# 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_n^{m}, 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 (labelled) intervals of the m-Tamari lattice, which we study in this paper. An interval [P,Q] of T_n^{m} is labelled if the north steps of Q are labelled 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 labelled intervals of T_n^{m} 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 labelled m-Tamari intervals of size n, is found to be $(m+1)^n(mn+1)^{n-2}.$ 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. The form of this equation is highly non-standard: it 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.
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https://hal.archives-ouvertes.fr/hal-00674492
Contributor : Mireille Bousquet-Mélou Connect in order to contact the contributor
Submitted on : Friday, March 23, 2012 - 9:57:05 AM
Last modification on : Monday, December 20, 2021 - 4:50:11 PM
Long-term archiving on: : Sunday, June 24, 2012 - 2:21:24 AM

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### Identifiers

• HAL Id : hal-00674492, version 2
• ARXIV : 1202.5925

### Citation

Mireille Bousquet-Mélou, Guillaume Chapuy, Louis-François Préville-Ratelle. The representation of the symmetric group on m-Tamari intervals. 2012. ⟨hal-00674492v2⟩

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