Identifier
Mp00058:
Perfect matchings
—to permutation⟶
Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Images
[(1,2)] => [2,1] => [2,1] => [[.,.],.]
[(1,2),(3,4)] => [2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
[(1,3),(2,4)] => [3,4,1,2] => [3,4,1,2] => [[.,[.,.]],[.,.]]
[(1,4),(2,3)] => [4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,1,6,5,4,3] => [[.,.],[[[[.,.],.],.],.]]
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [3,6,1,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
[(1,4),(2,3),(5,6)] => [4,3,2,1,6,5] => [4,3,2,1,6,5] => [[[[.,.],.],.],[[.,.],.]]
[(1,5),(2,3),(4,6)] => [5,3,2,6,1,4] => [5,3,2,6,1,4] => [[[[.,.],.],[.,.]],[.,.]]
[(1,6),(2,3),(4,5)] => [6,3,2,5,4,1] => [6,3,2,5,4,1] => [[[[.,.],.],[[.,.],.]],.]
[(1,6),(2,4),(3,5)] => [6,4,5,2,3,1] => [6,4,5,2,3,1] => [[[[.,.],[.,.]],[.,.]],.]
[(1,5),(2,4),(3,6)] => [5,4,6,2,1,3] => [5,4,6,2,1,3] => [[[[.,.],[.,.]],.],[.,.]]
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [4,6,5,1,3,2] => [[.,[[.,.],.]],[[.,.],.]]
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [3,6,1,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [2,1,6,5,4,3] => [[.,.],[[[[.,.],.],.],.]]
[(1,2),(3,6),(4,5)] => [2,1,6,5,4,3] => [2,1,6,5,4,3] => [[.,.],[[[[.,.],.],.],.]]
[(1,3),(2,6),(4,5)] => [3,6,1,5,4,2] => [3,6,1,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
[(1,4),(2,6),(3,5)] => [4,6,5,1,3,2] => [4,6,5,1,3,2] => [[.,[[.,.],.]],[[.,.],.]]
[(1,5),(2,6),(3,4)] => [5,6,4,3,1,2] => [5,6,4,3,1,2] => [[[[.,[.,.]],.],.],[.,.]]
[(1,6),(2,5),(3,4)] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],.]
[(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [2,1,8,7,6,5,4,3] => [[.,.],[[[[[[.,.],.],.],.],.],.]]
[(1,3),(2,4),(5,6),(7,8)] => [3,4,1,2,6,5,8,7] => [3,8,1,7,6,5,4,2] => [[.,[.,.]],[[[[[.,.],.],.],.],.]]
[(1,4),(2,3),(5,6),(7,8)] => [4,3,2,1,6,5,8,7] => [4,3,2,1,8,7,6,5] => [[[[.,.],.],.],[[[[.,.],.],.],.]]
[(1,5),(2,3),(4,6),(7,8)] => [5,3,2,6,1,4,8,7] => [5,3,2,8,1,7,6,4] => [[[[.,.],.],[.,.]],[[[.,.],.],.]]
[(1,6),(2,3),(4,5),(7,8)] => [6,3,2,5,4,1,8,7] => [6,3,2,8,7,1,5,4] => [[[[.,.],.],[[.,.],.]],[[.,.],.]]
[(1,7),(2,3),(4,5),(6,8)] => [7,3,2,5,4,8,1,6] => [7,3,2,8,6,5,1,4] => [[[[.,.],.],[[[.,.],.],.]],[.,.]]
[(1,8),(2,3),(4,5),(6,7)] => [8,3,2,5,4,7,6,1] => [8,3,2,7,6,5,4,1] => [[[[.,.],.],[[[[.,.],.],.],.]],.]
[(1,8),(2,4),(3,5),(6,7)] => [8,4,5,2,3,7,6,1] => [8,4,7,2,6,5,3,1] => [[[[.,.],[.,.]],[[[.,.],.],.]],.]
[(1,7),(2,4),(3,5),(6,8)] => [7,4,5,2,3,8,1,6] => [7,4,8,2,6,5,1,3] => [[[[.,.],[.,.]],[[.,.],.]],[.,.]]
[(1,6),(2,4),(3,5),(7,8)] => [6,4,5,2,3,1,8,7] => [6,4,8,2,7,1,5,3] => [[[[.,.],[.,.]],[.,.]],[[.,.],.]]
[(1,5),(2,4),(3,6),(7,8)] => [5,4,6,2,1,3,8,7] => [5,4,8,2,1,7,6,3] => [[[[.,.],[.,.]],.],[[[.,.],.],.]]
[(1,4),(2,5),(3,6),(7,8)] => [4,5,6,1,2,3,8,7] => [4,8,7,1,6,5,3,2] => [[.,[[.,.],.]],[[[[.,.],.],.],.]]
[(1,3),(2,5),(4,6),(7,8)] => [3,5,1,6,2,4,8,7] => [3,8,1,7,6,5,4,2] => [[.,[.,.]],[[[[[.,.],.],.],.],.]]
