Identifier
Values
[1] => [1] => [[1]] => 1
[2] => [1,1] => [[1],[2]] => 1
[1,1] => [2] => [[1,2]] => 2
[3] => [2,1] => [[1,3],[2]] => 2
[2,1] => [1,1,1] => [[1],[2],[3]] => 1
[1,1,1] => [3] => [[1,2,3]] => 3
[4] => [2,2] => [[1,2],[3,4]] => 3
[3,1] => [1,1,1,1] => [[1],[2],[3],[4]] => 1
[2,2] => [2,1,1] => [[1,4],[2],[3]] => 2
[2,1,1] => [3,1] => [[1,3,4],[2]] => 3
[1,1,1,1] => [4] => [[1,2,3,4]] => 4
[5] => [3,2] => [[1,2,5],[3,4]] => 4
[4,1] => [3,1,1] => [[1,4,5],[2],[3]] => 3
[3,2] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => 1
[3,1,1] => [2,1,1,1] => [[1,5],[2],[3],[4]] => 2
[2,2,1] => [2,2,1] => [[1,3],[2,5],[4]] => 3
[2,1,1,1] => [4,1] => [[1,3,4,5],[2]] => 4
[1,1,1,1,1] => [5] => [[1,2,3,4,5]] => 5
[6] => [3,3] => [[1,2,3],[4,5,6]] => 5
[5,1] => [3,2,1] => [[1,3,6],[2,5],[4]] => 4
[4,2] => [2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => 2
[4,1,1] => [2,2,1,1] => [[1,4],[2,6],[3],[5]] => 3
[3,3] => [3,1,1,1] => [[1,5,6],[2],[3],[4]] => 3
[3,2,1] => [1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => 1
[3,1,1,1] => [4,1,1] => [[1,4,5,6],[2],[3]] => 4
[2,2,2] => [2,2,2] => [[1,2],[3,4],[5,6]] => 4
[2,2,1,1] => [4,2] => [[1,2,5,6],[3,4]] => 5
[2,1,1,1,1] => [5,1] => [[1,3,4,5,6],[2]] => 5
[1,1,1,1,1,1] => [6] => [[1,2,3,4,5,6]] => 6
[7] => [4,3] => [[1,2,3,7],[4,5,6]] => 6
[6,1] => [3,3,1] => [[1,3,4],[2,6,7],[5]] => 5
[5,2] => [3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => 4
[5,1,1] => [4,2,1] => [[1,3,6,7],[2,5],[4]] => 5
[4,3] => [2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => 3
[4,2,1] => [1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => 1
[4,1,1,1] => [2,2,2,1] => [[1,3],[2,5],[4,7],[6]] => 4
[3,3,1] => [2,1,1,1,1,1] => [[1,7],[2],[3],[4],[5],[6]] => 2
[3,2,2] => [3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => 3
[3,2,1,1] => [4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => 4
[3,1,1,1,1] => [5,1,1] => [[1,4,5,6,7],[2],[3]] => 5
[2,2,2,1] => [3,2,2] => [[1,2,7],[3,4],[5,6]] => 5
[2,2,1,1,1] => [5,2] => [[1,2,5,6,7],[3,4]] => 6
[2,1,1,1,1,1] => [6,1] => [[1,3,4,5,6,7],[2]] => 6
[1,1,1,1,1,1,1] => [7] => [[1,2,3,4,5,6,7]] => 7
[8] => [4,4] => [[1,2,3,4],[5,6,7,8]] => 7
[7,1] => [4,3,1] => [[1,3,4,8],[2,6,7],[5]] => 6
[6,2] => [3,3,1,1] => [[1,4,5],[2,7,8],[3],[6]] => 5
[6,1,1] => [4,2,2] => [[1,2,7,8],[3,4],[5,6]] => 6
[5,3] => [2,2,2,1,1] => [[1,4],[2,6],[3,8],[5],[7]] => 4
