My problem is similar to this one, but different in some significant ways.
As in the above question, I have voting with $n$ voters and $m$ candidates. However, I care about which voter voted for which candidate. As such, there are a total of $m^n$ possible configurations.
Also, of these possible configurations, how many did candidate 1 have a plurality (see below). That is, how many did candidate 1 win (I choose candidate 1 arbitrarily. Of course it is symmetric across candidates). Further, how many configurations are ties in which candidate 1 participated.
For example, here are some situations ($n=6, m=5$) where candidate 1 wins.
v1 v2 v3 v4 v5 v6
-----------------
1 1 1 1 1 1
1 1 1 1 3 3
1 1 1 2 2 3
1 1 2 3 4 5
(Notice that what I want is a plurality and not a majority: a candidate does not need to have more than 50% of the votes, just more votes than anyone else.)
Here is a two-way tie:
v1 v2 v3 v4 v5 v6
-----------------
1 1 2 2 3 4
Candidates 1 and 2 are tied for the most number of votes. Here are some situations ($n=5, m=3$) where candidate 1 loses.
v1 v2 v3 v4 v5
--------------
1 2 2 2 3
1 2 2 3 3
2 2 2 2 2
This would just be stars and bars, except I care about the order. That is, I want to count the following situations as distinct:
v1 v2 v3 v4 v5
--------------
1 1 1 2 3
1 1 1 3 2
1 1 1 3 3
1 1 2 1 3
...
