Decomposition of Greedy Tamari Intervals and Bipartite Planar Maps

Abstract

The greedy Tamari poset, inspired by the well-studied Tamari lattice, was recently defined by Dermenjian in the more general setting of greedy $ν$-Tamari posets. Bousquet-Mélou and Chapoton counted intervals of the greedy $m$-Tamari poset in 2024 by solving a functional equation, and found that they are equi-enumerous to planar $(m+1)$-constellations. In this work, we give a combinatorial proof of this fact for the case $m = 1$, which also gives the refined enumeration conjectured by Bousquet-Mélou and Chapoton. This is done by establishing a recursive decomposition of greedy Tamari intervals isomorphic to that of bipartite planar maps. We also propose a more general and refined conjecture for the case of general $m$.