Combinatorial Theory

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 Bousquet-Mélou, Fusy and Préville-Ratelle, who were using a different catalytic parameter.

Mathematics Subject Classifications: 05A15, 06A07, 06A11

Keywords: Tamari posets, planar maps, enumeration, algebraic generating functions