THE PARTITION OF NUMBERS. 
107 
where 
F, (B) = X 
B ; 
ki 
l*‘2W.O. k x \ k,\ ... k ,! 
Thence we find 
(D OTl D OTi ... D m .F* (B)) b = i 
r F„ (w,; 1*'2*»... i ki ). F ft (m a ; 1*‘2*» ... i ki ) ... F (m s ; 1*'2*» ... i ki ) . 
4 l /c, 2 /£2 ... i kl . k 1 ! k 2 ! ... ! 
the solution of the problem of enumeration in respect of the multipartite number 
m l m 2 ... m s . 
If, in the function F /£ (B), we substitute 
1 —x 
l-X s 
3s 
for B s , 
we obtain 
( 1 — x k+1 ) ( 1 — x k+2 ) 
(1 — x) (1 —x 2 ) 
because this function enumerates unipartite partitions whose parts are limited in 
number by k and in magnitude by 2. 
As an example consider partitions into three parts. We have the symmetrical 
results 
D 6 i(B 1 3 +3B 1 B 2 +2B 3 ) = % (B 1 W3B 1 B, + 2B,), 
D 5 £ (Bj :J +3B,B_. + 2B ; .) = l(3B*+SB l B a ), 
T>A (B 1 8 + 3B 1 B 3 + 2B 8 ) = l (6B- : + 3.2BA), 
rbi(B 1 3 +3B 1 B_, + 2B,) = l (7B 1 3 + 3B 1 B 2 + 2B 3 ), 
and D 2 , Lfi, D u , yield the same results as D 4 , D & , I) G , respectively. 
The number m, not exceeding 6, I ),„ and D 6 _ ni produce upon the operand the 
same result. This symmetry naturally follows from the known property ot the 
function 
(1— x k+l ) (l—x k+2 ) 
{l-x) (1-ad) 
which on expansion is, as regards coefficients, centrally symmetrical. 
We now at once deduce that 
D»F S (B) = D,*F, (B) = i (B 1 *+3B,B 3 +2B > ) 
D;F s (B) = D,>F 3 (B) = i (3-B, 3 + 3B,B.) 
D;f 3 (B) = D/F 3 (B) = J (6^ + 3. 2‘B,B 3 ) 
D;F 3 (B) = i(7-B I - + 3B l B,+2B > ). 
Q 2 