[5,2,1] => [4,1,1,1,1] => [[1,6,7,8],[2],[3],[4],[5]] => 4
[5,1,1,1] => [2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => 5
[4,4] => [4,2,1,1] => [[1,4,7,8],[2,6],[3],[5]] => 5
[4,3,1] => [1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => 1
[4,2,2] => [2,1,1,1,1,1,1] => [[1,8],[2],[3],[4],[5],[6],[7]] => 2
[4,2,1,1] => [3,1,1,1,1,1] => [[1,7,8],[2],[3],[4],[5],[6]] => 3
[4,1,1,1,1] => [5,2,1] => [[1,3,6,7,8],[2,5],[4]] => 6
[3,3,2] => [2,2,1,1,1,1] => [[1,6],[2,8],[3],[4],[5],[7]] => 3
[3,3,1,1] => [3,2,1,1,1] => [[1,5,8],[2,7],[3],[4],[6]] => 4
[3,2,2,1] => [3,2,2,1] => [[1,3,8],[2,5],[4,7],[6]] => 5
[3,2,1,1,1] => [5,1,1,1] => [[1,5,6,7,8],[2],[3],[4]] => 5
[3,1,1,1,1,1] => [6,1,1] => [[1,4,5,6,7,8],[2],[3]] => 6
[2,2,2,2] => [3,3,2] => [[1,2,5],[3,4,8],[6,7]] => 6
[2,2,2,1,1] => [5,3] => [[1,2,3,7,8],[4,5,6]] => 7
[2,2,1,1,1,1] => [6,2] => [[1,2,5,6,7,8],[3,4]] => 7
[2,1,1,1,1,1,1] => [7,1] => [[1,3,4,5,6,7,8],[2]] => 7
[1,1,1,1,1,1,1,1] => [8] => [[1,2,3,4,5,6,7,8]] => 8
[9] => [5,4] => [[1,2,3,4,9],[5,6,7,8]] => 8
[8,1] => [4,4,1] => [[1,3,4,5],[2,7,8,9],[6]] => 7
[7,2] => [4,3,1,1] => [[1,4,5,9],[2,7,8],[3],[6]] => 6
[7,1,1] => [4,3,2] => [[1,2,5,9],[3,4,8],[6,7]] => 7
[6,3] => [4,2,2,1] => [[1,3,8,9],[2,5],[4,7],[6]] => 6
[6,2,1] => [4,2,1,1,1] => [[1,5,8,9],[2,7],[3],[4],[6]] => 5
[6,1,1,1] => [5,2,2] => [[1,2,7,8,9],[3,4],[5,6]] => 7
[5,4] => [2,2,2,2,1] => [[1,3],[2,5],[4,7],[6,9],[8]] => 5
[5,3,1] => [3,1,1,1,1,1,1] => [[1,8,9],[2],[3],[4],[5],[6],[7]] => 3
[5,2,2] => [3,2,1,1,1,1] => [[1,6,9],[2,8],[3],[4],[5],[7]] => 4
[5,2,1,1] => [3,3,1,1,1] => [[1,5,6],[2,8,9],[3],[4],[7]] => 5
[5,1,1,1,1] => [3,2,2,2] => [[1,2,9],[3,4],[5,6],[7,8]] => 6
[4,4,1] => [4,1,1,1,1,1] => [[1,7,8,9],[2],[3],[4],[5],[6]] => 4
[4,3,2] => [1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9]] => 1
[4,3,1,1] => [2,1,1,1,1,1,1,1] => [[1,9],[2],[3],[4],[5],[6],[7],[8]] => 2
[4,2,2,1] => [2,2,1,1,1,1,1] => [[1,7],[2,9],[3],[4],[5],[6],[8]] => 3
[4,2,1,1,1] => [5,1,1,1,1] => [[1,6,7,8,9],[2],[3],[4],[5]] => 5
[4,1,1,1,1,1] => [6,2,1] => [[1,3,6,7,8,9],[2,5],[4]] => 7
[3,3,3] => [3,2,2,1,1] => [[1,4,9],[2,6],[3,8],[5],[7]] => 5
[3,3,2,1] => [2,2,2,1,1,1] => [[1,5],[2,7],[3,9],[4],[6],[8]] => 4
[3,3,1,1,1] => [5,2,1,1] => [[1,4,7,8,9],[2,6],[3],[5]] => 6
[3,2,2,2] => [3,3,2,1] => [[1,3,6],[2,5,9],[4,8],[7]] => 6
[3,2,2,1,1] => [5,3,1] => [[1,3,4,8,9],[2,6,7],[5]] => 7
[3,2,1,1,1,1] => [6,1,1,1] => [[1,5,6,7,8,9],[2],[3],[4]] => 6
