838 
MAJOR P. A. MACMAHON ON THE THEORY 
An examination of tlie rules shows that they are reversible, and that the process 
gives a one-to-one correspondence between compositions of n into p and n — p -j- i 
parts. 
A composition in general has, when prepared for conjugation, four dlfiPerent 
forms, viz.— 
(1) 
( 2 ) 
( 3 ) a, 
( 4 ) . . . as_i 
in all the four forms a-^, ci^, . . . co may have any positive integral values superior to 
unity. The numbers, a-^, ao, . . . a^, may have any positive integral values, including 
zero, with the exceptions. 
In form (2) cannot be zero. 
(3) aj „ 
(4) and „ 
The conjugates of the forms are 
(1) 1“'-' . «! + 2 . . ag + 2 . . . . a,_i + 2 . V-'-\ 
(2) . «! + 2 . . ag -f 2 . . . 1“-^-- . a,_i + 2 . I"--" .«,+ !, 
(3) oL^+1 . . ^3 + 2 . . . . a,_i + 2 . V--\ 
(4) «! + 1 . . ^3 + 2 . . . . a,_i -h 2 . r--“ . as 4- 1. 
5. Two compositions are said to be inverse (the one of the other) when the parts of 
the one, read from left to right, are identical with those of the other when read from 
right to left. 
A composition may therefore be self-inverse. 
In the graph of a self-inverse composition, the nodes must be symmetrically placed 
with respect to the extremities of the graph. If the number be even, the number of 
segments of the graph is even, and the two central nodes (nodes nearest to the centre 
of the graph) may be coincident, or they may include 2, 4, or any even number of 
segments. A self-inverse composition of an even number, say 2m, into an even 
number, say 2p, of parts, can only occur when the two central nodes of the graph are 
coincident and, attending to one side only of this node, we find that the number of 
self-inverse compositions of the number 2?n, composed of 2p parts, is equal to the 
