Identifier
Values
[1] => [[1]] => [[1]] => [1] => 1
[2] => [[1,2]] => [[1],[2]] => [2,1] => 2
[1,1] => [[1],[2]] => [[1,2]] => [1,2] => 1
[3] => [[1,2,3]] => [[1],[2],[3]] => [3,2,1] => 3
[2,1] => [[1,2],[3]] => [[1,3],[2]] => [2,1,3] => 2
[1,1,1] => [[1],[2],[3]] => [[1,2,3]] => [1,2,3] => 1
[4] => [[1,2,3,4]] => [[1],[2],[3],[4]] => [4,3,2,1] => 4
[3,1] => [[1,2,3],[4]] => [[1,4],[2],[3]] => [3,2,1,4] => 3
[2,2] => [[1,2],[3,4]] => [[1,3],[2,4]] => [2,4,1,3] => 3
[2,1,1] => [[1,2],[3],[4]] => [[1,3,4],[2]] => [2,1,3,4] => 2
[1,1,1,1] => [[1],[2],[3],[4]] => [[1,2,3,4]] => [1,2,3,4] => 1
[5] => [[1,2,3,4,5]] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => 5
[4,1] => [[1,2,3,4],[5]] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => 4
[3,2] => [[1,2,3],[4,5]] => [[1,4],[2,5],[3]] => [3,2,5,1,4] => 5
[3,1,1] => [[1,2,3],[4],[5]] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 3
[2,2,1] => [[1,2],[3,4],[5]] => [[1,3,5],[2,4]] => [2,4,1,3,5] => 3
[2,1,1,1] => [[1,2],[3],[4],[5]] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 2
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [[1,2,3,4,5]] => [1,2,3,4,5] => 1
[6] => [[1,2,3,4,5,6]] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => 6
[5,1] => [[1,2,3,4,5],[6]] => [[1,6],[2],[3],[4],[5]] => [5,4,3,2,1,6] => 5
[4,2] => [[1,2,3,4],[5,6]] => [[1,5],[2,6],[3],[4]] => [4,3,2,6,1,5] => 7
[4,1,1] => [[1,2,3,4],[5],[6]] => [[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => 4
[3,3] => [[1,2,3],[4,5,6]] => [[1,4],[2,5],[3,6]] => [3,6,2,5,1,4] => 6
[3,2,1] => [[1,2,3],[4,5],[6]] => [[1,4,6],[2,5],[3]] => [3,2,5,1,4,6] => 5
[3,1,1,1] => [[1,2,3],[4],[5],[6]] => [[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => 3
[2,2,2] => [[1,2],[3,4],[5,6]] => [[1,3,5],[2,4,6]] => [2,4,6,1,3,5] => 4
[2,2,1,1] => [[1,2],[3,4],[5],[6]] => [[1,3,5,6],[2,4]] => [2,4,1,3,5,6] => 3
[2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => 2
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => 1
[7] => [[1,2,3,4,5,6,7]] => [[1],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1] => 7
[6,1] => [[1,2,3,4,5,6],[7]] => [[1,7],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1,7] => 6
[5,1,1] => [[1,2,3,4,5],[6],[7]] => [[1,6,7],[2],[3],[4],[5]] => [5,4,3,2,1,6,7] => 5
[4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => [[1,5,6,7],[2],[3],[4]] => [4,3,2,1,5,6,7] => 4
[3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => [[1,4,5,6,7],[2],[3]] => [3,2,1,4,5,6,7] => 3
[2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => [[1,3,5,6,7],[2,4]] => [2,4,1,3,5,6,7] => 3
[2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => [[1,3,4,5,6,7],[2]] => [2,1,3,4,5,6,7] => 2
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [[1,2,3,4,5,6,7]] => [1,2,3,4,5,6,7] => 1
[8] => [[1,2,3,4,5,6,7,8]] => [[1],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1] => 8
[7,1] => [[1,2,3,4,5,6,7],[8]] => [[1,8],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1,8] => 7
[6,1,1] => [[1,2,3,4,5,6],[7],[8]] => [[1,7,8],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1,7,8] => 6
[5,1,1,1] => [[1,2,3,4,5],[6],[7],[8]] => [[1,6,7,8],[2],[3],[4],[5]] => [5,4,3,2,1,6,7,8] => 5
[4,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8]] => [[1,5,6,7,8],[2],[3],[4]] => [4,3,2,1,5,6,7,8] => 4
