- FindStat

edit this statistic or download as text // json

Identifier

St000421: Dyck paths ⟶ ℤ (values match St000418The number of Dyck paths that are weakly below a Dyck path.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,1,0,1,0,1,0,0,0] => 33

[1,1,1,0,1,1,0,0,0,0] => 36

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

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

[1,1,1,1,0,0,1,0,0,0] => 36

[1,1,1,1,0,1,0,0,0,0] => 40

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

[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] => 1

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

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

[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] => 1

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 3

[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] => 7

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

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

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

[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] => 25

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

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

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

[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] => 19

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

[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] => 25

[1,0,1,1,1,0,1,0,1,0,0,0] => 33

[1,0,1,1,1,0,1,1,0,0,0,0] => 36

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

>>> Load all 1200 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

The number of Dyck paths that are weakly below a Dyck path, except for the path itself.

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) ) - 1

Created

Feb 26, 2016 at 10:38 by Christian Stump

Updated

Feb 26, 2016 at 10:38 by Christian Stump