- FindStat

Identifier

Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000460: Integer partitions ⟶ ℤ

Values

[1] => [[1]] => [1] => [1] => 1

[2] => [[1,2]] => [1,2] => [1,1] => 2

[1,1] => [[1],[2]] => [2,1] => [2] => 2

[3] => [[1,2,3]] => [1,2,3] => [1,1,1] => 3

[2,1] => [[1,2],[3]] => [3,1,2] => [2,1] => 3

[1,1,1] => [[1],[2],[3]] => [3,2,1] => [3] => 3

[4] => [[1,2,3,4]] => [1,2,3,4] => [1,1,1,1] => 4

[3,1] => [[1,2,3],[4]] => [4,1,2,3] => [2,1,1] => 4

[2,2] => [[1,2],[3,4]] => [3,4,1,2] => [2,1,1] => 4

[2,1,1] => [[1,2],[3],[4]] => [4,3,1,2] => [3,1] => 4

[1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => [4] => 4

[5] => [[1,2,3,4,5]] => [1,2,3,4,5] => [1,1,1,1,1] => 5

[4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => [2,1,1,1] => 5

[3,2] => [[1,2,3],[4,5]] => [4,5,1,2,3] => [2,1,1,1] => 5

[3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => [3,1,1] => 5

[2,2,1] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => [3,1,1] => 5

[2,1,1,1] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => [4,1] => 5

[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [5] => 5

[6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => [1,1,1,1,1,1] => 6

[5,1] => [[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => [2,1,1,1,1] => 6

[4,2] => [[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => [2,1,1,1,1] => 6

[4,1,1] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => [3,1,1,1] => 6

[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [2,1,1,1,1] => 6

[3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => [3,1,1,1] => 6

[3,1,1,1] => [[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => [4,1,1] => 6

[2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => [3,1,1,1] => 6

[2,2,1,1] => [[1,2],[3,4],[5],[6]] => [6,5,3,4,1,2] => [4,1,1] => 6

[2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [6,5,4,3,1,2] => [5,1] => 6

[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [6] => 6

[7] => [[1,2,3,4,5,6,7]] => [1,2,3,4,5,6,7] => [1,1,1,1,1,1,1] => 7

[6,1] => [[1,2,3,4,5,6],[7]] => [7,1,2,3,4,5,6] => [2,1,1,1,1,1] => 7

[5,2] => [[1,2,3,4,5],[6,7]] => [6,7,1,2,3,4,5] => [2,1,1,1,1,1] => 7

[5,1,1] => [[1,2,3,4,5],[6],[7]] => [7,6,1,2,3,4,5] => [3,1,1,1,1] => 7

[4,3] => [[1,2,3,4],[5,6,7]] => [5,6,7,1,2,3,4] => [2,1,1,1,1,1] => 7

[4,2,1] => [[1,2,3,4],[5,6],[7]] => [7,5,6,1,2,3,4] => [3,1,1,1,1] => 7

[4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => [7,6,5,1,2,3,4] => [4,1,1,1] => 7

[3,3,1] => [[1,2,3],[4,5,6],[7]] => [7,4,5,6,1,2,3] => [3,1,1,1,1] => 7

[3,2,2] => [[1,2,3],[4,5],[6,7]] => [6,7,4,5,1,2,3] => [3,1,1,1,1] => 7

[3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => [7,6,4,5,1,2,3] => [4,1,1,1] => 7

[3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => [7,6,5,4,1,2,3] => [5,1,1] => 7

[2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => [7,5,6,3,4,1,2] => [4,1,1,1] => 7

[2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => [7,6,5,3,4,1,2] => [5,1,1] => 7

[2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,1,2] => [6,1] => 7

[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1] => [7] => 7

[8] => [[1,2,3,4,5,6,7,8]] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1] => 8

