- FindStat

Identifier

Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000009: Standard tableaux ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[2,1,1,1,1,1,1] => [7,1] => [[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]] => 28

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[4,1,1,1,1,1] => [6,1,1,1] => [[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]] => 27

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

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

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

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

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

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

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

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

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

[2,1,1,1,1,1,1,1] => [8,1] => [[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]] => 36

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

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

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

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

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

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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 charge of a standard tableau.

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

initial tableau

Description

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

$1$ through

$n$ row by row.