  The combinatorial basis of entropy by Boltzmann can be written $H= {N}^{-1}
\ln \mathbb{W}$, where $H$ is the dimensionless entropy of a system, per unit
entity, $N$ is the number of entities and $\mathbb{W}$ is the number of ways in
which a given realization of the system can occur, known as its statistical
weight. Maximizing the entropy (``MaxEnt'') of a system, subject to its
constraints, is then equivalent to choosing its most probable (``MaxProb'')
realization. For a system of distinguishable entities and states, $\mathbb{W}$
is given by the multinomial weight, and $H$ asymptotically approaches the
Shannon entropy. In general, however, $\mathbb{W}$ need not be multinomial,
leading to different entropy measures. This work examines the allocation of
distinguishable entities to non-degenerate or equally degenerate,
indistinguishable states. The non-degenerate form converges to the Shannon
entropy in some circumstances, whilst the degenerate case gives a new entropy
measure, a function of a multinomial coefficient, coding parameters, and
Stirling numbers of the second kind.
