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Identifier

St001929: Dyck paths ⟶ ℤ

Values

[1,0] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,1,0,1,0,0] => 6

[1,0,1,0,1,1,1,0,0,0] => 6

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

[1,0,1,1,0,0,1,1,0,0] => 8

[1,0,1,1,0,1,0,0,1,0] => 6

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

[1,0,1,1,0,1,1,0,0,0] => 8

[1,0,1,1,1,0,0,0,1,0] => 6

[1,0,1,1,1,0,0,1,0,0] => 8

[1,0,1,1,1,0,1,0,0,0] => 6

[1,0,1,1,1,1,0,0,0,0] => 10

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

[1,1,0,0,1,0,1,1,0,0] => 8

[1,1,0,0,1,1,0,0,1,0] => 8

[1,1,0,0,1,1,0,1,0,0] => 6

[1,1,0,0,1,1,1,0,0,0] => 10

[1,1,0,1,0,0,1,0,1,0] => 6

[1,1,0,1,0,0,1,1,0,0] => 6

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

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

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

[1,1,0,1,1,0,0,0,1,0] => 8

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

[1,1,0,1,1,0,1,0,0,0] => 6

[1,1,0,1,1,1,0,0,0,0] => 6

[1,1,1,0,0,0,1,0,1,0] => 6

[1,1,1,0,0,0,1,1,0,0] => 10

[1,1,1,0,0,1,0,0,1,0] => 8

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

[1,1,1,0,0,1,1,0,0,0] => 8

[1,1,1,0,1,0,0,0,1,0] => 6

[1,1,1,0,1,0,0,1,0,0] => 6

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

[1,1,1,0,1,1,0,0,0,0] => 8

[1,1,1,1,0,0,0,0,1,0] => 10

[1,1,1,1,0,0,0,1,0,0] => 6

[1,1,1,1,0,0,1,0,0,0] => 8

[1,1,1,1,0,1,0,0,0,0] => 6

[1,1,1,1,1,0,0,0,0,0] => 10

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

[1,0,1,0,1,0,1,0,1,1,0,0] => 5

[1,0,1,0,1,0,1,1,0,0,1,0] => 5

[1,0,1,0,1,0,1,1,0,1,0,0] => 10

[1,0,1,0,1,0,1,1,1,0,0,0] => 8

[1,0,1,0,1,1,0,0,1,0,1,0] => 5

[1,0,1,0,1,1,0,0,1,1,0,0] => 13

[1,0,1,0,1,1,0,1,0,0,1,0] => 10

[1,0,1,0,1,1,0,1,0,1,0,0] => 10

[1,0,1,0,1,1,0,1,1,0,0,0] => 15

[1,0,1,0,1,1,1,0,0,0,1,0] => 8

[1,0,1,0,1,1,1,0,0,1,0,0] => 15

[1,0,1,0,1,1,1,0,1,0,0,0] => 9

[1,0,1,0,1,1,1,1,0,0,0,0] => 18

[1,0,1,1,0,0,1,0,1,0,1,0] => 5

[1,0,1,1,0,0,1,0,1,1,0,0] => 14

[1,0,1,1,0,0,1,1,0,0,1,0] => 13

[1,0,1,1,0,0,1,1,0,1,0,0] => 15

[1,0,1,1,0,0,1,1,1,0,0,0] => 19

[1,0,1,1,0,1,0,0,1,0,1,0] => 10

[1,0,1,1,0,1,0,0,1,1,0,0] => 15

[1,0,1,1,0,1,0,1,0,0,1,0] => 10

[1,0,1,1,0,1,0,1,0,1,0,0] => 5

[1,0,1,1,0,1,0,1,1,0,0,0] => 13

[1,0,1,1,0,1,1,0,0,0,1,0] => 15

[1,0,1,1,0,1,1,0,0,1,0,0] => 14

[1,0,1,1,0,1,1,0,1,0,0,0] => 15

[1,0,1,1,0,1,1,1,0,0,0,0] => 19

[1,0,1,1,1,0,0,0,1,0,1,0] => 8

[1,0,1,1,1,0,0,0,1,1,0,0] => 19

[1,0,1,1,1,0,0,1,0,0,1,0] => 15

[1,0,1,1,1,0,0,1,0,1,0,0] => 13

[1,0,1,1,1,0,0,1,1,0,0,0] => 20

[1,0,1,1,1,0,1,0,0,0,1,0] => 9

[1,0,1,1,1,0,1,0,0,1,0,0] => 15

[1,0,1,1,1,0,1,0,1,0,0,0] => 8

[1,0,1,1,1,0,1,1,0,0,0,0] => 19

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Description

The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path.

Code

def to_noncrossing_matching(D):
    return PerfectMatching([(a+1, b) for a,b in D.tunnels()])

def length_of_first_loop(D1, D2):
    m1 = to_noncrossing_matching(D1).to_permutation()
    m2 = to_noncrossing_matching(D2).to_permutation()
    n = len(D1)
    i = 1
    l = 0
    while True:
        i = m2(m1(i))
        l += 2
        if i == 1:
            break
    return l

@cached_function
def statistic(D):
    """
    The number of meanders with top half D.
    """
    D = DyckWord(D)
    n = D.semilength()
    return sum(1 for E in DyckWords(n) if length_of_first_loop(D, E) == 2*n)

Created

Oct 31, 2023 at 16:24 by Martin Rubey

Updated

Oct 31, 2023 at 16:24 by Martin Rubey