Balancing two armies

Build two armies of equal total value from the piece values pawn $1$, knight $3$, bishop $3$, rook $5$, queen $9$. Balancing them is solving an equation: the sum on the left equals the sum on the right. There are usually many solutions, because different collections of values can share a total. A bishop and two pawns ($3 + 1 + 1 = 5$) balance a lone rook ($5$), and once you have one such pair you can hunt for others by holding the total fixed and swapping pieces of equal value.

Counting how many distinct equal-value armies exist is a partition question: in how many ways can a target total be written as a sum of allowed values? Order does not matter, so $3 + 1$ and $1 + 3$ are the same army.