THE PARTITION OF NUMBERS. 
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we find 
Eh„J-(Qi 2 +Q 2 ) = | (2m+l)Q 1 2 +iQ 2 , 
D 2 „i + i|- (Qr + Q 2 ) = ^ (2m+ 2) Q,“; 
and, by reason of the important properties possessed by the Q products in their 
relations with the Hammond operators, we can at once proceed to the results 
D 2m = ■§■ (2m+l) s Qi 2 +-|-Q 2 , 
DL +1 =i(2m + 2)U 2 . 
Thence we derive, by putting Qj = Q 2 = 1, the coefficients of the symmetric 
functions 
(2m’), (2m + l s ) 
(the exponent s meaning the numbers 2m, 2m+1 respectively s times repeated) in the 
development of the function 
*(V+Q»). 
Thence we obtain the numbers 
\ (2m+ l) J + |-, \ (2m + 2) s 
which, respectively, enumerate the ways of partitioning the multipartite numbers 
(2m, 2m... repeated s times), (2m+1, 2m+1... repeated s times) 
into two or fewer parts. 
When the enumeration is concerned with exactly two parts we have clearly to 
subtract unity in each case. In fact the generating function is 
and 
DLQi = DL+iQi = Q l5 
showing that unity must be subtracted. 
The numbers then become 
i(2m + l) s -£, \ (2m + 2) s —1. 
These numbers also enumerate the ways of exhibiting the composite integers 
(p l p 2 ...p s f\ (pxP2---PsY m+ \ 
as the product of two factors. 
To obtain a general formula for the multipartite numbe'r 
