- FindStat

Identifier

Mp00085: Standard tableaux —Schützenberger involution⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000008: Integer compositions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

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

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

$F_{5} = 1 + q + 2\ q^{2} + 3\ q^{3} + 4\ q^{4} + 4\ q^{5} + 4\ q^{6} + 3\ q^{7} + 2\ q^{8} + q^{9} + q^{10}$

$F_{6} = 1 + q + 2\ q^{2} + 4\ q^{3} + 5\ q^{4} + 7\ q^{5} + 9\ q^{6} + 10\ q^{7} + 9\ q^{8} + 9\ q^{9} + 7\ q^{10} + 5\ q^{11} + 3\ q^{12} + 2\ q^{13} + q^{14} + q^{15}$

$F_{7} = 1 + q + 2\ q^{2} + 4\ q^{3} + 6\ q^{4} + 9\ q^{5} + 13\ q^{6} + 17\ q^{7} + 20\ q^{8} + 23\ q^{9} + 24\ q^{10} + 24\ q^{11} + 21\ q^{12} + 20\ q^{13} + 15\ q^{14} + 12\ q^{15} + 8\ q^{16} + 5\ q^{17} + 3\ q^{18} + 2\ q^{19} + q^{20} + q^{21}$

$F_{8} = 1 + q + 2\ q^{2} + 4\ q^{3} + 7\ q^{4} + 10\ q^{5} + 16\ q^{6} + 23\ q^{7} + 32\ q^{8} + 40\ q^{9} + 50\ q^{10} + 56\ q^{11} + 62\ q^{12} + 68\ q^{13} + 67\ q^{14} + 63\ q^{15} + 59\ q^{16} + 50\ q^{17} + 43\ q^{18} + 32\ q^{19} + 27\ q^{20} + 18\ q^{21} + 13\ q^{22} + 8\ q^{23} + 5\ q^{24} + 3\ q^{25} + 2\ q^{26} + q^{27} + q^{28}$

Description

The major index of the composition.
The descents of a composition

$[c_1,c_2,\dots,c_k]$ are the partial sums

$c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$ , excluding the sum of all parts. The major index of a composition is the sum of its descents.
For details about the major index see Permutations/Descents-Major.

Map

horizontal strip sizes

Description

The composition of horizontal strip sizes.
We associate to a standard Young tableau

$T$ the composition

$(c_1,\dots,c_k)$ , such that

$k$ is minimal and the numbers

$c_1+\dots+c_i + 1,\dots,c_1+\dots+c_{i+1}$ form a horizontal strip in

$T$ for all

$i$ .

Map

Schützenberger involution

Description

Sends a standard tableau to the standard tableau obtained via the Schützenberger involution.