- FindStat

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Identifier

St000038: Dyck paths ⟶ ℤ

Values

[1,0] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,1,0,1,0,0,0] => 18

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,1,1,0,1,0,0,0] => 18

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

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

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

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

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

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

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

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

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

[1,1,0,1,0,1,0,1,0,0] => 16

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

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

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

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

[1,1,0,1,1,1,0,0,0,0] => 48

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

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

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

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

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

[1,1,1,0,1,0,0,0,1,0] => 18

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

[1,1,1,0,1,0,1,0,0,0] => 54

[1,1,1,0,1,1,0,0,0,0] => 72

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

[1,1,1,1,0,0,0,1,0,0] => 48

[1,1,1,1,0,0,1,0,0,0] => 72

[1,1,1,1,0,1,0,0,0,0] => 96

[1,1,1,1,1,0,0,0,0,0] => 120

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

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

[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] => 4

[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] => 4

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

[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] => 12

[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] => 12

[1,0,1,0,1,1,1,0,1,0,0,0] => 18

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

[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] => 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] => 8

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

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

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

[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] => 16

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

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

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

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

[1,0,1,1,0,1,1,1,0,0,0,0] => 48

[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] => 12

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

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

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

[1,0,1,1,1,0,1,0,0,0,1,0] => 18

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

[1,0,1,1,1,0,1,0,1,0,0,0] => 54

[1,0,1,1,1,0,1,1,0,0,0,0] => 72

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Description

The product of the heights of the descending steps of a Dyck path.
A Dyck path with 2n letters defines a partition inside an [n] x [n] board. This statistic counts the number of placements of n non-attacking rooks on the board.
By the Gessel-Viennot theory of orthogonal polynomials this corresponds to the 0-moment of the Hermite polynomials.
Summing the values of the statistic over all Dyck paths of fixed size n the number of perfect matchings (2n+1)!! is obtained: up steps are openers, down steps closers and the rooks determine a pairing of openers and closers.

References

[1] Viennot, Gérard A combinatorial theory for general orthogonal polynomials with extensions and applications MathSciNet:0838979

Code

def statistic(x):
    stat = 1
    currht = 0
    for c in x:
        if c == 1:
            currht = currht + 1
        else:
            stat = stat*currht
            currht = currht -  1
    return stat

Created

Feb 18, 2013 at 18:06 by Alejandro Morales

Updated

May 29, 2015 at 15:59 by Martin Rubey