THEORY OF THE PARTITION OF NUMBERS. 
663 
appears to be of general application if the difficulties presented by the algebra can be 
surmounted. 
Art. 57. At this point we may enquire into the meaning of the reduced generating 
function which has been so happily and ingeniously established. We may write it as 
the product of two fractions ;— 
a + 
1 
(1 — x) (1 — X Z ) ... (1 — X m ) (1 — S3 2 ) (1 — SU 3 ) ... (1 — X m+l ) 
the indication being that every two-layer arrangement is derivable from a combination 
of two ordinary single-layer partitions whose parts are drawn from the two series of 
numbers, 
1, 2, 3, . . . m, 
2, 3, . . . rn, m + 1, 
respectively. Otherwise we may say that a number N possesses as many two-layer 
partitions ( 2 ; m ; co ) as there are modes of partitionment employing the parts 
di> 2 ls 2 2 , 3 1? 3 2 , . . . th i, w? 2 , m -f- 1 2 . 
Ex. gr. If N = 4 and m = 3, the graphs are 9 in number 
111 11 11 1 211 21 2 22 2 
1 1111 11 2 
1 1 1 
1 
and employing parts 
we can form 9 partitions, viz.:— 
(V), (2,1, 2 ), (2 S 1, 2 ), (2, 2 ), (2,2,), (2 S »), (3,1,), (3,1,), (4,).* 
Art. 58. The problem is therefore reduced to establishing a one-to-one corre¬ 
spondence, between the graphs and the partitions of the kind indicated, of general 
application. I will in part establish this correspondence, which is not very simple in 
character, later on. At present it is convenient to take a further survey of the 
general problems in order to obtain ideas concerning the difficulties that confront us. 
I form a tableau of algebraic factors. 
* Tlie solution thus shows that the two-layer graphs may be exhibited as a one-layer graph by nodes 
of two colours, say black and red; nodes of different colours not appearing in any single line. 
