An m-ballot path of size n is a path on the square grid consisting of north and east 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 Tn obtained when m = 1. We prove that the number of intervals in this lattice is m + 1 n(mn + 1) (m + 1) 2 n + m n − 1 .This formula was recently conjectured by Bergeron in connection with the study of diagonal coinvariant spaces. The case m = 1 was proved a few years ago by Chapoton. Our proof is based on a recursive description of intervals, which translates into a functional equation satisfied by the associated generating function. The solution of this equation is an algebraic series, obtained by a guess-and-check approach. Finding a bijective proof remains an open problem.

Abstract. We consider the graded S n -modules of higher diagonally harmonic polynomials in three sets of variables (the trivariate case), and show that they have interesting ties with generalizations of the Tamari poset and parking functions. In particular we get several nice formulas for the associated Hilbert series and graded Frobenius characteristics. This also leads to entirely new combinatorial formulas.

Let $v$ be a grid path made of north and east steps. The lattice $\rm{T{\scriptsize AM}}(v)$, based on all grid paths weakly above $v$ and sharing the same endpoints as $v$, was introduced by Pr\'eville-Ratelle and Viennot (2014) and corresponds to the usual Tamari lattice in the case $v=(NE)^n$. Our main contribution is that the enumeration of intervals in $\rm{T{\scriptsize AM}}(v)$, over all $v$ of length $n$, is given by $\frac{2 (3n+3)!}{(n+2)! (2n+3)!}$. This formula was first obtained by Tutte(1963) for the enumeration of non-separable planar maps. Moreover, we give an explicit bijection from these intervals in $\rm{T{\scriptsize AM}}(v)$ to non-separable planar maps.Comment: 19 pages, 11 figures. Title changed, originally titled "From generalized Tamari intervals to non-separable planar maps (extended abstract)", submitte

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.

For any finite path v v on the square grid consisting of north and east unit steps, starting at (0,0), we construct a poset Tam ( v ) (v) that consists of all the paths weakly above v v with the same number of north and east steps as v v . For particular choices of v v , we recover the traditional Tamari lattice and the m m -Tamari lattice. Let v ← \overleftarrow {v} be the path obtained from v v by reading the unit steps of v v in reverse order, replacing the east steps by north steps and vice versa. We show that the poset Tam ( v ) (v) is isomorphic to the dual of the poset Tam ( v ← ) (\overleftarrow {v}) . We do so by showing bijectively that the poset Tam ( v ) (v) is isomorphic to the poset based on rotation of full binary trees with the fixed canopy v v , from which the duality follows easily. This also shows that Tam ( v ) (v) is a lattice for any path v v . We also obtain as a corollary of this bijection that the usual Tamari lattice, based on Dyck paths of height n n , can be partitioned into the (smaller) lattices Tam ( v ) (v) , where the v v are all the paths on the square grid that consist of n − 1 n-1 unit steps. We explain possible connections between the poset Tam ( v ) (v) and (the combinatorics of) the generalized diagonal coinvariant spaces of the symmetric group.