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XLIII. On a Problem of Double Partitions. 

 By A. Cayley, Esq.* 



It «+&+c+ . . = m, a + /3 + 7+ . .=fju (the quantities being 

 all positive integer numbers, not excluding zero), then 

 (a, a) -f- (b, ft) + (c, 7) + . . is considered as a partition of (m, fj,) . 

 And the partible quantity (m, /ju) and parts (a, a), (b, ft), fee. 

 being each of them composed of two elements, such partition is 

 said to belong to the theory of Double Partitions. The subject 

 (so far as I am aware) has hardly been considered except by 

 Professor Sylvester, and it is greatly to be regretted that only an 

 outline of his valuable researches has been published : the pre- 

 sent paper contains the demonstration of a theorem, due to him, 

 by which (subject to certain restrictions) the question of Double 

 Partitions is made to depend upon the ordinary theory of Single 

 Partitions. 



Let the question be proposed, " In how many ways can (m, p) 

 be made up of the given parts (a, a), (b, ft), (c, 7), &c" under 

 the following conditions (which are, it will be seen, necessary in 

 the demonstration of the theorem constituting the solution), viz. 



-, 75, -, &c. are unequal fractions, each in its least terms, 

 a ft y 



and 



a, ft, 7, &c. each less than //,-}- 2. 



The number of partitions is 



= coeff. x m y fX in n „ . .- , fl ■ — — , 



* (1 — x a 7f)(l— x b i/){\ — x c yy)..' 



the fraction being developed in ascending powers of x, y. 



Considering the fraction as a function of y, it may be expressed 

 as a sum of partial fractions in the form 



A foy) 1 B fey ) - cfoy)' ., 



l—x a y x l—x^yP l—x c yv ' 



where 



A(x, y) is rational in x, rational and integral of degree ct — l'my, 



B (^2/) M » „ ft-1 „ 



c (*>y) ,> „ >, 7-1 >, 



&c. 



« 



To find A (a?, y) we have, when y=x~«, 



1 



A(x, y) = 



(l-x b f){I-x c yy)...' 

 * Communicated by the Author. 



