THE PARTITION OF NUMBERS. 
83 
This discovery involves the complete enumerative solution of the unrestricted 
partition of multipartite numbers into a given number of parts. The reason of this 
is that the enumerating symmetric function generating function can be expanded in 
ascending powers of the functions Q 1} Q 2 , Q 3 , .... On every term of this expansion 
the repeated performance of Hammond operators is practically effective and is 
successful in forcing out the sought numerical coefficients. When the magnitudes 
of the integer constituents of the multipartite parts are restricted in any manner 
there exists similarly an appropriate series of symmetric functions, 
U 1; u 2 ,... u t ..., 
the formation of which is explained in the paper, which in their properties are 
analogous to the series Q 1? Q 3 , ... Q„ ... . This circumstance involves the complete 
enumerative solution when the magnitudes of the constituents of the parts are 
restricted in any manner whatever. 
Section I. 
The Partition of Multipartite Numbers. 
Art. 1. One of the problems which has engaged the attention of writers on the 
combinatory analysis is that of enumerating the different modes of exhibiting a given 
composite integer as the product of a given number of factors. For instance, the 
number 30, which is the product of three unrepeated primes, can be given as the 
product of two factors in the three ways, 
2x15, 3x10, 5x6. 
When the given composite number is a product of different primes the question is 
very easy and is completely solved by means of the generating function 
1/(1—a:) (l — 2x) ... (l — kx). 
In the ascending expansions the coefficient of x q ~ k is the number of ways of 
factorizing a number, which is the product of q different primes, into exactly k 
factorsA 
Generating functions of the same character have also been obtained for some other 
simple forms of the composite number such as p*p 2 ... p q , p?p£p?,... p q \ Pi, p 2 , ••• 
denoting primes. 
It is, of course, obvious that the absolute magnitudes of the prime factors have 
nothing whatever to do with the question, which necessarily appertains to the 
exponents of the primes and to nothing else. 
* Compare ‘ Netto’s Combinatorik,’ 1901, pp. 168 et seq. 
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