669] ON A PROBLEM OF ARRANGEMENTS, 
247 
or, reducing into numbers, this is 
No. = 720 - 1440 +1296 - 672 + 210 - 36 + 2, 
The formula for the next succeeding case, n = 7, gives 
80. 
No. = 5040 -10080 + 9240 - 5040 + 1764 - 392 +49 -2, = 579. 
Those for the preceding cases, n = 3, 4, 5, respectively are 
No. = 6-12+ 9- 2 
1, 
No. = 24- 48+ 40- 16+ 2 
No. = 120 - 240 + 210 - 100 + 25-2 
13. 
We have in general [1] = %, [2]=n, [1, 1] = 2 71 ( n ~ 3); and in the several columns 
of the formulae the sums of the numbers thus represented are equal to the coefficients 
of (l+l) 2 : thus, when n = 6 as above, the sums are 6, 15, 20, 15, 6, 1. As regards 
the calculation of the numbers in question, any symbol [a, /3, y] is a sum of symbols 
[a — a' + /3 — ¡3' + y — y + ...], where a + /3' + y' + ... is any partition of n — (a + /3 + y + ...); 
read, of the series of numbers 1, 2, 3,..., n, taken in cyclical order beginning with any 
number, retain a, omit cl, retain /3, omit ¡3', retain y, omit y', .... Thus in particular, 
n = 6, [1, 1] is a sum of symbols [1 — 3 + 1 — 1] and [1 — 2 + 1 — 2]; it is clear that 
any such symbol [a — a + ¡3 — /3' +...] is = n or a submultiple of n (in particular, if n 
be prime, the symbol is always = n): and we thus in every case obtain the value 
of [a, /3, y,...] by taking for the negative numbers the several partitions of 
n — (a + /3 + y + ...), 
[a — a + /3 — /3' + y — y + ...], 
and for each symbol 
writing its value, = 11 or a given submultiple of n, as just mentioned. There would, 
I think, be no use in pursuing the matter further, by seeking to obtain an analytical 
expression for the symbols [a, /3, y,...]. 
For the actual formation of the required arrangements, it is of course easy, when 
all the arrangements are written down, to strike out those which do not satisfy the 
prescribed conditions, and so obtain the system in question. Or introducing the notion 
of substitutions*, and accordingly considering each arrangement as derived by a 
substitution from the primitive arrangement abcd...jk, we can write down the substitu 
tions which give the system of arrangements in which no letter occupies its original 
place: viz. we must for this purpose partition the n letters into parts, no part less 
than 2, and then in each set taking one letter (say the first in alphabetical order) 
as fixed, permute in every possible way the other letters of the set; we thus obtain 
* In explanation of the notation of substitutions, observe that (abode) means that a is to be changed 
into b, b into c, c into d, d into e, e into a; and similarly (ab) (ode) means that a is to be changed into b, 
b into a, c into d, d into e, e into c.