- FindStat

Identifier

Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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]] => [[1,2,3,4,5,6,7,8]] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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]] => [[1,2,3,4,5,6,7,8,9]] => 0

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

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

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

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

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

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

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

[5,3,1] => [3,2,2,1,1] => [[1,4,9],[2,6],[3,8],[5],[7]] => [[1,2,3,5,7],[4,6,8],[9]] => 23

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

[5,2,1,1] => [4,2,1,1,1] => [[1,5,8,9],[2,7],[3],[4],[6]] => [[1,2,3,4,6],[5,7],[8],[9]] => 25

[5,1,1,1,1] => [5,1,1,1,1] => [[1,6,7,8,9],[2],[3],[4],[5]] => [[1,2,3,4,5],[6],[7],[8],[9]] => 26

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

[4,3,2] => [3,3,2,1] => [[1,3,6],[2,5,9],[4,8],[7]] => [[1,2,4,7],[3,5,8],[6,9]] => 26

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

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

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

[4,1,1,1,1,1] => [6,1,1,1] => [[1,5,6,7,8,9],[2],[3],[4]] => [[1,2,3,4],[5],[6],[7],[8],[9]] => 30

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

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

[3,3,1,1,1] => [5,2,2] => [[1,2,7,8,9],[3,4],[5,6]] => [[1,3,5],[2,4,6],[7],[8],[9]] => 30

[3,2,2,2] => [4,4,1] => [[1,3,4,5],[2,7,8,9],[6]] => [[1,2,6],[3,7],[4,8],[5,9]] => 30

[3,2,2,1,1] => [5,3,1] => [[1,3,4,8,9],[2,6,7],[5]] => [[1,2,5],[3,6],[4,7],[8],[9]] => 31

[3,2,1,1,1,1] => [6,2,1] => [[1,3,6,7,8,9],[2,5],[4]] => [[1,2,4],[3,5],[6],[7],[8],[9]] => 32

[3,1,1,1,1,1,1] => [7,1,1] => [[1,4,5,6,7,8,9],[2],[3]] => [[1,2,3],[4],[5],[6],[7],[8],[9]] => 33

[2,2,2,2,1] => [5,4] => [[1,2,3,4,9],[5,6,7,8]] => [[1,5],[2,6],[3,7],[4,8],[9]] => 32

[2,2,2,1,1,1] => [6,3] => [[1,2,3,7,8,9],[4,5,6]] => [[1,4],[2,5],[3,6],[7],[8],[9]] => 33

[2,2,1,1,1,1,1] => [7,2] => [[1,2,5,6,7,8,9],[3,4]] => [[1,3],[2,4],[5],[6],[7],[8],[9]] => 34

[2,1,1,1,1,1,1,1] => [8,1] => [[1,3,4,5,6,7,8,9],[2]] => [[1,2],[3],[4],[5],[6],[7],[8],[9]] => 35

[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]] => [[1,2,3,4,5,6,7,8,9,10]] => 0

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

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

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

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

>>> Load all 142 entries. <<<

[7,2,1] => [3,2,1,1,1,1,1] => [[1,7,10],[2,9],[3],[4],[5],[6],[8]] => [[1,2,3,4,5,6,8],[7,9],[10]] => 23

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

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

[6,3,1] => [3,2,2,1,1,1] => [[1,5,10],[2,7],[3,9],[4],[6],[8]] => [[1,2,3,4,6,8],[5,7,9],[10]] => 27

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

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

[6,1,1,1,1] => [5,1,1,1,1,1] => [[1,7,8,9,10],[2],[3],[4],[5],[6]] => [[1,2,3,4,5,6],[7],[8],[9],[10]] => 30

[5,5] => [2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10]] => [[1,3,5,7,9],[2,4,6,8,10]] => 25

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

[5,3,2] => [3,3,2,1,1] => [[1,4,7],[2,6,10],[3,9],[5],[8]] => [[1,2,3,5,8],[4,6,9],[7,10]] => 31

[5,3,1,1] => [4,2,2,1,1] => [[1,4,9,10],[2,6],[3,8],[5],[7]] => [[1,2,3,5,7],[4,6,8],[9],[10]] => 32

[5,2,2,1] => [4,3,1,1,1] => [[1,5,6,10],[2,8,9],[3],[4],[7]] => [[1,2,3,4,7],[5,8],[6,9],[10]] => 33

[5,2,1,1,1] => [5,2,1,1,1] => [[1,5,8,9,10],[2,7],[3],[4],[6]] => [[1,2,3,4,6],[5,7],[8],[9],[10]] => 34

