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. In the ultra-relativistic regime (which I'm interested in), the momentum and (normalized) energy of this state are $$ P = \sum_{j=1}^N n_j m_j \quad,\quad E = \sum_{j=1}^N |n_j m_j| \quad. $$ I would like to estimate the total number of states having the property $$ \begin{cases}P\leq\Lambda\\ E\leq\Lambda\end{cases} \quad. $$ Importantly, a need a very ROUGH estimate of the upper bound. Any suggestions greatly appreciated. Login To add answer/comment