Integer partition asymptotics for a finite set of relatively prime integers.

I need to get approximations for partition functions in order to limit the expansion of the generating series used to work out the exact value.

The unrestricted partition function $$p(n)$$ counts the number of partitions of the positive integer n, and satisfies the asymptotic formula :

$$p(n) ∼ \frac{e^{c_0 \sqrt n}}{(4\sqrt 3)n}$$

where $$c_0 = \pi \sqrt{\frac{2}{3}}$$

Ok no problem so far.. (can work out $$p(n)$$ approximations from that).

Now, let $$p_A(n)$$ denote the number of partitions of n into parts belonging to a finite set $$A$$, with $$gcd(A) = 1$$. For this function we are given the following (cf. Elementary Methods in Number Theory, p. 455-461) :

$$p_A(n) = \left(\frac{1}{\prod_{a \in A}a}\right) \frac{n^{|A|-1}}{(|A|-1)!} + O\left(n^{|A|-2}\right)$$

The problem is that for a given integer n and a set of coprimes smaller than n, I always get values close to zero. I can't figure out how to get proper approximations from that. I don't know if I can safely ignore the big-O function, or what to do with it.

What did I miss ? Can the function $$p_A(n)$$ defined above actually be used to get correct values or proper approximations for finding the number of partitions of $$n$$ into parts belonging to $$A$$, and if yes how ? Or if not, why ?

• Where in the linked book can we find that expression? Commented Aug 21, 2021 at 17:40
• It is page 456 (more generally pages 455 to 461). Commented Aug 21, 2021 at 18:07
• Do note that this is an asymptotic result that holds for any fixed such set $A$, as $n$ increases. It seems that you are varying $A$ with $n$, i.e. your $A$ seems to depend on $n$. Then there is no reason to believe that the same asymptotic result will hold. Commented Aug 21, 2021 at 20:47
• You're right, thank you I've completely missed that point and I didn't test increasing n using a fixed set, but still it seems I got non sense values so far.. I will give it another try tomorrow and update accordingly. Commented Aug 21, 2021 at 22:08

For the extreme case $$A = \{ k \mid 1 \le k \le n \text{ with } \gcd(k,n) = 1 \}$$, I don't think you'll be able to do better than the asymptotics of the unrestricted $$p(n)$$. Writing $$p'(n)$$ for these partitions into relatively prime parts, $$p'(n) = p(n) - 1$$ when $$n$$ is prime. But $$p'(n)$$ can also be much less than $$p(n)$$, e.g., $$p'(12) = 6$$ while $$p(12) = 77$$. My guess is that any useful approximations would need to include some measure of the "compositeness" of $$n$$. (By the way, the $$p'(n)$$ sequence is in the On-line Encyclopedia of Integer Sequences as A057562.) As per Servaes's comment, the cited result doesn't apply is $$A$$ is not fixed.
There are, though, some related cases with asymptotic results. For instance, A000837 counts partitions of $$n$$ into parts that are relatively prime to each other (so $$3+1$$, $$2+1+1$$, and $$1+1+1+1$$ are counted while $$4$$ and $$2+2$$ are not) and A000607 counts partitions into prime parts; both of these entries include asymptotic formulas.
• It's quite funny or absurd because I got A000607 early in the process and while working out its generating function, I was looking for a way to restrict the expansion of the corresponding generating series and power sequence at some point where we "know" (approximately) from the start which value (or not) to reach. So I ended up here while the formula I was looking for was just above the Gf. definition... : $$a(n) = \frac{1}{n}\sum_{k=1}^n \operatorname{sopf}(k)*a(n-k)$$ where $\operatorname{sopf}(n)$ is the sum of the distinct primes dividing n (A008472). Commented Aug 23, 2021 at 12:23