Identifier
Values
[1] => [[1]] => [1] => [1] => 0
[2] => [[1,2]] => [1,2] => [2,1] => 2
[1,1] => [[1],[2]] => [2,1] => [1,2] => 0
[3] => [[1,2,3]] => [1,2,3] => [3,2,1] => 6
[2,1] => [[1,2],[3]] => [3,1,2] => [1,3,2] => 3
[1,1,1] => [[1],[2],[3]] => [3,2,1] => [1,2,3] => 0
[4] => [[1,2,3,4]] => [1,2,3,4] => [4,3,2,1] => 12
[3,1] => [[1,2,3],[4]] => [4,1,2,3] => [1,4,3,2] => 8
[2,2] => [[1,2],[3,4]] => [3,4,1,2] => [2,1,4,3] => 6
[2,1,1] => [[1,2],[3],[4]] => [4,3,1,2] => [1,2,4,3] => 4
[1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => [1,2,3,4] => 0
[5] => [[1,2,3,4,5]] => [1,2,3,4,5] => [5,4,3,2,1] => 20
[4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => [1,5,4,3,2] => 15
[3,2] => [[1,2,3],[4,5]] => [4,5,1,2,3] => [2,1,5,4,3] => 12
[3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => [1,2,5,4,3] => 10
[2,2,1] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => [1,3,2,5,4] => 8
[2,1,1,1] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => [1,2,3,5,4] => 5
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 30
[5,1] => [[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => [1,6,5,4,3,2] => 24
[4,2] => [[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => [2,1,6,5,4,3] => 20
[4,1,1] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => [1,2,6,5,4,3] => 18
[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [3,2,1,6,5,4] => 18
[3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => [1,3,2,6,5,4] => 15
[3,1,1,1] => [[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => [1,2,3,6,5,4] => 12
[2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => [2,1,4,3,6,5] => 12
[2,2,1,1] => [[1,2],[3,4],[5],[6]] => [6,5,3,4,1,2] => [1,2,4,3,6,5] => 10
[2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [6,5,4,3,1,2] => [1,2,3,4,6,5] => 6
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0
[5,1,1] => [[1,2,3,4,5],[6],[7]] => [7,6,1,2,3,4,5] => [1,2,7,6,5,4,3] => 28
[4,2,1] => [[1,2,3,4],[5,6],[7]] => [7,5,6,1,2,3,4] => [1,3,2,7,6,5,4] => 24
[4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => [7,6,5,1,2,3,4] => [1,2,3,7,6,5,4] => 21
[3,3,1] => [[1,2,3],[4,5,6],[7]] => [7,4,5,6,1,2,3] => [1,4,3,2,7,6,5] => 22
[3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => [7,6,4,5,1,2,3] => [1,2,4,3,7,6,5] => 18
[3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => [7,6,5,4,1,2,3] => [1,2,3,4,7,6,5] => 14
[2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => [7,5,6,3,4,1,2] => [1,3,2,5,4,7,6] => 15
[2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => [7,6,5,3,4,1,2] => [1,2,3,5,4,7,6] => 12
[2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,1,2] => [1,2,3,4,5,7,6] => 7
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => 0
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Description
The number of inversions plus the major index of a permutation.
This is, the sum of St000004The major index of a permutation. and St000018The number of inversions of a permutation..
Map
complement
Description
Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
Map
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.
Map
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.