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

  • 1
    $\begingroup$ 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. $\endgroup$ – Fedor Petrov Dec 12 '20 at 9:18

A great resource is the book Combinatorics of Compositions and Words, Silvia Heubach and Toufik Mansour, CRC Press, 2010. Chapter 8 is "Asymptotics for Compositions."

  • $\begingroup$ I agree that it is a great resource on compositions, but I checked that chapter and, unfortunately, there's no result about compositions with parts restricted to a set A. $\endgroup$ – Nik Mar 19 '20 at 8:26
  • $\begingroup$ 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. $\endgroup$ – Brian Hopkins Mar 20 '20 at 1:54

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