GENERATING FUNCTIONS IN THE THEORY OF NUMBERS. 
121 
Xy 
occurvS 
“i 
55 
55 
/3i 
55 
55 
X,2 
55 
55 
55 
^■2 
• 
• 
« 
55 
55 
^2 
55 
Oi,i 
55 
55 
55 
55 
Vu 
times in places 
? 5 
5 J 
? i 
55 
5? 
55 
originally occu 
s? 
55 
55 
55 
55 
55 
55 
55 
pied by 
an ajj 
55 
X2 
55 
55 
rt’i 
55 
X2 
55 
Xa 
5 5 
5 5 
X2 
55 
X/i. 
Accordingly the proper generating function for the enumeration of the permutations 
possessing this j)roperty is 
_ 1 _ 
I (1 • • • (1 I 
Art. 16. As an interesting particular case we can find the generating function for 
the enumeration of those permutations of the quantities in 
/y» cv» ^2 
1 *^2 • • • 
which possess the property that no quantity is in the place originally occupied ; that 
is, in the permutation, no is to occupy a position formerly occupied by an x„ s having 
all values from 1 to n. 
Clearly we have merely to put 
■= h.^ = — . . . = n„ = 0, 
and the remaining letters, a, h, c, . . . n equal to unity. The generating function 
involves the coaxial minors of the determinant of the order 
0 , 1 , 1 , . . 1 
1 , 0 , 1 , . . 1 
1 , 1 , 0 , . . 1 
1 , 1 , 1 , . . 0 
This determinant has the value 
i-Y {n - I), 
MDCCCXCIV.—A. 
R- 