[7,1] => [[1,2,3,4,5,6,7],[8]] => [8,1,2,3,4,5,6,7] => [2,1,1,1,1,1,1] => 8

[6,2] => [[1,2,3,4,5,6],[7,8]] => [7,8,1,2,3,4,5,6] => [2,1,1,1,1,1,1] => 8

[6,1,1] => [[1,2,3,4,5,6],[7],[8]] => [8,7,1,2,3,4,5,6] => [3,1,1,1,1,1] => 8

[5,3] => [[1,2,3,4,5],[6,7,8]] => [6,7,8,1,2,3,4,5] => [2,1,1,1,1,1,1] => 8

[5,2,1] => [[1,2,3,4,5],[6,7],[8]] => [8,6,7,1,2,3,4,5] => [3,1,1,1,1,1] => 8

[5,1,1,1] => [[1,2,3,4,5],[6],[7],[8]] => [8,7,6,1,2,3,4,5] => [4,1,1,1,1] => 8

[4,4] => [[1,2,3,4],[5,6,7,8]] => [5,6,7,8,1,2,3,4] => [2,1,1,1,1,1,1] => 8

[4,3,1] => [[1,2,3,4],[5,6,7],[8]] => [8,5,6,7,1,2,3,4] => [3,1,1,1,1,1] => 8

[4,2,2] => [[1,2,3,4],[5,6],[7,8]] => [7,8,5,6,1,2,3,4] => [3,1,1,1,1,1] => 8

[4,2,1,1] => [[1,2,3,4],[5,6],[7],[8]] => [8,7,5,6,1,2,3,4] => [4,1,1,1,1] => 8

[4,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8]] => [8,7,6,5,1,2,3,4] => [5,1,1,1] => 8

[3,3,2] => [[1,2,3],[4,5,6],[7,8]] => [7,8,4,5,6,1,2,3] => [3,1,1,1,1,1] => 8

[3,3,1,1] => [[1,2,3],[4,5,6],[7],[8]] => [8,7,4,5,6,1,2,3] => [4,1,1,1,1] => 8

[3,2,2,1] => [[1,2,3],[4,5],[6,7],[8]] => [8,6,7,4,5,1,2,3] => [4,1,1,1,1] => 8

[3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => [8,7,6,4,5,1,2,3] => [5,1,1,1] => 8

[3,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,1,2,3] => [6,1,1] => 8

[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [7,8,5,6,3,4,1,2] => [4,1,1,1,1] => 8

[2,2,2,1,1] => [[1,2],[3,4],[5,6],[7],[8]] => [8,7,5,6,3,4,1,2] => [5,1,1,1] => 8

[2,2,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8]] => [8,7,6,5,3,4,1,2] => [6,1,1] => 8

[2,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,1,2] => [7,1] => 8

[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] => 8

[9] => [[1,2,3,4,5,6,7,8,9]] => [1,2,3,4,5,6,7,8,9] => [1,1,1,1,1,1,1,1,1] => 9

[8,1] => [[1,2,3,4,5,6,7,8],[9]] => [9,1,2,3,4,5,6,7,8] => [2,1,1,1,1,1,1,1] => 9

[7,2] => [[1,2,3,4,5,6,7],[8,9]] => [8,9,1,2,3,4,5,6,7] => [2,1,1,1,1,1,1,1] => 9

[7,1,1] => [[1,2,3,4,5,6,7],[8],[9]] => [9,8,1,2,3,4,5,6,7] => [3,1,1,1,1,1,1] => 9

[6,3] => [[1,2,3,4,5,6],[7,8,9]] => [7,8,9,1,2,3,4,5,6] => [2,1,1,1,1,1,1,1] => 9

[6,2,1] => [[1,2,3,4,5,6],[7,8],[9]] => [9,7,8,1,2,3,4,5,6] => [3,1,1,1,1,1,1] => 9

[6,1,1,1] => [[1,2,3,4,5,6],[7],[8],[9]] => [9,8,7,1,2,3,4,5,6] => [4,1,1,1,1,1] => 9

[5,4] => [[1,2,3,4,5],[6,7,8,9]] => [6,7,8,9,1,2,3,4,5] => [2,1,1,1,1,1,1,1] => 9

[5,3,1] => [[1,2,3,4,5],[6,7,8],[9]] => [9,6,7,8,1,2,3,4,5] => [3,1,1,1,1,1,1] => 9

[5,2,2] => [[1,2,3,4,5],[6,7],[8,9]] => [8,9,6,7,1,2,3,4,5] => [3,1,1,1,1,1,1] => 9

[5,2,1,1] => [[1,2,3,4,5],[6,7],[8],[9]] => [9,8,6,7,1,2,3,4,5] => [4,1,1,1,1,1] => 9

