Identifier
-
Mp00058:
Perfect matchings
—to permutation⟶
Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001083: 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] => [3,2,4,1] => 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] => [6,5,4,2,3,1] => 0
[(1,4),(2,3),(5,6)] => [4,3,2,1,6,5] => [1,4,2,3,5,6] => [6,5,3,2,4,1] => 0
[(1,5),(2,3),(4,6)] => [5,3,2,6,1,4] => [1,5,2,3,4,6] => [6,4,3,2,5,1] => 0
[(1,6),(2,3),(4,5)] => [6,3,2,5,4,1] => [1,6,2,3,4,5] => [5,4,3,2,6,1] => 0
[(1,6),(2,4),(3,5)] => [6,4,5,2,3,1] => [1,6,2,4,3,5] => [5,3,4,2,6,1] => 0
[(1,5),(2,4),(3,6)] => [5,4,6,2,1,3] => [1,5,2,4,3,6] => [6,3,4,2,5,1] => 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] => [6,4,5,3,2,1] => 0
[(1,2),(3,6),(4,5)] => [2,1,6,5,4,3] => [1,2,3,6,4,5] => [5,4,6,3,2,1] => 0
[(1,3),(2,6),(4,5)] => [3,6,1,5,4,2] => [1,3,2,6,4,5] => [5,4,6,2,3,1] => 0
[(1,4),(2,6),(3,5)] => [4,6,5,1,3,2] => [1,4,2,6,3,5] => [5,3,6,2,4,1] => 1
[(1,5),(2,6),(3,4)] => [5,6,4,3,1,2] => [1,5,2,6,3,4] => [4,3,6,2,5,1] => 2
[(1,6),(2,5),(3,4)] => [6,5,4,3,2,1] => [1,6,2,5,3,4] => [4,3,5,2,6,1] => 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] => [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] => 0
[(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] => 1
[(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] => 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] => [8,6,4,3,7,2,5,1] => 2
[(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] => 0
[(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] => 0
[(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] => 0
[(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] => 1
[(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
[(1,2),(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] => 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] => [6,5,7,4,3,2,8,1] => 0
[(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] => 0
[(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] => 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] => [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 boxed occurrences of 132 in a permutation.
This is the number of occurrences of the pattern $132$ such that any entry between the three matched entries is either larger than the largest matched entry or smaller than the smallest matched entry.
This is the number of occurrences of the pattern $132$ such that any entry between the three matched entries is either larger than the largest matched entry or smaller than the smallest matched entry.
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 $\tau(i) = \sigma(n+1-i)$.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
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