Intervals in the greedy Tamari posets

Abstract

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.