Identifier
-
Mp00058:
Perfect matchings
—to permutation⟶
Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000218: Permutations ⟶ ℤ
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] => 0
[(1,4),(2,3)] => [4,3,2,1] => [1,4,2,3] => [4,1,3,2] => 0
[(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] => [6,4,5,3,2,1] => 0
[(1,4),(2,3),(5,6)] => [4,3,2,1,6,5] => [1,4,2,3,5,6] => [6,3,5,4,2,1] => 0
[(1,5),(2,3),(4,6)] => [5,3,2,6,1,4] => [1,5,2,3,4,6] => [6,2,5,4,3,1] => 0
[(1,6),(2,3),(4,5)] => [6,3,2,5,4,1] => [1,6,2,3,4,5] => [6,1,5,4,3,2] => 0
[(1,6),(2,4),(3,5)] => [6,4,5,2,3,1] => [1,6,2,4,3,5] => [6,1,5,3,4,2] => 0
[(1,5),(2,4),(3,6)] => [5,4,6,2,1,3] => [1,5,2,4,3,6] => [6,2,5,3,4,1] => 0
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [1,4,2,5,3,6] => [6,3,5,2,4,1] => 1
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [1,3,2,5,4,6] => [6,4,5,2,3,1] => 0
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [1,2,3,5,4,6] => [6,5,4,2,3,1] => 0
[(1,2),(3,6),(4,5)] => [2,1,6,5,4,3] => [1,2,3,6,4,5] => [6,5,4,1,3,2] => 0
[(1,3),(2,6),(4,5)] => [3,6,1,5,4,2] => [1,3,2,6,4,5] => [6,4,5,1,3,2] => 0
[(1,4),(2,6),(3,5)] => [4,6,5,1,3,2] => [1,4,2,6,3,5] => [6,3,5,1,4,2] => 1
[(1,5),(2,6),(3,4)] => [5,6,4,3,1,2] => [1,5,2,6,3,4] => [6,2,5,1,4,3] => 2
[(1,6),(2,5),(3,4)] => [6,5,4,3,2,1] => [1,6,2,5,3,4] => [6,1,5,2,4,3] => 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] => [8,7,6,5,4,3,2,1] => 0
[(1,8),(2,4),(3,5),(6,7)] => [8,4,5,2,3,7,6,1] => [1,8,2,4,3,5,6,7] => [8,1,7,5,6,4,3,2] => 0
[(1,3),(2,5),(4,6),(7,8)] => [3,5,1,6,2,4,8,7] => [1,3,2,5,4,6,7,8] => [8,6,7,4,5,3,2,1] => 0
[(1,6),(2,5),(3,4),(7,8)] => [6,5,4,3,2,1,8,7] => [1,6,2,5,3,4,7,8] => [8,3,7,4,6,5,2,1] => 0
[(1,7),(2,5),(3,4),(6,8)] => [7,5,4,3,2,8,1,6] => [1,7,2,5,3,4,6,8] => [8,2,7,4,6,5,3,1] => 0
[(1,5),(2,7),(3,4),(6,8)] => [5,7,4,3,1,8,2,6] => [1,5,2,7,3,4,6,8] => [8,4,7,2,6,5,3,1] => 2
[(1,2),(3,8),(4,6),(5,7)] => [2,1,8,6,7,4,5,3] => [1,2,3,8,4,6,5,7] => [8,7,6,1,5,3,4,2] => 0
[(1,2),(3,6),(4,7),(5,8)] => [2,1,6,7,8,3,4,5] => [1,2,3,6,4,7,5,8] => [8,7,6,3,5,2,4,1] => 1
[(1,2),(3,5),(4,7),(6,8)] => [2,1,5,7,3,8,4,6] => [1,2,3,5,4,7,6,8] => [8,7,6,4,5,2,3,1] => 0
[(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] => 0
[(1,5),(2,3),(4,8),(6,7)] => [5,3,2,8,1,7,6,4] => [1,5,2,3,4,8,6,7] => [8,4,7,6,5,1,3,2] => 0
[(1,8),(2,3),(4,7),(5,6)] => [8,3,2,7,6,5,4,1] => [1,8,2,3,4,7,5,6] => [8,1,7,6,5,2,4,3] => 0
[(1,2),(3,6),(4,8),(5,7)] => [2,1,6,8,7,3,5,4] => [1,2,3,6,4,8,5,7] => [8,7,6,3,5,1,4,2] => 1
[(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
[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => [12,11,10,9,8,7,6,5,4,3,2,1] => 0
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Description
The number of occurrences of the pattern 213 in a permutation.
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.
Map
complement
Description
Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
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