##
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:

- the lattice property of Yn, and
- 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.