- FindStat

Identifier

Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000018: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

[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] => 0

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

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

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

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

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

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

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

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

[5,1] => [2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [6,5,4,3,1,2] => 14

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

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

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

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

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

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

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

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

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

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

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

[5,2] => [2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => [7,6,5,3,4,1,2] => 19

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 28

[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] => 27

[6,2] => [2,2,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8]] => [8,7,6,5,3,4,1,2] => 26

[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] => 25

[5,3] => [2,2,2,1,1] => [[1,2],[3,4],[5,6],[7],[8]] => [8,7,5,6,3,4,1,2] => 25

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

[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] => 22

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

[4,3,1] => [3,2,2,1] => [[1,2,3],[4,5],[6,7],[8]] => [8,6,7,4,5,1,2,3] => 23

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

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

[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] => 18

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

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

[3,2,2,1] => [4,3,1] => [[1,2,3,4],[5,6,7],[8]] => [8,5,6,7,1,2,3,4] => 19

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

[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] => 13

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

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

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

[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] => 7

[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] => 0

[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] => 36

[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] => 35

[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] => 34

[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] => 33

[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] => 33

[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] => 32

[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] => 30

[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] => 32

[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] => 31

[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] => 30

[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] => 29

[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] => 26

[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] => 30

[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] => 29

[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] => 28

[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] => 27

[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] => 25

[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] => 21

[3,3,3] => [3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => [7,8,9,4,5,6,1,2,3] => 27

[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] => 26

[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] => 24

[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] => 24

[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] => 23

[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] => 20

[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] => 15

[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] => 20

[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] => 14

[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] => 8

[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] => 0

[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] => 45

[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] => 44

[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] => 43

[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] => 42

[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] => 42

>>> Load all 141 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] => 41

[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] => 39

[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] => 41

[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] => 40

[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] => 39

[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] => 38

[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] => 35

[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] => 40

[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] => 39

[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] => 38

[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] => 37

[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] => 36

[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] => 34

[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] => 30

[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] => 37

[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] => 36

[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] => 36

[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] => 35

[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] => 33

[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] => 33

[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] => 32

[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] => 29

[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] => 24

[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] => 33

[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] => 32

[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] => 31

[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] => 28

[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] => 29

[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] => 27

[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] => 23

[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] => 17

[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] => 25

[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] => 24

[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] => 21

[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] => 16

[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] => 9

[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] => 0

[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] => 66

[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] => 36

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

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$F_{1} = 1$

$F_{2} = 1 + q$

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

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

$F_{5} = 1 + q^{4} + q^{6} + q^{7} + q^{8} + q^{9} + q^{10}$

$F_{6} = 1 + q^{5} + q^{8} + 2\ q^{9} + q^{11} + 2\ q^{12} + q^{13} + q^{14} + q^{15}$

$F_{7} = 1 + q^{6} + q^{10} + q^{11} + q^{12} + q^{14} + 2\ q^{15} + q^{16} + q^{17} + 2\ q^{18} + q^{19} + q^{20} + q^{21}$

$F_{8} = 1 + q^{7} + q^{12} + q^{13} + q^{15} + q^{16} + q^{17} + q^{18} + q^{19} + q^{20} + 2\ q^{21} + 2\ q^{22} + q^{23} + 2\ q^{24} + 2\ q^{25} + q^{26} + q^{27} + q^{28}$

$F_{9} = 1 + q^{8} + q^{14} + q^{15} + q^{18} + 2\ q^{20} + q^{21} + q^{23} + 2\ q^{24} + q^{25} + 2\ q^{26} + 2\ q^{27} + q^{28} + 2\ q^{29} + 3\ q^{30} + q^{31} + 2\ q^{32} + 2\ q^{33} + q^{34} + q^{35} + q^{36}$

$F_{10} = 1 + q^{9} + q^{16} + q^{17} + q^{21} + q^{23} + 2\ q^{24} + q^{25} + q^{27} + q^{28} + 2\ q^{29} + q^{30} + q^{31} + 2\ q^{32} + 3\ q^{33} + q^{34} + 2\ q^{35} + 3\ q^{36} + 2\ q^{37} + 2\ q^{38} + 3\ q^{39} + 2\ q^{40} + 2\ q^{41} + 2\ q^{42} + q^{43} + q^{44} + q^{45}$

Description

The number of inversions of a permutation.
This equals the minimal number of simple transpositions

$(i,i+1)$ needed to write

$\pi$ . Thus, it is also the Coxeter length of

$\pi$ .

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.