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Statistic identifier: St000135

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Collection: Parking functions

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Description: The number of lucky cars of the parking function.

A lucky car is a car that was able to park in its prefered spot.

The generating function,
$$
q\prod_{i=1}^{n-1} (i + (n-i+1)q)
$$
was established in [1].

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References: [1]   Gessel, I. M., Seo, S. A refinement of Cayley's formula for trees [[MathSciNet:2224940]]

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Code:
def statistic(pf):
    return len(pf.lucky_cars())

def generating_function(n):
    R.<q> = ZZ[]
    if n:
        return q * prod(i + q*(n-i+1) for i in range(1, n))
    return R.one()


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Statistic values:

[1]       => 1
[1,1]     => 1
[1,2]     => 2
[2,1]     => 2
[1,1,1]   => 1
[1,1,2]   => 1
[1,2,1]   => 2
[2,1,1]   => 2
[1,1,3]   => 2
[1,3,1]   => 2
[3,1,1]   => 2
[1,2,2]   => 2
[2,1,2]   => 2
[2,2,1]   => 2
[1,2,3]   => 3
[1,3,2]   => 3
[2,1,3]   => 3
[2,3,1]   => 3
[3,1,2]   => 3
[3,2,1]   => 3
[1,1,1,1] => 1
[1,1,1,2] => 1
[1,1,2,1] => 1
[1,2,1,1] => 2
[2,1,1,1] => 2
[1,1,1,3] => 1
[1,1,3,1] => 2
[1,3,1,1] => 2
[3,1,1,1] => 2
[1,1,1,4] => 2
[1,1,4,1] => 2
[1,4,1,1] => 2
[4,1,1,1] => 2
[1,1,2,2] => 1
[1,2,1,2] => 2
[1,2,2,1] => 2
[2,1,1,2] => 2
[2,1,2,1] => 2
[2,2,1,1] => 2
[1,1,2,3] => 1
[1,1,3,2] => 2
[1,2,1,3] => 2
[1,2,3,1] => 3
[1,3,1,2] => 2
[1,3,2,1] => 3
[2,1,1,3] => 2
[2,1,3,1] => 3
[2,3,1,1] => 3
[3,1,1,2] => 2
[3,1,2,1] => 3
[3,2,1,1] => 3
[1,1,2,4] => 2
[1,1,4,2] => 2
[1,2,1,4] => 3
[1,2,4,1] => 3
[1,4,1,2] => 2
[1,4,2,1] => 3
[2,1,1,4] => 3
[2,1,4,1] => 3
[2,4,1,1] => 3
[4,1,1,2] => 2
[4,1,2,1] => 3
[4,2,1,1] => 3
[1,1,3,3] => 2
[1,3,1,3] => 2
[1,3,3,1] => 2
[3,1,1,3] => 2
[3,1,3,1] => 2
[3,3,1,1] => 2
[1,1,3,4] => 3
[1,1,4,3] => 3
[1,3,1,4] => 3
[1,3,4,1] => 3
[1,4,1,3] => 3
[1,4,3,1] => 3
[3,1,1,4] => 3
[3,1,4,1] => 3
[3,4,1,1] => 3
[4,1,1,3] => 3
[4,1,3,1] => 3
[4,3,1,1] => 3
[1,2,2,2] => 2
[2,1,2,2] => 2
[2,2,1,2] => 2
[2,2,2,1] => 2
[1,2,2,3] => 2
[1,2,3,2] => 3
[1,3,2,2] => 3
[2,1,2,3] => 2
[2,1,3,2] => 3
[2,2,1,3] => 2
[2,2,3,1] => 2
[2,3,1,2] => 3
[2,3,2,1] => 3
[3,1,2,2] => 3
[3,2,1,2] => 3
[3,2,2,1] => 3
[1,2,2,4] => 3
[1,2,4,2] => 3
[1,4,2,2] => 3
[2,1,2,4] => 3
[2,1,4,2] => 3
[2,2,1,4] => 3
[2,2,4,1] => 3
[2,4,1,2] => 3
[2,4,2,1] => 3
[4,1,2,2] => 3
[4,2,1,2] => 3
[4,2,2,1] => 3
[1,2,3,3] => 3
[1,3,2,3] => 3
[1,3,3,2] => 3
[2,1,3,3] => 3
[2,3,1,3] => 3
[2,3,3,1] => 3
[3,1,2,3] => 3
[3,1,3,2] => 3
[3,2,1,3] => 3
[3,2,3,1] => 3
[3,3,1,2] => 3
[3,3,2,1] => 3
[1,2,3,4] => 4
[1,2,4,3] => 4
[1,3,2,4] => 4
[1,3,4,2] => 4
[1,4,2,3] => 4
[1,4,3,2] => 4
[2,1,3,4] => 4
[2,1,4,3] => 4
[2,3,1,4] => 4
[2,3,4,1] => 4
[2,4,1,3] => 4
[2,4,3,1] => 4
[3,1,2,4] => 4
[3,1,4,2] => 4
[3,2,1,4] => 4
[3,2,4,1] => 4
[3,4,1,2] => 4
[3,4,2,1] => 4
[4,1,2,3] => 4
[4,1,3,2] => 4
[4,2,1,3] => 4
[4,2,3,1] => 4
[4,3,1,2] => 4
[4,3,2,1] => 4

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Created: Jun 20, 2013 at 11:13 by Viviane Pons

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Last Updated: Oct 11, 2024 at 10:55 by Martin Rubey