Identifier
Values
[(1,2)] => [2,1] => [1,2] => [2,1] => 0
[(1,2),(3,4)] => [2,1,4,3] => [1,2,3,4] => [2,3,4,1] => 0
[(1,3),(2,4)] => [3,4,1,2] => [1,3,2,4] => [3,2,4,1] => 0
[(1,4),(2,3)] => [4,3,2,1] => [1,4,2,3] => [3,4,2,1] => 0
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => 0
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [1,3,2,4,5,6] => [3,2,4,5,6,1] => 0
[(1,4),(2,3),(5,6)] => [4,3,2,1,6,5] => [1,4,2,3,5,6] => [3,4,2,5,6,1] => 0
[(1,5),(2,3),(4,6)] => [5,3,2,6,1,4] => [1,5,2,3,4,6] => [3,4,5,2,6,1] => 0
[(1,6),(2,3),(4,5)] => [6,3,2,5,4,1] => [1,6,2,3,4,5] => [3,4,5,6,2,1] => 0
[(1,6),(2,4),(3,5)] => [6,4,5,2,3,1] => [1,6,2,4,3,5] => [3,5,4,6,2,1] => 0
[(1,5),(2,4),(3,6)] => [5,4,6,2,1,3] => [1,5,2,4,3,6] => [3,5,4,2,6,1] => 0
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [1,4,2,5,3,6] => [3,5,2,4,6,1] => 1
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [1,3,2,5,4,6] => [3,2,5,4,6,1] => 0
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [1,2,3,5,4,6] => [2,3,5,4,6,1] => 0
[(1,2),(3,6),(4,5)] => [2,1,6,5,4,3] => [1,2,3,6,4,5] => [2,3,5,6,4,1] => 0
[(1,3),(2,6),(4,5)] => [3,6,1,5,4,2] => [1,3,2,6,4,5] => [3,2,5,6,4,1] => 0
[(1,4),(2,6),(3,5)] => [4,6,5,1,3,2] => [1,4,2,6,3,5] => [3,5,2,6,4,1] => 1
[(1,5),(2,6),(3,4)] => [5,6,4,3,1,2] => [1,5,2,6,3,4] => [3,5,6,2,4,1] => 2
[(1,6),(2,5),(3,4)] => [6,5,4,3,2,1] => [1,6,2,5,3,4] => [3,5,6,4,2,1] => 0
[(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [1,2,3,4,5,6,7,8] => [2,3,4,5,6,7,8,1] => 0
[(1,3),(2,4),(5,6),(7,8)] => [3,4,1,2,6,5,8,7] => [1,3,2,4,5,6,7,8] => [3,2,4,5,6,7,8,1] => 0
[(1,4),(2,3),(5,6),(7,8)] => [4,3,2,1,6,5,8,7] => [1,4,2,3,5,6,7,8] => [3,4,2,5,6,7,8,1] => 0
[(1,5),(2,3),(4,6),(7,8)] => [5,3,2,6,1,4,8,7] => [1,5,2,3,4,6,7,8] => [3,4,5,2,6,7,8,1] => 0
[(1,3),(2,5),(4,7),(6,8)] => [3,5,1,7,2,8,4,6] => [1,3,2,5,4,7,6,8] => [3,2,5,4,7,6,8,1] => 0
[(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [1,2,3,4,5,6,7,8,9,10] => [2,3,4,5,6,7,8,9,10,1] => 0
[(1,3),(2,4),(5,6),(7,8),(9,10)] => [3,4,1,2,6,5,8,7,10,9] => [1,3,2,4,5,6,7,8,9,10] => [3,2,4,5,6,7,8,9,10,1] => 0
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Description
The number of occurrences of the pattern 4231 in a permutation.
It is a necessary condition that a permutation $\pi$ avoids this pattern for the Schubert variety associated to $\pi$ to be smooth [2].
Map
Inverse Kreweras complement
Description
Sends the permutation $\pi \in \mathfrak{S}_n$ to the permutation $c\pi^{-1}$ where $c = (1,\ldots,n)$ is the long cycle.
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.