Combinatorial families of multilabelled increasing trees and hook-length formulas

TL;DR

This paper introduces and analyzes various multilabelled increasing tree families, deriving generating functions, explicit enumeration formulas, and hook-length formulas, including elliptic cases involving special functions.

Contribution

It generalizes increasing trees to multilabelled versions, provides differential equation characterizations, explicit enumeration results, and a method to derive hook-length formulas.

Findings

Explicit enumeration formulas for multilabelled increasing trees.

Generating functions characterized by differential equations.

Hook-length formulas derived from enumeration results.

Abstract

In this work we introduce and study various generalizations of the notion of increasingly labelled trees, where the label of a child node is always larger than the label of its parent node, to multilabelled tree families, where the nodes in the tree can get multiple labels. For all tree classes we show characterizations of suitable generating functions for the tree enumeration sequence via differential equations. Furthermore, for several combinatorial classes of multilabelled increasing tree families we present explicit enumeration results. We also present multilabelled increasing tree families of an elliptic nature, where the exponential generating function can be expressed in terms of the Weierstrass-p function or the lemniscate sine function. Furthermore, we show how to translate enumeration formulas for multilabelled increasing trees into hook-length formulas for trees and present a general "reverse engineering" method to discover hook-length formulas associated to such tree families.