ABSTRACT
Note also that, the enumeration of distributions of indistinguishable balls
into indistinguishable urns is merely a partition enumeration problem.
In this chapter, after the introduction of the notion of a partition of a
positive integer n, recurrence relations and generating functions for the to-
tal number of partitions of n and the numbers of partitions of n into k and
into at most k parts are derived. Then, a universal generating function for
the number of partitions of n into parts of specied or unspecied number,
whose number of parts of any specic size belongs to a subset of non-
negative integers, is obtained. As applications of this generating function
several interesting sequences of partition numbers are presented. Further-
more, relations connecting various partition numbers are deduced by using
their generating functions. Also, after introducing the Ferrers diagram of
a partition and the notion of a conjugate (and a self-conjugate) partition,
additional interrelations among certain partition numbers are derived. Eu-
ler's pentagonal theorem on the dierence of the number of partitions of n
into an even number of unequal parts and the number of partitions of n into
an odd number of unequal parts is obtained. The last section is devoted
to the derivation of combinatorial identities; the Euler and Gauss-Jacobi
identities are deduced. As a complement to this subject, a collection of
exercises on q-numbers, q-factorials and q-binomial coeÆcients, as well as
on q-Stirling numbers of the rst and second kind, is provided.
