454 Proceedings of Royal Society of Edinburgh. [sess. 
Table op Y n ,s, the Number op Permutations op 1, 2, 3, . . n 
which have S Inverted-pairs. 
n— 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
2 
II 
0 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
2 
2 
5 
9 
14 
20 
27 
35 
44 
54 
3 
1 
6 
15 
29 
49 
76 
111 
155 
209 
4 
5 
20 
49 
98 
174 
285 
440 
649 
5 
3 
22 
71 
169 
343 
628 
1068 
1717 
6 
1 
20 
90 
259 
602 
1230 
2298 
4015 
7 
15 
101 
359 
961 
2191 
4489 
8504 
8 
9 
101 
455 
1415 
3606 
8095 
16599 
9 
4 
90 
531 
1940 
5545 
13640 
30239 
10 
1 
71 
573 
2493 
8031 
21670 
51909 
11 
49 
573 
3017 
11021 
32683 
84591 
12 
29 
531 
3450 
14395 
47043 
131625 
13 
14 
455 
3736 
17957 
64889 
&c. 
14 
5 
359 
3836 
21540 
&c. 
15 
1 
259 
3736 
24584 
16 
169 
3450 
27073 
17 
98 
3017 
28675 
18 
49 
2493 
29228 
19 
20 
1940 
28675 
20 
6 
1415 
27073 
21 
1 
961 
24584 
22 
602 
21540 
23 
343 
17957 
etc. 
etc. 
etc. 
The practical rule corresponding to the difference-equation will 
he found to be : — To find any number in any column , say the 
column headed “ 5/’ add five consecutive numbers of the preceding 
column, beginning at the corresponding place in the latter column. 
For example, 
^ 5,8 = 0 + 0 + 1 + 3 + 5 , 
= 9. 
(17) Knowing that 
y =V +Y +Y + +V 
n,S ¥ »-l,5 ‘ n-1,8-1^ n-l,S~n+l 
we have, of course, 
V — V + V 4- -4- V 
v n,S - 1 w-1,5-1^ ¥ re- 1,8-2 ~ T re-1, 5-/t, 
and therefore by subtraction 
