- FindStat

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Identifier

St001035: Dyck paths ⟶ ℤ

Values

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

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

[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] => 0

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

[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] => 2

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

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

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

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

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

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

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

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

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

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

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

[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] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 1

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

[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] => 2

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

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

[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] => 3

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

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

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

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

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

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

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

[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] => 3

[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] => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 2

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

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

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

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

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Description

The convexity degree of the parallelogram polyomino associated with the Dyck path.
A parallelogram polyomino is $k$-convex if $k$ is the maximal number of turns an axis-parallel path must take to connect two cells of the polyomino.
For example, any rotation of a Ferrers shape has convexity degree at most one.
The (bivariate) generating function is given in Theorem 2 of [1].

References

[1] Battaglino, D., Fedou, J. M., Rinaldi, S., Socci, S. The number of $k$-parallelogram polyominoes MathSciNet:3091068

Code

def statistic(D):
    return ParallelogramPolyomino.from_dyck_word(D).degree_convexity()

Created

Nov 06, 2017 at 11:19 by Martin Rubey

Updated

Nov 06, 2017 at 16:06 by Martin Rubey