Tamari Lattice — binary trees, transfer systems, noncrossing partitions

Hover an element

Drag to pan, scroll to zoom. Hover an element to see its covers; click to pin it. The elements are the C_n+1 full binary trees with n+2 leaves, equivalently the transfer systems on the chain [n], equivalently the noncrossing partitions of [n+1]; the three buttons redraw the same set without moving anything. Trees and transfer systems carry the Tamari order: a right rotation (xy)z → x(yz) on trees is exactly adding relations to a transfer system, so those two drawings share one Hasse diagram. Noncrossing partitions carry the Kreweras order, refinement: coarser than Tamari (every Kreweras relation is a Tamari relation, not conversely), so the vertices stay put and only the edges change. The Hasse diagram of the Tamari lattice is the 1-skeleton of the associahedron, and that is how it is drawn: each tree sits at its vertex of Loday’s associahedron (the i-th coordinate is the product of the leaf counts of the two subtrees at the i-th node in in-order), projected to the plane by a vertical functional that every rotation increases and a horizontal one chosen to spread the vertices. For n = 2 that is the pentagon, for n = 3 the three-dimensional associahedron with its six pentagons and three squares; beyond that the picture is a shadow of a higher-dimensional polytope. Color is the number of blocks, which is also the number of peaks of the Dyck path and the number of tree nodes whose right child is a leaf.

Made for Math 372: Combinatorics at Reed College, fall 2026. The C_n+1 elements are drawn three ways: as full binary trees with n+2 leaves, as transfer systems on the chain [n] (Balchin, Barnes and Roitzheim, N_∞-operads and associahedra, showed these form the Tamari lattice under inclusion), and as noncrossing partitions of [n+1]. The first two carry Tamari's order, rotation (Tamari 1962; the lattice property is Huang–Tamari 1972); the third carries Kreweras' refinement order (1971), which is coarser: Bernardi and Bonichon (Intervals in Catalan lattices and realizers of triangulations, 2009) fix the bijections used here, under which every Kreweras relation is a Tamari relation and every Tamari relation is a Stanley relation on Dyck paths. The layout is a planar projection of Loday’s realization of the associahedron (Realization of the Stasheff polytope, 2004), whose 1-skeleton, oriented by a suitable linear functional, is the Hasse diagram of the Tamari lattice. Part of e-infinity.space/viz.