CHAP, ill] PARTITIONS. 133 



Let U r be the number of pairs from 1, , n whose sums are ^ r. Then 



U r = i {r(r - 2) + i[l - (- I)-]}, 3 ^ r ^ n + 1; 

 U r = %n(n - 1) - Uzn-r+i, w + l<r^ 2/1-1. 



Similar applications are made to the cases u = 3, u = 4. 



J. W. L. Glaisher 101 noted that, if P(w) is the number of partitions of u 

 into the elements 1, -, n, each partition containing exactly r parts, order 

 attended to and repetitions not excluded, then 



P(r + k) + P(r + n + k) + P(r + 2n + &) + = n r ~ l 



(k = 0, 1, >- t n - 1). 



E. A. A. David 102 noted that Arbogast's 850 law of derivatives gives 



Qj , ar 2 a 2 ar 8 fl ar 4 (a4 + a2/2) 



n\" (n- 2)!" (n - 3)! " (n - 4)1 



ar\a^z + as) _ of 1 a? 



~l ---- I ^^ / j - ' 



fa- 5)! pilfrl 



summed for all sets of positive integral solutions of 



Pi + 2p 2 -f 3p 3 + = n. 



The latter sets are all given by the exponents of the terms in the left member. 



A. Cayley 103 tabulated all partitions of 1, , 18, where in each partition 

 1, 2, are designated by a, 6, , so as to give the literal terms in the co- 

 efficients of any co variant of a binary quantic. 



G. B. Marsano 104 treated the number of combinations 2 or 3 at a time 

 of 1, 2, , m to give a sum =i C. Simpler and more general results were 

 given by Gigli. 181 



F. Franklin 105 proved Euler's formula (3). The coefficient of x w in the 

 left member is evidently the excess E of the number of partitions of w into 

 an even number of distinct parts over that into an odd number of parts. 

 To find E, write {a} for a number == a, and let the parts of each partition 

 be in ascending order. Consider a partition with r parts, the first being 1 ; 

 deleting 1 and adding 1 to the final part, we get a partition into r 1 

 parts, the first being {2}, and without two consecutive numbers at the end, 

 and conversely. These two types of partitions do not affect the required 

 E, one being of even order and one of odd order. Hence we need consider 

 only partitions commencing with {2} and ending with two consecutive 

 numbers. Consider any one of these with r parts, the first being 2 ; deleting 

 2 and adding 1 to each of the last two parts, we get a partition into r 1 



101 Messenger Math., 9, 1880, 47-8. 



102 Comptes Rendus Paris, 90, 1880, 1344-6; 91, 1880, 621-2; Jour, de Math., (3), 8, 1882, 



61-72. 



103 Amer. Jour. Math., 4, 1881, 248-255; Coll. Math. Papers, XI, 357-364. 



104 Giornale di Mat., 19, 1881, 156-170; 20, 1882, 249-270. 



105 Comptes Rendus Paris, 92, 1881, 448^50. Cf. Sylvester, 117 11-13. 



