- FindStat

Identifier

Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 215 entries. <<<

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$F_{1} = 1$

$F_{2} = 1 + q$

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

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

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

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

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

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

$F_{9} = 1 + 8\ q + 12\ q^{2} + 7\ q^{3} + 2\ q^{4}$

$F_{10} = 1 + 9\ q + 16\ q^{2} + 11\ q^{3} + 4\ q^{4} + q^{5}$

Description

The number of descents of a standard tableau.
Entry

$i$ of a standard Young tableau is a descent if

$i+1$ appears in a row below the row of

$i$ .

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.

Map

Bulgarian solitaire

Description

A move in Bulgarian solitaire.
Remove the first column of the Ferrers diagram and insert it as a new row.
If the partition is empty, return the empty partition.