[3,1,1,1,1,1,1] => [7,1,1] => [[1,4,5,6,7,8,9],[2],[3]] => 7
[2,2,2,2,1] => [3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => 7
[2,2,2,1,1,1] => [6,3] => [[1,2,3,7,8,9],[4,5,6]] => 8
[2,2,1,1,1,1,1] => [7,2] => [[1,2,5,6,7,8,9],[3,4]] => 8
[2,1,1,1,1,1,1,1] => [8,1] => [[1,3,4,5,6,7,8,9],[2]] => 8
[1,1,1,1,1,1,1,1,1] => [9] => [[1,2,3,4,5,6,7,8,9]] => 9
[10] => [5,5] => [[1,2,3,4,5],[6,7,8,9,10]] => 9
[9,1] => [5,4,1] => [[1,3,4,5,10],[2,7,8,9],[6]] => 8
[8,2] => [4,4,1,1] => [[1,4,5,6],[2,8,9,10],[3],[7]] => 7
[8,1,1] => [4,4,2] => [[1,2,5,6],[3,4,9,10],[7,8]] => 8
[7,3] => [4,3,2,1] => [[1,3,6,10],[2,5,9],[4,8],[7]] => 7
>>> Load all 190 entries. <<<
[7,2,1] => [4,3,1,1,1] => [[1,5,6,10],[2,8,9],[3],[4],[7]] => 6
[7,1,1,1] => [5,3,2] => [[1,2,5,9,10],[3,4,8],[6,7]] => 8
[6,4] => [2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10]] => 6
[6,3,1] => [4,2,1,1,1,1] => [[1,6,9,10],[2,8],[3],[4],[5],[7]] => 5
[6,2,2] => [3,3,2,1,1] => [[1,4,7],[2,6,10],[3,9],[5],[8]] => 6
[6,2,1,1] => [5,3,1,1] => [[1,4,5,9,10],[2,7,8],[3],[6]] => 7
[6,1,1,1,1] => [3,3,2,2] => [[1,2,7],[3,4,10],[5,6],[8,9]] => 7
[5,5] => [4,2,2,2] => [[1,2,9,10],[3,4],[5,6],[7,8]] => 7
[5,4,1] => [3,3,1,1,1,1] => [[1,6,7],[2,9,10],[3],[4],[5],[8]] => 5
[5,3,2] => [2,1,1,1,1,1,1,1,1] => [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]] => 2
[5,3,1,1] => [2,2,1,1,1,1,1,1] => [[1,8],[2,10],[3],[4],[5],[6],[7],[9]] => 3
[5,2,2,1] => [2,2,2,1,1,1,1] => [[1,6],[2,8],[3,10],[4],[5],[7],[9]] => 4
[5,2,1,1,1] => [3,2,2,2,1] => [[1,3,10],[2,5],[4,7],[6,9],[8]] => 6
[5,1,1,1,1,1] => [6,2,2] => [[1,2,7,8,9,10],[3,4],[5,6]] => 8
[4,4,2] => [3,1,1,1,1,1,1,1] => [[1,9,10],[2],[3],[4],[5],[6],[7],[8]] => 3
[4,4,1,1] => [3,2,1,1,1,1,1] => [[1,7,10],[2,9],[3],[4],[5],[6],[8]] => 4
[4,3,3] => [4,1,1,1,1,1,1] => [[1,8,9,10],[2],[3],[4],[5],[6],[7]] => 4
[4,3,2,1] => [1,1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]] => 1
[4,3,1,1,1] => [5,1,1,1,1,1] => [[1,7,8,9,10],[2],[3],[4],[5],[6]] => 5
[4,2,2,2] => [3,2,2,1,1,1] => [[1,5,10],[2,7],[3,9],[4],[6],[8]] => 5
[4,2,2,1,1] => [5,2,1,1,1] => [[1,5,8,9,10],[2,7],[3],[4],[6]] => 6
[4,2,1,1,1,1] => [6,1,1,1,1] => [[1,6,7,8,9,10],[2],[3],[4],[5]] => 6
[4,1,1,1,1,1,1] => [7,2,1] => [[1,3,6,7,8,9,10],[2,5],[4]] => 8
[3,3,3,1] => [2,2,2,2,1,1] => [[1,4],[2,6],[3,8],[5,10],[7],[9]] => 5
[3,3,2,2] => [4,2,2,1,1] => [[1,4,9,10],[2,6],[3,8],[5],[7]] => 6
