136 HlSTOKY OF THE THEORY OF NUMBERS. [CHAP. Ill 



0. H. Mitchell 109 wrote (to; i, j) for the number of partitions of w into j 

 or fewer parts each ^ i. Let <f>j(w) be the largest integer ^ (j l)w/j. 

 Then 



; i, j) = Z (x; w x,j 1). 

 By successive applications of this formula, j can be reduced to unity. Hence 



(w,i,j) = Z Z 



Xi=wi X2=2*i-wi 2:3-2*2 *i Xj-i=2Zj-i 



where the final 2(1) denotes 1 -f- fa&i-z) (2z/_ 2 z/_ 3 ), i. e., as many 

 units as values of the summation index. There is given the long expression 

 equivalent to the last two signs of summation. This is said to furnish a 

 proof of the final result by Sylvester. 96 



G. S. Ely 110 noted that Euler's 13 table of partitions 



0123456 



may be constructed by use of columns instead of rows: To get the iih 

 element in the jth column, add to the (i l)th element in the jth column 

 the ith element in the (j i)th column. Euler had noted that the number 

 (w; w, j} of partitions of w into j or fewer parts is given by the number in 

 line j and column w. The number (w; i, j) of partitions of w into j parts 

 =i i can be found from this table when the greater of i and .7 is = (w 4)/2 

 by the following rule: Since (w; i, j} = (w; j, i}, let i ^j. Then to get 

 (w, i, j) subtract from the tabulated value of (w; w, j) the sum of the 

 first w i elements in the (j l)th row and add to the result 0, 1 or 2, 

 according as i ^ (w - 2)/2, = (w - 3)/2 or = (w - 4)/2. Next, the 

 number of expressions (w; i, j} is 



w 2 2w + t [w n 2 n 



N = 2 -- SL ^n 



where t = 6 if w is even, t = 5 if w is odd. Let s = 24 or 27 in the respec- 

 tive cases. Then 



w-=l 



.\n-l~\\\ 

 ~* -T- j , 



L t J/ .' 



+ i-l~\ 



J- 



W. P. Durfee 111 defined a self-opposite or self-conjugate partition to be 

 one such that, if exhibited as an array of units (an element n being repre- 



109 Johns Hopkins Univ. Circ., 1, 1882, 210. 



110 Ibid,, 211 (in full). 



111 Ibid., 2, Dec., 1882, 23 (in full). 



