Complete state counting for Gentile's generalization of the Pauli exclusion principle

In the present work, we perform the complete state counting for Gentile's approach to the generalized Pauli exclusion principle (GPEP), which has been lacking in the literature. We count the total number of ways to allocate n identical particles occupying a group of g states with up to q   particles in each state, in order to derive an exact expression for the statistical weight. Our obtained expression for the statistical weight gives the fermionic one for q=1q=1; and for q>1q>1, it tends fast to a bosonic weight. Moreover, we perform a numerical comparison between our state counting and Wu's (corresponding to the Haldane–Wu's formulation of the GPEP), which implies that Gentile's formulation gives rise to more boson-like behavior while Haldane–Wu's approach to more fermion-like behavior; this difference lies on the fact that each formulation has its own state-occupation rules on which correlation plays a key role.