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Theodore Kolokolnikov

2009-06-04
14:10:00 - 15:10:00

Combinatorics and PDEs

308 , Mathematics Research Center Building (ori. New Math. Bldg.)

Let a(n) denote the number of sign choices + and - such that +/-1 +/- 2 +/- 3 +- ・ ・ ・+/- n = 0. For example when n=3 we have 1+2-3=0 and -1-2+3=0 so a(3)=2. We are interested to know how a(n) grows as a function of n. In the limit of large n, we will derive an asymptotic formula for a(n) by using the fundamental solution of the heat equation. We will also investigate a more general question: given integers n,m, let b(m,n) be the number of partitions of the set {0, 1, 2, ..., n} that add up to m. We derive an asymptotic formula for b(m,n) when n>>1 and m=O(n^2).