OF THE COMPOSITIONS OF NUMBERS. 
855 
It is useful to recognise four species of contact between adjacent parts of a 
composition 
in . . . . . . 
if qi is zero and zero we have a zero-zero contact 
qi „ Pa positive ,, zero-positive ,, 
g'i positive pjg zero ,, positive-zero ,, 
qi „ Pa positive ,, positive-positive ,, 
In this nomenclature we may say that the graph of a composition possesses as 
many essential nodes as the composition itself possesses positive-jDOsitive contacts. 
20. The theorem arrived at may be stated as follows :— 
“ Of compositions of pq possessing s positive-positive contacts, there is a one-to-one 
correspondence between those of r parts and those ofp-f-g' — 1 parts.” 
An essential node corresponding to a positive-positive contact in the composition 
occurs at a point of the graph where there is a change from a /3 direction to an a 
direction ; we may say that this is a /3a point on the line of route. Similarly to 
positive-zero, zero-positive, zero-zero contacts in the composition correspond /3/3, aa, 
a/3 points respectively on the line of route. 
The number of different lines of route that can be traced on the graph of the 
bipartite number is the number of permutations of p symbols a, and q symbols /S, for 
this is the number of ways in which the p a-segments and the q /3-segments, which 
make up a line of route, can form a succession. 
Hence the number of lines of route through the reticulation is 
(P + 2 \ 
\ P ) ' 
The whole of the compositions oi pq can be arranged in conjugate pairs. 
E.g., The correspondence in regard to the compositions of the bipartite 22 is shown 
in parallel columns. 
(22) 
5 = 0, '/• = 1 
(10 10 01 01) 
(To TI ol)' 
(21 01) 
>. 5 = 0, r = 2 
(To To 02 ) 
(To T 2 )^ 
(20 oT dT)_ 
5 = 0, r = 4 
s = 0, r = 3 
