CHAP, ill] PARTITIONS. 139 



where the El and OD, each taken in order, pair with each other, and 

 similarly for the 01 and ED. Of course for the exceptional numbers 1, 2, 5, 

 7, 12, , there is just one partition which is neither / nor D, and, according 

 as it is or E, we have in the product a coefficient 1 or + 1. 



J. J. Sylvester 117 called a partition regularized if its parts be written in 

 their order of magnitude, represented each part p by p points (nodes) in a 

 horizontal line, and noted that the conjugate partition is obtained by 

 counting the nodes by columns [Ferrers 35 ]. There is given (pp. 4-7) a 

 method due to Franklin to construct the partitions which are to be elimi- 

 nated from the indefinite partitions of n into j parts, including zero, so as to 

 obtain the partitions of n intoj parts si i, and hence to obtain the generating 

 function enumerating the latter partitions; also (pp. 18-21) his constructive 

 proof for the generating functions for partitions into repeated or unrepeated 

 parts limited in number and magnitude. Sylvester (p. 7) gave his own 

 construction of partitions of n into j parts chosen from 0,1, , i by em- 

 ploying a square matrix MI of order j in which the diagonal elements are 

 all i -f- 1, the elements below the diagonal are all unity and those above the 

 diagonal all zero. For 1 si q si j, let M q be the matrix whose ( J 9 ) rows are 

 obtained by adding the rows of M i in sets of q. Denote the rth row of 

 M q by (r, q) and the sum of its elements by [r, q~\. To each regularized 

 partition of n [r, q~] into j parts ^ 0, add (r, q) term to term. The 

 partitions of n into j parts so obtained from M q for all values of r are said 

 to form the system P q . If P is the system of all partitions of n into j 

 parts, the complete system of partitions of n into j parts si i is 



where the minus sign denotes cancellation, and the system may involve 

 duplicates as well as non-regularized partitions. It remained to prove that 

 a partition of n, in which the number /JL of different parts is > i, occurs () 

 times in P q and hence (1 1) M times in S' } this was proved later by M. 

 Jenkins. 118 Hence the number of partitions of n into j parts Si i is the 

 coefficient of x n in 



(1 - z i+1 )(l - z i+2 ) (!.- z f+ ')/{(l - x) -(I - a')}. 



Any integer N can be expressed (p. 15) as a sum of consecutive integers 

 in as many ways as N has odd factors; Sylvester 119 also stated this else- 

 where. Cf. Barbette, 201 Agronomov, 204 and Mason. 207 



The subsequent topics treated are : generating functions, correspondence 

 (p. 24, p. 38) between partitions into odd parts and partitions into distinct 

 parts, 119 " and graphical conversion of continued products into series. Then 

 he noted (p. 60) that if in Jacobi's 30 formula we use the lower signs and take 



117 A Constructive Theory of Partitions . . ., Amer. Jour. Math., 5, 1882, 251-330; 6, 



1884, 334-6 (for list of errata noted by M. Jenkins). Coll. Math. Papers, IV, 1-83 

 (with the errata noted by Jenkins corrected in the text), to which the page citations 

 refer. 



118 Ibid., 6, 1884, 331-3. 



119 Comptes Rendus Paris, 96, 1883, 674-5; Coll. Math. Papers, IV, 92. Math. Quest. 



Educ. Times, 39, 1883, 122; 48, 1888, 48-49. 

 9a Comptes Rendus Paris, 96, 1883, 1110-2; Coll. Math. Papers, IV, 95-96. 



