- FindStat

Identifier

Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St000391: Binary words ⟶ ℤ

Values

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

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

[3] => [1,1,1] => [[1],[2],[3]] => 11 => 3

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

[1,1,1] => [3] => [[1,2,3]] => 00 => 0

[4] => [1,1,1,1] => [[1],[2],[3],[4]] => 111 => 6

[3,1] => [2,1,1] => [[1,4],[2],[3]] => 110 => 3

[2,2] => [2,2] => [[1,2],[3,4]] => 010 => 2

[2,1,1] => [3,1] => [[1,3,4],[2]] => 100 => 1

[1,1,1,1] => [4] => [[1,2,3,4]] => 000 => 0

[5] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => 1111 => 10

[4,1] => [2,1,1,1] => [[1,5],[2],[3],[4]] => 1110 => 6

[3,2] => [2,2,1] => [[1,3],[2,5],[4]] => 1010 => 4

[3,1,1] => [3,1,1] => [[1,4,5],[2],[3]] => 1100 => 3

[2,2,1] => [3,2] => [[1,2,5],[3,4]] => 0100 => 2

[2,1,1,1] => [4,1] => [[1,3,4,5],[2]] => 1000 => 1

[1,1,1,1,1] => [5] => [[1,2,3,4,5]] => 0000 => 0

[6] => [1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => 11111 => 15

[5,1] => [2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => 11110 => 10

[4,2] => [2,2,1,1] => [[1,4],[2,6],[3],[5]] => 11010 => 7

[4,1,1] => [3,1,1,1] => [[1,5,6],[2],[3],[4]] => 11100 => 6

[3,3] => [2,2,2] => [[1,2],[3,4],[5,6]] => 01010 => 6

[3,2,1] => [3,2,1] => [[1,3,6],[2,5],[4]] => 10100 => 4

[3,1,1,1] => [4,1,1] => [[1,4,5,6],[2],[3]] => 11000 => 3

[2,2,2] => [3,3] => [[1,2,3],[4,5,6]] => 00100 => 3

[2,2,1,1] => [4,2] => [[1,2,5,6],[3,4]] => 01000 => 2

[2,1,1,1,1] => [5,1] => [[1,3,4,5,6],[2]] => 10000 => 1

[1,1,1,1,1,1] => [6] => [[1,2,3,4,5,6]] => 00000 => 0

[7] => [1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => 111111 => 21

[6,1] => [2,1,1,1,1,1] => [[1,7],[2],[3],[4],[5],[6]] => 111110 => 15

[5,2] => [2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => 111010 => 11

[5,1,1] => [3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => 111100 => 10

[4,3] => [2,2,2,1] => [[1,3],[2,5],[4,7],[6]] => 101010 => 9

[4,2,1] => [3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => 110100 => 7

[4,1,1,1] => [4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => 111000 => 6

