Identifier
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [3,1,2] => [3,1,2] => 1
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,2,1] => [3,2,1] => [3,2,1] => 2
[3,1,2] => [3,2,1] => [3,2,1] => [3,2,1] => 2
[3,2,1] => [3,2,1] => [3,2,1] => [3,2,1] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 1
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => [3,1,2,4] => 1
[1,3,4,2] => [1,4,3,2] => [4,3,1,2] => [4,3,1,2] => 2
[1,4,2,3] => [1,4,3,2] => [4,3,1,2] => [4,3,1,2] => 2
[1,4,3,2] => [1,4,3,2] => [4,3,1,2] => [4,3,1,2] => 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [2,4,1,3] => [2,4,1,3] => 1
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[2,3,4,1] => [4,2,3,1] => [2,4,3,1] => [2,4,3,1] => 2
[2,4,1,3] => [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 2
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[3,1,4,2] => [4,2,3,1] => [2,4,3,1] => [2,4,3,1] => 2
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[3,2,4,1] => [4,2,3,1] => [2,4,3,1] => [2,4,3,1] => 2
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[4,1,2,3] => [4,2,3,1] => [2,4,3,1] => [2,4,3,1] => 2
[4,1,3,2] => [4,2,3,1] => [2,4,3,1] => [2,4,3,1] => 2
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
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Description
The number of descents in a parking function.
This is the number of indices $i$ such that $p_i > p_{i+1}$.
Map
Demazure product with inverse
Description
This map sends a permutation $\pi$ to $\pi^{-1} \star \pi$ where $\star$ denotes the Demazure product on permutations.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.
Map
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00067Foata bijection.
Map
parking function
Description
Interpret the permutation as a parking function.