- FindStat

Identifier

Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000169: 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]] => 1

[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]] => 3

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

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

[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]] => 6

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

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

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

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

[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]] => 10

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

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

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

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

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

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

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

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

[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]] => 15

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

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

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

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

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

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

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

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

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

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

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

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

[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]] => 21

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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]] => 28

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

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

[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]] => 16

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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]] => 36

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

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

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

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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 cocharge of a standard tableau.
The cocharge of a standard tableau

$T$ , denoted

$\mathrm{cc}(T)$ , is defined to be the cocharge of the reading word of the tableau. The cocharge of a permutation

$w_1 w_2\cdots w_n$ can be computed by the following algorithm:
1) Starting from

$w_n$ , scan the entries right-to-left until finding the entry

$1$ with a superscript

$0$ .
2) Continue scanning until the

$2$ is found, and label this with a superscript

$1$ . Then scan until the

$3$ is found, labeling with a

$2$ , and so on, incrementing the label each time, until the beginning of the word is reached. Then go back to the end and scan again from right to left, and *do not* increment the superscript label for the first number found in the next scan. Then continue scanning and labeling, each time incrementing the superscript only if we have not cycled around the word since the last labeling.
3) The cocharge is defined as the sum of the superscript labels on the letters.

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.