836 
MAJOR P. A. MACMA.HON ON THE THEORY 
and herein the coefficient of is 
" - _ (V 
V - 1/ 
!n — 
n — s 
+ 
p 
n — 2s — 1 
n — 2s — p 
(IJ \n — s — 'p I 
The number of parts being unrestricted the generating function is 
'1 — x^\p x{\ — a?®) 
S ‘xJP 
p 
X 
1 — 2 * + 
The expression ^ is unchanged by the substitution oi n -- p \ for^ ; hence 
the numbers of compositions of n into p parts and into n — p -{■ 1 parts are identical. 
3, The graph of a number n is taken to be a straight line divided at — 1 points 
into n equal segments. 
The graph of a composition of the number n is obtained by placing nodes at certain 
of these n — 1 points of division. 
AB being the graph of the number 7, for the representation of the composi¬ 
tion (214), nodes are placed at the points P, Q, so that in moving from ^ to P by 
steps proceeding from node to node, 2, 1, and 4 segments of the line are passed over 
in succession. Although strictly speaking the initial and final points A, B are nodes 
on the graphs of all the compositions, it is only the inter-terminal nodes that will be 
considered in what follows, as appertaining to the graph. 
The number of parts in the composition exceeds by unity the number of nodes on 
the graph. 
For a composition of n into p parts we can place nodes at anyp — 1 out of the 
n — 1 points of the graph of the number. The number of such composition graphs 
is at once seen to be 
and further, since each of the n — 1 points of the number graph is or is not the 
position of a node, the total number of composition graphs is 
4. Associated with any one graph, there is another graph obtained by obliterating 
the nodes and placing nodes at the points not previously occupied. 
These graphs are said to be conjugate. 
If a graph denotes a composition of n into p parts, the conjugate graph denotes a 
composition of n into ?i — + 1 parts, 
