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 ½(3j² + 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 obviously dissected into a square, containing say θ² 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 θ parts, but is subject to no other condition. Hence the number of self-conjugate partitions of n which are associated with a square of θ² nodes is clearly equal to the number of partitions of ½(n - θ² into θ or few parts, i.e. it is the coefficient of x½(n-θ²) in

1 ,
1 - x. 1 - x². 1 - x³. ... 1 - xθ.

or of xn in

xθ2 ,
1 - x2. 1 - x4. 1 - x6. ... 1 - x2θ

and the whole generating function is

1 + Σ θ = ∞θ = 1xθ2 .
1 - x2. 1 - x4. 1 - x6. ... 1 - x2θ

Now the graph is also composed of θ angles of nodes, each angle containing an uneven number of nodes; hence the partition is transformable into one containing θ 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 θ² nodes is the coefficient of aθxn in the product (1 + ax) (1 + ax3) (1 + ax5) ... (1 + ax2s+1) ..., and thence the coefficient of aθ in this product is xθ2 / (1 - x2. 1 - x4. 1 - x6 ... 1 - x2θ), and we have the expansion

(1 + ax) (1 + ax3) (1 + ax5) ... ad inf. = 1 +x a + x4 a2 +x9 a3 + ...
1 - x21 - x2. 1 - x41 - x2. 1 - x4. 1 - x6