Noam Zeilberger, University of Birmingham

Abstract

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 (via the OEIS) 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, this formula was already computed by Bill Tutte in the early 1960s...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. (Based on this paper)