THE PARTITION OE NUMBERS. 
103 
When D m operates upon any product 
A/'A / 5 ... A/ Ci 
the demonstration proceeds as with the Q function ante. 
Writing out the A product at length and underneath it any composition of m into 
k x + k 2 + ... +k { parts, zero counting as a part, we note that if the composition operator 
is to have the effect of multiplying the product by unity and not by zero, every part 
under the first k y factors of the operand must be zero or unity; every part under the 
next k 2 factors must be zero or 2; every part under the next k 3 factors must be zero 
or 3; and so on, until finally every part under the last k H factors must be zero or i. 
The number of compositions of m which possess these properties is equal to the 
coefficients of x m in the developments of 
(1 + x)'" (1 + x 2 ) k 2 (1 + x A )' i3 ... (1 + X i ) / ', 
which may be written 
/1 — x 2 \ kl fl—x*\ k -/1 — x 6 \ k3 (1 — cc 2l \ /l1 
\1 — x) \1 — x 2 J \l—x 3 ) \l—x l ) 
or, in Cayley’s notation, 
(2)* 1 (4) t2 (6) A ' 3 ... (2 i) ki 
(l) /Cl (2)* 2 (3)* 3 ... (i) ki ' 
Let this coefficient be denoted by 
F a (?)i; 1 *' 2 * 2 ... i h ) 
D m A/'A / 2 ... A ki - F a (m; ... ... A/b 
so that 
Looking to the symmetric function expressions of A 1; A 2 ... A, ..., it will be noted 
that the only portion of the product 
A/ Cl A/' 2 ... A/', 
that is free from the elements a, /3, y ... , is unity. 
Hence the portion of 
d,a /c, a/ 2 ... A/b 
that is free from the elements, is 
F a (in ; l 4 l 2 * 2 ... i h< ); 
and we may write, as before, 
(DAM / 2 • • • A/‘)a = 1 = F a (m ; D‘ 2 fe ... i ki ). 
