- FindStat

Identifier

Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000336: Standard tableaux ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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 leg major index of a standard tableau.
The leg length of a cell is the number of cells strictly below in the same column. This statistic is the sum of all leg lengths. Therefore, this is actually a statistic on the underlying integer partition.
It happens to coincide with the (leg) major index of a tabloid restricted to standard Young tableaux, defined as follows: the descent set of a tabloid is the set of cells, not in the top row, whose entry is strictly larger than the entry directly above it. The leg major index is the sum of the leg lengths of the descents plus the number of descents.

Map

initial tableau

Description

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

$1$ through

$n$ row by row.