(m)
|
| (l + 2) (l + 3) (l + m + 1)
| ------- . ------- ... -----------
| (2) (3) (m + 1)
| . . ... .
| . . ... .
| . . ... .
| (l + n) (l + n + 1) (l + m + n - 1)
| ------- . ----------- ... ---------------
| (n) (n + 1) (m + n - 1)
y

one factor appearing at each point of the lattice.

In general, partition problems present themselves which depend upon the
solution of a number of simultaneous relations in integers of the form

[lambda]_1.[alpha]_1 + [lambda]_2.[alpha]_2 +
[lambda]_3.[alpha]_3 + ... >= 0,

the coefficients [lambda] being given positive or negative integers, and
in some cases the generating function has been determined in a form
which exhibits the fundamental solutions of the problems from which all
other solutions are derivable by addition. (See MacMahon, _Phil. Trans._
vol. cxcii. (1899), pp. 351-401; and _Trans. Camb. Phil. Soc._ vol.
xviii. (1899), pp. 12-34.)

Method of symmetric functions.

The number of distributions of n objects (p1p2p3 ...) into parcels (m)
is the coefficient of b^m(p1p2p3 ...)x^n in the development of the
fraction

1
----------------------------------------------------------------------------
(1 - b[alpha]x. 1 - b[beta]x. 1 - b[gamma]x ... )
X (1 - b[alpha]^2x^2. 1 - b[alpha][beta]x^2. 1 - b[beta]^2x^2 ... )
X (1 - b[alpha]^3x^3. 1 - b[alpha]^2[beta]x^3. 1 - b[alpha][beta][gamma]x^3 ...)

. . . . . .

and if we write the expansion of that portion which involves products of
the letters [alpha], [beta], [gamma], ... of degree r in the form

1 + h_r1.bx^r + h_r2.b^2x^2r + ...,

we may write the development

r=[oo]
[Pi] (1 + h_r1.bx^r + h_r2.b^2x^2r + ...),
r=1

and picking out the coefficient of b^m x^n we find

[Sigma] h_[tau]1.h_[tau]2.h_[tau]3 ...,
t1 t2 t3

where [Sigma][tau] = m, [Sigma][tau]t = n.

The quantities h are symmetric functions of the quantities [alpha],
[beta], [gamma], ... which in simple cases can be calculated without
difficulty, and then the distribution function can be formed.

_Ex. Gr._--Required the enumeration of the partitions of