  The Euler theorem in partition theory and its generalization are derived from
a non-interacting quantum field theory in which each bosonic mode with a given
frequency is equivalent to a sum of bosonic mode whose frequency is twice
($s$-times) as much, and a fermionic (parafermionic) mode with the same
frequency. Explicit formulas for the graded parafermionic partition functions
are obtained, and the inverse of the graded partition function (IGPPF), turns
out to be bosonic (fermionic) partition function depending on the parity of the
order $s$ of the parafermions. It is also shown that these partition functions
are generating functions of partitions of integers with restrictions. If the
parity of the order $s$ is even, then mixing a system of parafermions with a
system whose partition function is (IGPPF), results in a system of fermions and
bosons. On the other hand, if the parity of $s$ is odd, then, the system we
obtain is still a mixture of fermions and bosons but the corresponding Fock
space of states is truncated. It turns out that these partition functions are
given in terms of the Jacobi theta function $\theta_{4}$, and generate
sequences in partition theory. Our partition functions coincide with the
overpartitions, and jagged partitions in conformal field theory. Also, The
partition functions obtained are related to the Ramond characters of the
superconformal minimal models, and in the counting of the Moore-Read edge
spectra that appear in the fractional quantum Hall effect. The different
partition functions for the Riemann gas that are the counter parts of the Euler
gas are obtained by a simple change of variables. In particular the counter
part of the Jacobi Theta function is $\frac{\zeta(2t)}{\zeta(t)^2}$.
