Restricted Dumont permutations, Dyck paths, and noncrossing partitions

We complete the enumeration of Dumont permutations of the second kind avoiding a pattern of length 4 which is itself a Dumont permutation of the second kind. We also consider some combinatorial statistics on Dumont permutations avoiding certain patterns of length 3 and 4 and give a natural bijection between 3142-avoiding Dumont permutations of the second kind and noncrossing partitions that uses cycle decomposition, as well as bijections between 132-, 231- and 321-avoiding Dumont permutations and Dyck paths. Finally, we enumerate Dumont permutations of the first kind simultaneously avoiding certain pairs of 4-letter patterns and another pattern of arbitrary length.Comment: 20 pages, 5 figure

Generating trees for permutations avoiding generalized patterns

We construct generating trees with one, two, and three labels for some classes of permutations avoiding generalized patterns of length 3 and 4. These trees are built by adding at each level an entry to the right end of the permutation, which allows us to incorporate the adjacency condition about some entries in an occurrence of a generalized pattern. We use these trees to find functional equations for the generating functions enumerating these classes of permutations with respect to different parameters. In several cases we solve them using the kernel method and some ideas of Bousquet-M\'elou. We obtain refinements of known enumerative results and find new ones.Comment: 17 pages, to appear in Ann. Com

Enumerative perspectives on chord diagrams

The topic of this thesis is enumerating certain classes of chord diagrams, perfect matchings of the interval $\{1, 2, \ldots, 2n\}$. We consider hereditary classes of chord diagrams that are restricted to satisfy one of several connectedness properties: connectivity, 1-terminality, and 1-sym-terminality (in order of increasing restrictedness). Such classes are defined by a set of minimal forbidden subdiagrams or patterns, and we focus on forbidding graphically-defined subdiagrams, in particular those whose intersection graph is isomorphic to a cycle. There are exactly two cycle diagrams of size $n$: the top cycle $T_{n}$ and bottom cycle $B_{n}$. The class \mc{D}(T_{\geqslant 3}) of diagrams avoiding a top cycle of size three or greater was previously shown to be equinumerous with the class of \includesvg[scale=.34]{graphics/D_213.svg}-free diagrams by Jel\'{i}nek, while the connected version of this class was put in bijection with planar bridgeless combinatorial maps by Courtiel, Yeats, and Zeilberger. We begin by extending the recently developed automated enumeration framework Combinatorial Exploration of Albert, Baen, Claesson, Nadeau, Pantone, and Ulfarsson for enumerating combinatorial classes to chord diagrams. This framework algorithmically searches for a combinatorial specification for a given class by decomposing the class using a fixed set of decomposition strategies. Building off of their work, we construct a geometric version of chord diagrams amenable to Combinatorial Exploration and then describe a series of decomposition strategies for these geometric chord diagram classes. Most of these strategies are based on those developed for permutation classes by Albert, Baen, Claesson, Nadeau, Pantone, and Ulfarsson, but several appear to be new. We then manually apply this framework to successfully enumerate a handful of diagram classes, including \mc{C}(\includesvg[scale=.34]{graphics/D_123.svg}, \includesvg[scale=.34]{graphics/D_132.svg}), \mc{C}(T_{\geqslant 3}, B_{\geqslant 3}), \mc{D}(\includesvg[scale=.34]{graphics/D_123.svg}, \includesvg[scale=.34]{graphics/D_132.svg}, \includesvg[scale=.34]{graphics/D_213.svg}), \mc{C}(B_{\geqslant 3}), and \mc{T}(B_{\geqslant 3}). All but the second class have not previously been enumerated, and we give explicit closed-form formulas for each of them. As a corollary it follows that the number |\mc{C}_{n+1}(B_{\geqslant 3})| of bottom-cycle-free diagrams of size $n+1$ is equal to |\mc{D}_{n}(\includesvg[scale=.34]{graphics/D_123.svg}, \includesvg[scale=.34]{graphics/D_132.svg})|, while |\mc{C}_{n+1}(T_{\geqslant 3}, B_{\geqslant 3})| = |\mc{D}_{n}(\includesvg[scale=.34]{graphics/D_123.svg}, \includesvg[scale=.34]{graphics/D_132.svg}, \includesvg[scale=.34]{graphics/D_213.svg})|. This appears to be a universal offset phenomenon---where connected classes are enumeratively equivalent to not-necessarily connected classes, the counting sequences are offset by 1. This points to a general map \mc{C}_{n+1} \to \mc{D}_{n} restricting to bijections on these connected classes. The restriction $\psi$ of such a map can be explicitly obtained between 1-terminal diagrams \mc{T}_{n+1} of size $n+1$ and diagrams \mc{D}_{n} of size $n$, and we give a novel description $\psi$, prove that it is a bijection, and show that it restricts to a bijection between 1-terminal tree diagrams \mc{T}(T_{\geqslant 3}) = \mc{T}(T_{\geqslant 3}, B_{\geqslant 3}) and noncrossing diagrams \mc{D}(\includesvg[scale=.34]{graphics/D_12.svg}), thereby counting the former. We then investigate the relationship between the map $\psi$ and notion of higher terminality analogous to higher connectivity, as well as relate it to increasing trees and Stirling permutations. Finally, we obtain a characterization of closure under subdiagram avoidance for $\psi$ and its inverse, giving bijections for an infinite set of pairs of restricted hereditary classes. We then obtain related results in a short study of 1-sym-terminal classes. Diagram classes defined by forbidding top cycles require alternative methods to those used above. For this, we construct a novel tree-like decomposition for connected chord diagrams. This gives a recurrence relation for the number of connected diagrams counted by size and the index of the first so-called terminal chord in a total order known as the intersection order. Applying the decomposition to connected top-cycle-free diagrams gives a similar recurrence. We then use this decomposition to construct recursive bijections between between \mc{C}(T_{\geqslant 3}) and the class of connected \includesvg[scale=.34]{graphics/D_213.svg}-free diagrams, as well as triangulations of a disk. Via prior work of Brown, the latter leads to an explicit formula for the counting sequence of these diagram classes. The recurrence relation for connected chord diagrams was previously implicitly obtained in work of Marie and Yeats giving chord diagram expansion solutions to certain Dyson-Schwinger equations from quantum field theory. Their proof was technically complex and passed to certain recursively-defined binary trees. We generalize this work using our connected diagram decomposition to solve a larger family of Dyson-Schwinger equations via weighted generating functions for weighted connected chord diagrams. We then discuss several conjectures towards obtaining similar solutions for more general and physically-realistic Dyson-Schwinger equations

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