Title: A George Szekeres formula for restricted partitions
|Affiliation:||University of Waterloo|
We give asymptotic formulas for the number of integer partitions of n with at most j parts and having largest part at most r, say A(n,j,r). We do this also for the number of partitions of n with largest part equal to r and having exactly j parts, say C(n,j,r). The rank of a partition, defined by F. J. Dyson, is the largest part minus the number of parts. The k-th rank of a partition, defined by A. O. L. Atkin, is the k-th largest part minus the number of parts greater than or equal to k. A graphical partition of of an even integer n is a partition of n whose parts form the degree sequence of a simple graph. We show that the number of graphical partitions of n is bounded by a constant times one over the logarithm of n to the power 1/2. B. Pittel showed that the fraction of partitions of n that are graphical tends to zero with n but did not give an estimate of how quickly this fraction goes to zero. The George Szekeres circle method for partitions is crucial for the estimation of A(n,j,r) and C(n,j,r). The Nash-Williams' characterization of graphical partitions is also crucial. The number of partitions of n having rank r was estimated by P. Erdős and myself, it is now possible to estimate the number of partitions of n that have k-th rank equal to r. The techniques used will only be briefly sketched and some comments will be made describing collaborating with P. Erdős and G. Szekeres. In particular both Erdős and Szekeres had strong social consciences.