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Identifier

St000558: Set partitions ⟶ ℤ

Values

{{1,2}} => 1

{{1},{2}} => 0

{{1,2,3}} => 3

{{1,2},{3}} => 1

{{1,3},{2}} => 1

{{1},{2,3}} => 1

{{1},{2},{3}} => 0

{{1,2,3,4}} => 6

{{1,2,3},{4}} => 3

{{1,2,4},{3}} => 3

{{1,2},{3,4}} => 2

{{1,2},{3},{4}} => 1

{{1,3,4},{2}} => 3

{{1,3},{2,4}} => 2

{{1,3},{2},{4}} => 1

{{1,4},{2,3}} => 2

{{1},{2,3,4}} => 3

{{1},{2,3},{4}} => 1

{{1,4},{2},{3}} => 1

{{1},{2,4},{3}} => 1

{{1},{2},{3,4}} => 1

{{1},{2},{3},{4}} => 0

{{1,2,3,4,5}} => 10

{{1,2,3,4},{5}} => 6

{{1,2,3,5},{4}} => 6

{{1,2,3},{4,5}} => 4

{{1,2,3},{4},{5}} => 3

{{1,2,4,5},{3}} => 6

{{1,2,4},{3,5}} => 4

{{1,2,4},{3},{5}} => 3

{{1,2,5},{3,4}} => 4

{{1,2},{3,4,5}} => 4

{{1,2},{3,4},{5}} => 2

{{1,2,5},{3},{4}} => 3

{{1,2},{3,5},{4}} => 2

{{1,2},{3},{4,5}} => 2

{{1,2},{3},{4},{5}} => 1

{{1,3,4,5},{2}} => 6

{{1,3,4},{2,5}} => 4

{{1,3,4},{2},{5}} => 3

{{1,3,5},{2,4}} => 4

{{1,3},{2,4,5}} => 4

{{1,3},{2,4},{5}} => 2

{{1,3,5},{2},{4}} => 3

{{1,3},{2,5},{4}} => 2

{{1,3},{2},{4,5}} => 2

{{1,3},{2},{4},{5}} => 1

{{1,4,5},{2,3}} => 4

{{1,4},{2,3,5}} => 4

{{1,4},{2,3},{5}} => 2

{{1,5},{2,3,4}} => 4

{{1},{2,3,4,5}} => 6

{{1},{2,3,4},{5}} => 3

{{1,5},{2,3},{4}} => 2

{{1},{2,3,5},{4}} => 3

{{1},{2,3},{4,5}} => 2

{{1},{2,3},{4},{5}} => 1

{{1,4,5},{2},{3}} => 3

{{1,4},{2,5},{3}} => 2

{{1,4},{2},{3,5}} => 2

{{1,4},{2},{3},{5}} => 1

{{1,5},{2,4},{3}} => 2

{{1},{2,4,5},{3}} => 3

{{1},{2,4},{3,5}} => 2

{{1},{2,4},{3},{5}} => 1

{{1,5},{2},{3,4}} => 2

{{1},{2,5},{3,4}} => 2

{{1},{2},{3,4,5}} => 3

{{1},{2},{3,4},{5}} => 1

{{1,5},{2},{3},{4}} => 1

{{1},{2,5},{3},{4}} => 1

{{1},{2},{3,5},{4}} => 1

{{1},{2},{3},{4,5}} => 1

{{1},{2},{3},{4},{5}} => 0

{{1,2,3,4,5,6}} => 15

{{1,2,3,4,5},{6}} => 10

{{1,2,3,4,6},{5}} => 10

{{1,2,3,4},{5,6}} => 7

{{1,2,3,4},{5},{6}} => 6

{{1,2,3,5,6},{4}} => 10

{{1,2,3,5},{4,6}} => 7

{{1,2,3,5},{4},{6}} => 6

{{1,2,3,6},{4,5}} => 7

{{1,2,3},{4,5,6}} => 6

{{1,2,3},{4,5},{6}} => 4

{{1,2,3,6},{4},{5}} => 6

{{1,2,3},{4,6},{5}} => 4

{{1,2,3},{4},{5,6}} => 4

{{1,2,3},{4},{5},{6}} => 3

{{1,2,4,5,6},{3}} => 10

{{1,2,4,5},{3,6}} => 7

{{1,2,4,5},{3},{6}} => 6

{{1,2,4,6},{3,5}} => 7

{{1,2,4},{3,5,6}} => 6

{{1,2,4},{3,5},{6}} => 4

{{1,2,4,6},{3},{5}} => 6

{{1,2,4},{3,6},{5}} => 4

{{1,2,4},{3},{5,6}} => 4

{{1,2,4},{3},{5},{6}} => 3

{{1,2,5,6},{3,4}} => 7

{{1,2,5},{3,4,6}} => 6

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Description

The number of occurrences of the pattern {{1,2}} in a set partition.

Code

def Klazar_occurrences(pi, pattern):
    """
    Return the number of occurrences of pattern in pi.

    EXAMPLES:

    From Mansour 2013, page 86, Example 3.24

    The set partition π = 135/26/4 contains four occurrences of
    the pattern 12/3, namely, 13/6, 13/4, 15/6, and 35/6. On the other hand, the
    set partition π avoids the pattern 12/34.

    sage: Klazar_occurrences([[1,3,5],[2,6],[4]], [[1,2],[3]])
    4
    sage: Klazar_occurrences([[1,3,5],[2,6],[4]], [[1,2],[3,4]])
    0
    """
    pi = SetPartition(pi).standardization()
    pattern = SetPartition(pattern).standardization()
    count = 0
    for s in Subsets(range(1,1+pi.size()), pattern.size()):
        if pi.restriction(s).standardization() == pattern:
            count += 1
    return count


def statistic(pi):
    return Klazar_occurrences(pi, [[1,2]])

Created

Aug 05, 2016 at 09:31 by Martin Rubey

Updated

Aug 05, 2016 at 09:31 by Martin Rubey