- FindStat

Identifier

Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000059: Standard tableaux ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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]] => 9

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

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

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

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

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

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

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

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

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

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

[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]] => 16

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

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

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

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

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

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

[1,1,1,1,1,1,1] => [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]] => 28

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

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

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

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

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

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

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

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

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

[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]] => 18

[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]] => 20

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

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

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

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

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

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

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

[1,1,1,1,1,1,1,1] => [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]] => 36

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

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

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

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

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

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

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

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

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

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

[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]] => 30

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

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

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

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

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

[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]] => 26

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

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

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

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

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

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

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

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

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

[1,1,1,1,1,1,1,1,1] => [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]] => 45

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

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

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

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

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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 inversion number of a standard tableau as defined by Haglund and Stevens.
Their inversion number is the total number of inversion pairs for the tableau. An inversion pair is defined as a pair of cells (a,b), (x,y) such that the content of (x,y) is greater than the content of (a,b) and (x,y) is north of the inversion path of (a,b), where the inversion path is defined in detail in [1].

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.