[5,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9]] => [9,8,7,6,1,2,3,4,5] => [5,1,1,1,1] => 9

[4,4,1] => [[1,2,3,4],[5,6,7,8],[9]] => [9,5,6,7,8,1,2,3,4] => [3,1,1,1,1,1,1] => 9

[4,3,2] => [[1,2,3,4],[5,6,7],[8,9]] => [8,9,5,6,7,1,2,3,4] => [3,1,1,1,1,1,1] => 9

[4,3,1,1] => [[1,2,3,4],[5,6,7],[8],[9]] => [9,8,5,6,7,1,2,3,4] => [4,1,1,1,1,1] => 9

[4,2,2,1] => [[1,2,3,4],[5,6],[7,8],[9]] => [9,7,8,5,6,1,2,3,4] => [4,1,1,1,1,1] => 9

[4,2,1,1,1] => [[1,2,3,4],[5,6],[7],[8],[9]] => [9,8,7,5,6,1,2,3,4] => [5,1,1,1,1] => 9

[4,1,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,1,2,3,4] => [6,1,1,1] => 9

[3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => [7,8,9,4,5,6,1,2,3] => [3,1,1,1,1,1,1] => 9

[3,3,2,1] => [[1,2,3],[4,5,6],[7,8],[9]] => [9,7,8,4,5,6,1,2,3] => [4,1,1,1,1,1] => 9

[3,3,1,1,1] => [[1,2,3],[4,5,6],[7],[8],[9]] => [9,8,7,4,5,6,1,2,3] => [5,1,1,1,1] => 9

[3,2,2,2] => [[1,2,3],[4,5],[6,7],[8,9]] => [8,9,6,7,4,5,1,2,3] => [4,1,1,1,1,1] => 9

[3,2,2,1,1] => [[1,2,3],[4,5],[6,7],[8],[9]] => [9,8,6,7,4,5,1,2,3] => [5,1,1,1,1] => 9

[3,2,1,1,1,1] => [[1,2,3],[4,5],[6],[7],[8],[9]] => [9,8,7,6,4,5,1,2,3] => [6,1,1,1] => 9

[3,1,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,4,1,2,3] => [7,1,1] => 9

[2,2,2,2,1] => [[1,2],[3,4],[5,6],[7,8],[9]] => [9,7,8,5,6,3,4,1,2] => [5,1,1,1,1] => 9

[2,2,2,1,1,1] => [[1,2],[3,4],[5,6],[7],[8],[9]] => [9,8,7,5,6,3,4,1,2] => [6,1,1,1] => 9

[2,2,1,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,3,4,1,2] => [7,1,1] => 9

[2,1,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,4,3,1,2] => [8,1] => 9

[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] => 9

[10] => [[1,2,3,4,5,6,7,8,9,10]] => [1,2,3,4,5,6,7,8,9,10] => [1,1,1,1,1,1,1,1,1,1] => 10

[9,1] => [[1,2,3,4,5,6,7,8,9],[10]] => [10,1,2,3,4,5,6,7,8,9] => [2,1,1,1,1,1,1,1,1] => 10

[8,2] => [[1,2,3,4,5,6,7,8],[9,10]] => [9,10,1,2,3,4,5,6,7,8] => [2,1,1,1,1,1,1,1,1] => 10

[8,1,1] => [[1,2,3,4,5,6,7,8],[9],[10]] => [10,9,1,2,3,4,5,6,7,8] => [3,1,1,1,1,1,1,1] => 10

[7,3] => [[1,2,3,4,5,6,7],[8,9,10]] => [8,9,10,1,2,3,4,5,6,7] => [2,1,1,1,1,1,1,1,1] => 10

>>> Load all 145 entries. <<<

[7,2,1] => [[1,2,3,4,5,6,7],[8,9],[10]] => [10,8,9,1,2,3,4,5,6,7] => [3,1,1,1,1,1,1,1] => 10

[7,1,1,1] => [[1,2,3,4,5,6,7],[8],[9],[10]] => [10,9,8,1,2,3,4,5,6,7] => [4,1,1,1,1,1,1] => 10

[6,4] => [[1,2,3,4,5,6],[7,8,9,10]] => [7,8,9,10,1,2,3,4,5,6] => [2,1,1,1,1,1,1,1,1] => 10

