## A proof-theoretic analysis of the rotation lattice of binary trees

### Noam Zeilberger

The classical Tamari lattice Yn is defined as the set of binary trees with n internal nodes, with the partial ordering induced by the (right) rotation operation. It is not obvious why Yn is a lattice, but this was first proved by Haya Friedman and Dov Tamari in the late 1950s. More recently, Frédéric Chapoton discovered another surprising fact about the rotation ordering, namely that Yn contains exactly 2(4n+1)! / ((n+1)!(3n+2)!) pairs of related trees. Even more surprisingly, the same formula was already computed by Bill Tutte in the 1960s but in an entirely different context, namely enumeration of triangulations of the plane! And...this is starting to get spooky...the same formula also counts a certain natural family of lambda terms!??

In the talk I will describe a new way of looking at the rotation ordering that is motivated by all this and some old ideas in proof theory. This will lead us to systematic ways of thinking about:

1. the lattice property of Yn, and
2. the Tutte-Chapoton formula for the number of intervals in Yn.
No advanced background in either proof theory or combinatorics will be assumed.

Based on the paper "A sequent calculus for a semi-associative law", LMCS 15:1, https://lmcs.episciences.org/5167.