Two Problems in Asymptotic Combinatorics

Combinatorics Seminar
Friday, April 2, 2010 - 15:05
1 hour (actually 50 minutes)
Skiles 255
Professor, University of Georgia, Athens, GA
I will divide the talk between two topics. The first is Stirling numbers of the second kind, $S(n,k)$.  For each $n$ the maximum $S(n,k)$ is achieved either at a unique $k=K_n$, or is achieved twice consecutively at $k=K_n,K_n+1$.  Call those $n$ of the second kind {\it exceptional}.  Is $n=2$ the only exceptional integer? The second topic is $m\times n$ nonnegative integer matrices all of whose rows sum to $s$ and all of whose columns sum to $t$, $ms=nt$.  We have an asymptotic formula for the number of these matrices, valid for various ranges of $(m,s;n,t)$.  Although obtained by a lengthy calculation, the final formula is succinct and has an interesting probabilistic interpretation.  The work presented here is collaborative with Carl Pomerance and Brendan McKay, respectively.