Combinatorial families of multilabelled increasing trees and hook-length formulas

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.Comment: 37 page

Embedded Trees and the Support of the ISE ⋆

Abstract. Embedded trees are labelled rooted trees, where the root has zero label and where the labels of adjacent vertices differ by±1. Recently it was proved by Chassaing and Schaeffer, and Janson and Marckert that the distribution of the maximum and minimum label are closely related to the support of the density of the integrated superbrownian excursion (ISE). The purpose of this paper is make this probabilistic limiting relation more explicit by using a generating function approach due to Bouttier, Di Francesco, and Guitter that is based on properties of Jacobi’s θ-functions. In particular we derive an integral representation of the joint distribution function of the supremum and infimum of the support of the ISE in terms of the Weierstrass ℘-function.

Embedded Trees and the Support of the ISE

Embedded trees are labelled rooted trees, where the root has zero label and where the labels of adjacent vertices differ (at most) by ±1. Recently it was proved (see [6] and [9]) that the distribution of the maximum and minimum label are closely related to the support of the density of the integrated superbrownian excursion (ISE). The purpose of this paper is make this probabilistic limiting relation more explicit by using a generating function approach due to Bouttier, Di Francesco, and Guitter [5] that is based on properties of Jacobi’s θ-functions. In particular we derive an integral representation of the joint distribution function of the supremum and infimum of the support of the ISE in terms of the Weierstrass ℘-function. Furthermore we re-derive the limiting radius distribution in random quadrangulations (by Chassaing and Schaeffer [6]) with the help of exact counting generating functions.