I don't really know how to approach a combinatorial problem arising from the physics context. Here's the setting. The state of a bosonic system in ${(1+1)}$ dimensions is described by a vector of the form $$ |\psi\rangle = |n_1^{m_1},n_2^{m_2},\ldots,n_N^{m^{N}}\rangle \quad,\quad n_j=\pm1,\pm2,\cdots,\pm\Lambda\quad. $$ This states contains $m_1$ particles of momentum $n_1$, $m_2$ particles of momentum $n_2$, etc.
The momentum and of this state is $$ P = \sum_{j=1}^N n_j m_j \quad. $$
I would like to estimate the total number of states having the property $$ P = 0 \quad. $$ Importantly, a need a very ROUGH estimate of the upper bound. Any suggestions greatly appreciated.