Mathematical_Recreations-Kraitchik-2e

Games 805
aspects of the theory of permutations that may be explained
very simply. Suppose we have any set of objects to which
some normal order has been assigned. By way of illustration
we may use the numbers from 1 through 5 in the order 12345.
Let them be put in an arbitrary
order, say 25143. Any pair of the
objects that is now not in its normal
relative order is called an inve:rsion.
(21 and 53 are inversions in 25143.)
The total number of inversions in
any permutation of the objects is
most easily found by determining
for each object how many objects FIGURE 167. The 15
that normally precede it now follow Puzzle; Start.
it. In the permutation 25143, for
example, there are the five inversions 21,51,54, 53,43. According
as the number of inversions is even or odd the permutation
is called even or odd, and the evenness or oddness
of a permutation is called its parity. Thus the parity of a permutation
depends solely on the order of the objects permuted,
not on the manner in which the permutation was obtained.
A fundamental property of permutations is the fact that
the parity of a permutation is changed whenever any two of
the objects are transposed. Thus the odd permutation 25143
becomes the even permutation 23145 when 5 and 3 are interchanged.
This is obviously true when the objects transposed
are adjacent, since only the relative order of these two objects
is affected. If the objects transposed are not adjacent,
one can obtain the result of the transposition by a sequence
of transpositions of adjacent pairs. If object P in position p
is to be interchanged with object Q in position q, (q - p =
k > 1), then k successive transpositions of P with the object
next to it on the right will bring P to position q. The last of
these transpositions, however, moves Q one position to the