[3,3,2,1,1] => [5,2,2,1] => [[1,3,8,9,10],[2,5],[4,7],[6]] => 7
[3,3,1,1,1,1] => [6,2,1,1] => [[1,4,7,8,9,10],[2,6],[3],[5]] => 7
[3,2,2,2,1] => [3,3,3,1] => [[1,3,4],[2,6,7],[5,9,10],[8]] => 7
[3,2,2,1,1,1] => [6,3,1] => [[1,3,4,8,9,10],[2,6,7],[5]] => 8
[3,2,1,1,1,1,1] => [7,1,1,1] => [[1,5,6,7,8,9,10],[2],[3],[4]] => 7
[3,1,1,1,1,1,1,1] => [8,1,1] => [[1,4,5,6,7,8,9,10],[2],[3]] => 8
[2,2,2,2,2] => [4,3,3] => [[1,2,3,10],[4,5,6],[7,8,9]] => 8
[2,2,2,2,1,1] => [6,4] => [[1,2,3,4,9,10],[5,6,7,8]] => 9
[2,2,2,1,1,1,1] => [7,3] => [[1,2,3,7,8,9,10],[4,5,6]] => 9
[2,2,1,1,1,1,1,1] => [8,2] => [[1,2,5,6,7,8,9,10],[3,4]] => 9
[2,1,1,1,1,1,1,1,1] => [9,1] => [[1,3,4,5,6,7,8,9,10],[2]] => 9
[1,1,1,1,1,1,1,1,1,1] => [10] => [[1,2,3,4,5,6,7,8,9,10]] => 10
[9,2] => [5,4,1,1] => [[1,4,5,6,11],[2,8,9,10],[3],[7]] => 8
[9,1,1] => [5,4,2] => [[1,2,5,6,11],[3,4,9,10],[7,8]] => 9
[8,3] => [4,4,2,1] => [[1,3,6,7],[2,5,10,11],[4,9],[8]] => 8
[8,2,1] => [4,4,1,1,1] => [[1,5,6,7],[2,9,10,11],[3],[4],[8]] => 7
[8,1,1,1] => [5,3,3] => [[1,2,3,10,11],[4,5,6],[7,8,9]] => 9
[7,4] => [5,2,2,2] => [[1,2,9,10,11],[3,4],[5,6],[7,8]] => 8
[7,2,2] => [4,3,2,1,1] => [[1,4,7,11],[2,6,10],[3,9],[5],[8]] => 7
[7,2,1,1] => [5,3,2,1] => [[1,3,6,10,11],[2,5,9],[4,8],[7]] => 8
[6,2,2,1] => [5,2,2,1,1] => [[1,4,9,10,11],[2,6],[3,8],[5],[7]] => 7
[6,2,1,1,1] => [3,3,2,2,1] => [[1,3,8],[2,5,11],[4,7],[6,10],[9]] => 7
[6,1,1,1,1,1] => [3,3,3,2] => [[1,2,5],[3,4,8],[6,7,11],[9,10]] => 8
[5,5,1] => [5,3,1,1,1] => [[1,5,6,10,11],[2,8,9],[3],[4],[7]] => 7
[4,2,2,2,1] => [3,3,3,1,1] => [[1,4,5],[2,7,8],[3,10,11],[6],[9]] => 7
[3,3,3,1,1] => [4,2,2,2,1] => [[1,3,10,11],[2,5],[4,7],[6,9],[8]] => 7
[3,3,2,2,1] => [4,3,2,2] => [[1,2,7,11],[3,4,10],[5,6],[8,9]] => 8
[3,2,2,2,2] => [4,3,3,1] => [[1,3,4,11],[2,6,7],[5,9,10],[8]] => 8
[2,2,2,2,2,1] => [4,4,3] => [[1,2,3,7],[4,5,6,11],[8,9,10]] => 9
[12] => [6,6] => [[1,2,3,4,5,6],[7,8,9,10,11,12]] => 11
[9,3] => [5,4,2,1] => [[1,3,6,7,12],[2,5,10,11],[4,9],[8]] => 9
[9,2,1] => [5,4,1,1,1] => [[1,5,6,7,12],[2,9,10,11],[3],[4],[8]] => 8
[9,1,1,1] => [5,4,3] => [[1,2,3,7,12],[4,5,6,11],[8,9,10]] => 10
[8,4] => [4,4,2,2] => [[1,2,7,8],[3,4,11,12],[5,6],[9,10]] => 9
[8,2,2] => [4,4,2,1,1] => [[1,4,7,8],[2,6,11,12],[3,10],[5],[9]] => 8
[8,2,1,1] => [5,3,3,1] => [[1,3,4,11,12],[2,6,7],[5,9,10],[8]] => 9
[7,4,1] => [5,3,2,1,1] => [[1,4,7,11,12],[2,6,10],[3,9],[5],[8]] => 8
