- FindStat

Identifier

Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000246: Permutations ⟶ ℤ (values match St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.)

Values

[1] => [1] => [[1]] => [1] => 0

[2] => [1,1] => [[1],[2]] => [2,1] => 0

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

[3] => [1,1,1] => [[1],[2],[3]] => [3,2,1] => 0

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

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

[4] => [1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 0

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

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

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

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

[5] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => 0

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

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

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

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

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

[1,1,1,1,1] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 10

[6] => [1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => 0

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

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

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

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

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

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

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

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

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

[1,1,1,1,1,1] => [6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => 15

[7] => [1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

[2,1,1,1,1,1] => [6,1] => [[1,2,3,4,5,6],[7]] => [7,1,2,3,4,5,6] => 15

[1,1,1,1,1,1,1] => [7] => [[1,2,3,4,5,6,7]] => [1,2,3,4,5,6,7] => 21

[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] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[3,1,1,1,1,1] => [6,1,1] => [[1,2,3,4,5,6],[7],[8]] => [8,7,1,2,3,4,5,6] => 15

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

[2,2,2,1,1] => [5,3] => [[1,2,3,4,5],[6,7,8]] => [6,7,8,1,2,3,4,5] => 13

[2,2,1,1,1,1] => [6,2] => [[1,2,3,4,5,6],[7,8]] => [7,8,1,2,3,4,5,6] => 16

[2,1,1,1,1,1,1] => [7,1] => [[1,2,3,4,5,6,7],[8]] => [8,1,2,3,4,5,6,7] => 21

[1,1,1,1,1,1,1,1] => [8] => [[1,2,3,4,5,6,7,8]] => [1,2,3,4,5,6,7,8] => 28

[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] => 0

[8,1] => [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] => 1

[7,2] => [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] => 2

[7,1,1] => [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] => 3

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

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

[6,1,1,1] => [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

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

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

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

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

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

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

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

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

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

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

[4,1,1,1,1,1] => [6,1,1,1] => [[1,2,3,4,5,6],[7],[8],[9]] => [9,8,7,1,2,3,4,5,6] => 15

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

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

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

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

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

[3,2,1,1,1,1] => [6,2,1] => [[1,2,3,4,5,6],[7,8],[9]] => [9,7,8,1,2,3,4,5,6] => 16

[3,1,1,1,1,1,1] => [7,1,1] => [[1,2,3,4,5,6,7],[8],[9]] => [9,8,1,2,3,4,5,6,7] => 21

[2,2,2,2,1] => [5,4] => [[1,2,3,4,5],[6,7,8,9]] => [6,7,8,9,1,2,3,4,5] => 16

[2,2,2,1,1,1] => [6,3] => [[1,2,3,4,5,6],[7,8,9]] => [7,8,9,1,2,3,4,5,6] => 18

[2,2,1,1,1,1,1] => [7,2] => [[1,2,3,4,5,6,7],[8,9]] => [8,9,1,2,3,4,5,6,7] => 22

[2,1,1,1,1,1,1,1] => [8,1] => [[1,2,3,4,5,6,7,8],[9]] => [9,1,2,3,4,5,6,7,8] => 28

[1,1,1,1,1,1,1,1,1] => [9] => [[1,2,3,4,5,6,7,8,9]] => [1,2,3,4,5,6,7,8,9] => 36

[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] => 0

[9,1] => [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] => 1

[8,2] => [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] => 2

[8,1,1] => [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] => 3

[7,3] => [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] => 3

>>> Load all 140 entries. <<<

[7,2,1] => [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] => 4

[7,1,1,1] => [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] => 6

[6,4] => [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] => 4

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

[6,2,2] => [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

[6,2,1,1] => [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] => 7

[6,1,1,1,1] => [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] => 10

[5,5] => [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

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

[5,3,2] => [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] => 7

[5,3,1,1] => [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] => 8

[5,2,2,1] => [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] => 9

[5,2,1,1,1] => [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] => 11

[5,1,1,1,1,1] => [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] => 15

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

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

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

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

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

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

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

[4,2,1,1,1,1] => [6,2,1,1] => [[1,2,3,4,5,6],[7,8],[9],[10]] => [10,9,7,8,1,2,3,4,5,6] => 16

