OF THE COMPOSITIONS OF NUMBERS. 
895 
and 
The coefficient of 
in 
H' = h\ + + /I'g + ... 
Virt n 
a 
2>n 
{1-. — ] Y 
enumerates, in respect of the multipartite PiP-i ■ ■ ■ p,i, the number of combinations 
having m parts. 
From previous work this coefficient is 
where 
/(piPs • • • Pn, w) = 
{k — + • • • V"> 
'p-^ + m — I\ /pg + m — I\ [pn +9)1—1 
Pi 
Pi 
/m\ /pj + m — 2\ /pi + 7)1 — 2\ fp^ + m — 2\ 
“\1/V Pi )\ Pi Pn ) 
+ &c., 
to m terms. 
f 
81. Hence the whole number of combinations is 
[k - l)-^-^f{p\P^ ■ ■ • Vn, 1) + (/- - 1)-''"V(Pii?2 • • • Pn, 2) + . 
a result which is immediately obtainable from the reticulation. 
82. There exists a very interesting correspondence between the compositions of the 
multipartite 
1 
into k parts, zeros not excluded, and the combinations of order k of the unipartite 
number 
zeros excluded. 
The generating function for the compositions of multipartite numbers into k parts 
zeros not excluded, is 
(1 4- + /ij + . . .)^' = (1 — (I — ... (1 — 
hence the number in the case of 'Pip .2 • • • Jhi is 
p + Pi - 1\ fk + 1^2 - l-\ p + Ps - l'\ p' + Pn - 1\ 
\ Pi / \ Pi J \ Ps -* \ Pn J 
