Algebraic Combinatorics Seminar

Title: Counting Partitions Inside a Rectangle

Abstract:

The study of integer partitions is a classic subject with applications ranging from number theory to representation theory and combinatorics. More recently, interest in the subject has been driven by a connection to Kronecker coefficients, which are at the heart of certain geometric complexity theoretic approaches to resolving P vs NP.  This talk examines the number N_n(l,m) of integer partitions of n fitting inside an m × l rectangle, equal to the q^n coefficient of the q-binomial (m+l,l)_q. We give an exact asymptotic formula when l = Am and n = Bm^2 for constants A and B; our approach uses a carefully chosen probability distribution on partitions to apply a local central limit theorem.

In 1856, Cayley conjectured that for fixed l and m the sequence N_n(l,m) is unimodal in n (i.e., increases until hitting a maximum at n=lm/2 and then decreases). This was proven by Sylvester in 1878 via the representation theory of sl_2, however the sequence was not known to be strictly unimodal until 2013, when Panova and Pak used the fact that consecutive differences of terms in the sequence are Kronecker coefficients. We give the first asymptotic proof of unmodality, and the first effective bounds for consecutive differences of terms.

Joint work with Greta Panova and Robin Pemantle.