The representation of the symmetric group on m -Tamari intervals

Abstract

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, called the m-Tamari lattice and denoted by T (m) n , which generalizes the usual Tamari lattice T n obtained when m = 1. This lattice was introduced by F. Bergeron in connection with the study of diagonal coinvariant spaces in three sets of n variables. The representation of the symmetric group S n on these spaces is conjectured to be closely related to the natural representation of S n on (labeled) intervals of the m-Tamari lattice, which we study in this paper.Anis labeled if the north steps of Q are labeled from 1 to n in such a way the labels increase along any sequence of consecutive north steps. The symmetric group S n acts on labeled intervals of T (m) n by permutation of the labels. We prove an explicit formula, conjectured by F. Bergeron and the third author, for the character of the associated representation of S n . In particular, the dimension of the representation, that is, the number of labeled m-Tamari intervals of size n, is found to be ✩ G. 310M. Bousquet-Mélou et al. / Advances in Mathematics 247 (2013) (m + 1) n (mn + 1) n−2 .These results are new, even when m = 1.The form of these numbers suggests a connection with parking functions, but our proof is not bijective. The starting point is a recursive description of m-Tamari intervals. It yields an equation for an associated generating function, which is a refined version of the Frobenius series of the representation. This equation involves two additional variables x and y, a derivative with respect to y and iterated divided differences with respect to x. The hardest part of the proof consists in solving it, and we develop original techniques to do so, partly inspired by previous work on polynomial equations with "catalytic" variables.

Combinatorial mappings of exclusion processes

“…It is also 'nearly' a lowerdiagonal matrix, and a simple example of a Hessenberg matrix. Taking (55), this gives a simplified recursive determinant formula (adapted from Theorem 2.1 of [33]):…”

“…The enumeration of these upper quadrant walks for a general step set is is a wellresearched topic, and a technique known as the kernel method finds the generating function for most walks [55]. The step set we are dealt with here proves to be one of the more stubborn, and is solved by a more involved obstinate kernel method.…”

“…The step set we are dealt with here proves to be one of the more stubborn, and is solved by a more involved obstinate kernel method. While the details of the calculation are beyond the scope of this review, the symmetry of the sixstep walk is exploited in order to solve what turns out to be a two-parameter recursion relation of the generating function [55].…”

A simplicial complex splitting associativity

“…We describe basic notions bout m-Dyck paths. For more detailed constructions and the proofs of the results we refer to [7], [8] and [9]. Definition 4.1.1.…”

Chains of maximum length in the Tamari lattice

“…[BFP11] show that T n (m) is a lattice and is in fact isomorphic to a sublattice of T nm . At the top of the lattice is the empty partition and at the bottom is the partition ((n − 1)m, .…”