Identifier
Values
[1] => [[1]] => [1] => 1
[2] => [[1,2]] => [1,2] => 1
[1,1] => [[1],[2]] => [2,1] => 2
[3] => [[1,2,3]] => [1,2,3] => 1
[2,1] => [[1,2],[3]] => [3,1,2] => 3
[1,1,1] => [[1],[2],[3]] => [3,2,1] => 6
[4] => [[1,2,3,4]] => [1,2,3,4] => 1
[3,1] => [[1,2,3],[4]] => [4,1,2,3] => 4
[2,2] => [[1,2],[3,4]] => [3,4,1,2] => 6
[2,1,1] => [[1,2],[3],[4]] => [4,3,1,2] => 12
[1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 24
[5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 1
[4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => 5
[3,2] => [[1,2,3],[4,5]] => [4,5,1,2,3] => 10
[3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => 20
[2,2,1] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => 30
[2,1,1,1] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => 60
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => 120
[6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => 1
[5,1] => [[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => 6
[4,2] => [[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => 15
[4,1,1] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => 30
[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => 20
[3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => 60
[3,1,1,1] => [[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => 120
[2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => 90
[2,2,1,1] => [[1,2],[3,4],[5],[6]] => [6,5,3,4,1,2] => 180
[2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [6,5,4,3,1,2] => 360
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => 720
[7] => [[1,2,3,4,5,6,7]] => [1,2,3,4,5,6,7] => 1
[6,1] => [[1,2,3,4,5,6],[7]] => [7,1,2,3,4,5,6] => 7
[5,2] => [[1,2,3,4,5],[6,7]] => [6,7,1,2,3,4,5] => 21
[5,1,1] => [[1,2,3,4,5],[6],[7]] => [7,6,1,2,3,4,5] => 42
[4,3] => [[1,2,3,4],[5,6,7]] => [5,6,7,1,2,3,4] => 35
[4,2,1] => [[1,2,3,4],[5,6],[7]] => [7,5,6,1,2,3,4] => 105
[4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => [7,6,5,1,2,3,4] => 210
[3,3,1] => [[1,2,3],[4,5,6],[7]] => [7,4,5,6,1,2,3] => 140
[3,2,2] => [[1,2,3],[4,5],[6,7]] => [6,7,4,5,1,2,3] => 210
[3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => [7,6,4,5,1,2,3] => 420
[3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => [7,6,5,4,1,2,3] => 840
[2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => [7,5,6,3,4,1,2] => 630
[2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => [7,6,5,3,4,1,2] => 1260
[2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,1,2] => 2520
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1] => 5040
[8] => [[1,2,3,4,5,6,7,8]] => [1,2,3,4,5,6,7,8] => 1
[7,1] => [[1,2,3,4,5,6,7],[8]] => [8,1,2,3,4,5,6,7] => 8
[6,2] => [[1,2,3,4,5,6],[7,8]] => [7,8,1,2,3,4,5,6] => 28
[6,1,1] => [[1,2,3,4,5,6],[7],[8]] => [8,7,1,2,3,4,5,6] => 56
[5,3] => [[1,2,3,4,5],[6,7,8]] => [6,7,8,1,2,3,4,5] => 56
[5,2,1] => [[1,2,3,4,5],[6,7],[8]] => [8,6,7,1,2,3,4,5] => 168
[5,1,1,1] => [[1,2,3,4,5],[6],[7],[8]] => [8,7,6,1,2,3,4,5] => 336
[4,4] => [[1,2,3,4],[5,6,7,8]] => [5,6,7,8,1,2,3,4] => 70
[4,3,1] => [[1,2,3,4],[5,6,7],[8]] => [8,5,6,7,1,2,3,4] => 280
[4,2,2] => [[1,2,3,4],[5,6],[7,8]] => [7,8,5,6,1,2,3,4] => 420
[4,2,1,1] => [[1,2,3,4],[5,6],[7],[8]] => [8,7,5,6,1,2,3,4] => 840
[4,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8]] => [8,7,6,5,1,2,3,4] => 1680
[3,3,2] => [[1,2,3],[4,5,6],[7,8]] => [7,8,4,5,6,1,2,3] => 560
[3,3,1,1] => [[1,2,3],[4,5,6],[7],[8]] => [8,7,4,5,6,1,2,3] => 1120
[3,2,2,1] => [[1,2,3],[4,5],[6,7],[8]] => [8,6,7,4,5,1,2,3] => 1680
[3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => [8,7,6,4,5,1,2,3] => 3360
[3,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,1,2,3] => 6720
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [7,8,5,6,3,4,1,2] => 2520
[2,2,2,1,1] => [[1,2],[3,4],[5,6],[7],[8]] => [8,7,5,6,3,4,1,2] => 5040
[2,2,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8]] => [8,7,6,5,3,4,1,2] => 10080
[2,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,1,2] => 20160
[1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1] => 40320
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Description
The number of permutations less than or equal to a permutation in left weak order.
This is the same as the number of permutations less than or equal to the given permutation in right weak order.
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.
Map
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.