For integers 1≤m≤n, let s(n,m) denote the number of m-tuples (k1,…,km)
Nearby in the stack
of nonnegative integers satisfying
n=j=1∑mjkj.
We obtain a complete asymptotic expansion for
logs(n,m)
, uniformly for all
n≥m
as
m→∞
. The coefficients are given explicitly in terms of limiting prime-block functions arising from the residue structure of the problem. We also determine the full hierarchy of multiplicative corrections in the sparse regime, identifying explicit constants at every fixed order and, in particular, the first correction constants
1/4
and
(1−log2)/4
. The results give uniform two-parameter asymptotics as both the target and the number of allowed reciprocal parts grow.