Partitions with an odd number of standard tableaux

The set of all integer partitions of all integers can be arranged into a graph known as Young's lattice; a partition of n is connected by an edge to a partition of n+1 if the latter can be obtained from the former by increasing one of its parts by one.

A fundamental invariant of a partition is its f-statistic, which is the number of geodesic paths from the partition in question to the unique partition of 1 in Young's lattice. This statistic also appears as the dimension of the irreducible representation of a symmetric group that is associated to the partition.

Using the idea of the core and quotient of a partition, Macdonald showed how to enumerate partitons with odd f-statistic in 1971. In this talk I will announce a surprising result about how these partitions sit in Young's lattice: they form a binary tree.

This talk is based on ongoing collaboration with Arvind Ayyer and Steven Spallone.