- FindStat

Identifier

Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The last entry in the first row of a standard tableau.

Map

initial tableau

Description

Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.

Map

to partition

Description

Sends a composition to the partition obtained by sorting the entries.