10 
Proceedings of Royal Society of Edinburgh. [sess. 
As there is evidently 1 self-conjugate permutation when n= 1, 
and 2 when n = 2, the first ten values U n are 
U,= 1 , 
u 6 
= 76, 
u 2 = 2, 
= 232, 
U 3 = 4, 
u 8 
= 764, 
u 4 =10, 
u 9 
= 2620, 
U 5 = 26, 
U 10 
= 9496. 
4. The expression first got for the number of self-conjugate 
permutations ought, of course, to satisfy this difference-equation. 
To show directly that it does, it is best in the first place to simplify 
it a little. The typical term 
1 
cu c 
r ! 
n , 2 v - / n- 2,2 4,2 
C„. 
\ n(n-\) (n-2)(n-S) (^-4)(w-5) (n - 2r + 2)(n - 2r + 1) 
= ri 1.2 * 172 ‘ T72 172 
l)(w- 2) . . . (n-2r+l) 
r\ 2 r ’ 
w(rc-l)(w-2) . . . (n-2r+l) (2r)\ 
(2 r)\ X r\ 2 r 
= C„,2, (1.3.5. . .2,-1). 
The whole expression consequently becomes 
1 + 1 . C W(2 + 1 . 3C n>4 + 1\ 3 . 5C„ >6 + 1 .3.5. 7C„ >8 + . . . 
Taking this for JJ n we have 
+ (n - 1)U m _2 = 1 + 1 . C w _ 1>2 + 1 . 3C n _ lt4 + 1 . 3 . 5C OT _ li6 + .... 
+ (n — 1) { 1 + 1 . C w _ 2i2 + 1 . 3C w _ 2>4 + 1 . 3. 5C w _ 2>6 + . . .) 
=i+i.|c„. 1 , 2 +(»-i) } +i.3 
+ 1.3.5 { C„_, 6 + ^ C„- 2 .4 } + 
But the general term on the right here 
= 1 . 3 . 5 . . . . (2r- 1) | C„_ llSr + C n .^. t | ( 
✓ 
= 1.3.5. . . . (2r — 1) | C„_ lt2) . + C j" 
= 1.3.5. . . . (2r— 1) C„, 2r . 
