Counting smaller elements in the Tamari and m-Tamari lattices

We introduce new combinatorial objects, the interval- posets, that encode intervals of the Tamari lattice. We then find a combinatorial interpretation of the bilinear operator that appears in the functional equation of Tamari intervals described by Chapoton. Thus, we retrieve this functional equation and prove that the polynomial recursively computed from the bilinear operator on each tree T counts the number of trees smaller than T in the Tamari order. Then we show that a similar m + 1-linear operator is also used in the functionnal equation of m-Tamari intervals. We explain how the m-Tamari lattices can be interpreted in terms of m+1-ary trees or a certain class of binary trees. We then use the interval-posets to recover the functional equation of m-Tamari intervals and to prove a generalized formula that counts the number of elements smaller than or equal to a given tree in the m-Tamari lattice.Comment: 46 pages + 3 pages of code appendix, 27 figures. Long version of arXiv:1212.0751. To appear in Journal of Combinatorial Theory, Series

The number of intervals in the m-Tamari lattices

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, which generalizes the usual Tamari lattice obtained when m=1. We prove that the number of intervals in this lattice is $$\frac {m+1}{n(mn+1)} {(m+1)^2 n+m\choose n-1}.$$ This formula was recently conjectured by Bergeron in connection with the study of 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.Comment: 19 page

Intervals in the greedy Tamari posets

We consider a greedy version of the $m$-Tamari order defined on $m$-Dyck paths, recently introduced by Dermenjian. Inspired by intriguing connections between intervals in the ordinary 1-Tamari order and planar triangulations, and more generally by the existence of simple formulas counting intervals in the ordinary $m$-Tamari orders, we investigate the number of intervals in the greedy order on $m$-Dyck paths of fixed size. We find again a simple formula, which also counts certain planar maps (of prescribed size) called $(m+1)$-constellations. For instance, when $m=1$ the number of intervals in the greedy order on 1-Dyck paths of length $2n$ is proved to be $\frac{3\cdot 2^{n-1}}{(n+1)(n+2)} \binom{2n}{n}$, which is also the number of bipartite maps with $n$ edges. Our approach is recursive, and uses a ``catalytic'' parameter, namely the length of the final descent of the upper path of the interval. The resulting bivariate generating function is algebraic for all $m$. We show that the same approach can be used to count intervals in the ordinary $m$-Tamari lattices as well. We thus recover the earlier result of the first author, Fusy and Pr\'eville-Ratelle, who were using a different catalytic parameter.Comment: 23 page

An extension of Tamari lattices

For any finite path $v$ on the square grid consisting of north and east unit steps, starting at (0,0), we construct a poset Tam$(v)$ that consists of all the paths weakly above $v$ with the same number of north and east steps as $v$. For particular choices of $v$, we recover the traditional Tamari lattice and the $m$-Tamari lattice. Let $\overleftarrow{v}$ be the path obtained from $v$ by reading the unit steps of $v$ in reverse order, replacing the east steps by north steps and vice versa. We show that the poset Tam$(v)$ is isomorphic to the dual of the poset Tam$(\overleftarrow{v})$. We do so by showing bijectively that the poset Tam$(v)$ is isomorphic to the poset based on rotation of full binary trees with the fixed canopy $v$, from which the duality follows easily. This also shows that Tam$(v)$ is a lattice for any path $v$. We also obtain as a corollary of this bijection that the usual Tamari lattice, based on Dyck paths of height $n$, is a partition of the (smaller) lattices Tam$(v)$, where the $v$ are all the paths on the square grid that consist of $n-1$ unit steps. We explain possible connections between the poset Tam$(v)$ and (the combinatorics of) the generalized diagonal coinvariant spaces of the symmetric group.Comment: 18 page

Geometric realizations of Tamari interval lattices via cubic coordinates

We introduce cubic coordinates, which are integer words encoding intervals in the Tamari lattices. Cubic coordinates are in bijection with interval-posets, themselves known to be in bijection with Tamari intervals. We show that in each degree the set of cubic coordinates forms a lattice, isomorphic to the lattice of Tamari intervals. Geometric realizations are naturally obtained by placing cubic coordinates in space, highlighting some of their properties. We consider the cellular structure of these realizations. Finally, we show that the poset of cubic coordinates is shellable

The representation of the symmetric group on m-Tamari intervals

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. 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.Comment: 29 pages --- This paper subsumes the research report arXiv:1109.2398, which will not be submitted to any journa