308 Mr. J. W. L. Glaisher on Partitions. 



2x—2r into even parts, subject to the condition that the num- 

 ber of these latter (even) parts must not exceed 2r (the number 

 of units in the top line). Thus, writing for the moment 

 Y' n (a, b, c, . . . ) to represent the number of ways of partitioning 

 x into the parts a,b 3 c . . . , no more than n of such parts ap- 

 pearing in any one partition, we see that 



P(l, 3, 5 . . . )2# = 1 + P' 8 (2 3 4, 6 . . .) (2a? -2) 



+ P' 4 (2, 4, 6 . . .)'(2*-i) . . . +P' 2 ,_ 2 (2, 4, 6 . . .)2, 

 the first term in the latter expression corresponding to the single 

 partition of 2x as the sum of 2x units. Now obviously the 

 number of ways of partitioning 2n into the parts 2, 4, 6 . . . is 

 equal to the number of ways of partitioning n into the parts 

 1, 2, 3 ... , so that 



P(l, 3, 5 . . .)2a?= 1 + P' a (l, 2, 3 . . .) (a-1) 



+ P' 4 (l, 2, 3 . . .jfr-2) . . . +Fv_,(l, 2, 3 . ..Jl, 



which, transformed by means of the well-known theorem 



Y' n {\,2,3...)x=V(l,2,3...n)x, . . .• (I) 

 becomes 



P(l, 3, 5 . . 02o?=l + P(l, 2)(*-l) +P(l, 2, 3, 4)(*-2) . . . 



+ P(1, 2... 2^-2)1 . (2) 



By a similar method we easily arrive at the expression for the 

 decomposition of an uneven number, viz. 



P(l,3,5...)(2.r + 1)=2 + P(l,2,3)(a"-1) 



+ P(l,2,3,4,5)(a?-2) ... + P(1, 2, 3 . . . 2z- 1)1, (3) 



formulae expressing the value of P(l, 3, 5 . . .)x in terms of par- 

 titions into the elements 1,2; 1, 2, 3, &c, and which have thus 

 been obtained by general reasoning without analysis : they can, 

 of course, be derived more directly by means of the identity 



1 _ x x 2 



1— x.l — x 3 .l — x 5 ...~ 1—x 2 1— x^.l — x 4 



x 3 



+ l-x*.l-x*.l-x 6 + C * 



The proposition (1) that has been used (viz. that the number 

 of partitions of x into parts not exceeding n in number is equal to 

 the number of partitions of x into the parts 1, 2, 3 ... n) admits 

 of almost intuitive proof by Mr. Ferrers's method of breaking up 

 and arranging the parts in a partition, so that when read as lines 

 we obtain the number of partitions into parts not exceeding n 

 in number, and when read as columns the number of partitions 

 into 1, 2, 3 . . . n : see Phil. Mag. S. 4. vol. v. p. 201 (1853). 