[5,1,1,1,1,1] => [6,1,1,1,1] => [[1,6,7,8,9,10],[2],[3],[4],[5]] => [[1,2,3,4,5],[6],[7],[8],[9],[10]] => 35

[4,4,2] => [3,3,2,2] => [[1,2,7],[3,4,10],[5,6],[8,9]] => [[1,3,5,8],[2,4,6,9],[7,10]] => 32

[4,4,1,1] => [4,2,2,2] => [[1,2,9,10],[3,4],[5,6],[7,8]] => [[1,3,5,7],[2,4,6,8],[9],[10]] => 33

[4,3,3] => [3,3,3,1] => [[1,3,4],[2,6,7],[5,9,10],[8]] => [[1,2,5,8],[3,6,9],[4,7,10]] => 33

[4,3,2,1] => [4,3,2,1] => [[1,3,6,10],[2,5,9],[4,8],[7]] => [[1,2,4,7],[3,5,8],[6,9],[10]] => 35

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

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

[4,2,2,1,1] => [5,3,1,1] => [[1,4,5,9,10],[2,7,8],[3],[6]] => [[1,2,3,6],[4,7],[5,8],[9],[10]] => 37

[4,2,1,1,1,1] => [6,2,1,1] => [[1,4,7,8,9,10],[2,6],[3],[5]] => [[1,2,3,5],[4,6],[7],[8],[9],[10]] => 38

[4,1,1,1,1,1,1] => [7,1,1,1] => [[1,5,6,7,8,9,10],[2],[3],[4]] => [[1,2,3,4],[5],[6],[7],[8],[9],[10]] => 39

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

[3,3,2,2] => [4,4,2] => [[1,2,5,6],[3,4,9,10],[7,8]] => [[1,3,7],[2,4,8],[5,9],[6,10]] => 37

[3,3,2,1,1] => [5,3,2] => [[1,2,5,9,10],[3,4,8],[6,7]] => [[1,3,6],[2,4,7],[5,8],[9],[10]] => 38

[3,3,1,1,1,1] => [6,2,2] => [[1,2,7,8,9,10],[3,4],[5,6]] => [[1,3,5],[2,4,6],[7],[8],[9],[10]] => 39

[3,2,2,2,1] => [5,4,1] => [[1,3,4,5,10],[2,7,8,9],[6]] => [[1,2,6],[3,7],[4,8],[5,9],[10]] => 39

[3,2,2,1,1,1] => [6,3,1] => [[1,3,4,8,9,10],[2,6,7],[5]] => [[1,2,5],[3,6],[4,7],[8],[9],[10]] => 40

[3,2,1,1,1,1,1] => [7,2,1] => [[1,3,6,7,8,9,10],[2,5],[4]] => [[1,2,4],[3,5],[6],[7],[8],[9],[10]] => 41

[3,1,1,1,1,1,1,1] => [8,1,1] => [[1,4,5,6,7,8,9,10],[2],[3]] => [[1,2,3],[4],[5],[6],[7],[8],[9],[10]] => 42

[2,2,2,2,2] => [5,5] => [[1,2,3,4,5],[6,7,8,9,10]] => [[1,6],[2,7],[3,8],[4,9],[5,10]] => 40

[2,2,2,2,1,1] => [6,4] => [[1,2,3,4,9,10],[5,6,7,8]] => [[1,5],[2,6],[3,7],[4,8],[9],[10]] => 41

[2,2,2,1,1,1,1] => [7,3] => [[1,2,3,7,8,9,10],[4,5,6]] => [[1,4],[2,5],[3,6],[7],[8],[9],[10]] => 42

[2,2,1,1,1,1,1,1] => [8,2] => [[1,2,5,6,7,8,9,10],[3,4]] => [[1,3],[2,4],[5],[6],[7],[8],[9],[10]] => 43

[2,1,1,1,1,1,1,1,1] => [9,1] => [[1,3,4,5,6,7,8,9,10],[2]] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]] => 44

[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

[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]] => [[1,2,3,4,5,6,7,8,9,10,11,12]] => 0

[6,6] => [2,2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 36

[2,2,2,2,2,2] => [6,6] => [[1,2,3,4,5,6],[7,8,9,10,11,12]] => [[1,7],[2,8],[3,9],[4,10],[5,11],[6,12]] => 60

[] => [] => [] => [] => 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 (standard) major index of a standard tableau.
A descent of a standard tableau

$T$ is an index

$i$ such that

$i+1$ appears in a row strictly below the row of

$i$ . The (standard) major index is the the sum of the descents.

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.

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

conjugate

Description

Sends a standard tableau to its conjugate tableau.