839 
OF THE COMPOSITIONS OF NUMBERS. 
number of compositions of m composed of p parts. In a notation, which is self- 
explanatory, we may write 
SIC (2m. ip) = C (», p) = . 
Next consider the self-inverse compositions of 2m into an uneven number, 2y> — 1, 
of parts. The two central nodes must be distinct, and may include any even number 
of segments. If this even number be 2/c the corresponding number of self-inverse 
compositions is equal to the number of compositions o^ m — k into p — 1 parts. 
Hence 
SIC (2m, 2p — 1) = C (m — I, p — 1) + C {m — 2, y) — 2) + . . C {p —I, p — 1)> 
or 
SIC (2m, 2;, - 1) = C (m, 2 .) = (“ : . 
Self-inverse compositions of uneven numbers occur only when the number of parts 
is uneven, and it is easy to prove that 
SIC (2m — 1, 2p — 1) = C (m, p) = • 
Hence, without restriction of the number of parts, 
SIC (2m) = SIC (2m -p ]) = C (m 1) = 2'". 
This completes the enumeration of the self-inverse compositions. 
6. Two compositions which are at once conjugate and inverse, may be termed 
“inverse conjugates.” 
A composition whose conjugate is its own inverse is said to be “ inversely 
conjugate.” 
Inversely conjugate compositions of a number which have p parts, can occur only 
wheny> — n — p 1, or n = 2p> — 1, an uneven number. 
The inversely conjugate compositions of 2m + 1 are composed of m -p 1 parts. 
Consider a graph in which white and black nodes have reference to the two 
inverse conjugates respectively. 
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The black nodes are placed to the right and left of the centre of the graph in 
a manner similar to the white nodes to the left and right. If on the right there are 
