- FindStat

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Identifier

St001348: Dyck paths ⟶ ℤ

Values

[1,0] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 4

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 7

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

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

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

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

[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] => 8

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

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

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

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

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

[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] => 12

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

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

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

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

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

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

[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] => 11

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

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

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

[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] => 9

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

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

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

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

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

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

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$F_{1} = q$

$F_{2} = 2\ q^{2}$

$F_{3} = 4\ q^{3} + q^{4}$

$F_{4} = 8\ q^{4} + 4\ q^{5} + 2\ q^{6}$

$F_{5} = 16\ q^{5} + 12\ q^{6} + 9\ q^{7} + 4\ q^{8} + q^{9}$

$F_{6} = 32\ q^{6} + 32\ q^{7} + 30\ q^{8} + 20\ q^{9} + 12\ q^{10} + 4\ q^{11} + 2\ q^{12}$

$F_{7} = 64\ q^{7} + 80\ q^{8} + 88\ q^{9} + 73\ q^{10} + 56\ q^{11} + 34\ q^{12} + 20\ q^{13} + 9\ q^{14} + 4\ q^{15} + q^{16}$

Description

The bounce of the parallelogram polyomino associated with the Dyck path.
A bijection due to Delest and Viennot [1] associates a Dyck path with a parallelogram polyomino. The bounce statistic is defined in [2].

References

[1] Delest, M.-P., Viennot, Gérard Algebraic languages and polyominoes enumeration MathSciNet:0774044
[2] Aval, J.-C., D'Adderio, M., Dukes, M., Hicks, A., Le Borgne, Y. Statistics on parallelogram polyominoes and a q,t-analogue of the Narayana numbers arXiv:1301.4803

Code

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

Created

Jan 26, 2019 at 16:12 by Martin Rubey

Updated

Jan 26, 2019 at 16:12 by Martin Rubey