310 



identity 



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



1 +xz . \ ^- x 2 z . I + x 3 z . . . =1 + 



+ 



a*z* 



xz 



1 — X 1 — X . 1 



- +&C. 



•X" 



The value of P(l, 3, 5 . . ,)x given by (4) is in fact that found by 

 Euler {Opera minora collect a, vol. i. p. 93). The class of parti- 

 tions without omissions is one which has not, I believe, been par- 

 ticularly noticed before ; and it is curious that it should be iden- 

 tical with the partitionment into uneven numbers. 



There is also another mode of transforming this latter class of 

 partitions, which can be best made clear by an example. Thus, 

 consider the partition 1 +3 + 3 + 7, and arrange the parts as in 

 the following scheme : — 



1 



1 



1 



1 



1 







2 



2 





1 



1 



2 





1 



the 1 occupying the left-hand lower corner, the three squares 

 surrounding it being occupied by 2's, as there are two 3's in the 

 partition, the next five squares being left vacant as there is no 5, 

 and the next seven squares being filled by l's as there is one 7. 

 Then dividing the square into similar belts to those which re- 

 presented the different parts in its formation, only beginning at 

 the opposite (viz. the upper right-hand) corner, we have the parts 

 1, 1 + 1, 1 + 2 + 1, 1 + 2 + 1 + 2 + 1, that is 1, 2,4, 7, which are 

 always thus given in order of magnitude, and are subject to the 

 following laws of formation, viz. : — that any number of parts (ex- 

 cept the first) may be equal ; but that, taking no account of these 

 repetitions, i. e. regarding, for example, a+b+b+c simply as 

 a + b + c, then the parts a, b, c, d, &c. are such that Z>=2# + «, 

 c = b-\-ot-\-/3, ^=c + yQ + 7, &c. (so that the second part must be 

 at least the double of the first). The same decomposition may 

 also be derived without a diagram by observing that, for example, 

 0_j_36 + 7d =c ?+(2d) + (2d+6)+(2d + 26 + ft); but the mode of 

 formation is too complicated to render the transformation one of 

 much interest. 



The number of transformations of P(l, 3, 5. . . )#, how- 

 ever, is noteworthy, as we have seen the equivalence of the 

 numbers of (i) partitions into the uneven elements 1, 3, 5 . . ., 

 repetitions not excluded ; (ii) partitions into the elements 

 J, 2, 3... without repetitions; (iii) partitions into the parts 



