246
ON A PROBLEM OF ARRANGEMENTS.
[669
Numbering the conditions 1, 2, 3according to the places to which they relate,
a single condition is called [1] ; two conditions are called [2] or [1, 1], according as
the numbers are consecutive or non-consecutive : three conditions are called [3], [2, 1],
or [1, 1, 1], according as the numbers are all three consecutive, two consecutive and
one not consecutive, or all non-consecutive ; and so on : the numbers which refer to
the conditions being always written in their natural order, and it being understood
that they follow each other cyclically, so that 1 is consecutive to n. Thus, n = 6, the
set 126 of conditions is [3], as consisting of 3 consecutive conditions; and similarly
1346 is [2, 2].
Consider a single condition [1], say this is 1 ; the arrangements which fail in
regard to this condition are those which contain in the first place a or b ; whichever
it be, the other n — 1 letters may be arranged in any form whatever ; and there are
thus 211 (n — 1) failing arrangements.
Next for two conditions; these may be [2], say the conditions are 1 and 2: or
else [1> 1], say they are 1 and 3. In the former case, the arrangements which fail
are those which contain in the first and second places ab, ac, or be : and for each of
these, the other n — 2 letters may be arranged in any order whatever ; there are thus
311 (n — 2) failing arrangements. In the latter case, the failing arrangements have in
the first place a or b, and in the third place c or d,—viz. the letters in these two
places are a.c, a. d,b.c, or b.d, and in each case the other n— 2 letters may be arranged
in any order whatever : the number of failing arrangements is thus = 2.2. II (n — 2).
And so, in general, when the conditions are [a, ¡3, y,...], the number of failing arrange
ments is
= (a+ l)(/3 + l) (7 + 1).. .11 (w — a — /3 — y...).
But for [w], that is, for the entire system of the n conditions, the number of failing
arrangements is (not as by the rule it should be = n +1, but) = 2,—viz. the only
arrangements which fail in regard to each of the n conditions are (as is at once
seen), cibc...jk, and bc...jka.
Changing now the notation so that [1], [2], [1, 1], &c., shall denote the number
of the conditions [1], [2], [1, 1], &c., respectively, it is easy to see the form of the
general result. If, for greater clearness, we write n — 6, we have
1 -2(1) +2(12)
No. = 720- {([1]= 6)2} 120+ f ([2] =6)3
(+([1, 1] = 9) 2.2
- 2(123)
24- j' ([3] =6)4
+ ([2,1] =12) 3.2
+ ([1,1,1]= 2)2.2.2
+ 2(1234)
+ ( ([4]
+ ([3, 1]
+ ([2, 2]
= 6) 5
= 6)4.
= 3)3.
- 2(12345)
-{([5] = 6)6}1
+(123456)
+ {([6] = 1)2};
-
