Dyson's conjecture on asymptotics for partition cranks
Speaker: Jehanne Dousse (Paris 7)
Time: 4.00PM
Date: Monday April 28th 2014
Location: Mathematics Seminar Room, Room AG 1.01, First Floor, Agriculture Building, UCD Belfield
Abstract:
Dyson's crank was introduced to explain Ramanujan's famous partition congruences p(5n+4) = 0 mod 5, p(7n+5) = 0 mod 7, and p(11n+6) = 0 mod 11 combinatorially. If for a partition lambda, o(lambda) denotes the number of ones in lambda, and l(lambda) denotes the number of parts strictly larger than o(lambda), then the crank of lambda is defined as the largest part of lambda if o(lambda)=0 and l(lambda)-o(lambda) if o(lambda)>0. Let M(m,n) denote the number of partitions of n with crank m. In 1987, Dyson conjectured the following asymptotic equivalent when n and m both go to infinity: M(m,n) ~ 1/4 \beta sech²(\beta m) p(n), where \beta = \pi / (6n)^{1/2}, but didn't conjecture the exact condition of validity on n and m. First I will give some background on the Hardy-Ramanujan circle method and its variant due to Wright which are useful to prove the asymptotic equivalent of the number of partitions of n, and then I will show how extending this method to two variables allows to prove Dyson's conjecture.
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