Séminaire Lotharingien de Combinatoire, B79d (2020), 23 pp.
Baptiste Rognerud
Exceptional and Modern Intervals of the Tamari Lattice
Abstract.
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.
Received: January 11, 2018.
Revised: March 12, 2019.
Accepted: June 7, 2020.
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