-------------------------------------- in ascending powers of
1 - a. 1 - ax. 1 - ax^2. 1 - ax^3...ad inf.

a; for the coefficient of a^j.x^n in the expansion is the number of ways
of composing n with j or fewer parts, and this we have seen in the
coefficients of x^n in the ascending expansion of

1
------------------------.
1 - x. 1 - x^2...1 - x^j

Therefore

1 a a^2
--------------------------- = 1 + ----- + -------------- + ...
1 - a. 1 - ax. 1 - ax^2.... 1 - x 1 - x. 1 - x^2

a^j
+ ------------------------ + ....
1 - x. 1 - x^2...1 - x^j

The coefficient of a^j.x^n in the expansion of

1
-------------------------------------
1 - a. 1 - ax. 1 - ax^2. ... 1 - ax^i

denotes the number of ways of composing n with j or fewer parts, none of
which are greater than i. The expansion is known to be

1 - x^(j+1). 1 - x^(j+2). ... 1 - x^(j+i)
[Sigma]-----------------------------------------a^j.
1 - x. 1 - x^2. ... 1 - x^i

It has been established by the constructive method by F. Franklin
(_Amer. Jour. of Math._ v. 254), and shows that the generating function
for the partitions in question is

1 - x^(j+1). 1 - x^(j+2). ... 1 - x^(j+i)
-----------------------------------------,
1 - x. 1 - x^2. ... 1 - x^i

which, observe, is unaltered by interchange of i and j.

Franklin has also similarly established the identity of Euler

j=-[oo]
(1 - x)(1 - x^2)(1 - x^3)...ad inf. = [Sigma](-)jx^{1/2(3j^2+j)},
j=+[oo]

known as the "pentagonal number theorem," which on interpretation shows
that the number of ways of partitioning n into an even number of
unrepeated parts is equal to that into an uneven number, except when n
has the pentagonal form 1/2(3j^2 + j), j positive or negative, when the
difference between the numbers of the partitions is (-)^j.

+----------+
|. . . .| . . . . .
|. . . .| . .
|. . . .| .
|. . . .|
+----------+
. . .
. .
.
.
.

To illustrate an important dissection of the graph we will consider
those graphs which read the same by columns as by lines; these are
called self-conjugate. Such a graph may be o