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

对于给定的非负实数序列\（b_k \），具有m个部分\（c_ {n，m} \）的正整数n的加权分区数具有等于\（\ prod _的双变量生成函数{k = 1} ^ \ infty（1-yz ^ k）^ {-b_k} \）。假设\（b_k \ sim Ck ^ {r-1} \），\（r> 0 \）以及权重\（b_k \）的Dirichlet生成函数的相关条件，我们找到\（b c_ {n，m} \）当\（m = m（n）\）满足\（m = o \ left（n ^ \ frac {r} {r + 1} \ right）\）和\（\ lim _ {n \ rightarrow \ infty} m / \ log ^ {3+ \ epsilon} n = \ infty \），\（\ epsilon> 0 \）。