arXiv subjects
Publications and source records attributed to Clément Chenevière.
We show that the rowmotion operator from dynamical algebraic combinatorics behaves the same on all alt $ν$-Tamari lattices for a fixed lattice path $ν$. We use this invariance of rowmotion to establish cyclic sieving and homomesy results for rational Tamari lattices. We also conjecture that this invariance extends to the more general cross Tamari lattices.
This document is an extended abstract for two articles in preparation. Recently, framing lattices were introduced to generalize many classical lattices such as the Tamari lattice and the weak order on the symmetric group. We define bricks and brick cliques as a combinatorial model for join-irreducible elements and canonical join representations in all framing lattices, generalizing noncrossing arc diagrams of (Reading, 2015) for the weak order on the symmetric group. Our model captures the natural bijection between join and meet canonical representations, as well as duality upon reflections of the framed graph. The proof is bijective, with an explicit reconstruction algorithm in two steps. A useful intermediate construction in our bijective proof is our new definition of cornered cliques. These enable us to define cubical coordinates on a framing lattice, generalizing bracket vectors for the Tamari lattice and providing an efficient comparison criterion.
The Cambrian lattices, introduced in (Reading, 2006), generalize the Tamari lattice to any choice of Coxeter element in any finite Coxeter group. They are further generalized to the m-Cambrian lattices (Stump, Thomas, Williams, 2015). However, their definitions do not provide a practical setup to work with combinatorially. In this paper, we provide a new equivalent definition of the m-Cambrian lattices on simple objects called m-noncrossing partitions, using a simple and effective comparison criterion. It is obtained by showing that each interval has a unique maximal chain that is c-increasing, which is computed by a greedy algorithm. Our proof is uniform, involving all Coxeter groups and all choices of Coxeter element at the same time. This work has been accepted as an extended abstract for the FPSAC 2025 conference. A long version of this work will be available later.
Given a lattice path $ν$, the $ν$-Tamari lattice and the $ν$-Dyck lattice are two natural examples of partial order structures on the set of lattice paths that lie weakly above $ν$. In this paper, we introduce a more general family of lattices, called alt $ν$-Tamari lattices, which contains these two examples as particular cases. Unexpectedly, we show that all these lattices have the same number of linear intervals.
We count the number of linear intervals in the Tamari and the Dyck lattices according to their height, using generating series and Lagrange inversion. Surprisingly, these numbers are the same in both lattices. We define a new family of posets on Dyck paths, which we call alt-Tamari posets. Each alt-Tamari poset depends on the choice of an increment function delta in {0,1}^n. We recover the Tamari and the Dyck lattices as extreme cases with delta = 1 and delta = 0, respectively. We prove that all the alt-Tamari posets have the same number of linear intervals of any given height.