I have a question related with number of additive partition or method similar like this: $$p(5)=1+4=2+3=1+1+1+1+1=1+1+1+2=1+2+2=1+1+3$$
For a given number $n$,if we are trying to calculate number of additive partition of $p(n\cdot k)$ where $k$ is some integer,or $p(n+a+c)$ , $a,c \in \mathbb Z$, is there commutative rules or something similar to calculate number of partition of sum of some numbers?or number of partition of some number multiplied by another number? For example like this $$p(n+k)\overset{?}{=}p(n)+p(k)?$$
Please help me
There are certainly no "rules" for the partition function such as which you speak of. However recently Ken Ono and Jan Bruinier recently proved a finite formula for computing the partition number.
Also, Folsom, Kent, and Ono show that the partition sequence exhibit fractal behavior.