraic structure via Bell polynomials and Lagrange inversion.
Abstract
The number of binary increasing trees is the number of alternating permutations. Riordan found the formulas for the number of ternary and quaternary increasing trees. We obtain an explicit formula for the number of $\text{$m$-ary}$ increasing trees for any $m$, which can be expressed in terms of the degree-chromatic polynomial of the complete graph.