Identifier
Values
[1,0] => [2,1] => [1,2] => 0
[1,0,1,0] => [3,1,2] => [1,3,2] => 0
[1,1,0,0] => [2,3,1] => [3,1,2] => 0
[1,0,1,0,1,0] => [4,1,2,3] => [1,3,4,2] => 0
[1,0,1,1,0,0] => [3,1,4,2] => [4,2,1,3] => 0
[1,1,0,0,1,0] => [2,4,1,3] => [3,1,4,2] => 0
[1,1,0,1,0,0] => [4,3,1,2] => [1,2,4,3] => 0
[1,1,1,0,0,0] => [2,3,4,1] => [3,4,1,2] => 1
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [1,3,4,5,2] => 0
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [5,2,3,1,4] => 1
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [4,2,1,5,3] => 0
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [1,3,2,5,4] => 0
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [4,2,5,1,3] => 1
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [3,1,4,5,2] => 0
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [3,5,2,1,4] => 0
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [1,5,3,4,2] => 1
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [1,2,4,5,3] => 0
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [5,4,2,1,3] => 0
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [3,4,1,5,2] => 1
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [3,1,2,5,4] => 0
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [1,5,2,4,3] => 0
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [3,4,5,1,2] => 3
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [1,3,4,5,6,2] => 0
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [6,2,3,4,1,5] => 3
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [5,2,3,1,6,4] => 1
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [1,3,4,2,6,5] => 0
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [5,2,3,6,1,4] => 2
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [4,2,1,5,6,3] => 0
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [4,2,6,3,1,5] => 1
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [1,3,6,4,5,2] => 1
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [1,3,2,5,6,4] => 0
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [6,2,5,3,1,4] => 2
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [4,2,5,1,6,3] => 1
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [4,2,1,3,6,5] => 0
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [1,3,6,2,5,4] => 0
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [4,2,5,6,1,3] => 3
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [3,1,4,5,6,2] => 0
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [3,6,2,4,1,5] => 1
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [3,5,2,1,6,4] => 0
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [3,1,4,2,6,5] => 0
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [3,5,2,6,1,4] => 1
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [1,5,3,4,6,2] => 1
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [6,4,2,3,1,5] => 2
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [1,6,3,4,5,2] => 3
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [6,1,3,4,5,2] => 3
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [6,5,2,3,1,4] => 2
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [5,4,2,1,6,3] => 0
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [1,5,3,2,6,4] => 0
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [1,6,3,2,5,4] => 0
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [5,4,2,6,1,3] => 2
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [3,4,1,5,6,2] => 1
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [3,4,6,2,1,5] => 0
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [3,1,6,4,5,2] => 1
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [3,1,2,5,6,4] => 0
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [3,6,5,2,1,4] => 0
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [1,5,6,3,4,2] => 3
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [1,5,2,4,6,3] => 0
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [1,2,3,5,6,4] => 0
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [6,4,5,2,1,3] => 5
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [3,4,5,1,6,2] => 3
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [3,4,1,2,6,5] => 1
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [3,1,6,2,5,4] => 0
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [1,5,6,2,4,3] => 2
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [3,4,5,6,1,2] => 6
[1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [1,3,4,5,6,7,2] => 0
[1,0,1,0,1,0,1,1,0,1,0,0] => [7,1,2,3,6,4,5] => [1,3,4,5,2,7,6] => 0
[1,0,1,0,1,1,0,1,0,0,1,0] => [7,1,2,5,3,4,6] => [1,3,4,7,5,6,2] => 1
[1,0,1,0,1,1,0,1,0,1,0,0] => [7,1,2,6,3,4,5] => [1,3,4,2,6,7,5] => 0
[1,0,1,0,1,1,1,0,1,0,0,0] => [7,1,2,5,6,3,4] => [1,3,4,7,2,6,5] => 0
[1,0,1,1,0,1,0,0,1,0,1,0] => [7,1,4,2,3,5,6] => [1,3,6,4,5,7,2] => 1
[1,0,1,1,0,1,0,1,0,0,1,0] => [7,1,5,2,3,4,6] => [1,3,7,4,5,6,2] => 3
[1,0,1,1,0,1,1,0,0,1,0,0] => [7,1,4,2,6,3,5] => [1,3,6,4,2,7,5] => 0
[1,0,1,1,0,1,1,0,1,0,0,0] => [7,1,5,2,6,3,4] => [1,3,7,4,2,6,5] => 0
[1,0,1,1,1,0,1,0,0,0,1,0] => [7,1,4,5,2,3,6] => [1,3,6,7,4,5,2] => 3
[1,0,1,1,1,0,1,0,0,1,0,0] => [7,1,4,6,2,3,5] => [1,3,6,2,5,7,4] => 0
[1,0,1,1,1,0,1,0,1,0,0,0] => [7,1,6,5,2,3,4] => [1,3,2,4,6,7,5] => 0
[1,0,1,1,1,1,0,1,0,0,0,0] => [7,1,4,5,6,2,3] => [1,3,6,7,2,5,4] => 2
[1,1,0,1,0,1,0,1,0,1,0,0] => [7,6,1,2,3,4,5] => [1,2,4,5,6,7,3] => 0
[1,1,1,1,0,1,0,1,0,0,0,0] => [7,6,4,5,1,2,3] => [1,2,7,3,5,6,4] => 1
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Description
The number of occurrences of the patterns 2143 or 4231 in a permutation.
It is a necessary and sufficient condition that a permutation $\pi$ avoids these two patterns for the Schubert variety associated to $\pi$ to be smooth [1].
Map
Lehmer code rotation
Description
Sends a permutation $\pi$ to the unique permutation $\tau$ (of the same length) such that every entry in the Lehmer code of $\tau$ is cyclically one larger than the Lehmer code of $\pi$.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.