Identifier
Values
[1] => [[1]] => [1] => [1] => 0
[2] => [[1,2]] => [1,2] => [1,2] => 1
[1,1] => [[1],[2]] => [2,1] => [2,1] => 0
[3] => [[1,2,3]] => [1,2,3] => [1,2,3] => 3
[2,1] => [[1,3],[2]] => [2,1,3] => [2,1,3] => 2
[1,1,1] => [[1],[2],[3]] => [3,2,1] => [3,2,1] => 0
[4] => [[1,2,3,4]] => [1,2,3,4] => [1,2,3,4] => 6
[3,1] => [[1,3,4],[2]] => [2,1,3,4] => [2,1,3,4] => 5
[2,2] => [[1,2],[3,4]] => [3,4,1,2] => [1,4,2,3] => 4
[2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => [3,2,1,4] => 3
[1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => [4,3,2,1] => 0
[5] => [[1,2,3,4,5]] => [1,2,3,4,5] => [1,2,3,4,5] => 10
[4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => [2,1,3,4,5] => 9
[3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => [1,4,2,3,5] => 8
[3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => [3,2,1,4,5] => 7
[2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => [1,4,5,2,3] => 6
[2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => [4,3,2,1,5] => 4
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 15
[5,1] => [[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => 14
[4,2] => [[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => [1,4,2,3,5,6] => 13
[4,1,1] => [[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => [3,2,1,4,5,6] => 12
[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [1,2,6,3,4,5] => 12
[3,2,1] => [[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => [1,4,5,2,3,6] => 11
[3,1,1,1] => [[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => [4,3,2,1,5,6] => 9
[2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => [1,6,2,5,3,4] => 9
[2,2,1,1] => [[1,4],[2,6],[3],[5]] => [5,3,2,6,1,4] => [1,5,4,6,2,3] => 8
[2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => 5
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 0
[7] => [[1,2,3,4,5,6,7]] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 21
[6,1] => [[1,3,4,5,6,7],[2]] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => 20
[5,1,1] => [[1,4,5,6,7],[2],[3]] => [3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => 18
[4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => 15
[3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => [5,4,3,2,1,6,7] => [5,4,3,2,1,6,7] => 11
[2,1,1,1,1,1] => [[1,7],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => 6
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => 0
[8] => [[1,2,3,4,5,6,7,8]] => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => 28
[7,1] => [[1,3,4,5,6,7,8],[2]] => [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => 27
[6,2] => [[1,2,5,6,7,8],[3,4]] => [3,4,1,2,5,6,7,8] => [1,4,2,3,5,6,7,8] => 26
[6,1,1] => [[1,4,5,6,7,8],[2],[3]] => [3,2,1,4,5,6,7,8] => [3,2,1,4,5,6,7,8] => 25
[5,1,1,1] => [[1,5,6,7,8],[2],[3],[4]] => [4,3,2,1,5,6,7,8] => [4,3,2,1,5,6,7,8] => 22
[4,4] => [[1,2,3,4],[5,6,7,8]] => [5,6,7,8,1,2,3,4] => [1,2,3,8,4,5,6,7] => 24
[4,3,1] => [[1,3,4,8],[2,6,7],[5]] => [5,2,6,7,1,3,4,8] => [1,4,2,7,3,5,6,8] => 23
[4,2,2] => [[1,2,7,8],[3,4],[5,6]] => [5,6,3,4,1,2,7,8] => [1,6,2,5,3,4,7,8] => 22
[4,1,1,1,1] => [[1,6,7,8],[2],[3],[4],[5]] => [5,4,3,2,1,6,7,8] => [5,4,3,2,1,6,7,8] => 18
[3,3,1,1] => [[1,4,5],[2,7,8],[3],[6]] => [6,3,2,7,8,1,4,5] => [1,5,4,2,8,3,6,7] => 20
[3,1,1,1,1,1] => [[1,7,8],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1,7,8] => [6,5,4,3,2,1,7,8] => 13
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [7,8,5,6,3,4,1,2] => [1,8,2,7,3,6,4,5] => 16
[2,2,1,1,1,1] => [[1,6],[2,8],[3],[4],[5],[7]] => [7,5,4,3,2,8,1,6] => [1,7,6,5,4,8,2,3] => 12
[2,1,1,1,1,1,1] => [[1,8],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1,8] => 7
[1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 0
[9] => [[1,2,3,4,5,6,7,8,9]] => [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => 36
[8,1] => [[1,3,4,5,6,7,8,9],[2]] => [2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => 35
[7,1,1] => [[1,4,5,6,7,8,9],[2],[3]] => [3,2,1,4,5,6,7,8,9] => [3,2,1,4,5,6,7,8,9] => 33
[6,1,1,1] => [[1,5,6,7,8,9],[2],[3],[4]] => [4,3,2,1,5,6,7,8,9] => [4,3,2,1,5,6,7,8,9] => 30
[5,1,1,1,1] => [[1,6,7,8,9],[2],[3],[4],[5]] => [5,4,3,2,1,6,7,8,9] => [5,4,3,2,1,6,7,8,9] => 26
[4,1,1,1,1,1] => [[1,7,8,9],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1,7,8,9] => [6,5,4,3,2,1,7,8,9] => 21
[3,1,1,1,1,1,1] => [[1,8,9],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1,8,9] => [7,6,5,4,3,2,1,8,9] => 15
[2,1,1,1,1,1,1,1] => [[1,9],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1,9] => [8,7,6,5,4,3,2,1,9] => 8
[1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => 0
[10] => [[1,2,3,4,5,6,7,8,9,10]] => [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => 45
[9,1] => [[1,3,4,5,6,7,8,9,10],[2]] => [2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9,10] => 44
[8,1,1] => [[1,4,5,6,7,8,9,10],[2],[3]] => [3,2,1,4,5,6,7,8,9,10] => [3,2,1,4,5,6,7,8,9,10] => 42
[7,1,1,1] => [[1,5,6,7,8,9,10],[2],[3],[4]] => [4,3,2,1,5,6,7,8,9,10] => [4,3,2,1,5,6,7,8,9,10] => 39
[6,1,1,1,1] => [[1,6,7,8,9,10],[2],[3],[4],[5]] => [5,4,3,2,1,6,7,8,9,10] => [5,4,3,2,1,6,7,8,9,10] => 35
[5,1,1,1,1,1] => [[1,7,8,9,10],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1,7,8,9,10] => [6,5,4,3,2,1,7,8,9,10] => 30
[4,1,1,1,1,1,1] => [[1,8,9,10],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1,8,9,10] => [7,6,5,4,3,2,1,8,9,10] => 24
[3,1,1,1,1,1,1,1] => [[1,9,10],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1,9,10] => [8,7,6,5,4,3,2,1,9,10] => 17
[2,1,1,1,1,1,1,1,1] => [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,4,3,2,1,10] => [9,8,7,6,5,4,3,2,1,10] => 9
[1,1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 0
[] => [] => [] => [] => 0
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Description
The number of non-inversions of a permutation.
For a permutation of $\{1,\ldots,n\}$, this is given by $\operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi)$.
Map
reading tableau
Description
Return the RSK recording tableau of the reading word of the (standard) tableau $T$ labeled down (in English convention) each column to the shape of a partition.
Map
major-index to inversion-number bijection
Description
Return the permutation whose Lehmer code equals the major code of the preimage.
This map sends the major index to the number of inversions.
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.