相关论文: Catalan's intervals and realizers of triangulation…

Using the notion of series parallel interval order, we propose a unified setting to describe Dyck lattices and Tamari lattices (two well known lattice structures on Catalan objects) in terms of basic notions of the theory of posets. As a…

We present a direct bijection between planar 3-connected triangulations and bridgeless planar maps, which were first enumerated by Tutte (1962) and Walsh and Lehman (1975) respectively. Previously known bijections by Wormald (1980) and Fusy…

The Tamari lattice, defined on Catalan objects such as binary trees and Dyck paths, is a well-studied poset in combinatorics. It is thus natural to try to extend it to other families of lattice paths. In this article, we fathom such a…

The dual of a map is a fundamental construction on combinatorial maps, but many other combinatorial objects also possess their notion of duality. For instance, the Tamari lattice is isomorphic to its order dual, which induces an involution…

In this paper, we exploit the combinatorics and geometry of triangulations of products of simplices to derive new results in the context of Catalan combinatorics of $\nu$-Tamari lattices. In our framework, the main role of "Catalan objects"…

We introduce a simple bijection between Tamari intervals and the blossoming trees (Poulalhon and Schaeffer, 2006) encoding planar triangulations, using a new meandering representation of such trees. Its specializations to the families of…

There are three classical lattices on the Catalan numbers: the Tamari lattice, the lattice of noncrossing partitions and the lattice of Dyck paths. The first is known to be isomorphic to the lattice of torsion classes of the path algebra of…

The usual, or type A_n, Tamari lattice is a partial order on T_n^A, the triangulations of an (n+3)-gon. We define a partial order on T_n^B, the set of centrally symmetric triangulations of a (2n+2)-gon. We show that it is a lattice, and…

Stanley considered Dyck paths where each maximal run of down-steps to the $x$-axis has odd length; they are also enumerated by (shifted) Catalan numbers. Prefixes of these combinatorial objects are enumerated using the kernel method. A more…

We introduce $\delta$-cliffs, a generalization of permutations and increasing trees depending on a range map $\delta$. We define a first lattice structure on these objects and we establish general results about its subposets. Among them, we…

We introduce a new poset structure on Dyck paths where the covering relation is a particular case of the relation inducing the Tamari lattice. We prove that the transitive closure of this relation endows Dyck paths with a lattice structure.…

We use a recently introduced combinatorial object, the interval-poset, to describe two bijections on intervals of the Tamari lattice. Both bijections give a combinatorial proof of some previously known results. The first one is an inner…

Generalized Tamari intervals have been recently introduced by Pr\'eville-Ratelle and Viennot, and have been proved to be in bijection with (rooted planar) non-separable maps by Fang and Pr\'eville-Ratelle. We present two new bijections…

The Tamari order is a central object in algebraic combinatorics and many other areas. Defined as the transitive closure of an associativity law, the Tamari order possesses a surprisingly rich structure: it is a congruence-uniform lattice.…

An m-ballot path of size n is a path on the square grid consisting of north and east unit steps, starting at (0,0), ending at (mn,n), and never going below the line {x=my. The set of these paths can be equipped with a lattice structure,…

In this paper, we explain how the classical Catalan families of objects involving paths, tableaux, triangulations, parentheses configurations and more generalize canonically to a three-dimensional version. In particular, we present…

In the Stanley lattice defined on Dyck paths of size $n$, cover relations are obtained by replacing a valley $DU$ by a peak $UD$. We investigate a greedy version of this lattice, first introduced by Chenevi\`ere, where cover relations…

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…

Given a lattice path $\nu$, the $\nu$-Tamari lattice and the $\nu$-Dyck lattice are two natural examples of partial order structures on the set of lattice paths that lie weakly above $\nu$. In this paper, we introduce a more general family…

This paper develops a Cambrian extension of the work of C. Ceballos, A. Padrol and C. Sarmiento on $\nu$-Tamari lattices and their tropical realizations. For any signature $\varepsilon \in \{\pm\}^n$, we consider a family of…