142 HISTORY OF THE THEORY OF NUMBERS. [CHAP, in 



where (k) = 1 x k , for application to semin variants. 



M. Jenkins 131 gave a method to examine bends of a graph of a partition 

 without actually constructing the graphs (cf. Sylvester 117 ), and discussed 

 the addition of two regularized graphs, row to row, in order. 



J. B. Pomey 132 wrote A for the number of sets of values X = or 1 

 satisfying Xi + 2X 2 + + m\ m n. Then 



f(x) = (1+ a;)(l + * 2 ) (1 + * m ) = XU?**', M = m(m 



t=0 



It follows readily that 



A: = A:_ n , A: = AT" + A^ E ^r = 2-v, E AT = 2-, 



<=0 t=0 



<=o 



summed for all positive solutions of Xi + 2X 2 + + m\ m = i. Thus 

 C7 is the excess of the number of partitions into an even number of parts 

 over that into an odd number. Also, 



Z CA_. = o, c: + Z ci_ y = o. 



j=0 j=l 



D. Bancroft 133 considered the (w, i, j] partitions of w into j parts ^ i. 

 Then 



; i, j) = (w - j; i - 1, j) + (w, i, j - 1). 



Taking j = w k and summing f or k = 0, , k, we get 



& 

 (w; i, w) = (w; i, w k 1) + S (x m t i l,w x). 



Hence, if k ^ w/2, (w; i, w k 1) is expressed in terms of r,- = (r; r, j). 

 If k = f w + a, where w is even and < a ^ (w + 4)/6, 



& 



(w; i, |w a 1) = Wi ^xXi-i + a(0;_ 2 + 1*_ 2 ) 



x=0 



+ (a - l)(2 t -_ 2 + 3i_ 2 ) + - + (2o - 2),-_ 2 + (2o - !)<_,. 



This and a like formula include the rule by Ely. 110 



E. Catalan 134 noted that, if (N, p) is the number of partitions of ^V into 

 p distinct parts, and r(k) is the number of divisors of k, 



(N, 1) - 2(N, 2) + 3(N, 3) ---- 



= r(N) - T (N - 1) - r(N - 2) + r(N - 5) + r(N - 7) - . 



131 Amcr. Jour. Math., 7, 1885, 74-81. 



132 Nouv. Ann. Math., (3), 4, 1885, 408-417. 



133 Johns Hopkins Univ. Circ., 5, 1886, 64. 



134 Assoc. fran$. av. sc., 15, 1886, I, 86. 



