bviously dissected into a
square, containing say [theta]^2 nodes, and into two graphs, one lateral
and one subjacent, the latter being the conjugate of the former. The
former graph is limited to contain not more than [theta] parts, but is
subject to no other condition. Hence the number of self-conjugate
partitions of n which are associated with a square of [theta]^2 nodes is
clearly equal to the number of partitions of 1/2(n = [theta]^2) into
[theta] or few parts, i.e. it is the coefficient of x^{1/2(n-[theta]^2)}
in

1
-------------------------------------------,
1 - x. 1 - x^2. 1 - x^3. ... 1 - x^[theta].

x^[theta]^2
or of x^n in ---------------------------------------------.
1 - x^2. 1 - x^4. 1 - x^6. ... 1 - x^2[theta]

and the whole generating function is

[theta]=[oo] x^[theta]^2
1 + [Sigma] ---------------------------------------------.
[theta]=1 1 - x^2. 1 - x^4. 1 - x^6. ... 1 - x^2[theta]

Now the graph is also composed of [theta] angles of nodes, each angle
containing an uneven number of nodes; hence the partition is
transformable into one containing [theta] unequal uneven numbers. In the
case depicted this partition is (17, 9, 5, 1). Hence the number of the
partitions based upon a square of [theta]^2 nodes is the coefficient of
a^[theta].x^n in the product (1 + ax)(1 + ax^3)(1 + ax^5)...(1 +
ax^{2s-1}), and thence the coefficient of a^[theta] in this product is

x^[theta]^2
---------------------------------------------, and we have the expansion
1 - x^2. 1 - x^4. 1 - x^6. ... 1 - x^2[theta]

(1 + ax)(1 + ax^3)(1 + ax^5)...ad inf.

x x^4 x^9
= 1 + ------- a + ---------------- a^2 + ----------------------- a^3 + ...
1 - x^2 1 - x^2. 1 - x^4 1 - x^2. 1 - x^4. - x^6

Again, if we restrict the part magnitude to i, the largest angle of
nodes contains at most 2i - 1 nodes, and based upon a square of
[theta]^2 nodes we have partitions enumerated by the coefficient of
a^[theta].x^n in the product (1 + ax)(1 + ax^3)(1 + ax^5)...(1 +
ax^{2i-1}); moreover the same number enumerates the partition of 1/2(n -
[theta]^2) into [theta] or fewer parts, of which the largest part is
equal to or less than i -[theta], and is thus given by the coefficient
of x^{1/2(n-[theta]^2)} in the expansion of

1 - x^{i-[t]+1}. 1 - x^{i-[t]