- FindStat

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Identifier

St000376: 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] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 4

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

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

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

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

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

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

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

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

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

[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] => 4

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 0

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

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

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

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

[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] => 4

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

[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] => 6

[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] => 6

[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] => 5

[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] => 4

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

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

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

[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] => 5

[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] => 7

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

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

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

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

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

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

[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] => 5

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

[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] => 6

[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] => 5

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

[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] => 4

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

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

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

>>> Load all 1200 entries. <<<

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$F_{2} = 2$

$F_{3} = 4 + q$

$F_{4} = 7 + 4\ q + 3\ q^{2}$

$F_{5} = 11 + 8\ q + 12\ q^{2} + 8\ q^{3} + 3\ q^{4}$

$F_{6} = 16 + 13\ q + 22\ q^{2} + 27\ q^{3} + 28\ q^{4} + 15\ q^{5} + 10\ q^{6} + q^{7}$

$F_{7} = 22 + 19\ q + 34\ q^{2} + 45\ q^{3} + 66\ q^{4} + 66\ q^{5} + 70\ q^{6} + 52\ q^{7} + 36\ q^{8} + 15\ q^{9} + 4\ q^{10}$

Description

The bounce deficit of a Dyck path.
For a Dyck path

$D$ of semilength

$n$ , this is defined as

$\binom{n}{2} - \operatorname{area}(D) - \operatorname{bounce}(D).$
The zeta map Mp00032inverse zeta map sends this statistic to the dinv deficit St000369The dinv deficit of a Dyck path., both are thus equidistributed.

References

[1] Lee, K., Li, L., Loehr, N. A. A Combinatorial Approach to the Symmetry of $q,t$ -Catalan Numbers arXiv:1602.01126

Code

def statistic(D):
    return binomial(len(D)/2,2)-D.area()-D.bounce()

Created

Feb 05, 2016 at 21:03 by Christian Stump

Updated

Feb 05, 2016 at 21:03 by Christian Stump