The partition function P(x) is defined in the sense that it counts the number of permutations that exist to sum up to x.
eg.
4 = 4
= 3+1
= 2+2
= 2+1+1
= 1+1+1+1
Thus P(4) = 4 as there are 4 ways to permute.

Is there a P_odd(x) where only odd integer permutations are allowed.
ie
4 = 1+3
= 1+1+1+1
P_odd(4) = 2