104 
MAJOR P. A. MacMAHON : SEVENTH MEMOIR ON 
The number of partitions of the unipartite number m into k or fewer parts, restricted 
not to exceed unity, is therefore 
D F (A) — y (f• • • A,- 1, )a = i 
m kK )a=1 4 1 * 2 **...**.^! k 2 \...k\ ’ 
= x 
F n (m; l kl 2 k2 ... i ki ) 
1 k '2 k *...i ki . k,\ k ,!... Jc l !’ 
= coefficients of x m in 
1 — x 2 \ kl (1 — a ,4 Y’ 2 /l — x 2l ^ /ci 
\1— x) \l—x 2 j 
1 -a;’ > 
l k as...i ki .k l \ k,\... k t ! 
the summation being for every partition 
ki P 2&.> -f-... + iki 
of the number k. 
Now, obviously, the number we seek is also the coefficient of x m in 1 + x + x 2 + ... +x*. 
Hence the formula 
when k is 3 the identity is 
-§■ {(l + a;) 3 + 3 (1 +x) (1 +x 2 ) + 2 (l -fa; 3 )} = l+x + x 2 + x s . 
We have now the result 
ki 
D„ u D„ i2 ... D mi . A/'A/ 2 ... A- 
= F„ (nii ; 1*'2* 2 ... %"). F a (m 2 ; 1 /Cl 2* 2 ... i") ... F (m s ; H2* 2 ... i’") . A/ 1 A/ 2 ... A ki ; 
and the number of partitions of the multipartite number 
m{m 2 ...m s 
into k or fewer parts, no integer constituent of the multipartite parts exceeding unity, 
is 
v F„ (m, ; F'2* 2 ... Y). F„ (m 2 ; H2 A ' 2 ... i ki ) ... F a (rn s ; 
^ 1 *‘ 2 * 2 ...^. k 1 \\k 2 \...k i \ 
the general solution of the problem. 
Some examples are now given. 
For the partitions into two, or fewer parts, it is only necessary to consider the cases 
m = 1 and m = 2, since there are no partitions of the nature examined when m >2. 
D?|(A 1 2 + A 2 ) = i(A’ + AJ, 
