Identifier
Values
[1,0] => [2,1] => [1,2] => [1,2] => 0
[1,0,1,0] => [3,1,2] => [1,2,3] => [1,2,3] => 0
[1,1,0,0] => [2,3,1] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0] => [4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,1,0,0] => [3,1,4,2] => [1,4,2,3] => [2,1,4,3] => 1
[1,1,0,0,1,0] => [2,4,1,3] => [1,3,2,4] => [2,3,1,4] => 1
[1,1,0,1,0,0] => [4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,1,0,0,0] => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [1,2,5,3,4] => [2,3,1,5,4] => 1
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [1,5,2,4,3] => [3,2,5,4,1] => 2
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [1,4,2,3,5] => [2,1,4,3,5] => 1
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [1,4,5,2,3] => [2,1,5,3,4] => 1
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [1,3,4,2,5] => [2,4,1,3,5] => 1
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [1,5,2,4,3] => [3,2,5,4,1] => 2
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [1,2,4,3,5] => [2,3,4,1,5] => 1
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [1,5,2,3,4] => [2,1,3,5,4] => 1
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [1,4,2,3,5] => [2,1,4,3,5] => 1
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [1,3,2,5,4] => [3,4,2,5,1] => 2
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [1,2,3,6,4,5] => [2,3,4,1,6,5] => 1
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [1,2,6,3,5,4] => [3,4,2,6,5,1] => 2
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [1,2,5,3,4,6] => [2,3,1,5,4,6] => 1
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [1,2,5,6,3,4] => [2,3,1,6,4,5] => 1
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [1,6,2,4,5,3] => [3,2,6,1,5,4] => 2
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [1,5,2,6,3,4] => [3,4,1,6,2,5] => 1
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [1,4,2,3,5,6] => [2,1,4,3,5,6] => 1
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [1,5,2,3,4,6] => [2,1,3,5,4,6] => 1
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [1,4,2,6,3,5] => [3,4,1,5,6,2] => 1
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [1,4,6,2,5,3] => [3,2,6,4,5,1] => 2
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [1,6,2,4,3,5] => [3,2,5,1,6,4] => 2
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [1,4,5,2,3,6] => [2,1,5,3,4,6] => 1
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [1,4,5,6,2,3] => [2,1,6,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [1,3,4,5,2,6] => [2,5,1,3,4,6] => 1
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [1,3,6,2,5,4] => [3,2,5,6,4,1] => 2
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [1,6,2,4,3,5] => [3,2,5,1,6,4] => 2
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [1,5,2,6,3,4] => [3,4,1,6,2,5] => 1
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [1,5,6,2,4,3] => [3,2,6,5,1,4] => 2
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [1,2,4,5,3,6] => [2,3,5,1,4,6] => 1
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [1,2,6,3,4,5] => [2,3,1,4,6,5] => 1
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [1,2,3,5,4,6] => [2,3,4,5,1,6] => 1
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [1,2,6,3,4,5] => [2,3,1,4,6,5] => 1
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [1,6,2,5,3,4] => [3,2,1,6,5,4] => 2
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [1,5,2,4,3,6] => [3,2,5,4,1,6] => 2
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [1,5,2,3,4,6] => [2,1,3,5,4,6] => 1
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [1,5,6,2,3,4] => [2,1,3,6,4,5] => 1
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [1,4,5,2,3,6] => [2,1,5,3,4,6] => 1
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [1,6,2,3,5,4] => [3,2,4,6,5,1] => 2
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [1,3,5,2,6,4] => [3,5,2,6,1,4] => 2
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [1,3,4,2,6,5] => [3,5,2,4,6,1] => 2
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [1,6,2,5,3,4] => [3,2,1,6,5,4] => 2
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [1,2,5,3,4,6] => [2,3,1,5,4,6] => 1
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [1,2,4,3,5,6] => [2,3,4,1,5,6] => 1
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [1,6,2,3,4,5] => [2,1,3,4,6,5] => 1
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [1,5,2,3,4,6] => [2,1,3,5,4,6] => 1
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [1,4,2,3,6,5] => [3,2,5,4,6,1] => 2
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [1,3,2,6,4,5] => [3,4,2,1,6,5] => 2
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[1,1,0,1,0,1,0,1,0,1,0,0] => [7,6,1,2,3,4,5] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[1,1,1,0,1,0,1,0,1,0,0,0] => [6,7,5,1,2,3,4] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[1,1,1,1,0,1,0,1,0,0,0,0] => [7,6,4,5,1,2,3] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[1,1,1,1,1,0,1,0,0,0,0,0] => [7,3,4,5,6,1,2] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
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Description
The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right.
For a permutation $\pi$ of length $n$, this is the number of indices $2 \leq j \leq n$ such that for all $1 \leq i < j$, the pair $(i,j)$ is an inversion of $\pi$.
Map
runsort
Description
The permutation obtained by sorting the increasing runs lexicographically.
Map
major-index to inversion-number bijection
Description
Return the permutation whose Lehmer code equals the major code of the preimage.
This map sends the major index to the number of inversions.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.