A Sequent Calculus for a Semi-Associative Law

We introduce a sequent calculus with a simple restriction of Lambek\u27s product rules that precisely captures the classical Tamari order, i.e., the partial order on fully-bracketed words (equivalently, binary trees) induced by a semi-associative law (equivalently, tree rotation). We establish a focusing property for this sequent calculus (a strengthening of cut-elimination), which yields the following coherence theorem: every valid entailment in the Tamari order has exactly one focused derivation. One combinatorial application of this coherence theorem is a new proof of the Tutte-Chapoton formula for the number of intervals in the Tamari lattice Y_n. Elsewhere, we have also used the sequent calculus and the coherence theorem to build a surprising bijection between intervals of the Tamari order and a natural fragment of lambda calculus, consisting of the beta-normal planar lambda terms with no closed proper subterms

Intervals in the greedy Tamari posets

We consider a greedy version of the $m$-Tamari order defined on $m$-Dyck paths, recently introduced by Dermenjian. Inspired by intriguing connections between intervals in the ordinary 1-Tamari order and planar triangulations, and more generally by the existence of simple formulas counting intervals in the ordinary $m$-Tamari orders, we investigate the number of intervals in the greedy order on $m$-Dyck paths of fixed size. We find again a simple formula, which also counts certain planar maps (of prescribed size) called $(m+1)$-constellations. For instance, when $m=1$ the number of intervals in the greedy order on 1-Dyck paths of length $2n$ is proved to be $\frac{3\cdot 2^{n-1}}{(n+1)(n+2)} \binom{2n}{n}$, which is also the number of bipartite maps with $n$ edges. Our approach is recursive, and uses a ``catalytic'' parameter, namely the length of the final descent of the upper path of the interval. The resulting bivariate generating function is algebraic for all $m$. We show that the same approach can be used to count intervals in the ordinary $m$-Tamari lattices as well. We thus recover the earlier result of the first author, Fusy and Pr\'eville-Ratelle, who were using a different catalytic parameter.Comment: 23 page

Unified bijections for maps with prescribed degrees and girth

This article presents unified bijective constructions for planar maps, with control on the face degrees and on the girth. Recall that the girth is the length of the smallest cycle, so that maps of girth at least $d=1,2,3$ are respectively the general, loopless, and simple maps. For each positive integer $d$, we obtain a bijection for the class of plane maps (maps with one distinguished root-face) of girth $d$ having a root-face of degree $d$. We then obtain more general bijective constructions for annular maps (maps with two distinguished root-faces) of girth at least $d$. Our bijections associate to each map a decorated plane tree, and non-root faces of degree $k$ of the map correspond to vertices of degree $k$ of the tree. As special cases we recover several known bijections for bipartite maps, loopless triangulations, simple triangulations, simple quadrangulations, etc. Our work unifies and greatly extends these bijective constructions. In terms of counting, we obtain for each integer $d$ an expression for the generating function $F_d(x_d,x_{d+1},x_{d+2},...)$ of plane maps of girth $d$ with root-face of degree $d$, where the variable $x_k$ counts the non-root faces of degree $k$. The expression for $F_1$ was already obtained bijectively by Bouttier, Di Francesco and Guitter, but for $d\geq 2$ the expression of $F_d$ is new. We also obtain an expression for the generating function \G_{p,q}^{(d,e)}(x_d,x_{d+1},...) of annular maps with root-faces of degrees $p$ and $q$, such that cycles separating the two root-faces have length at least $e$ while other cycles have length at least $d$. Our strategy is to obtain all the bijections as specializations of a single "master bijection" introduced by the authors in a previous article. In order to use this approach, we exhibit certain "canonical orientations" characterizing maps with prescribed girth constraints

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

Combinatorics of the Permutahedra, Associahedra, and Friends

I present an overview of the research I have conducted for the past ten years in algebraic, bijective, enumerative, and geometric combinatorics. The two main objects I have studied are the permutahedron and the associahedron as well as the two partial orders they are related to: the weak order on permutations and the Tamari lattice. This document contains a general introduction (Chapters 1 and 2) on those objects which requires very little previous knowledge and should be accessible to non-specialist such as master students. Chapters 3 to 8 present the research I have conducted and its general context. You will find: * a presentation of the current knowledge on Tamari interval and a precise description of the family of Tamari interval-posets which I have introduced along with the rise-contact involution to prove the symmetry of the rises and the contacts in Tamari intervals; * my most recent results concerning q, t-enumeration of Catalan objects and Tamari intervals in relation with triangular partitions; * the descriptions of the integer poset lattice and integer poset Hopf algebra and their relations to well known structures in algebraic combinatorics; * the construction of the permutree lattice, the permutree Hopf algebra and permutreehedron; * the construction of the s-weak order and s-permutahedron along with the s-Tamari lattice and s-associahedron. Chapter 9 is dedicated to the experimental method in combinatorics research especially related to the SageMath software. Chapter 10 describes the outreach efforts I have participated in and some of my approach towards mathematical knowledge and inclusion.Comment: 163 pages, m\'emoire d'Habilitation \`a diriger des Recherche

Enumerative Combinatorics of Intervals in the Dyck Pattern Poset

We initiate the study of the enumerative combinatorics of the intervals in the Dyck pattern poset. More specifically, we find some closed formulas to express the size of some specific intervals, as well as the number of their covering relations. In most of the cases, we are also able to refine our formulas by rank. We also provide the first results on the Möbius function of the Dyck pattern poset, giving for instance a closed expression for the Möbius function of initial intervals whose maximum is a Dyck path having exactly two peaks