[3,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8]] => [[1,4,5,6,7,8],[2],[3]] => [3,2,1,4,5,6,7,8] => 3
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [[1,3,5,7],[2,4,6,8]] => [2,4,6,8,1,3,5,7] => 5
[2,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8]] => [[1,3,4,5,6,7,8],[2]] => [2,1,3,4,5,6,7,8] => 2
[1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => [[1,2,3,4,5,6,7,8]] => [1,2,3,4,5,6,7,8] => 1
[9] => [[1,2,3,4,5,6,7,8,9]] => [[1],[2],[3],[4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,4,3,2,1] => 9
[8,1] => [[1,2,3,4,5,6,7,8],[9]] => [[1,9],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1,9] => 8
[7,1,1] => [[1,2,3,4,5,6,7],[8],[9]] => [[1,8,9],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1,8,9] => 7
[6,1,1,1] => [[1,2,3,4,5,6],[7],[8],[9]] => [[1,7,8,9],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1,7,8,9] => 6
[5,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9]] => [[1,6,7,8,9],[2],[3],[4],[5]] => [5,4,3,2,1,6,7,8,9] => 5
[4,1,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8],[9]] => [[1,5,6,7,8,9],[2],[3],[4]] => [4,3,2,1,5,6,7,8,9] => 4
[3,1,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8],[9]] => [[1,4,5,6,7,8,9],[2],[3]] => [3,2,1,4,5,6,7,8,9] => 3
[2,1,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8],[9]] => [[1,3,4,5,6,7,8,9],[2]] => [2,1,3,4,5,6,7,8,9] => 2
[1,1,1,1,1,1,1,1,1] => [[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] => 1
[10] => [[1,2,3,4,5,6,7,8,9,10]] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,5,4,3,2,1] => 10
[9,1] => [[1,2,3,4,5,6,7,8,9],[10]] => [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,4,3,2,1,10] => 9
[8,1,1] => [[1,2,3,4,5,6,7,8],[9],[10]] => [[1,9,10],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1,9,10] => 8
[7,1,1,1] => [[1,2,3,4,5,6,7],[8],[9],[10]] => [[1,8,9,10],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1,8,9,10] => 7
[6,1,1,1,1] => [[1,2,3,4,5,6],[7],[8],[9],[10]] => [[1,7,8,9,10],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1,7,8,9,10] => 6
[5,1,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9],[10]] => [[1,6,7,8,9,10],[2],[3],[4],[5]] => [5,4,3,2,1,6,7,8,9,10] => 5
[4,1,1,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8],[9],[10]] => [[1,5,6,7,8,9,10],[2],[3],[4]] => [4,3,2,1,5,6,7,8,9,10] => 4
[3,1,1,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8],[9],[10]] => [[1,4,5,6,7,8,9,10],[2],[3]] => [3,2,1,4,5,6,7,8,9,10] => 3
[2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10]] => [[1,3,5,7,9],[2,4,6,8,10]] => [2,4,6,8,10,1,3,5,7,9] => 6
[2,1,1,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]] => [[1,3,4,5,6,7,8,9,10],[2]] => [2,1,3,4,5,6,7,8,9,10] => 2
[1,1,1,1,1,1,1,1,1,1] => [[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] => 1
[12] => [[1,2,3,4,5,6,7,8,9,10,11,12]] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]] => [12,11,10,9,8,7,6,5,4,3,2,1] => 12
[2,2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => [2,4,6,8,10,12,1,3,5,7,9,11] => 7
[] => [] => [] => [] => 1
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Description
The number of longest increasing subsequences of a permutation.
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.
Map
conjugate
Description
Sends a standard tableau to its conjugate tableau.