CHAP, ill] PARTITIONS. 141 



G. S. Ely 127 noted that the partitions of n -f- 1 can be derived from those 

 of n by adding unity to each of the parts in turn or adding a new part unity. 

 Hence every partition of n into parts of which v are distinct gives v + 1 

 partitions of n + 1. If the total number of partitions of n be of parity 

 opposite to that of the number of partitions of n + 1, there has been a gain 

 in the self-conjugate partitions of n + 1 over those of n, if n > 1. 



A. Cayley 127 " wrote the article on partitions in the Encyclopaedia Britan- 

 nica. The article on combinatory analysis was by P. A. MacMahon. 1276 



G. S. Ely 128 called a compound 54 partition of N, 



a a 



regular if a* = 6, : ^ = ei for every i. A graph is obtained by repre- 

 senting each portion by an array of points in a plane and superimposing 

 the planes in order. Thus any compound partition may be read in six 

 ways. If (w; n; i, j) is the number of regular compound partitions of w, 

 the number of portions being ^= n, and each portion being partitioned into 

 i or fewer parts ^ j, the symbol is unaltered by any of the six rearrange- 

 ments of n, i, j. 



G. Chrystal 129 gave a recursion formula which may be used to form 

 mechanically a double entry table for the number n P T of partitions of r 

 obtained from 2, 3, , n. Since 



we see by changing n to n + 1 that 



(1 - Z" +1 )(l + n+l P,X + n 



whence 



n+lPs = nPs (S = 1, ' ' ', n), n+lPn+l = jJn+l + 1, 



n+l-in+r == n* n+r ~T~ n-fn r+1 (T = ) 



He noted that Tait 138 had recently communicated similar results. 



J. J. Sylvester stated and W. J. C. Sharp 1290 proved the double theorem 

 that, if v [and vi] is the number of ways n is a sum of i distinct positive 

 integers [and ^ j~\, then 



M. Jenkins 1296 evaluated the number of partitions of n into three parts. 



A. Cayley 130 employed non-unitary partitions (into parts > 1) and gave 



the developments up to z 100 of the reciprocals of (2), (2) (3), -, (2) (6), 



127 Johns Hopkins Univ. Circ., 3, 1884, 76-7. 



1278 Ed. 9, 17, 1884, 614; ed. 11, 19, 1911, 865. Coll. Math. Papers, XI, 589-91. 

 127b Ed. 11, 6, 1911, 752-8; ed. 9, Supplement, 3 (= ed. 10, vol. 27), 1902, 152-9. 



128 Amer. Jour. Math., 6, 1884, 382-4. 



129 Proc. Edinburgh Math. Soc., 2, 1884, 49-50. 

 129ft Math. Quest. Educ. Times, 41, 1884, 66-7. 

 129b Ibid., 107. 



130 Amer. Jour. Math., 7, 1885, 57-8; Coll. Math. Papers, XII, 273-4. 



