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Identifier

Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

Return the inverse of the Kreweras complement of a Dyck path, regarded as a noncrossing set partition.
To identify Dyck paths and noncrossing set partitions, this maps uses the following classical bijection. The number of down steps after the $i$-th up step of the Dyck path is the size of the block of the set partition whose maximal element is $i$. If $i$ is not a maximal element of a block, the $(i+1)$-st step is also an up step.

Properties

bijective, graded

Sage code

def mapping(d):
    r"""
    Compute the Kreweras complement of a Dyck path, regarded as a
    noncrossing partition.
    """
    n = len(d) / 2
    if not n:
        return d
    D = DyckWords(n)
    pi = D(d).to_noncrossing_partition().to_permutation()
    c = Permutation(list(range(2,n+1))+[1])
    p = SetPartition((c*pi.inverse()).to_cycles())
    return D.from_noncrossing_partition(p)

Weight

Created

Jan 26, 2025 at 17:20 by Martin Rubey

Updated

Jan 26, 2025 at 17:20 by Martin Rubey