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 ?
Best Answer
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.