[6,3,1] => [[1,2,3,4,5,6],[7,8,9],[10]] => [10,7,8,9,1,2,3,4,5,6] => [3,1,1,1,1,1,1,1] => 10

[6,2,2] => [[1,2,3,4,5,6],[7,8],[9,10]] => [9,10,7,8,1,2,3,4,5,6] => [3,1,1,1,1,1,1,1] => 10

[6,2,1,1] => [[1,2,3,4,5,6],[7,8],[9],[10]] => [10,9,7,8,1,2,3,4,5,6] => [4,1,1,1,1,1,1] => 10

[6,1,1,1,1] => [[1,2,3,4,5,6],[7],[8],[9],[10]] => [10,9,8,7,1,2,3,4,5,6] => [5,1,1,1,1,1] => 10

[5,5] => [[1,2,3,4,5],[6,7,8,9,10]] => [6,7,8,9,10,1,2,3,4,5] => [2,1,1,1,1,1,1,1,1] => 10

[5,4,1] => [[1,2,3,4,5],[6,7,8,9],[10]] => [10,6,7,8,9,1,2,3,4,5] => [3,1,1,1,1,1,1,1] => 10

[5,3,2] => [[1,2,3,4,5],[6,7,8],[9,10]] => [9,10,6,7,8,1,2,3,4,5] => [3,1,1,1,1,1,1,1] => 10

[5,3,1,1] => [[1,2,3,4,5],[6,7,8],[9],[10]] => [10,9,6,7,8,1,2,3,4,5] => [4,1,1,1,1,1,1] => 10

[5,2,2,1] => [[1,2,3,4,5],[6,7],[8,9],[10]] => [10,8,9,6,7,1,2,3,4,5] => [4,1,1,1,1,1,1] => 10

[5,2,1,1,1] => [[1,2,3,4,5],[6,7],[8],[9],[10]] => [10,9,8,6,7,1,2,3,4,5] => [5,1,1,1,1,1] => 10

[5,1,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,1,2,3,4,5] => [6,1,1,1,1] => 10

[4,4,2] => [[1,2,3,4],[5,6,7,8],[9,10]] => [9,10,5,6,7,8,1,2,3,4] => [3,1,1,1,1,1,1,1] => 10

[4,4,1,1] => [[1,2,3,4],[5,6,7,8],[9],[10]] => [10,9,5,6,7,8,1,2,3,4] => [4,1,1,1,1,1,1] => 10

[4,3,3] => [[1,2,3,4],[5,6,7],[8,9,10]] => [8,9,10,5,6,7,1,2,3,4] => [3,1,1,1,1,1,1,1] => 10

[4,3,2,1] => [[1,2,3,4],[5,6,7],[8,9],[10]] => [10,8,9,5,6,7,1,2,3,4] => [4,1,1,1,1,1,1] => 10

[4,3,1,1,1] => [[1,2,3,4],[5,6,7],[8],[9],[10]] => [10,9,8,5,6,7,1,2,3,4] => [5,1,1,1,1,1] => 10

[4,2,2,2] => [[1,2,3,4],[5,6],[7,8],[9,10]] => [9,10,7,8,5,6,1,2,3,4] => [4,1,1,1,1,1,1] => 10

[4,2,2,1,1] => [[1,2,3,4],[5,6],[7,8],[9],[10]] => [10,9,7,8,5,6,1,2,3,4] => [5,1,1,1,1,1] => 10

[4,2,1,1,1,1] => [[1,2,3,4],[5,6],[7],[8],[9],[10]] => [10,9,8,7,5,6,1,2,3,4] => [6,1,1,1,1] => 10

[4,1,1,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,5,1,2,3,4] => [7,1,1,1] => 10

[3,3,3,1] => [[1,2,3],[4,5,6],[7,8,9],[10]] => [10,7,8,9,4,5,6,1,2,3] => [4,1,1,1,1,1,1] => 10

[3,3,2,2] => [[1,2,3],[4,5,6],[7,8],[9,10]] => [9,10,7,8,4,5,6,1,2,3] => [4,1,1,1,1,1,1] => 10

[3,3,2,1,1] => [[1,2,3],[4,5,6],[7,8],[9],[10]] => [10,9,7,8,4,5,6,1,2,3] => [5,1,1,1,1,1] => 10

