- FindStat

edit this statistic or download as text // json

Identifier

St001669: Dyck paths ⟶ ℤ

Values

[1,0] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 6

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

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

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

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

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

[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] => 4

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

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

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

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

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

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

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

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

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

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

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

[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] => 3

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

[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] => 4

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 8

[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] => 8

[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] => 8

[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] => 5

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

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

[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] => 4

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

[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] => 4

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

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

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

[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] => 8

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

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

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

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

[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] => 6

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

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

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

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

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

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

[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

>>> 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 single rises in a Dyck path.
A single rise is a step which is neither preceded nor followed by a step of the same kind.

Code

def statistic(D):
    m = sum(1 for i in range(1, len(D)-1) if D[i-1] == D[i+1] == 1-D[i])
    if len(D) > 1:
        return m + 1 - D[1] + D[-2]
    return m

Created

Jan 14, 2021 at 13:45 by Martin Rubey

Updated

Jan 14, 2021 at 13:45 by Martin Rubey