OF THE COMPOSITION'S OP NUMBERS. 
891 
78. Pursuing this chain of reasoning it is completely manifest that the generating 
function of m-part combinations is 
{h, + {Ic - 1 ) h, + (^■ - 1)2 /,3 + . . . + (rf = 
00 
and hence the complete generating function of combinations of the multipartite 
number 
is 
Pilh • • 
Ai + (^- - 1) ^3 + ■ ■ ■ + (k - 
1 _ /ij _ _ 1) A, _ . . . _ (A- _ 
which is effectively the same as 
1 _ 1 _ 
k 1 — Z; 2 (A: — 1) S Ci^a.,2 ~ ^ — 1)2 S + . . . + ( —)'d' (A — . . . a.,P 
for the latter is obtained by adding the fraction l/A to tlie former and then transforming 
from homogeneous product sums to elementary symmetric functions. 
Just as in the case of A; = 2, corresponding to compositions, this generating function 
admits of an imi^ortant transformation to a factorized redundant form. 
74. In the above fraction put for a.^, . . . respectively; it 
may then be written 
A - 1 _ 1 _ 1 ^ 
A ’ - 1 + A {1 - (A - 1) } {1 _ (A - 1 ) s,ci2] ... {1 - (A - 1) s„«4 ~ AN ' 
For brevity put 
== S( {ka^ -h /lUo + . . . + ka-i + 1 + . . . + a,;) = 5/ (A,; + Aa^) 
l>.2 — Ac., 
jVI = (l (l . . . (l kSiPik^, 
and 
1 1 1 
A ■ (1 - Si) (1 - S 3 ) ... (1 - S„) ~ AD ■ 
It will be shown that — is a generating function equivalent to the former in regard 
to terms which are products of powers of 
5 X 2 
