Counting smaller trees in the Tamari order

Grégory Chatel (1) , Viviane Pons (1)

1 LIGM - Laboratoire d'Informatique Gaspard-Monge

We introduce new objects, the interval-posets, that encode intervals of the Tamari lattice. We then find a combinatorial interpretation of the bilinear form 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 form on each tree $T$ counts the number of trees smaller than $T$ in the Tamari order.

Mots clés: binary trees; Tamari lattice; Tamari intervals

Domaines: Mathématique discrète [cs.DM]; Combinatoire [math.CO]

Grégory Chatel, Viviane Pons. Counting smaller trees in the Tamari order. FPSAC 2013, Jun 2013, Paris, France. pp.463-474. ⟨hal-00824154v1⟩