It is clear that a large class of problems in permutations can be solved in a similar manner, viz. by giving special values to the elements of the determinant of the matrix. The redundant product leads uniquely to the real generating function, but the latter has generally more than one representation as a redundant product, in the cases in which it is representable at all. For the existence of a redundant form, the coefficients of x1, x2, ... x1x2 ... in the denominator of the real generating function must satisfy 2n - n² + n - 2 conditions, and assuming this to be the case, a redundant form can be constructed which involves n - 1 undetermined quantities. We are thus able to pass from any particular 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.)]
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 Case III. 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 (321³). Euler’s pioneering work in the subject rests on the observation that the algebraic multiplication
xa × xb × xc × ... xa+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 ζpxn in the ascending expansion of the fraction
| 1 | , |
| 1 - ζxa. 1 - ζxb. 1 - ζxc. ... |
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 - ζ). Similarly, if the parts are to be unrepeated, the generating function is the algebraic product
(1 + ζxa) (1 + ζxb) (1 + ζxc) ...;
if each part may occur at most twice,
(1 + ζxa + ζ2x2a) (1 + ζxb + ζ2x2b) (1 + ζxc + ζ2x2c) ...;