[4,1,1,1,1,1,1] => [7,1,1,1] => [[1,2,3,4,5,6,7],[8],[9],[10]] => [10,9,8,1,2,3,4,5,6,7] => 21

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

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

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

[3,3,1,1,1,1] => [6,2,2] => [[1,2,3,4,5,6],[7,8],[9,10]] => [9,10,7,8,1,2,3,4,5,6] => 17

[3,2,2,2,1] => [5,4,1] => [[1,2,3,4,5],[6,7,8,9],[10]] => [10,6,7,8,9,1,2,3,4,5] => 16

[3,2,2,1,1,1] => [6,3,1] => [[1,2,3,4,5,6],[7,8,9],[10]] => [10,7,8,9,1,2,3,4,5,6] => 18

[3,2,1,1,1,1,1] => [7,2,1] => [[1,2,3,4,5,6,7],[8,9],[10]] => [10,8,9,1,2,3,4,5,6,7] => 22

[3,1,1,1,1,1,1,1] => [8,1,1] => [[1,2,3,4,5,6,7,8],[9],[10]] => [10,9,1,2,3,4,5,6,7,8] => 28

[2,2,2,2,2] => [5,5] => [[1,2,3,4,5],[6,7,8,9,10]] => [6,7,8,9,10,1,2,3,4,5] => 20

[2,2,2,2,1,1] => [6,4] => [[1,2,3,4,5,6],[7,8,9,10]] => [7,8,9,10,1,2,3,4,5,6] => 21

[2,2,2,1,1,1,1] => [7,3] => [[1,2,3,4,5,6,7],[8,9,10]] => [8,9,10,1,2,3,4,5,6,7] => 24

[2,2,1,1,1,1,1,1] => [8,2] => [[1,2,3,4,5,6,7,8],[9,10]] => [9,10,1,2,3,4,5,6,7,8] => 29

[2,1,1,1,1,1,1,1,1] => [9,1] => [[1,2,3,4,5,6,7,8,9],[10]] => [10,1,2,3,4,5,6,7,8,9] => 36

[1,1,1,1,1,1,1,1,1,1] => [10] => [[1,2,3,4,5,6,7,8,9,10]] => [1,2,3,4,5,6,7,8,9,10] => 45

[2,2,2,2,2,2] => [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] => 30

[] => [] => [] => [] => 0

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + q + q^{3}$

$F_{4} = 1 + q + q^{2} + q^{3} + q^{6}$

$F_{5} = 1 + q + q^{2} + q^{3} + q^{4} + q^{6} + q^{10}$

$F_{6} = 1 + q + q^{2} + 2\ q^{3} + q^{4} + 2\ q^{6} + q^{7} + q^{10} + q^{15}$

$F_{7} = 1 + q + q^{2} + 2\ q^{3} + q^{4} + q^{5} + 2\ q^{6} + q^{7} + q^{9} + q^{10} + q^{11} + q^{15} + q^{21}$

$F_{8} = 1 + q + q^{2} + 2\ q^{3} + 2\ q^{4} + q^{5} + 2\ q^{6} + 2\ q^{7} + q^{8} + q^{9} + q^{10} + q^{11} + q^{12} + q^{13} + q^{15} + q^{16} + q^{21} + q^{28}$

$F_{9} = 1 + q + q^{2} + 2\ q^{3} + 2\ q^{4} + q^{5} + 3\ q^{6} + 2\ q^{7} + q^{8} + 2\ q^{9} + 2\ q^{10} + q^{11} + 2\ q^{12} + q^{13} + q^{15} + 2\ q^{16} + q^{18} + q^{21} + q^{22} + q^{28} + q^{36}$

$F_{10} = 1 + q + q^{2} + 2\ q^{3} + 2\ q^{4} + 2\ q^{5} + 3\ q^{6} + 2\ q^{7} + 2\ q^{8} + 3\ q^{9} + 2\ q^{10} + q^{11} + 3\ q^{12} + 2\ q^{13} + q^{14} + q^{15} + 2\ q^{16} + q^{17} + q^{18} + q^{20} + 2\ q^{21} + q^{22} + q^{24} + q^{28} + q^{29} + q^{36} + q^{45}$

Description

The number of non-inversions of a permutation.
For a permutation of

$\{1,\ldots,n\}$ , this is given by

$\operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi)$ .

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.

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.