Exceptional and Modern Intervals of the Tamari Lattice

Baptiste Rognerud

Séminaire lotharingien de combinatoire, Tome 79 (2018-2023)

Résumé

In this article we use the theory of interval-posets recently introduced by Châtel and Pons in order to describe some interesting families of intervals in the Tamari lattices. These families are defined as interval-posets avoiding specific configurations. At first, we consider exceptional interval-posets and we show that they correspond to the intervals which are obtained as images of non-crossing trees in the Dendriform operad. We also show that the exceptional intervals are exactly the intervals of the Tamari lattices induced by intervals in the posets of non-crossing partitions. In the second part, we introduce the notion of modern and infinitely modern interval-posets. We show that the modern intervals are in bijection with the new intervals of the Tamari lattices in the sense of Chapoton. Finally, we consider the family of infinitely modern intervals and we prove that there are as many infinitely modern interval-posets of size n as there are ternary trees with n inner vertices.