72] 
423 
72. 
NOTE ON THE THEORY OF PERMUTATIONS. 
[From the Philosophical Magazine, vol. xxxiv. (1849), pp. 527—529.] 
It seems worth inquiring whether the distinction made use of in the theory of 
determinants, of the permutations of a series of things all of them different, into 
positive and negative permutations, can be made in the case of a series of things 
not all of them different. The ordinary rule is well known, viz. permutations are con 
sidered as positive or negative according as they are derived from the primitive 
arrangement by an even or an odd number of inversions (that is, interchanges of 
two things) ; and it is obvious that this rule fails when two or more of the series of 
things become identical, since in this case any given permutation can be derived 
indifferently by means of an even or an odd number of inversions. To state the rule 
in a different form, it will be convenient to enter into some preliminary explanations. 
Consider a series of n things, all of them different, and let abc ... be the primitive 
arrangement; imagine a symbol such as (xyz) (it) (vw) ... where x, y, &c., are the entire 
series of n things, and which symbol is to be considered as furnishing a rule by which 
a permutation is to be derived from the primitive arrangement abc... as follows, viz. 
the (xyz) of the symbol denotes that the letters x, y, z in the primitive arrange 
ment abc ... are to be interchanged x into y, y into z, z into x. The (u) of the 
symbol denotes that the letter u in the primitive arrangement abc ... is to remain 
unaltered. The (vw) of the symbol denotes that the letters v, w in the primitive 
arrangement are to be interchanged v into w and w into v, and so on. It is easily 
seen that any permutation whatever can be derived (and derived in one manner only) 
from the primitive arrangement by means of a rule such as is furnished by the symbol 
in question 1 ; and moreover that the number of inversions requisite in order to obtain 
the permutation by means of the rule in question, is always the smallest number of 
1 See on this subject Cauchy’s “Mémoire sur les Arrangemens Ac.”, Exercises d’Analyse et de Physique 
Mathématique, t. hi. [1844], p. 151.