CHAP. Hi] PARTITIONS. 125 



sums in which the preceding summand is multiplied by n(a l)y a or 



*)y a - 



Fergola 64 stated that the number of sets of positive integral solutions of 



H- a n x n = n 

 is A/(n!), where 



O n G\ "2 0"3 ' ' * ~ 



<T n i n 1 0"i ff% '.. 0" n _ 4 ffn 3 



<T n _2 n 2 0"i '.. <7 n _ 5 0"n 4 



A = 



<r n _3 n - 3 



> 



while o> is the sum of those divisors of r which occur among the positive 

 integers ai, a 2 > - When a,- = i (i = 1, , n), o> becomes the sum <r(r) 

 of all the divisors of r. If in A we change the sign of the second components 

 in the first column and change the sign before each a above the main 

 diagonal, we obtain a determinant equal to ( l) k nl when n is of the form 

 k(3k d= l)/2, but equal to zero when n is not of that form. 



C. Sardi 65 proved the preceding theorems. 



N. Trudi 66 proved Sylvester's formula 2PF a for the number P n (or, , X) 

 of partitions of n into elements a, , X. He also showed that W q = SF(p), 

 summed for the primitive qih roots p of unity, where F(p) is the coefficient 

 of 1/t in 



- (P + O-'-MI - (P + 0-}- 1 - {i - (P + o x r. 



Let d, ., a r be those of the numbers a, -, X which are divisible by q, 

 and 61, , b s the remaining numbers. Let 



e nt 



- p b e~ bt ) ' t r a : - -a r n(l - p 6 ) ' 



upon writing the denominator on the left as the exponential of its logarithm 

 and expanding the exponentials. Laws are given to determine the A's. 

 From the coefficient of t~ l we see that P n = 27 r> g , summed for the various 

 divisors q of the various elements a, - , X, where 



" * " ai ... a^a - p 61 )- -a -P 6 r 



summed for all the primitive <?th roots p of unity. Simplifications are given 

 in three cases: m = l,m = 2, r=l. He tabulated results for 



P n (l, 3, 6, 8), P n (l, 2, 3, 6, 8, 10), P n (l, 2, . ., g ), P n (2, 3, ., g), g < 8. 



"Giornale di Mat., 1, 1863, 63-64. 



65 Ibid., 3, 1865, 94-99, 377-380. 



66 Atti Accad. Sc. Fis. e Mat. Napoli, 2, 1865, No. 23, 50 pp. 



