Reverse permutation

A reverse permutation in combinatorics is a permutation that you get by inserting the position of an element into the position indicated by the value of the element in the numeric set. If the inverse permutation π is applied to a numerical series, and then the inverse to it π^-1 then in the end we will get such a result as if we did not use these permutations at all, this rule helps to check the correctness of the permutation performed.

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Reverse permutation

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What is the difference between permutation and reverse permutation

From any inversion table d₁,d₂,…d_n it is possible to uniquely restore the permutation that generates this table by sequentially determining the relative location of elements n, n-1,….,1 ( in this order). For example, a permutation corresponding to the inversion table (2,3,6,4,0,2,2,1,0) = (d₁,d₂,d₃,d₄,d₅,d₆,d₇,d₈,d₉), it can be constructed as follows: we write out the number 9, since d₈=1, then 8 is to the right of 9. Because d₇=2, to 7 stands to the right of 8 and 9. Since d₆=2, then 6 is to the right of the two numbers already written out, thus the arrangement of the numbers is obtained 9,8,6,7. Now we write 5 on the left, because d5=0, we put 4 after the four numbers already written out, 3 after 6 numbers written out (i.e. at the right end) and get 5,9,8,6,4,7,3. Inserting 2 and 1 in the same way, we come to the permutation (5,9,1,8,2,6,4,7,3).

Reverse permutations

One should not confuse “inversions” of permutations with inverse permutations. Let a₁,a₂,….a_n – various balls, the indexes of which we associate with the numbers of the balls. Then the original arrangement of the balls is uniquely determined by the identical permutation (e=1,2,…n)

The reverse of a permutation is a permutation that is obtained if the rows are swapped in the original permutation, and then the columns are ordered in ascending order by the upper elements, i.e. it is clear that a sequential change in the order of the balls according to the permutations and the reverse leads to their original location, i.e. to an identical permutation.