cular redundant generating function to one equivalent to it,
but involving n - 1 undetermined quantities. Assuming these quantities
at pleasure we obtain a number of different algebraic products, each of
which may have its own meaning in arithmetic, and thus the number of
arithmetical correspondences obtainable is subject to no finite limit
(cf. MacMahon, _loc. cit._ pp. 125 et seq.)]

Case III.

3. _The Theory of Partitions. Parcels defined by (m)._--When an ordinary
unipartite number n is broken up into other numbers, and the order of
occurrence of the numbers is immaterial, the collection of numbers is
termed a partition of the number n. It is usual to arrange the numbers
comprised in the collection, termed the parts of the partition, in
descending order of magnitude, and to indicate repetitions of the same
part by the use of exponents. Thus (32111), a partition of 8, is written
(321^3). Euler's pioneering work in the subject rests on the observation
that the algebraic multiplication

x^a X x^b X x^c X ... = x^(a+b+c+...)

is equivalent to the arithmetical addition of the exponents a, b, c, ...
He showed that the number of ways of composing n with p integers drawn
from the series a, b, c, ..., repeated or not, is equal to the
coefficient of [zeta]^p.x^n in the ascending expansion of the fraction

1
------------------------------------------------,
1 - [zeta]x^a. 1 - [zeta]x^b. 1 - [zeta]x^c. ...

which he termed the generating function of the partitions in question.

If the partitions are to be composed of p, or fewer parts, it is merely
necessary to multiply this fraction by 1/(1 - [zeta]). Similarly, if the
parts are to be unrepeated, the generating function is the algebraic
product

(1 + [zeta]x^a)(1 + [zeta]x^b)(1 + [zeta]x^c)...;

if each part may occur at most twice,

(1 + [zeta]x^a + [zeta]^2x^2a)(1 + [zeta]x^b + [zeta]^2x^2b)
(1 + [zeta]x^c + [zeta]^2x^2c)...;

and generally if each part may occur at most k - 1 times it is

1 - [zeta]^k.x^ka 1 - [zeta]^k.x^kb 1 - [zeta]^k.x^kc
----------------- . ----------------- . ----------------- . ...
1 - [zeta]x^a 1 - [zeta]x^b 1 - [zeta]x^c

It is thus easy to form generating functions for the partitions of
numbers into parts subject to various restrictions. If there be no
restriction in regard to the numbers of the parts, the generating
function is

1
-----------