843 
OF THE COMPOSITIONS OF NUMBERS. 
10. Taking for the present the general multipartite number to be 
P 1 P 2 P 3 • • • 
it is easily seen that the enumeration of the compositions into r parts is the same 
problem as the enumeration of the distributions of +^^ 3+^53 + . . . things, of 
which Pi are of one sort, p^ of a second, of a third, and so forth into r different 
parcels. 
This number'"' is the coefficient of . . . in the expansion of 
{hi + h.2 + + . . .)'■ 
wherein hs denotes the sum of the homogeneous products, of degree s, of the quantities 
«!, ao, a3, . . 
Hence, the generating function of the total number of compositions is 
+ ho 4-/43 + ... 
1 — hi — h^ — Ag — . . . 
The coefficient of ... in the expansion of (/q + /q + /q + . • •)'’ is readily 
found to be 
M r -l\fpo-\- r - 1\ /p 3 + r - 
\ Pi / \ P-2 J\lhr'' 
'A (ih + r — ■>\ [iJo + r — 2\ [p.^ + r — 
1 
d" 
Ih / \ Ih J \ P-i 
r\ fpi + r - 3\ (p 2 + r - ‘d\ (p. + »■ - 3 
Pi 
P2 
Pi i 
— ... to r terms. 
and the enumeration is analytically complete. 
11 . This method is laborious in the case of high multipartite numbers since v may 
have all values from unity to pi + Ps + ^>3 + . . . ; but fortunately the series, above 
written, possesses some remarkable properties which can be utilized so as greatly to 
abridge the necessary labour. 
Let F {pipzP^ . . .) and f {p\Pz%>-^ . . . , r) denote, respectively, the total number of 
* See the Author, “ Symmetric Functions and the Theory of Uistiibatioiis,” ‘ Proceedings of the 
London Mathematical Society,’ vol. 19, Nos. 318-320. 
5 P 2 
