- FindStat

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Identifier

St000947: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 3

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

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

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

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

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

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

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

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

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

[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] => 6

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

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

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

[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] => 6

[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] => 6

[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] => 13

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

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

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

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

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

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

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

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

[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] => 8

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The major index east count of a Dyck path.
The descent set $\operatorname{des}(D)$ of a Dyck path $D = D_1 \cdots D_{2n}$ with $D_i \in \{N,E\}$ is given by all indices $i$ such that $D_i = E$ and $D_{i+1} = N$. This is, the positions of the valleys of $D$.
The major index of a Dyck path is then the sum of the positions of the valleys, $\sum_{i \in \operatorname{des}(D)} i$, see St000027The major index of a Dyck path..
The major index east count is given by $\sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = E\}$.

References

[1] Stump, C. On bijections between $231$-avoiding permutations and Dyck paths MathSciNet:2734176
[2] Zhao, H., Zhong, Z. Two statistics linking Dyck paths and non-crossing partitions MathSciNet:2788700

Code

def statistic(D):
    vals = D.valleys()
    return sum(1 for d in vals for i in range(d+1) if D[i] == 0)

Created

Aug 23, 2017 at 00:53 by Martin Rubey

Updated

Apr 25, 2018 at 11:56 by Christian Stump