- FindStat

Identifier

St000135: Parking functions ⟶ ℤ

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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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].

References

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

Code

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

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

Created

Jun 20, 2013 at 11:13 by Viviane Pons

Updated

Oct 11, 2024 at 10:55 by Martin Rubey