- FindStat

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Identifier

St000979: Dyck paths ⟶ ℤ

Values

[] => 0

[1,0] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,0,0,1,1,1,0,0,0] => 16

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,1,0,1,1,0,0,0,0] => 16

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

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

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

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

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

[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] => 10

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

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

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

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

[1,0,1,0,1,1,0,0,1,1,0,0] => 16

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

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

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

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

[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] => 16

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

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

[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] => 12

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

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

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

[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] => 6

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

[1,0,1,1,0,1,0,1,1,0,0,0] => 16

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

[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] => 23

[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] => 20

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

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

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

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

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

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

>>> Load all 1746 entries. <<<

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Description

Half of MacMahon's equal index of a Dyck path.
This is half the sum of the positions of double (up- or down-)steps of a Dyck path, see [1, p. 135].

References

[1] MacMahon, P. A. Combinatory analysis MathSciNet:0141605

Code

def statistic(p):
    return sum( i+1 for i in range(len(p)-1) if p[i] == p[i+1] ) / 2

Created

Sep 08, 2017 at 12:22 by Christian Stump

Updated

Mar 10, 2021 at 15:23 by Martin Rubey