- FindStat

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Identifier

St000059: Standard tableaux ⟶ ℤ

Values

[] => 0

[[1]] => 0

[[1,2]] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[1,2],[3],[4],[5]] => 9

[[1],[2],[3],[4],[5]] => 10

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[1,2,3],[4,5],[6]] => 11

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

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

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

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

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

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

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

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

[[1,2,4],[3],[5],[6]] => 11

[[1,2,3],[4],[5],[6]] => 12

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

>>> Load all 2137 entries. <<<

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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].

References

[1] Haglund, J., Stevens, L. An extension of the Foata map to standard Young tableaux MathSciNet:2264944

Code

def statistic(T):
    """
    sage: T = StandardTableau([[1,4],[2,5],[3]])
    sage: statistic(T)
    [(2, 1), (5, 4), (3, 1), (3, 2)]
    """
    def inversion_path(T, i, j):
        """Return the inversion path of the cell i, j, as a list of column
        indices.
    
        Given this list ``p``, cells below the path are those in
        ``T[k][l]`` with ``0<=k= p[k]``.
    
        EXAMPLES::
    
            sage: T = StandardTableau([[1,2,4,8],[3,7,11,12],[5,9,13,14],[6,10,15,16]])
            sage: inversion_path(T, 3, 3)
            [0, 2, 2]
        
            sage: T = StandardTableau([[1,2],[3,4]])
            sage: inversion_path(T, 1, 1)
            [0]
        
            sage: T = StandardTableau([[1,3],[2,4]])
            sage: inversion_path(T, 1, 1)
            [1]
    
        """
        columns = []
        while True:
            if j == 0:
                columns.extend([0]*i)
                return columns[::-1]
            elif i == 0:
                return columns[::-1]
    
            elif T[i-1][j] > T[i][j-1]:
                columns.append(j)
                i -= 1
    
            elif T[i-1][j] < T[i][j-1]:
                j -= 1

    I = []
    for i,j in T.cells():
        c = T[i][j]
        p = inversion_path(T, i, j)
        for row, col in enumerate(p):
            for d in T[row][col:]:
                if d < c:
                    I.append((c, d))
    return len(I)

Created

Apr 09, 2013 at 21:30 by Bryan D'Amico

Updated

May 12, 2020 at 15:18 by Martin Rubey