From generalized Tamari intervals to non-separable planar maps

Wenjie Fang (1, 2) , Louis-François Préville-Ratelle (3)

1 IRIF (UMR_8243) - Institut de Recherche en Informatique Fondamentale
2 LaBRI - Laboratoire Bordelais de Recherche en Informatique
3 Instituto de Matemática y Física - Universidad de Talca

Let v be a grid path made of north and east steps. The lattice TAM(v), based on all grid paths weakly above the grid path v sharing the same endpoints as v, was introduced by Pre ́ville-Ratelle and Viennot (2014) and corresponds to the usual Tamari lattice in the case v = (NE)n. They showed that TAM(v) is isomorphic to the dual of TAM(←−v ), where ←−v is the reverse of v with N and E exchanged. Our main contribution is a bijection from intervals in TAM(v) to non-separable planar maps. It follows that the number of intervals in TAM(v) over all v of length n is 2(3n+3)! (n+2)!(2n+3)! . This formula was first obtained by Tutte(1963) for non-separable planar maps.

Domaines: Combinatoire [math.CO]

Wenjie Fang, Louis-François Préville-Ratelle. From generalized Tamari intervals to non-separable planar maps. 28-th International Conference on Formal Power Series and Algebraic Combinatorics, Simon Fraser University, Jul 2016, Vancouver, Canada. ⟨10.46298/dmtcs.6421⟩. ⟨hal-02173377⟩