Identifier
-
Mp00042:
Integer partitions
—initial tableau⟶
Standard tableaux
St000059: Standard tableaux ⟶ ℤ
Values
[1] => [[1]] => 0
[2] => [[1,2]] => 0
[1,1] => [[1],[2]] => 1
[3] => [[1,2,3]] => 0
[2,1] => [[1,2],[3]] => 2
[1,1,1] => [[1],[2],[3]] => 3
[4] => [[1,2,3,4]] => 0
[3,1] => [[1,2,3],[4]] => 3
[2,2] => [[1,2],[3,4]] => 4
[2,1,1] => [[1,2],[3],[4]] => 5
[1,1,1,1] => [[1],[2],[3],[4]] => 6
[5] => [[1,2,3,4,5]] => 0
[4,1] => [[1,2,3,4],[5]] => 4
[3,2] => [[1,2,3],[4,5]] => 6
[3,1,1] => [[1,2,3],[4],[5]] => 7
[2,2,1] => [[1,2],[3,4],[5]] => 8
[2,1,1,1] => [[1,2],[3],[4],[5]] => 9
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => 10
[6] => [[1,2,3,4,5,6]] => 0
[5,1] => [[1,2,3,4,5],[6]] => 5
[4,2] => [[1,2,3,4],[5,6]] => 8
[4,1,1] => [[1,2,3,4],[5],[6]] => 9
[3,3] => [[1,2,3],[4,5,6]] => 9
[3,2,1] => [[1,2,3],[4,5],[6]] => 11
[3,1,1,1] => [[1,2,3],[4],[5],[6]] => 12
[2,2,2] => [[1,2],[3,4],[5,6]] => 12
[2,2,1,1] => [[1,2],[3,4],[5],[6]] => 13
[2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => 14
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => 15
[7] => [[1,2,3,4,5,6,7]] => 0
[6,1] => [[1,2,3,4,5,6],[7]] => 6
[5,2] => [[1,2,3,4,5],[6,7]] => 10
[5,1,1] => [[1,2,3,4,5],[6],[7]] => 11
[4,3] => [[1,2,3,4],[5,6,7]] => 12
[4,2,1] => [[1,2,3,4],[5,6],[7]] => 14
[4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 15
[3,3,1] => [[1,2,3],[4,5,6],[7]] => 15
[3,2,2] => [[1,2,3],[4,5],[6,7]] => 16
[3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => 17
[3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => 18
[2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => 18
[2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => 19
[2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => 20
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => 21
[8] => [[1,2,3,4,5,6,7,8]] => 0
[7,1] => [[1,2,3,4,5,6,7],[8]] => 7
[6,2] => [[1,2,3,4,5,6],[7,8]] => 12
[6,1,1] => [[1,2,3,4,5,6],[7],[8]] => 13
[5,3] => [[1,2,3,4,5],[6,7,8]] => 15
[5,2,1] => [[1,2,3,4,5],[6,7],[8]] => 17
[5,1,1,1] => [[1,2,3,4,5],[6],[7],[8]] => 18
[4,4] => [[1,2,3,4],[5,6,7,8]] => 16
[4,3,1] => [[1,2,3,4],[5,6,7],[8]] => 19
[4,2,2] => [[1,2,3,4],[5,6],[7,8]] => 20
[4,2,1,1] => [[1,2,3,4],[5,6],[7],[8]] => 21
[4,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8]] => 22
[3,3,2] => [[1,2,3],[4,5,6],[7,8]] => 21
[3,3,1,1] => [[1,2,3],[4,5,6],[7],[8]] => 22
[3,2,2,1] => [[1,2,3],[4,5],[6,7],[8]] => 23
[3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 24
[3,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8]] => 25
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => 24
[2,2,2,1,1] => [[1,2],[3,4],[5,6],[7],[8]] => 25
[2,2,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8]] => 26
[2,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8]] => 27
[1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => 28
[9] => [[1,2,3,4,5,6,7,8,9]] => 0
[8,1] => [[1,2,3,4,5,6,7,8],[9]] => 8
[7,2] => [[1,2,3,4,5,6,7],[8,9]] => 14
[7,1,1] => [[1,2,3,4,5,6,7],[8],[9]] => 15
[6,3] => [[1,2,3,4,5,6],[7,8,9]] => 18
[6,2,1] => [[1,2,3,4,5,6],[7,8],[9]] => 20
[6,1,1,1] => [[1,2,3,4,5,6],[7],[8],[9]] => 21
[5,4] => [[1,2,3,4,5],[6,7,8,9]] => 20
[5,3,1] => [[1,2,3,4,5],[6,7,8],[9]] => 23
[5,2,2] => [[1,2,3,4,5],[6,7],[8,9]] => 24
[5,2,1,1] => [[1,2,3,4,5],[6,7],[8],[9]] => 25
[5,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9]] => 26
[4,4,1] => [[1,2,3,4],[5,6,7,8],[9]] => 24
[4,3,2] => [[1,2,3,4],[5,6,7],[8,9]] => 26
[4,3,1,1] => [[1,2,3,4],[5,6,7],[8],[9]] => 27
[4,2,2,1] => [[1,2,3,4],[5,6],[7,8],[9]] => 28
[4,2,1,1,1] => [[1,2,3,4],[5,6],[7],[8],[9]] => 29
[4,1,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8],[9]] => 30
[3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => 27
[3,3,2,1] => [[1,2,3],[4,5,6],[7,8],[9]] => 29
[3,3,1,1,1] => [[1,2,3],[4,5,6],[7],[8],[9]] => 30
[3,2,2,2] => [[1,2,3],[4,5],[6,7],[8,9]] => 30
[3,2,2,1,1] => [[1,2,3],[4,5],[6,7],[8],[9]] => 31
[3,2,1,1,1,1] => [[1,2,3],[4,5],[6],[7],[8],[9]] => 32
[3,1,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8],[9]] => 33
[2,2,2,2,1] => [[1,2],[3,4],[5,6],[7,8],[9]] => 32
[2,2,2,1,1,1] => [[1,2],[3,4],[5,6],[7],[8],[9]] => 33
[2,2,1,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8],[9]] => 34
[2,1,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8],[9]] => 35
[1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9]] => 36
[10] => [[1,2,3,4,5,6,7,8,9,10]] => 0
[9,1] => [[1,2,3,4,5,6,7,8,9],[10]] => 9
[8,2] => [[1,2,3,4,5,6,7,8],[9,10]] => 16
[8,1,1] => [[1,2,3,4,5,6,7,8],[9],[10]] => 17
[7,3] => [[1,2,3,4,5,6,7],[8,9,10]] => 21
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Description
The inversion number of a standard tableau as defined by Haglund and Stevens.
Their inversion number is the total number of inversion pairs for the tableau. An inversion pair is defined as a pair of cells (a,b), (x,y) such that the content of (x,y) is greater than the content of (a,b) and (x,y) is north of the inversion path of (a,b), where the inversion path is defined in detail in [1].
Their inversion number is the total number of inversion pairs for the tableau. An inversion pair is defined as a pair of cells (a,b), (x,y) such that the content of (x,y) is greater than the content of (a,b) and (x,y) is north of the inversion path of (a,b), where the inversion path is defined in detail in [1].
Map
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.
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