The asymptotic number of weighted partitions with a given number of parts

Abstract

AbstractFor a given sequence $$b_k$$ b k of non-negative real numbers, the number of weighted partitions of a positive integer n having m parts $$c_{n,m}$$ c n , m has bivariate generating function equal to $$\prod _{k=1}^\infty (1-yz^k)^{-b_k}$$ ∏ k = 1 ∞ ( 1 - y z k ) - b k . Under the assumption that $$b_k\sim Ck^{r-1}$$ b k ∼ C k r - 1 , $$r>0$$ r > 0 , and related conditions on the Dirichlet generating function of the weights $$b_k$$ b k , we find asymptotics for $$c_{n,m}$$ c n , m when $$m=m(n)$$ m = m ( n ) satisfies $$m=o\left( n^\frac{r}{r+1}\right) $$ m = o n r r + 1 and $$\lim _{n\rightarrow \infty }m/\log ^{3+\epsilon }n=\infty $$ lim n → ∞ m / log 3 + ϵ n = ∞ , $$\epsilon >0$$ ϵ > 0 .