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Figure 2.4.1
From this example we see that there are 24 permutations of {1, 2, 3, 4}. This result could have been anticipated without actually
listing the permutations by arguing as follows. Since the first position can be filled in four ways and then the second position in
three ways, there are ways of filling the first two positions. Since the third position can then be filled in two ways, there are
ways of filling the first three positions. Finally, since the last position can then be filled in only one way, there are
ways of filling all four positions. In general, the set will have different
permutations.
We will denote a general permutation of the set by . Here, is the first integer in the permutation,
is the second, and so on. An inversion is said to occur in a permutation whenever a larger integer precedes a
smaller one. The total number of inversions occurring in a permutation can be obtained as follows: (1) find the number of integers
that are less than and that follow in the permutation; (2) find the number of integers that are less than and that follow in
the permutation. Continue this counting process for . The sum of these numbers will be the total number of inversions
in the permutation.
EXAMPLE 3 Counting Inversions
Determine the number of inversions in the following permutations:
(a) (6, 1, 3, 4, 5, 2)
(b) (2, 4, 1, 3)
(c) (1, 2, 3, 4)
Solution .
(a) The number of inversions is
