  In the quantum theory, using the notion of partial supersymmetry, in which
some, but not all, operators have superpartners we derive the Euler theorem in
partition theory. The paraferminic partition function gives another identity in
partition theory with restrictions. Also an explicit formula for the graded
parafermionic partition function is obtained. It turns out that the ratio of
the former partition function to the latter is given in terms of the Jacobi
Theta function, $\theta_{4}$. The inverted graded parafermionic partition
function is shown to be a generating function of partitions of numbers with
restriction that generalizes the Euler generating function and as a result we
obtain new sequences of partitions of numbers with given restrictions.
