Identifier
Values
[(1,2)] => [2,1] => [1,2] => [2,1] => 0
[(1,2),(3,4)] => [2,1,4,3] => [1,2,3,4] => [4,3,2,1] => 0
[(1,3),(2,4)] => [3,4,1,2] => [1,3,2,4] => [4,2,3,1] => 1
[(1,4),(2,3)] => [4,3,2,1] => [1,4,2,3] => [3,2,4,1] => 2
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 0
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [1,3,2,4,5,6] => [6,5,4,2,3,1] => 1
[(1,4),(2,3),(5,6)] => [4,3,2,1,6,5] => [1,4,2,3,5,6] => [6,5,3,2,4,1] => 2
[(1,5),(2,3),(4,6)] => [5,3,2,6,1,4] => [1,5,2,3,4,6] => [6,4,3,2,5,1] => 3
[(1,6),(2,3),(4,5)] => [6,3,2,5,4,1] => [1,6,2,3,4,5] => [5,4,3,2,6,1] => 4
[(1,6),(2,4),(3,5)] => [6,4,5,2,3,1] => [1,6,2,4,3,5] => [5,3,4,2,6,1] => 5
[(1,5),(2,4),(3,6)] => [5,4,6,2,1,3] => [1,5,2,4,3,6] => [6,3,4,2,5,1] => 4
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [1,4,2,5,3,6] => [6,3,5,2,4,1] => 3
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [1,3,2,5,4,6] => [6,4,5,2,3,1] => 2
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [1,2,3,5,4,6] => [6,4,5,3,2,1] => 1
[(1,2),(3,6),(4,5)] => [2,1,6,5,4,3] => [1,2,3,6,4,5] => [5,4,6,3,2,1] => 2
[(1,3),(2,6),(4,5)] => [3,6,1,5,4,2] => [1,3,2,6,4,5] => [5,4,6,2,3,1] => 3
[(1,4),(2,6),(3,5)] => [4,6,5,1,3,2] => [1,4,2,6,3,5] => [5,3,6,2,4,1] => 4
[(1,5),(2,6),(3,4)] => [5,6,4,3,1,2] => [1,5,2,6,3,4] => [4,3,6,2,5,1] => 5
[(1,6),(2,5),(3,4)] => [6,5,4,3,2,1] => [1,6,2,5,3,4] => [4,3,5,2,6,1] => 6
[(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => 0
[(1,8),(2,3),(4,5),(6,7)] => [8,3,2,5,4,7,6,1] => [1,8,2,3,4,5,6,7] => [7,6,5,4,3,2,8,1] => 6
[(1,4),(2,5),(3,6),(7,8)] => [4,5,6,1,2,3,8,7] => [1,4,2,5,3,6,7,8] => [8,7,6,3,5,2,4,1] => 3
[(1,3),(2,5),(4,6),(7,8)] => [3,5,1,6,2,4,8,7] => [1,3,2,5,4,6,7,8] => [8,7,6,4,5,2,3,1] => 2
[(1,5),(2,7),(3,4),(6,8)] => [5,7,4,3,1,8,2,6] => [1,5,2,7,3,4,6,8] => [8,6,4,3,7,2,5,1] => 6
[(1,2),(3,8),(4,5),(6,7)] => [2,1,8,5,4,7,6,3] => [1,2,3,8,4,5,6,7] => [7,6,5,4,8,3,2,1] => 4
[(1,8),(2,7),(3,4),(5,6)] => [8,7,4,3,6,5,2,1] => [1,8,2,7,3,4,5,6] => [6,5,4,3,7,2,8,1] => 10
[(1,2),(3,5),(4,7),(6,8)] => [2,1,5,7,3,8,4,6] => [1,2,3,5,4,7,6,8] => [8,6,7,4,5,3,2,1] => 2
[(1,6),(2,3),(4,7),(5,8)] => [6,3,2,7,8,1,4,5] => [1,6,2,3,4,7,5,8] => [8,5,7,4,3,2,6,1] => 5
[(1,3),(2,4),(5,7),(6,8)] => [3,4,1,2,7,8,5,6] => [1,3,2,4,5,7,6,8] => [8,6,7,5,4,2,3,1] => 2
[(1,2),(3,4),(5,8),(6,7)] => [2,1,4,3,8,7,6,5] => [1,2,3,4,5,8,6,7] => [7,6,8,5,4,3,2,1] => 2
[(1,8),(2,3),(4,7),(5,6)] => [8,3,2,7,6,5,4,1] => [1,8,2,3,4,7,5,6] => [6,5,7,4,3,2,8,1] => 8
[(1,2),(3,8),(4,7),(5,6)] => [2,1,8,7,6,5,4,3] => [1,2,3,8,4,7,5,6] => [6,5,7,4,8,3,2,1] => 6
[(1,8),(2,7),(3,6),(4,5)] => [8,7,6,5,4,3,2,1] => [1,8,2,7,3,6,4,5] => [5,4,6,3,7,2,8,1] => 12
[(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] => [10,9,8,7,6,5,4,3,2,1] => 0
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Description
The number of non-inversions of a permutation.
For a permutation of $\{1,\ldots,n\}$, this is given by $\operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi)$.
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.
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.