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

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