Partitions of n into tn parts

Szekeres proved, using complex analysis, an asymptotic formula for the number of partitions of n into at most k parts. Canfield discovered a simplification of the formula, and proved it without complex analysis. We re-prove the formula, in the asymptotic regime when k is at least a constant times n, by showing that it is equivalent to a local central limit theorem in Fristedt’s model for random partitions. We then apply the formula to derive asymptotics for the number of minimal difference d partitions with a given number of parts. As a corollary, we find (explicitly computable) constants cd,βd,γd,σd such that the number of minimal difference d partitions of n is (1+o(1))cdn−3/4exp(βdn) (a result of Meinardus), almost all of them (fraction a(1+o(1))) have approximately γdn parts, and the distribution of the number of parts in a random such partition is asymptotically normal with standard deviation (1+o(1))σdn1/4. In particular, γ2=15log[(1+5)/2]/π.