[3,3,1,1,1,1] => [[1,2,3],[4,5,6],[7],[8],[9],[10]] => [10,9,8,7,4,5,6,1,2,3] => [6,1,1,1,1] => 10

[3,2,2,2,1] => [[1,2,3],[4,5],[6,7],[8,9],[10]] => [10,8,9,6,7,4,5,1,2,3] => [5,1,1,1,1,1] => 10

[3,2,2,1,1,1] => [[1,2,3],[4,5],[6,7],[8],[9],[10]] => [10,9,8,6,7,4,5,1,2,3] => [6,1,1,1,1] => 10

[3,2,1,1,1,1,1] => [[1,2,3],[4,5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,4,5,1,2,3] => [7,1,1,1] => 10

[3,1,1,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,5,4,1,2,3] => [8,1,1] => 10

[2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10]] => [9,10,7,8,5,6,3,4,1,2] => [5,1,1,1,1,1] => 10

[2,2,2,2,1,1] => [[1,2],[3,4],[5,6],[7,8],[9],[10]] => [10,9,7,8,5,6,3,4,1,2] => [6,1,1,1,1] => 10

[2,2,2,1,1,1,1] => [[1,2],[3,4],[5,6],[7],[8],[9],[10]] => [10,9,8,7,5,6,3,4,1,2] => [7,1,1,1] => 10

[2,2,1,1,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,5,3,4,1,2] => [8,1,1] => 10

[2,1,1,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,5,4,3,1,2] => [9,1] => 10

[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] => 10

[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] => [1,1,1,1,1,1,1,1,1,1,1,1] => 12

[6,6] => [[1,2,3,4,5,6],[7,8,9,10,11,12]] => [7,8,9,10,11,12,1,2,3,4,5,6] => [2,1,1,1,1,1,1,1,1,1,1] => 12

[6,4,2] => [[1,2,3,4,5,6],[7,8,9,10],[11,12]] => [11,12,7,8,9,10,1,2,3,4,5,6] => [3,1,1,1,1,1,1,1,1,1] => 12

[4,4,2,2] => [[1,2,3,4],[5,6,7,8],[9,10],[11,12]] => [11,12,9,10,5,6,7,8,1,2,3,4] => [4,1,1,1,1,1,1,1,1] => 12

[4,2,2,2,2] => [[1,2,3,4],[5,6],[7,8],[9,10],[11,12]] => [11,12,9,10,7,8,5,6,1,2,3,4] => [5,1,1,1,1,1,1,1] => 12

[2,2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]] => [11,12,9,10,7,8,5,6,3,4,1,2] => [6,1,1,1,1,1,1] => 12

[1,1,1,1,1,1,1,1,1,1,1,1] => [[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] => 12

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$F_{1} = q$

$F_{2} = 2\ q^{2}$

$F_{3} = 3\ q^{3}$

$F_{4} = 5\ q^{4}$

$F_{5} = 7\ q^{5}$

$F_{6} = 11\ q^{6}$

$F_{7} = 15\ q^{7}$

$F_{8} = 22\ q^{8}$

$F_{9} = 30\ q^{9}$

$F_{10} = 42\ q^{10}$

Description

The hook length of the last cell along the main diagonal of an integer partition.

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

LLPS

Description

The Lewis-Lyu-Pylyavskyy-Sen shape of a permutation.
An ascent in a sequence

$u = (u_1, u_2, \ldots)$ is an index

$i$ such that

$u_i < u_{i+1}$ . Let

$\mathrm{asc}(u)$ denote the number of ascents of

$u$ , and let

$\mathrm{asc}^{*}(u) := \begin{cases} 0 &\textrm{if u is empty}, \\ 1 + \mathrm{asc}(u) &\textrm{otherwise}.\end{cases}$
Given a permutation

$w$ in the symmetric group

$\mathfrak{S}_n$ , define

$A'_k := \max_{u_1, \ldots, u_k} (\mathrm{asc}^{*}(u_1) + \cdots + \mathrm{asc}^{*}(u_k))$
where the maximum is taken over disjoint subsequences

${u_i}$ of

$w$ .
Then

$A'_1, A'_2-A'_1, A'_3-A'_2,\dots$ is a partition of

$n$ . Its conjugate is the Lewis-Lyu-Pylyavskyy-Sen shape of a permutation.