1876.] 



Verification in the Partition of Numbers. 



251 



of 2's, z the number of 3's, and the S extends to all the partitions of the 

 given number n; so that 2l=N,the total number of the partitions of n. 

 A single formula of verification, however, is rarely sufficient to afford a 

 perfect test of accuracy, as the effects produced by certain omissions may 

 cancel one another. In the present case, for example, as the terms are 

 alternately positive and negative, if the omitted partition contained one 

 1 and no 2 it would appear as 1 in the first term and as 1 in the second 

 term, and its omission would thus not be pointed out. Having had 

 occasion several times to employ Sylvester's equation I., and having 

 found the need of some additional formula to be used in connexion with 

 it, I was led to seek for identities which would afford similar verifica- 

 tions. It seemed natural first to investigate what I. became when all 

 the signs were positive. 

 Starting from the identity 



1+ i^+r 



^+ 



.i_^ ■ i_f.i_ji.i_f 



■+&c.=l + *.l + * 2 .l-M 3 



and dividing throughout by 1— t . 1— f . 1 — t 3 . . . we have 

 1 



+ 



,t.l-f.l-t 

 + <fcc. 

 1 + *.1 + * 2 .1 + * 3 



(l-*) a l-tM- 



+ 



(l-*) a (l-* a ) a l-? 



1-*.1-* 2 .1-* 3 ... 



= (l + 2t+2f + &c.) (l + 2t 2 + 2t i + &c.) (l + 2t 3 + 2? + &c.) . . . ; 



whence, equating the coefficients of t n , 



II. 2(1 + so + xy + csyz + &c.) = 22% 



where r is the number of different elements contained in a partition. 

 Take as an example n=9 ; the partitions are : — 



9 8+1 7+1+1 6+1+1+1 5+1+1+1+1 



7+2 6+2+1 5+2+1+1 4+2+1+1+1 



4+3+1+1 3+3+1+1+1 



4+2+2+1 3+2+2+1+1 



3+3+2+1 2+2+2+2+1 

 3+2+2+2 



6 + 3 

 5 + 4 



5 + 3 + 1 

 5 + 2 + 2 

 4+4 + 1 

 4 + 3 + 2 

 3 + 3 + 3 



4+1+1+1+1+1 

 3+2+1+1+1+1 

 2+2+2+1+1+1 



1+1+1+1+1+1+1+1+1 



3+1+1+1+1+1+1 

 2+2+1+1+1+1+1 

 2+1+1+1+1+1+1+1 



Here N (the number of partitions) =30, _v (the number of l's) =67, 

 2^y = 47, and 1/xyz= 10. Also the number of partitions involving only 



u2 



