Exact ground-state number fluctuations of trapped ideal and interacting fermions

We consider a small and fixed number of fermions in an isolated one-dimensional trap (microcanonical ensemble). The ground state of the system is defined at T = 0, with the lowest single-particle levels occupied. The number of particles in this ground state fluctuates as a function of excitation energy. By breaking up the energy spectrum into particle and hole sectors, and mapping the problem onto the classic number partitioning theory, we formulate a new method of calculating the exact particle number fluctuation more efficiently than the direct combinatorics method. The exact groundstate number fluctuation for particles interacting via an inverse-square pairwise interaction is also calculated.