Generating functions for generating trees

Certain families of combinatorial objects admit recursive descriptions in terms of generating trees: each node of the tree corresponds to an object, and the branch leading to the node encodes the choices made in the construction of the object. Generating trees lead to a fast computation of enumeration sequences (sometimes, to explicit formulae as well) and provide efficient random generation algorithms. We investigate the links between the structural properties of the rewriting rules defining such trees and the rationality, algebraicity, or transcendence of the corresponding generating function.Comment: This article corresponds, up to minor typo corrections, to the article submitted to Discrete Mathematics (Elsevier) in Nov. 1999, and published in its vol. 246(1-3), March 2002, pp. 29-5

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

Enumerative combinatorics, continued fractions and total positivity

Determining whether a given number is positive is a fundamental question in mathematics. This can sometimes be answered by showing that the number counts some collection of objects, and hence, must be positive. The work done in this dissertation is in the field of enumerative combinatorics, the branch of mathematics that deals with exact counting. We will consider several problems at the interface between enumerative combinatorics, continued fractions and total positivity. In our first contribution, we exhibit a lower-triangular matrix of polynomials in six indeterminates that appears empirically to be coefficientwise totally positive, and which includes as a special case the Eulerian triangle. This generalises Brenti’s conjecture from 1996. We prove the coefficientwise total positivity of a three-variable case which includes the reversed Stirling subset triangle. Our next contribution is the study of two sequences whose Stieltjes-type continued fraction coefficients grow quadratically; we study the Genocchi and median Genocchi numbers. We find Stieltjes-type and Thron-type continued fractions for some multivariate polynomials that enumerate D-permutations, a class of permutations of 2n, with respect to a very large (sometimes infinite) number of simultaneous statistics that measure cycle status, record status, crossings and nestings. After this, we interpret the Foata–Zeilberger bijection in terms of Laguerre digraphs, which enables us to count cycles in permutations. Using this interpretation, we obtain Jacobi-type continued fractions for multivariate polynomials enumerating permutations, and also Thron-type and Stieltjes-type continued fractions for multivariate polynomials enumerating D-permutations, in both cases including the counting of cycles. This enables us to prove some conjectured continued fractions due to Sokal–Zeng from 2022, and Randrianarivony–Zeng from 1996. Finally, we introduce the higher-order Stirling cycle and subset numbers; these generalise the Stirling cycle and subset numbers, respectively. We introduce some conjectures which involve different total-positivity questions for these triangular arrays and then answer some of them