[3,3,1] => [3,2,2] => [[1,2,7],[3,4],[5,6]] => 010100 => 6

[3,2,2] => [3,3,1] => [[1,3,4],[2,6,7],[5]] => 100100 => 5

[3,2,1,1] => [4,2,1] => [[1,3,6,7],[2,5],[4]] => 101000 => 4

[3,1,1,1,1] => [5,1,1] => [[1,4,5,6,7],[2],[3]] => 110000 => 3

[2,2,2,1] => [4,3] => [[1,2,3,7],[4,5,6]] => 001000 => 3

[2,2,1,1,1] => [5,2] => [[1,2,5,6,7],[3,4]] => 010000 => 2

[2,1,1,1,1,1] => [6,1] => [[1,3,4,5,6,7],[2]] => 100000 => 1

[1,1,1,1,1,1,1] => [7] => [[1,2,3,4,5,6,7]] => 000000 => 0

[8] => [1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => 1111111 => 28

[7,1] => [2,1,1,1,1,1,1] => [[1,8],[2],[3],[4],[5],[6],[7]] => 1111110 => 21

[6,2] => [2,2,1,1,1,1] => [[1,6],[2,8],[3],[4],[5],[7]] => 1111010 => 16

[6,1,1] => [3,1,1,1,1,1] => [[1,7,8],[2],[3],[4],[5],[6]] => 1111100 => 15

[5,3] => [2,2,2,1,1] => [[1,4],[2,6],[3,8],[5],[7]] => 1101010 => 13

[5,2,1] => [3,2,1,1,1] => [[1,5,8],[2,7],[3],[4],[6]] => 1110100 => 11

[5,1,1,1] => [4,1,1,1,1] => [[1,6,7,8],[2],[3],[4],[5]] => 1111000 => 10

[4,4] => [2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => 0101010 => 12

[4,3,1] => [3,2,2,1] => [[1,3,8],[2,5],[4,7],[6]] => 1010100 => 9

[4,2,2] => [3,3,1,1] => [[1,4,5],[2,7,8],[3],[6]] => 1100100 => 8

[4,2,1,1] => [4,2,1,1] => [[1,4,7,8],[2,6],[3],[5]] => 1101000 => 7

[4,1,1,1,1] => [5,1,1,1] => [[1,5,6,7,8],[2],[3],[4]] => 1110000 => 6

[3,3,2] => [3,3,2] => [[1,2,5],[3,4,8],[6,7]] => 0100100 => 7

[3,3,1,1] => [4,2,2] => [[1,2,7,8],[3,4],[5,6]] => 0101000 => 6

[3,2,2,1] => [4,3,1] => [[1,3,4,8],[2,6,7],[5]] => 1001000 => 5

[3,2,1,1,1] => [5,2,1] => [[1,3,6,7,8],[2,5],[4]] => 1010000 => 4

[3,1,1,1,1,1] => [6,1,1] => [[1,4,5,6,7,8],[2],[3]] => 1100000 => 3

[2,2,2,2] => [4,4] => [[1,2,3,4],[5,6,7,8]] => 0001000 => 4

[2,2,2,1,1] => [5,3] => [[1,2,3,7,8],[4,5,6]] => 0010000 => 3

[2,2,1,1,1,1] => [6,2] => [[1,2,5,6,7,8],[3,4]] => 0100000 => 2

[2,1,1,1,1,1,1] => [7,1] => [[1,3,4,5,6,7,8],[2]] => 1000000 => 1

[1,1,1,1,1,1,1,1] => [8] => [[1,2,3,4,5,6,7,8]] => 0000000 => 0

[9] => [1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9]] => 11111111 => 36

[8,1] => [2,1,1,1,1,1,1,1] => [[1,9],[2],[3],[4],[5],[6],[7],[8]] => 11111110 => 28

[7,2] => [2,2,1,1,1,1,1] => [[1,7],[2,9],[3],[4],[5],[6],[8]] => 11111010 => 22

[7,1,1] => [3,1,1,1,1,1,1] => [[1,8,9],[2],[3],[4],[5],[6],[7]] => 11111100 => 21

[6,3] => [2,2,2,1,1,1] => [[1,5],[2,7],[3,9],[4],[6],[8]] => 11101010 => 18

[6,2,1] => [3,2,1,1,1,1] => [[1,6,9],[2,8],[3],[4],[5],[7]] => 11110100 => 16

[6,1,1,1] => [4,1,1,1,1,1] => [[1,7,8,9],[2],[3],[4],[5],[6]] => 11111000 => 15

[5,4] => [2,2,2,2,1] => [[1,3],[2,5],[4,7],[6,9],[8]] => 10101010 => 16

[5,3,1] => [3,2,2,1,1] => [[1,4,9],[2,6],[3,8],[5],[7]] => 11010100 => 13

[5,2,2] => [3,3,1,1,1] => [[1,5,6],[2,8,9],[3],[4],[7]] => 11100100 => 12

[5,2,1,1] => [4,2,1,1,1] => [[1,5,8,9],[2,7],[3],[4],[6]] => 11101000 => 11

[5,1,1,1,1] => [5,1,1,1,1] => [[1,6,7,8,9],[2],[3],[4],[5]] => 11110000 => 10

[4,4,1] => [3,2,2,2] => [[1,2,9],[3,4],[5,6],[7,8]] => 01010100 => 12

[4,3,2] => [3,3,2,1] => [[1,3,6],[2,5,9],[4,8],[7]] => 10100100 => 10

[4,3,1,1] => [4,2,2,1] => [[1,3,8,9],[2,5],[4,7],[6]] => 10101000 => 9

[4,2,2,1] => [4,3,1,1] => [[1,4,5,9],[2,7,8],[3],[6]] => 11001000 => 8

[4,2,1,1,1] => [5,2,1,1] => [[1,4,7,8,9],[2,6],[3],[5]] => 11010000 => 7

[4,1,1,1,1,1] => [6,1,1,1] => [[1,5,6,7,8,9],[2],[3],[4]] => 11100000 => 6

[3,3,3] => [3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => 00100100 => 9

