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

-- Noam Zeilberger (Birmingham University)

Time: 14:00 -- Location: Salle Poincaré du LIX

Join seminar with the Parsifal team

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 surprising was how Chapoton discovered this formula: via the Online Encyclopedia of Integer Sequences, because the formula had already been computed in the early 1960s by Bill Tutte, but for a completely different family of objects!)

In the talk I will describe a new way of looking at the rotation ordering that is motivated by 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 following paper: https://arxiv.org/abs/1803.10080.)

Category: seminars
Tags: Combi seminar combinatorics