- FindStat

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Identifier

St000790: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1200 entries. <<<

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$F_{2} = 1 + q$

$F_{3} = 4 + q^{3}$

$F_{4} = 10 + 3\ q + q^{6}$

$F_{5} = 31 + 6\ q + 4\ q^{3} + q^{10}$

$F_{6} = 97 + 21\ q + 8\ q^{3} + 5\ q^{6} + q^{15}$

$F_{7} = 316 + 66\ q + 30\ q^{3} + 10\ q^{6} + 6\ q^{10} + q^{21}$

Description

The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path.
Apparently, the total number of these is given in [1].
The statistic counting all pairs of distinct tunnels is the area of a Dyck path St000012The area of a Dyck path..

References

[1] oeis A0000245

Code

def statistic(D):
    D = DyckWord(D)
    n = len(D)    
    tunnels = D.tunnels()
    t = [(i, j) for (i, j) in tunnels if i + j == n]
    res = 0
    for (a1, b1) in t:
        for (a2, b2) in t:
            if a1 < a2 < b2 < b1:
                res += 1
    return res

Created

Apr 23, 2017 at 15:11 by Martin Rubey

Updated

Apr 23, 2017 at 15:11 by Martin Rubey