Distribution Function for $n \ge g$ Quantum Particles

A new quantum mechanical distribution function $n^I(\varepsilon)$, is derived for the condition $n \ge g$, where in contrast to the exclusion principle $n \le g$ for fermions, each energy state must be populated by at least one particle. Although the particles share many features with bosons, the anomalous behavior of $n^I(\varepsilon)$ precludes Bose-Einstein condensation (BEC) due to the required occupancy of the excited states, which creates a permanently pressurized background at $T=0$, similar to the degeneracy pressure of fermions. An exhaustive classification scheme is presented for both distinguishable and indistinguishable, particles and energy levels based on Richard Stanley's twelvefold way in combinatorics.