- FindStat

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Identifier

St000418: Dyck paths ⟶ ℤ (values match St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 8

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

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

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

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

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

[1,1,0,1,1,1,0,0,0,0] => 28

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

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

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

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

[1,1,1,0,0,1,1,0,0,0] => 25

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

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

[1,1,1,0,1,0,1,0,0,0] => 34

[1,1,1,0,1,1,0,0,0,0] => 37

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

[1,1,1,1,0,0,0,1,0,0] => 28

[1,1,1,1,0,0,1,0,0,0] => 37

[1,1,1,1,0,1,0,0,0,0] => 41

[1,1,1,1,1,0,0,0,0,0] => 42

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

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

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

[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] => 5

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

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

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

[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] => 10

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

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

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

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

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

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

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

[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] => 10

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

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

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

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

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

[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] => 20

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

[1,0,1,1,0,1,1,1,0,0,0,0] => 28

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

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

[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] => 20

[1,0,1,1,1,0,0,1,1,0,0,0] => 25

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

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

[1,0,1,1,1,0,1,0,1,0,0,0] => 34

[1,0,1,1,1,0,1,1,0,0,0,0] => 37

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

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Description

The number of Dyck paths that are weakly below a Dyck path.

Code

def contains(A,B):
     Aa = A.to_area_sequence()
     Bb = B.to_area_sequence()
     return all( Aa[i] >= Bb[i] for i in range(len(Aa)) )

def statistic(D):
    n = len(D)/2
    return sum( 1 for A in DyckWords(n) if contains(D,A) )

Created

Feb 26, 2016 at 10:33 by Christian Stump

Updated

Feb 26, 2016 at 10:33 by Christian Stump