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



E. Catalan 25 proved that Xi + + x n = m has ( n+ ~ 1 ) sets of solu- 

 tions is 0. 



0. Rodrigues 26 noted that the number Z n , ; of ways of permuting n 

 letters, such that there are i inversions in each permutation, is the number 

 of sets of solutions of X Q + Xi + + x n -\ = i, where x k takes only the 

 values 0,1, , k and where the value of x k for each permutation is the 

 number of inversions produced by x k +i. Thus Z n , i is the coefficient of 

 ^ in the expansion of 



(1 + 0(1 + < + V (i + *+ + i 71 ' 1 ) = (i - 0-"^, 



where P = (1 - t)(l - F) (1 - t n ). Let E n . i be the coefficient of t* 

 in the expansion of P. Thus E n , i = E n -i, i En-i. i- n , E n , i E if i, and 



Z n , i = E n , ' + ( 1 ) En, i-l + ' ' ' + ( j ) E n , 0- 



Here E n , ,- equals the excess of the number of partitions of i in an even 

 number of distinct integers < n + 1 over the number in an odd, the number 

 of parts being also < n -f- 1. 



M. A. Stern 27 wrote n C q (or B C,) for the number of combinations without 

 (or with) repetitions with the sum n and class q (i. e., q at a tune), meaning 

 the number of partitions of n into q distinct parts (or equal or distinct 

 parts). Evidently n C' 2 = [ft/2]. Hence, by (2), we get 



nl n 1 n 1 n 1 



nc;= i: z zi;[j{n-(afc+i)- (4*1+1) 



*j_S= V4= *1= ft = 



Since ^Cg = m C' q if m = n q(q l)/2, we get by (1), 



Again, 



If C(n) is the number of all partitions of n into distinct parts, 



In 



Z (- l) z C(n - T//2) = (- 1)' or (y = W =F 2), 



^=0 



according as n is or is not of the form 3r 2 T r. This follows by expanding 



1 - x z 1 - x 4 I - x 6 



' ' - ' - ' ' - * 



1 x 1 x 2 1 x 3 



25 Jour, de Math., 3, 1838, 111-2. 



26 Jour, de Math., 4, 1839, 236-240. 



27 Jour, fur Math., 21, 1840, 91-97, 177-9. Further results were quoted under Stern 16 " of 



Ch. X in Vol. I of this History. 



