To get started you can use [oeis.org][1] to investigate this. For instance, from plugging in the numerators corresponding to some of your data it seems that
$$\#P_n^{(2)}(1)=n(n+1)$$
("the oblong numbers")
and
$$\#P_n^{(6)}(1)=n(n+1)(n-1)^4$$
("the number of $n$-colorings of the Triangle Graph of order 3").

  [1]: http://oeis.org