THE PARTITION OF NUMBERS. 
89 
The value is clearly equal to the number of compositions of rn which do not have 
the effect of multiplication by zero. Suppose that we write out the operand at length 
QiQiQ, ••• Q 2 Q 2 Q-.Q.Q.Q, • •• 
in i blocks, containing Jc u k 2 ,...k t factors respectively. 
Underneath the k factors we will suppose written any composition of m into/; parts, 
zero being included as a part. 
In order that the corresponding operation may result in unity and not in zero we 
have the conditions :— 
(i.) Any number, including zero, may occur underneath any of the k x factors Q, ; 
(ii.) Zero, or any multiple of 2, may occur underneath any of the k 2 factors Q L , ; 
(iii.) Zero, or any multiple of 3, may occur underneath any of the k 2 factors Q :i ; 
(iii. ...) And lastly, zero, or any multiple of i, may occur underneath any of the k { 
factors Q t . 
How many such compositions exist ? 
We have merely to find the coefficient of x m in the expansion of 
(l +£C + SC 2 + ...)*' (l +x 2 + x i + ...) ki (l +£C 3 + CC 6 + ...)* 3 .(l + X l + X 2i + ] 
for this is the function which enumerates the compositions which possess these 
properties. In fact to form the composition we take a power of x from each of the 
first k x factors; then a power of x from each of the next k 2 factors, observing that 
the exponents of x are all zero or multiples of two; then a power of x from the next 
k 3 factors, observing that the exponents are all zero or multiples of three ; and so on, 
until finally in the k { factors we find that the exponents are all zero or multiples of i. 
Hence it follows that the operation 
DAA* 2 ••• Q* 
has the effect of multiplying Q/A* 2 ... Q/‘ by a number which is given by the coefficient 
of x m in the expansion of 
(1 -x)~ k ' (1 -a; 2 )-*’ (1 -x 3 )~ k > ... (1 -a*)"* ; 
an elegant theorem. 
Let this coefficient be denoted by 
' F 9 (m; U‘2 /C2 ... i ki ); 
so that 
D m Q/A* 2 ... Q/‘ = F 2 {m; l k '2 k > ... U) . ... Q/‘ 
Art. 8. Looking to the symmetric function expressions of Q t , ••• Q t if w ifi b e 
noted that the only portion of the product 
QAQA... W 
VOL. CCXVII.—A. 
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