- FindStat

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Identifier

St000978: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,1,1,0,0,0] => 17

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

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

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

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

[1,0,1,1,0,1,1,0,0,0] => 17

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

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

[1,0,1,1,1,0,1,0,0,0] => 17

[1,0,1,1,1,1,0,0,0,0] => 24

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

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

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

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

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

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

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

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

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

[1,1,0,1,0,1,1,0,0,0] => 17

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

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

[1,1,0,1,1,0,1,0,0,0] => 17

[1,1,0,1,1,1,0,0,0,0] => 24

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

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

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

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

[1,1,1,0,0,1,1,0,0,0] => 21

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

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

[1,1,1,0,1,0,1,0,0,0] => 17

[1,1,1,0,1,1,0,0,0,0] => 24

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

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

[1,1,1,1,0,0,1,0,0,0] => 22

[1,1,1,1,0,1,0,0,0,0] => 24

[1,1,1,1,1,0,0,0,0,0] => 30

[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] => 11

[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] => 11

[1,0,1,0,1,0,1,1,1,0,0,0] => 21

[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] => 18

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

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

[1,0,1,0,1,1,0,1,1,0,0,0] => 21

[1,0,1,0,1,1,1,0,0,0,1,0] => 17

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

[1,0,1,0,1,1,1,0,1,0,0,0] => 21

[1,0,1,0,1,1,1,1,0,0,0,0] => 30

[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] => 16

[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] => 16

[1,0,1,1,0,0,1,1,1,0,0,0] => 26

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

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

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

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

[1,0,1,1,0,1,0,1,1,0,0,0] => 21

[1,0,1,1,0,1,1,0,0,0,1,0] => 17

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

[1,0,1,1,0,1,1,0,1,0,0,0] => 21

[1,0,1,1,0,1,1,1,0,0,0,0] => 30

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

[1,0,1,1,1,0,0,0,1,1,0,0] => 24

[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] => 17

[1,0,1,1,1,0,0,1,1,0,0,0] => 27

[1,0,1,1,1,0,1,0,0,0,1,0] => 17

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

[1,0,1,1,1,0,1,0,1,0,0,0] => 21

[1,0,1,1,1,0,1,1,0,0,0,0] => 30

[1,0,1,1,1,1,0,0,0,0,1,0] => 24

>>> Load all 1200 entries. <<<

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Description

The sum of the positions of double down-steps of a Dyck path.
This is part of MacMahon's equal index of a word, see [1, p. 135].

References

[1] MacMahon, P. A. Combinatory analysis. Vol. I, II (bound in one volume) MathSciNet:2417935

Code

def statistic(p):
    return sum( i+1 for i in range(len(p)-1) if p[i] == p[i+1] == 0 )

Created

Sep 08, 2017 at 12:19 by Christian Stump

Updated

Sep 08, 2017 at 12:19 by Christian Stump