The partition function for $n$, $P(n)$ gives the number of partitions that exist for $n$.
I've been trying to find a function that gives the number of partitions where order matters, e.g. $1+2+3$ is considered a different entry from $2+3+1$. If a sum consists of only one element such as $2+2+2$, it is only counted once.
Does such a function exist (has any extensive research been conducted into it)?
What about an approximation for such a function?
Thanks.
These are called the compositions of a number $n$. There are $2^{n-1}$ of them, as the Wikipedia article notes, simply because there are $n-1$ spaces in the expression $$ (1 \square 1 \square 1 \dotsb \square 1) $$ containing $n$ "$1$"s, into which one can insert either a "$,$" or a "$+$", to give any list which sums to $n$, and these are all distinct.