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Identifier

St000946: 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] => 2

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

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

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

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

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

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

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

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

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

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

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

[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] => 1

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

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

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

[1,0,1,0,1,1,0,0,1,0] => 7

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,0,0,1,0,1,1,0,0] => 7

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

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

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

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

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

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

[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] => 7

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

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

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

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

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

[1,1,1,0,0,1,0,0,1,0] => 7

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

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

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

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

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

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

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

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

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

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

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

[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] => 5

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

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

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

[1,0,1,0,1,1,0,0,1,0,1,0] => 7

[1,0,1,0,1,1,0,0,1,1,0,0] => 12

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

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

[1,0,1,0,1,1,0,1,1,0,0,0] => 7

[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] => 7

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

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

[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] => 10

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

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

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

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

[1,0,1,1,0,1,0,0,1,1,0,0] => 11

[1,0,1,1,0,1,0,1,0,0,1,0] => 7

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

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

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

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

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

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

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

[1,0,1,1,1,0,0,0,1,1,0,0] => 11

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

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

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

[1,0,1,1,1,0,1,0,0,0,1,0] => 7

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

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

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

[1,0,1,1,1,1,0,0,0,0,1,0] => 7

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Description

The sum of the skew hook positions in a Dyck path.
A skew hook is an occurrence of a down step followed by two up steps or of an up step followed by a down step.
Write $U_i$ for the $i$-th up step and $D_j$ for the $j$-th down step in the Dyck path. Then the skew hook set is the set $$H = \{j: U_{i−1} U_i D_j \text{ is a skew hook}\} \cup \{i: D_{i−1} D_i U_j\text{ is a skew hook}\}.$$
This statistic is the sum of all elements in $H$.

References

[1] Zhao, H., Zhong, Z. Two statistics linking Dyck paths and non-crossing partitions MathSciNet:2788700

Code

def statistic(P):
    # record for each step how many steps of the same type there were before
    i = j = 0
    S = [None]*len(P)
    for k, s in enumerate(P):
        if s == 0:
            j += 1
            S[k] = j
        else:
            i += 1
            S[k] = i

    H = []
    for k in range(2, len(P)):
        if P[k-2] == P[k-1] == 1 and P[k] == 0:
            H.append(S[k])
        if P[k-2] == P[k-1] == 0 and P[k] == 1:
            H.append(S[k-1])

    return sum(H)

Created

Aug 23, 2017 at 01:00 by Martin Rubey

Updated

Aug 23, 2017 at 01:00 by Martin Rubey