edit this statistic or download as text // json
Identifier
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
>>> Load all 145 entries. <<<
[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
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
click to show known generating functions       
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