Bose-Einstein, Fermi-Dirac and Boltzmann Counting

 
Figure 1: Bose-Einstein and Fermi-Dirac Counting

give the number of Bose-Einstein, Fermi-Dirac, and Boltzmann complections depicted, which would contribute to the overall counting problem in statistical thermodynamics.
In the following, g (below) is s_i and n (below) is n_i in Dykstra's Physical Chemistry, a Modern Introduction, pages 484-5.

One has for the number of (distinguishable) Boltzmann states

but the number of indistinguishable states is

which is the Boltzmann value corrected for indistinguishability. The number of Bose-Einstein states is

and the number of Fermi-Dirac states is