258 On Verification in the Partition of Numbers. [Feb. 3, 



according as n is or is not a triangular number. The general term on 

 the left-hand side is (_)»Q(1, 3, 5 . . .)(n-6m 2 ±2m). For n=Q, the 

 formula gives 



2-1-1 = 0. 



There is a proposition of Euler's which follows immediately from (e), 

 viz. that the number of partitions of a number into the uneven elements 

 1, 3, 5 ... is equal to the number of partitions without repetitions into the 

 elements 1, 2, 3 . . ., or, in the notation already used, that P(l, 3, 5 . . ,)n 

 =Q(1, 2,3.. .)n. It is natural to suppose that this striking theorem 

 should admit of a simple demonstration by converting the partitions of 

 the one system into those of the other according to some rule, such as 

 in Mr. Ferrers's method of showing the identity of the numbers of parti- 

 tionments into the elements 1, 2, 3 . . .m, and into parts not exceeding 

 m in number (Phil. Mag. S. 4. vol. v. p. 201, 1853). The conversion, 

 though not so simple as that just referred to, is nevertheless elegant and 

 elementary ; it in effect consists of a transformation into the binary scale. 

 Thus consider a partition into uneven numbers containing a l's, /3 

 3's, y 5's, &c. This is transformable at once into 1 . a+3./3 + 5.y 

 + &c. Now express a, ft y ... in the binary scale so that a=2 a +2 a '+ 

 &c, /3 = 2 h + 2 b ' + &c, y = 2° + 2°' + &c, then the partition becomes 

 l(2 a +&c.)+3(2 6 +&c.)+5(2 c +&c.) + &c., viz. becomes 2 a +2 a '+&c. 

 + 3 . 2 5 + 3 . 2 6 ' + &c. + 5 . 2 c +5 . 2 c ' + &c, in which no two parts are iden- 

 tical, since a number can be expressed in only one way in the form 2 m A, 

 A being uneven. 



To illustrate this take Euler's example of n=10 ; the partitions into 

 uneven elements are 



(i) 9 + 1, (vi) 5+3 + 1 + 1, 



(ii) 7 + 3, (vii) 3 + 3 + 3 + 1, 



(iii) 7 + 1 + 1 + 1, (viii) 3 + 3 + 1 + 1 + 1 + 1, 



(iv) 5 + 5, (ix) 3 + 1 + 1 + 1 + 1 + 1 + 1 + 1, 



( V ) 5 + 1 + 1 + 1 + 1 + 1, (x) 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1. 



By the conversion (i) and (ii) remain unaltered; (iii) =7 + 1.3=7 

 + l(2 + l) = 7 + 2 + l; (iv) =5.2 = 10; (v) =5 + 1 . 5=5 + l(2 2 +l) 

 =5 + 4 + 1; (vi) =5 + 3 + 1.2 = 5 + 3 + 2; (vii) =3.3 + 1=3(2 + 1) 

 + 1 = 6 + 3 + 1; (viii) =3.2 + 1.4 = 6 + 4; (ix) =3 + 1 . 7 = 3 + 1 (2 2 

 + 2 + l) = 3 + 4 + 2 + l; (x) =1 . 10 = l(2 3 +2)=8 + 2; thus giving the 

 partitions without repetitions, viz. 



9 + 1, 



5+3 + 2, 



7+3, 



6+3 + 1, 



7 + 2 + 1, 



6 + 4, 



10, 



3 + 4+2 + 1, 



5 + 4+1, 



8 + 2. 



