Let’s say there are $n$ voters who vote on $m$ candidates. Each voter creates a list where they rank the candidates from most favorite to least favorite. There are $m!$ different possible lists each of them could have made, so there are $m!^n$ possible events. From each event you can define a unique Smith set.
Let $f(n,m,k)$ be the amount of events with $n$ voters and $m$ candidates where the Smith set has cardinality $k$, how can I define this function?
Some additional info:
$f(n,m,1) + f(n,m,2) + \ldots + f(n,m,m) = m!^n$
$f(n,m,a)=0$ when $a>m$ or $a<1$
The motivation behind the question is that $f(n,m,k)/m!^n$ is the probability of ending up with $k$ winners in a ranked choice vote where you consider the candidates in the Smith set as winners. I want to choose an $n$ and $m$ and plot $f(n,m,k)/m!^n$ where $k$ varies from 1 to $m$ to see the probability of getting different amount of winners depending on the $n$ and $m$ I choose.