[7,3,1,1] => [3,3,2,2,1,1] => [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]] => 7
[7,2,2,1] => [5,2,2,2,1] => [[1,3,10,11,12],[2,5],[4,7],[6,9],[8]] => 8
[6,6] => [5,3,2,2] => [[1,2,7,11,12],[3,4,10],[5,6],[8,9]] => 9
[6,2,2,1,1] => [4,3,2,2,1] => [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]] => 8
[6,2,1,1,1,1] => [3,3,3,2,1] => [[1,3,6],[2,5,9],[4,8,12],[7,11],[10]] => 8
[4,2,2,2,2] => [4,3,3,1,1] => [[1,4,5,12],[2,7,8],[3,10,11],[6],[9]] => 8
[3,3,3,2,1] => [2,2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]] => 7
[3,3,2,2,2] => [4,3,3,2] => [[1,2,5,12],[3,4,8],[6,7,11],[9,10]] => 9
[3,3,2,2,1,1] => [6,4,2] => [[1,2,5,6,11,12],[3,4,9,10],[7,8]] => 10
[3,2,2,2,2,1] => [4,4,3,1] => [[1,3,4,8],[2,6,7,12],[5,10,11],[9]] => 9
[1,1,1,1,1,1,1,1,1,1,1,1] => [12] => [[1,2,3,4,5,6,7,8,9,10,11,12]] => 12
[9,4] => [5,4,2,2] => [[1,2,7,8,13],[3,4,11,12],[5,6],[9,10]] => 10
[9,2,2] => [5,4,2,1,1] => [[1,4,7,8,13],[2,6,11,12],[3,10],[5],[9]] => 9
[9,2,1,1] => [5,4,3,1] => [[1,3,4,8,13],[2,6,7,12],[5,10,11],[9]] => 10
[8,5] => [5,3,3,2] => [[1,2,5,12,13],[3,4,8],[6,7,11],[9,10]] => 10
[8,4,1] => [5,3,3,1,1] => [[1,4,5,12,13],[2,7,8],[3,10,11],[6],[9]] => 9
[7,2,2,2] => [4,3,3,2,1] => [[1,3,6,13],[2,5,9],[4,8,12],[7,11],[10]] => 9
[6,6,1] => [5,3,2,2,1] => [[1,3,8,12,13],[2,5,11],[4,7],[6,10],[9]] => 9
[6,2,2,1,1,1] => [4,4,2,2,1] => [[1,3,8,9],[2,5,12,13],[4,7],[6,11],[10]] => 9
[4,2,2,2,2,1] => [4,4,3,1,1] => [[1,4,5,9],[2,7,8,13],[3,11,12],[6],[10]] => 9
[3,3,2,2,2,1] => [4,4,3,2] => [[1,2,5,9],[3,4,8,13],[6,7,12],[10,11]] => 10
[9,5] => [5,4,3,2] => [[1,2,5,9,14],[3,4,8,13],[6,7,12],[10,11]] => 11
[9,4,1] => [5,4,3,1,1] => [[1,4,5,9,14],[2,7,8,13],[3,11,12],[6],[10]] => 10
[9,2,2,1] => [5,4,2,2,1] => [[1,3,8,9,14],[2,5,12,13],[4,7],[6,11],[10]] => 10
[8,5,1] => [5,3,3,2,1] => [[1,3,6,13,14],[2,5,9],[4,8,12],[7,11],[10]] => 10
[5,2,2,2,2,1] => [4,4,3,2,1] => [[1,3,6,10],[2,5,9,14],[4,8,13],[7,12],[11]] => 10
[9,5,1] => [5,4,3,2,1] => [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]] => 11
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Description
The number of ascents of a standard tableau.
Entry $i$ of a standard Young tableau is an ascent if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Map
Loehr-Warrington inverse
Description
Return a partition whose length is the diagonal inversion number of the preimage.
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.