e p1 + q1 + r1 + ... = m. We may thus denote the
classes either by the multipartite numbers

_________
p1q1r1...,

or by the partition (p1q1r1...) of the unipartite number m. The
distributions to be considered are such that any number of objects may
be in any one class subject to the restriction that no class is empty.
Two cases arise. If the order of the objects in a particular class is
immaterial, the class is termed a _parcel_; if the order is material,
the class is termed a _group_. The distribution into parcels is alone
considered here, and the main problem is the enumeration of the
distributions of objects defined by the partition (pqr...) of the number
n into parcels defined by the partition (p1q1r1...) of the number m.
(See "Symmetric Functions and the Theory of Distributions," _Proc.
London Mathematical Society_, vol. xix.) Three particular cases are of
great importance. Case I. is the "one-to-one distribution," in which the
number of parcels is equal to the number of objects, and one object is
distributed in each parcel. Case II. is that in which the parcels are
all different, being defined by the partition (1111...), conveniently
written (1^m); this is the theory of the compositions of unipartite and
multipartite numbers. Case III. is that in which the parcels are all
similar, being defined by the partition (m); this is the theory of the
partitions of unipartite and multipartite numbers. Previous to
discussing these in detail, it is necessary to describe the method of
symmetric functions which will be largely utilized.

The distribution function.

Let [alpha], [beta], [gamma], ... be the roots of the equation

x^n - a1x^(n-1) + a2x^(n-2) - ... = 0.

The symmetric function [Sigma][alpha]^p[beta]^q[gamma]^r..., where p
+ q + r + ... = n is, in the partition notation, written (pqr...). Let

A_[(pqr...), (p1q1r1...)]

denote the number of ways of distributing the n objects defined by the
partition (pqr...) into the m parcels defined by the partition
(p1q1r1...). The expression

[Sigma]A_[(pqr...), (p1q1r1...)].(pqr...),

where the numbers p1, q1, r1 ... are fixed and assumed to be in
descending order of magnitude, the summation being for every partition
(pqr...) of the number n, is defined to be the distribution function of
the objects defined by (pqr...) into the parcels defined by (p1q1r1...).
It gives a complete enumeration of n objects of whatever species into
parcels of the given s