THE PARTITION OF NUMBERS. 
Ill 
where F u (m l ; F'2* 2 ... i ki ) is equal to the coefficient of x mx in 
Y *2 Y hi 
-^~u 2u • • • -^‘w • 
D,.D„...D,.F t (U) 
_ y F„ (m i ; l*'2* a ... i kt ) . F„ (m 2 ; 1*'2* 2 ... A) ... F (m s ; 1*'2*-.. 
_ 4 F. 2 ki ... i !ii .k x \k 2 \... k { ! 
• * ‘) TT *iTT A- 
• U/'U/ 2 ... U/‘ ; ; 
and thence the number of partitions of the multipartite number 
m l m 2 ... m s , 
into k or fewer parts, such parts being drawn exclusively from the series 
is 
X 
u 2 , ... u s , 
F u (m 1 ; 1 *‘2 Aa ... F). F u (m 2 ; l Ai 2*-... A')... F„ (m s ; l Al 2 A ’ 2 ... F) 
F.2 /l2 ... F. k x ! k 2 \... F! 
the summation being in regard to the partitions of £. 
As an example, I will consider partitions of multipartite numbers where the 
numbers which are constituents of the multipartite parts are limited to be either 3, 
5, or 7. For the partitions into three or fewer parts we have the function 
i(U 1 3 + 3U 1 U 2 + 2U 3 ), 
and we have to find the coefficients of x' n in the three functions 
( 1 + X 3 + X b + X 1 ) 3 , 
(l + x 3 + x h + x i ) (l + £C 6 + a: 10 + a; 14 ), 
(l + a: 9 + a: 15 + a; 21 ). 
Thence, as particular cases, 
DJJ, 3 = 6U, 3 ; D n Uj 3 = 3U, 3 ; D 10 Uj 3 = 9Uj 3 ; 
D 12 U,U 2 = 0 ; D n U,U 2 = 11,17,; = 0 ; 
fhAT, = D u U 3 = D 10 U 3 = 0. 
Thence 
D-i(U 1 3 +3U 1 U 2 +2U 3 ) = i.6-U 1 3 , 
D--HU 1 3 +3U 1 U 2 +2U 3 ) = £ (6 <r -U 1 3 +3 XJiU 2 ), 
(n i 3 +3U 1 U 2 +2U 3 ) = i . 9-U, 3 , 
showing that the multipartite numbers 
12"‘, IF 2 , 10"*, 
have 6"‘ \ ^ (3 ff2 + 3), jt. IF 3 partitions respectively into three or fewer parts. 
