- FindStat

Identifier

Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000765: Integer compositions ⟶ ℤ

Values

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

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$F_{1} = q$

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

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

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

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

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

$F_{7} = 11\ q + 2\ q^{2} + q^{3} + q^{7}$

$F_{8} = 15\ q + 4\ q^{2} + q^{3} + q^{4} + q^{8}$

$F_{9} = 22\ q + 4\ q^{2} + 2\ q^{3} + q^{4} + q^{9}$

Description

The number of weak records in an integer composition.
A weak record is an element

$a_i$ such that

$a_i \geq a_j$ for all

$j < i$ .

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

$i$ .