- FindStat

Identifier

Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000766: Integer compositions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 2 + q$

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

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

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

$F_{7} = 2 + 3\ q + 3\ q^{2} + 3\ q^{3} + q^{4} + 2\ q^{5} + q^{6}$

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

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

Description

The number of inversions of an integer composition.
This is the number of pairs

$(i,j)$ such that

$i < j$ and

$c_i > c_j$ .

Map

initial tableau

Description

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

$1$ through

$n$ row by row.

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

$i$ .