When the inverse ordinal number is even, it is a positive number. This is a property in permutations.
In mathematics, for a given permutation (a sequential arrangement of a set of numbers), an inverse ordinal number is the number in that permutation that is larger than the preceding number but the number that precedes it is smaller. If the inverse ordinal number of an arrangement is even, it is called an even permutation; If it is an odd number, it is called an odd arrangement.
There is a theorem in mathematics that for any arrangement of integers, the parity of its inverse ordinal determines the positivity and negativity of the arrangement. If the inverse ordinal number is even, the permutation is positive; If the inverse ordinal number is odd, then the permutation is negative.
Therefore, when the inverse ordinal number is even, it corresponds to a positive permutation. This has some application in the theory of permutations and combinations.
If the inverse ordinal number of a permutation is odd, we can use some concrete examples to illustrate that its corresponding permutation is negative.
Consider a simple permutation:.
The reverse pairs of this arrangement are: (1, 3) and (1, 2). Therefore, the inverse ordinal number is 2 and is even.
Now, if we consider the case where the inverse ordinal number is odd, such as permutation.
In reverse order, there are: (3, 2), (3, 1), (2, 1). So the inverse ordinal number is 3, which is an odd number.
In mathematics, these properties often involve the commutation of permutations, which are related to areas such as sorting algorithms, algebra, and combinatorics. This property is sometimes used to explain the parity of permutations and to calculate the sign of permutations.