GENERATING FUNCTIONS IN THE THEORY OF NUMBERS. 
123 
For the true general (or condensed) generating function we have thus to evaluate 
the coaxial minors of the chess-hoard pattern determinant of the order, 
0 , 1 , 0 , 1 , 0 . . . 
1 , 0 , 1 , 0 , 1 . . . 
0 , 1 , 0 , 1 , 0 . . . 
1 , 0 , 1 , 0 , 1 . . . 
0 , 1 , 0 , 1 , 0 . . . 
Here, all the minors of Order 1 are zero. 
A minor (coaxial) of Order 2 has either the value zero or negative unity. If the 
minor be formed by deletion of all rows except the and and all columns except 
the and {q > p) the value will be zero, if = 0 mod 2, and will he 
negative unity in all other cases. 
Coaxial minors of Order > 2 as well as the whole determinant vanish, because in 
every case two rows are found to be identical. 
Hence the true generating function is 
_1_ 
1 — (ajg + + . . .) — + ocg + x,’ 
which may be written 
1 
a 'III 
I ^ '^XgXa,0,11 + 1 
Art. 18. Again for the enumeration of the permutations which are such that no 
quantity with an uneven suffix is in a compartment with an even suffix, and also no 
quantity with an even suffix is in a compartment with an uneven suffix, we are led to 
the complementary chess-board pattern determinant: — 
1, 0, 1, 0, 1 . . . 
0, 1, 0, 1, 0 . . . 
1, 0, I, 0, 1 . . . 
0, 1, 0, 1, 0 . . . 
1, 0, 1, 0, 1 . . . 
• • • • • • 
and thence to the true generating function 
R 2 
