Identifier
Values
[1] => 0
[1,1] => 1
[1,2] => 0
[2,1] => 0
[1,1,1] => 3
[1,1,2] => 2
[1,2,1] => 2
[2,1,1] => 2
[1,1,3] => 1
[1,3,1] => 1
[3,1,1] => 1
[1,2,2] => 1
[2,1,2] => 1
[2,2,1] => 1
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,1,1,1] => 6
[1,1,1,2] => 5
[1,1,2,1] => 5
[1,2,1,1] => 5
[2,1,1,1] => 5
[1,1,1,3] => 4
[1,1,3,1] => 4
[1,3,1,1] => 4
[3,1,1,1] => 4
[1,1,1,4] => 3
[1,1,4,1] => 3
[1,4,1,1] => 3
[4,1,1,1] => 3
[1,1,2,2] => 4
[1,2,1,2] => 4
[1,2,2,1] => 4
[2,1,1,2] => 4
[2,1,2,1] => 4
[2,2,1,1] => 4
[1,1,2,3] => 3
[1,1,3,2] => 3
[1,2,1,3] => 3
[1,2,3,1] => 3
[1,3,1,2] => 3
[1,3,2,1] => 3
[2,1,1,3] => 3
[2,1,3,1] => 3
[2,3,1,1] => 3
[3,1,1,2] => 3
[3,1,2,1] => 3
[3,2,1,1] => 3
[1,1,2,4] => 2
[1,1,4,2] => 2
[1,2,1,4] => 2
[1,2,4,1] => 2
[1,4,1,2] => 2
[1,4,2,1] => 2
[2,1,1,4] => 2
[2,1,4,1] => 2
[2,4,1,1] => 2
[4,1,1,2] => 2
[4,1,2,1] => 2
[4,2,1,1] => 2
[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] => 1
[1,1,4,3] => 1
[1,3,1,4] => 1
[1,3,4,1] => 1
[1,4,1,3] => 1
[1,4,3,1] => 1
[3,1,1,4] => 1
[3,1,4,1] => 1
[3,4,1,1] => 1
[4,1,1,3] => 1
[4,1,3,1] => 1
[4,3,1,1] => 1
[1,2,2,2] => 3
[2,1,2,2] => 3
[2,2,1,2] => 3
[2,2,2,1] => 3
[1,2,2,3] => 2
[1,2,3,2] => 2
[1,3,2,2] => 2
[2,1,2,3] => 2
[2,1,3,2] => 2
[2,2,1,3] => 2
[2,2,3,1] => 2
[2,3,1,2] => 2
[2,3,2,1] => 2
[3,1,2,2] => 2
[3,2,1,2] => 2
[3,2,2,1] => 2
[1,2,2,4] => 1
[1,2,4,2] => 1
[1,4,2,2] => 1
[2,1,2,4] => 1
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Description
The area of the Dyck path corresponding to a parking function and the total displacement of a parking function.
The area Dyck path corresponding to a parking function for a parking function $p_1,\ldots,p_n$ of length $n$, is given by $\binom{n+1}{2} - \sum_i p_i$.
The total displacement of a parking function $ p \in PF_n $ is defined by
$$ \operatorname{disp}(p) := \sum_{i=1}^{n} d_i, $$
where the displacement vector $ d := (d_1, d_2, \ldots, d_n) $ and $ d_i := \pi^{-1}(i) - p_i $ for all $ i \in [n] $,
such that each $ d_i $ is the positive difference between the actual spot a car parks and its preferred spot.
The area Dyck path corresponding to a parking function for a parking function $p_1,\ldots,p_n$ of length $n$, is given by $\binom{n+1}{2} - \sum_i p_i$.
The total displacement of a parking function $ p \in PF_n $ is defined by
$$ \operatorname{disp}(p) := \sum_{i=1}^{n} d_i, $$
where the displacement vector $ d := (d_1, d_2, \ldots, d_n) $ and $ d_i := \pi^{-1}(i) - p_i $ for all $ i \in [n] $,
such that each $ d_i $ is the positive difference between the actual spot a car parks and its preferred spot.
References
[1] Triangle of coefficients of polynomials enumerating trees with n labeled nodes by inversions. OEIS:A052121
[2] Kang, Y., Selig, T., Yang, G., Zhang, Y., Zhu, H. On friendship and cyclic parking functions arXiv:2310.06560
[2] Kang, Y., Selig, T., Yang, G., Zhang, Y., Zhu, H. On friendship and cyclic parking functions arXiv:2310.06560
Code
def statistic(p):
return p.area()
Created
May 08, 2014 at 23:40 by Joanna Heppner
Updated
Jul 16, 2024 at 20:06 by Marvin Gandhi
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