- FindStat

Identifier

Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 210 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[13] => [7,6] => [2,2,2,2,2,2,1] => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13]] => 2

[12,1] => [6,6,1] => [3,2,2,2,2,2] => [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12,13]] => 3

[10,1,1,1] => [5,5,3] => [3,3,3,2,2] => [[1,2,3],[4,5,6],[7,8,9],[10,11],[12,13]] => 3

[9,4] => [5,4,2,2] => [4,4,2,2,1] => [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]] => 4

[9,2,2] => [5,4,2,1,1] => [5,3,2,2,1] => [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]] => 5

[9,2,1,1] => [5,4,3,1] => [4,3,3,2,1] => [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13]] => 4

[8,5] => [5,3,3,2] => [4,4,3,1,1] => [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]] => 4

[8,4,1] => [5,3,3,1,1] => [5,3,3,1,1] => [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]] => 5

[7,2,2,2] => [4,3,3,2,1] => [5,4,3,1] => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13]] => 5

[6,6,1] => [5,3,2,2,1] => [5,4,2,1,1] => [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]] => 5

[6,2,2,1,1,1] => [4,4,2,2,1] => [5,4,2,2] => [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13]] => 5

[4,2,2,2,2,1] => [4,4,3,1,1] => [5,3,3,2] => [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13]] => 5

[3,3,2,2,2,1] => [4,4,3,2] => [4,4,3,2] => [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13]] => 4

[3,2,2,2,2,2] => [4,4,4,1] => [4,3,3,3] => [[1,2,3,4],[5,6,7],[8,9,10],[11,12,13]] => 4

[2,2,2,2,2,2,1] => [5,4,4] => [3,3,3,3,1] => [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13]] => 3

[14] => [7,7] => [2,2,2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14]] => 2

[12,1,1] => [6,6,2] => [3,3,2,2,2,2] => [[1,2,3],[4,5,6],[7,8],[9,10],[11,12],[13,14]] => 3

[9,5] => [5,4,3,2] => [4,4,3,2,1] => [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]] => 4

[9,4,1] => [5,4,3,1,1] => [5,3,3,2,1] => [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]] => 5

[9,2,2,1] => [5,4,2,2,1] => [5,4,2,2,1] => [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]] => 5

[8,5,1] => [5,3,3,2,1] => [5,4,3,1,1] => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]] => 5

[6,2,2,2,2] => [2,2,2,2,2,2,2] => [7,7] => [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]] => 7

[5,2,2,2,2,1] => [4,4,3,2,1] => [5,4,3,2] => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]] => 5

[3,3,2,2,2,2] => [4,4,4,2] => [4,4,3,3] => [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13,14]] => 4

[2,2,2,2,2,2,2] => [5,5,4] => [3,3,3,3,2] => [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14]] => 3

[12,1,1,1] => [6,6,3] => [3,3,3,2,2,2] => [[1,2,3],[4,5,6],[7,8,9],[10,11],[12,13],[14,15]] => 3

[9,5,1] => [5,4,3,2,1] => [5,4,3,2,1] => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]] => 5

[2,2,2,2,2,2,2,1] => [5,5,5] => [3,3,3,3,3] => [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15]] => 3

[16] => [8,8] => [2,2,2,2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16]] => 2

[12,1,1,1,1] => [6,6,4] => [3,3,3,3,2,2] => [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14],[15,16]] => 3

[9,1,1,1,1,1,1,1] => [4,4,4,4] => [4,4,4,4] => [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]] => 4

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$F_{1} = q$

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

$F_{3} = q + q^{2} + q^{3}$

$F_{4} = q + 2\ q^{2} + q^{3} + q^{4}$

$F_{5} = q + 2\ q^{2} + 2\ q^{3} + q^{4} + q^{5}$

$F_{6} = q + 3\ q^{2} + 3\ q^{3} + 2\ q^{4} + q^{5} + q^{6}$

$F_{7} = q + 3\ q^{2} + 4\ q^{3} + 3\ q^{4} + 2\ q^{5} + q^{6} + q^{7}$

$F_{8} = q + 4\ q^{2} + 5\ q^{3} + 5\ q^{4} + 3\ q^{5} + 2\ q^{6} + q^{7} + q^{8}$

$F_{9} = q + 4\ q^{2} + 7\ q^{3} + 6\ q^{4} + 5\ q^{5} + 3\ q^{6} + 2\ q^{7} + q^{8} + q^{9}$

$F_{10} = q + 5\ q^{2} + 8\ q^{3} + 9\ q^{4} + 7\ q^{5} + 5\ q^{6} + 3\ q^{7} + 2\ q^{8} + q^{9} + q^{10}$

Description

The last entry in the first row of a standard tableau.

Map

Loehr-Warrington inverse

Description

Return a partition whose length is the diagonal inversion number of the preimage.

Map

initial tableau

Description

Sends an integer partition to the standard tableau obtained by filling the numbers

$1$ through

$n$ row by row.

Map

conjugate

Description

Return the conjugate partition of the partition.
The conjugate partition of the partition

$\lambda$ of

$n$ is the partition

$\lambda^*$ whose Ferrers diagram is obtained from the diagram of

$\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.