Identifier
Values
[1,0] => [1,0] => [2,1] => [2,1] => 1
[1,0,1,0] => [1,1,0,0] => [2,3,1] => [2,3,1] => 2
[1,1,0,0] => [1,0,1,0] => [3,1,2] => [3,1,2] => 2
[1,0,1,0,1,0] => [1,1,1,0,0,0] => [2,3,4,1] => [2,4,3,1] => 2
[1,0,1,1,0,0] => [1,1,0,0,1,0] => [2,4,1,3] => [2,4,1,3] => 3
[1,1,0,0,1,0] => [1,0,1,1,0,0] => [3,1,4,2] => [3,1,4,2] => 4
[1,1,0,1,0,0] => [1,0,1,0,1,0] => [4,1,2,3] => [4,1,3,2] => 2
[1,1,1,0,0,0] => [1,1,0,1,0,0] => [4,3,1,2] => [4,3,1,2] => 2
[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [2,5,4,3,1] => 2
[1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [2,5,4,1,3] => 3
[1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [2,5,1,4,3] => 3
[1,0,1,1,0,1,0,0] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [2,5,1,4,3] => 3
[1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [2,5,4,1,3] => 3
[1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [3,1,5,4,2] => 4
[1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [3,1,5,4,2] => 4
[1,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [4,1,5,3,2] => 4
[1,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [5,1,4,3,2] => 2
[1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [5,1,4,3,2] => 2
[1,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [4,3,1,5,2] => 6
[1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [5,3,1,4,2] => 4
[1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [5,4,1,3,2] => 2
[1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [5,3,4,1,2] => 4
[1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [2,6,5,4,3,1] => 2
[1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [2,6,5,4,1,3] => 3
[1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [2,6,5,1,4,3] => 3
[1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [2,6,5,1,4,3] => 3
[1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [2,6,5,4,1,3] => 3
[1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [2,6,1,5,4,3] => 3
[1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [2,6,1,5,4,3] => 3
[1,0,1,1,0,1,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [2,6,1,5,4,3] => 3
[1,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [2,6,1,5,4,3] => 3
[1,0,1,1,0,1,1,0,0,0] => [1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [2,6,1,5,4,3] => 3
[1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [2,6,5,1,4,3] => 3
[1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [2,6,5,1,4,3] => 3
[1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [2,6,5,1,4,3] => 3
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [2,6,5,4,1,3] => 3
[1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [3,1,6,5,4,2] => 4
[1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [3,1,6,5,4,2] => 4
[1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [3,1,6,5,4,2] => 4
[1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [3,1,6,5,4,2] => 4
[1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [3,1,6,5,4,2] => 4
[1,1,0,1,0,0,1,0,1,0] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [4,1,6,5,3,2] => 4
[1,1,0,1,0,0,1,1,0,0] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [4,1,6,5,3,2] => 4
[1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [5,1,6,4,3,2] => 4
[1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [6,1,5,4,3,2] => 2
[1,1,0,1,0,1,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [6,1,5,4,3,2] => 2
[1,1,0,1,1,0,0,0,1,0] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [5,1,6,4,3,2] => 4
[1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [6,1,5,4,3,2] => 2
[1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [6,1,5,4,3,2] => 2
[1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [6,1,5,4,3,2] => 2
[1,1,1,0,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [4,3,1,6,5,2] => 6
[1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [4,3,1,6,5,2] => 6
[1,1,1,0,0,1,0,0,1,0] => [1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [5,3,1,6,4,2] => 8
[1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [6,3,1,5,4,2] => 4
[1,1,1,0,0,1,1,0,0,0] => [1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [6,3,1,5,4,2] => 4
[1,1,1,0,1,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [5,4,1,6,3,2] => 6
[1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [6,4,1,5,3,2] => 4
[1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [5,6,1,4,3,2] => 4
[1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [6,4,1,5,3,2] => 4
[1,1,1,1,0,0,0,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [5,3,6,1,4,2] => 8
[1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [6,3,5,1,4,2] => 4
[1,1,1,1,0,0,1,0,0,0] => [1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [6,3,5,1,4,2] => 4
[1,1,1,1,0,1,0,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [6,5,4,1,3,2] => 2
[1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [6,3,5,4,1,2] => 4
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Description
The number of parking functions that give the same permutation.
A parking function $(a_1,\dots,a_n)$ is a list of preferred parking spots of $n$ cars entering a one-way street. Once the cars have parked, the order of the cars gives a permutation of $\{1,\dots,n\}$. This statistic records the number of parking functions that yield the same permutation of cars.
Map
peaks-to-valleys
Description
Return the path that has a valley wherever the original path has a peak of height at least one.
More precisely, the height of a valley in the image is the height of the corresponding peak minus $2$.
This is also (the inverse of) rowmotion on Dyck paths regarded as order ideals in the triangular poset.
Map
Simion-Schmidt map
Description
The Simion-Schmidt map sends any permutation to a $123$-avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.