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NOTE ON THE THEORY OF PERMUTATIONS.
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inversions by which the permutation can be derived. Let a, /3 ... be the number of
letters in the components (xyz), (u) (vw), &c., A the number of these components. The
number of inversions in question is evidently a — 1 + /3 — 1+ &c., or what comes to the
same thing, this number is (n — A). It will be convenient to term this number A the
exponent of irregularity of the permutation, and then (n — A) may be termed the
supplement of the exponent of irregularity. The rule in the case of a series of things,
all of them different, may consequently be stated as follows: “ a permutation is positive
or negative according as the supplement of the exponent of irregularity is even or
odd.” Consider now a series of things, not all of them different, and suppose that
this is derived from the system of the same number of things abc ... all of them
originally different, by supposing for instance a = b = &c., f=g = &c. A given permuta
tion of the system of things not all of them different, is of course derivable under
the supposition in question from several different permutations of the series abc
Considering the supplements of the exponents of irregularity of these last-mentioned
several permutations, we may consider the given permutation as positive or negative
according as the least of these numbers is even or odd. Hence we obtain the rule,
“a permutation of a series of things not all of them different, is positive or negative
according as the minimum supplement of irregularity of the permutation is even or
odd, the system being considered as a particular case of a system of the same number
of things all of them different, and the given permutation being successively considered
as derived from the different permutations which upon this supposition reduce them
selves to the given permutation.” This only differs from the rule, “a permutation of
a series of things, not all of them different, is positive or negative according as the
minimum number of inversions by which it can be obtained is even or odd, the
system being considered &c.,” inasmuch as the former enunciation is based upon and
indicates a direct method of determining the minimum number of inversions requisite in
order to obtain a given permutation; but the latter is, in simple cases, of the easier
application. As a very simple example, treated by the former rule, we may consider
the permutation 1212 derived from the primitive arrangement 1122. Considering this
primitive arrangement as a particular case of abed, there are four permutations which,
on the suppositions a — b — 1, c = d = 2, reduce themselves to 1212, viz. acbd, bead, adbc,
bdac, which are obtained by means of the respective symbols (a) (be) (d); (abc) (d);
(a) (bde); (abdc), the supplements of the exponents of irregularity being therefore
1, 2, 2, 3, or the permutation being negative; in fact it is obviously derivable by
means of an inversion of the two mean terms.
