# Asymptotic for restricted compositions into k parts

For every set of natural numbers $$A$$ and for all positive integers $$n$$, $$k$$, let $$c_k^A(n)$$ be the number of compositions of $$n$$ into $$k$$ parts from $$A$$, that is, the number of $$(a_1, \dots, a_k) \in A^k$$ such that $$a_1 + \cdots + a_k = n$$.

I'm looking for asymptotic formulas for $$c_k^A(n)$$ as $$n \to +\infty$$ and $$k \to +\infty$$ (of course, in order to say anything interesting, some hypotheses on the relative grown of $$n$$ and $$k$$, and on the set $$A$$, are needed).

Where can I find such kind of results? I guess that something is surely known, but I keep finding only results with $$A = \mathbb{N}$$ and/or $$k$$ fixed.

Thank for any suggestion

• That's the coefficient of $x^n$ in the polynomial $(\sum_{a\in A} x^a)^k$. Saddle point method is usually applied for asymptotics of such coefficients. – Fedor Petrov Dec 12 '20 at 9:18

• Oops. They introduce restricted parts $A$ in $\S$3.3 and carefully prove things in that generality through chapter 5, but then most of chapter 8 is about unrestricted compositions or Carlitz compositions. Exceptions are Example 8.26 with $A=\{1,2\}$ and parts of $\S$8.4 which might be relevant, e.g., the asymptotics of the largest part in Theorem 8.39. – Brian Hopkins Mar 20 '20 at 1:54