Contact Info
Combinatorics & Optimization
University of Waterloo
Waterloo, Ontario
Canada N2L 3G1
Phone: 519-888-4567, ext 33038
PDF files require Adobe Acrobat Reader.
Title: Counting Partitions Inside a Rectangle
Speaker: | Steve Melczer |
Affiliation: | University of Pennsylvania |
Room: | MC 6486 |
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.
Combinatorics & Optimization
University of Waterloo
Waterloo, Ontario
Canada N2L 3G1
Phone: 519-888-4567, ext 33038
PDF files require Adobe Acrobat Reader.
The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land granted to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is co-ordinated within the Office of Indigenous Relations.