[3,3,2,1] => [4,3,2] => [[1,2,5,9],[3,4,8],[6,7]] => 01001000 => 7

[3,3,1,1,1] => [5,2,2] => [[1,2,7,8,9],[3,4],[5,6]] => 01010000 => 6

[3,2,2,2] => [4,4,1] => [[1,3,4,5],[2,7,8,9],[6]] => 10001000 => 6

[3,2,2,1,1] => [5,3,1] => [[1,3,4,8,9],[2,6,7],[5]] => 10010000 => 5

[3,2,1,1,1,1] => [6,2,1] => [[1,3,6,7,8,9],[2,5],[4]] => 10100000 => 4

[3,1,1,1,1,1,1] => [7,1,1] => [[1,4,5,6,7,8,9],[2],[3]] => 11000000 => 3

[2,2,2,2,1] => [5,4] => [[1,2,3,4,9],[5,6,7,8]] => 00010000 => 4

[2,2,2,1,1,1] => [6,3] => [[1,2,3,7,8,9],[4,5,6]] => 00100000 => 3

[2,2,1,1,1,1,1] => [7,2] => [[1,2,5,6,7,8,9],[3,4]] => 01000000 => 2

[2,1,1,1,1,1,1,1] => [8,1] => [[1,3,4,5,6,7,8,9],[2]] => 10000000 => 1

[1,1,1,1,1,1,1,1,1] => [9] => [[1,2,3,4,5,6,7,8,9]] => 00000000 => 0

[10] => [1,1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]] => 111111111 => 45

[9,1] => [2,1,1,1,1,1,1,1,1] => [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]] => 111111110 => 36

[8,2] => [2,2,1,1,1,1,1,1] => [[1,8],[2,10],[3],[4],[5],[6],[7],[9]] => 111111010 => 29

[8,1,1] => [3,1,1,1,1,1,1,1] => [[1,9,10],[2],[3],[4],[5],[6],[7],[8]] => 111111100 => 28

[7,3] => [2,2,2,1,1,1,1] => [[1,6],[2,8],[3,10],[4],[5],[7],[9]] => 111101010 => 24

[7,2,1] => [3,2,1,1,1,1,1] => [[1,7,10],[2,9],[3],[4],[5],[6],[8]] => 111110100 => 22

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$F_{2} = 1 + q$

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

$F_{4} = 1 + q + q^{2} + q^{3} + q^{6}$

$F_{5} = 1 + q + q^{2} + q^{3} + q^{4} + q^{6} + q^{10}$

$F_{6} = 1 + q + q^{2} + 2\ q^{3} + q^{4} + 2\ q^{6} + q^{7} + q^{10} + q^{15}$

$F_{7} = 1 + q + q^{2} + 2\ q^{3} + q^{4} + q^{5} + 2\ q^{6} + q^{7} + q^{9} + q^{10} + q^{11} + q^{15} + q^{21}$

$F_{8} = 1 + q + q^{2} + 2\ q^{3} + 2\ q^{4} + q^{5} + 2\ q^{6} + 2\ q^{7} + q^{8} + q^{9} + q^{10} + q^{11} + q^{12} + q^{13} + q^{15} + q^{16} + q^{21} + q^{28}$

$F_{9} = 1 + q + q^{2} + 2\ q^{3} + 2\ q^{4} + q^{5} + 3\ q^{6} + 2\ q^{7} + q^{8} + 2\ q^{9} + 2\ q^{10} + q^{11} + 2\ q^{12} + q^{13} + q^{15} + 2\ q^{16} + q^{18} + q^{21} + q^{22} + q^{28} + q^{36}$

$F_{10} = 1 + q + q^{2} + 2\ q^{3} + 2\ q^{4} + 2\ q^{5} + 3\ q^{6} + 2\ q^{7} + 2\ q^{8} + 3\ q^{9} + 2\ q^{10} + q^{11} + 3\ q^{12} + 2\ q^{13} + q^{14} + q^{15} + 2\ q^{16} + q^{17} + q^{18} + q^{20} + 2\ q^{21} + q^{22} + q^{24} + q^{28} + q^{29} + q^{36} + q^{45}$

Description

The sum of the positions of the ones in a binary word.

Map

descent word

Description

The descent word of a standard Young tableau.
For a standard Young tableau of size

$n$ we set

$w_i=1$ if

$i+1$ is in a lower row than

$i$ , and

$0$ otherwise, for

$1\leq i < n$ .

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 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.