[(1,2),(3,5),(4,6),(7,8)] => [2,1,5,6,3,4,8,7] => [2,1,8,7,6,5,4,3] => [[.,.],[[[[[[.,.],.],.],.],.],.]]
[(1,2),(3,6),(4,5),(7,8)] => [2,1,6,5,4,3,8,7] => [2,1,8,7,6,5,4,3] => [[.,.],[[[[[[.,.],.],.],.],.],.]]
[(1,3),(2,6),(4,5),(7,8)] => [3,6,1,5,4,2,8,7] => [3,8,1,7,6,5,4,2] => [[.,[.,.]],[[[[[.,.],.],.],.],.]]
[(1,4),(2,6),(3,5),(7,8)] => [4,6,5,1,3,2,8,7] => [4,8,7,1,6,5,3,2] => [[.,[[.,.],.]],[[[[.,.],.],.],.]]
[(1,5),(2,6),(3,4),(7,8)] => [5,6,4,3,1,2,8,7] => [5,8,4,3,1,7,6,2] => [[[[.,[.,.]],.],.],[[[.,.],.],.]]
[(1,6),(2,5),(3,4),(7,8)] => [6,5,4,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => [[[[[[.,.],.],.],.],.],[[.,.],.]]
[(1,7),(2,5),(3,4),(6,8)] => [7,5,4,3,2,8,1,6] => [7,5,4,3,2,8,1,6] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
[(1,8),(2,5),(3,4),(6,7)] => [8,5,4,3,2,7,6,1] => [8,5,4,3,2,7,6,1] => [[[[[[.,.],.],.],.],[[.,.],.]],.]
[(1,8),(2,6),(3,4),(5,7)] => [8,6,4,3,7,2,5,1] => [8,6,4,3,7,2,5,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
[(1,7),(2,6),(3,4),(5,8)] => [7,6,4,3,8,2,1,5] => [7,6,4,3,8,2,1,5] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
[(1,6),(2,7),(3,4),(5,8)] => [6,7,4,3,8,1,2,5] => [6,8,4,3,7,1,5,2] => [[[[.,[.,.]],.],[.,.]],[[.,.],.]]
[(1,5),(2,7),(3,4),(6,8)] => [5,7,4,3,1,8,2,6] => [5,8,4,3,1,7,6,2] => [[[[.,[.,.]],.],.],[[[.,.],.],.]]
[(1,4),(2,7),(3,5),(6,8)] => [4,7,5,1,3,8,2,6] => [4,8,7,1,6,5,3,2] => [[.,[[.,.],.]],[[[[.,.],.],.],.]]
[(1,3),(2,7),(4,5),(6,8)] => [3,7,1,5,4,8,2,6] => [3,8,1,7,6,5,4,2] => [[.,[.,.]],[[[[[.,.],.],.],.],.]]
[(1,2),(3,7),(4,5),(6,8)] => [2,1,7,5,4,8,3,6] => [2,1,8,7,6,5,4,3] => [[.,.],[[[[[[.,.],.],.],.],.],.]]
[(1,2),(3,8),(4,5),(6,7)] => [2,1,8,5,4,7,6,3] => [2,1,8,7,6,5,4,3] => [[.,.],[[[[[[.,.],.],.],.],.],.]]
[(1,3),(2,8),(4,5),(6,7)] => [3,8,1,5,4,7,6,2] => [3,8,1,7,6,5,4,2] => [[.,[.,.]],[[[[[.,.],.],.],.],.]]
[(1,4),(2,8),(3,5),(6,7)] => [4,8,5,1,3,7,6,2] => [4,8,7,1,6,5,3,2] => [[.,[[.,.],.]],[[[[.,.],.],.],.]]
[(1,5),(2,8),(3,4),(6,7)] => [5,8,4,3,1,7,6,2] => [5,8,4,3,1,7,6,2] => [[[[.,[.,.]],.],.],[[[.,.],.],.]]
[(1,6),(2,8),(3,4),(5,7)] => [6,8,4,3,7,1,5,2] => [6,8,4,3,7,1,5,2] => [[[[.,[.,.]],.],[.,.]],[[.,.],.]]
[(1,7),(2,8),(3,4),(5,6)] => [7,8,4,3,6,5,1,2] => [7,8,4,3,6,5,1,2] => [[[[.,[.,.]],.],[[.,.],.]],[.,.]]
[(1,8),(2,7),(3,4),(5,6)] => [8,7,4,3,6,5,2,1] => [8,7,4,3,6,5,2,1] => [[[[[[.,.],.],.],[[.,.],.]],.],.]
[(1,8),(2,7),(3,5),(4,6)] => [8,7,5,6,3,4,2,1] => [8,7,5,6,3,4,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
[(1,7),(2,8),(3,5),(4,6)] => [7,8,5,6,3,4,1,2] => [7,8,5,6,3,4,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
[(1,6),(2,8),(3,5),(4,7)] => [6,8,5,7,3,1,4,2] => [6,8,5,7,3,1,4,2] => [[[[.,[.,.]],[.,.]],.],[[.,.],.]]
[(1,5),(2,8),(3,6),(4,7)] => [5,8,6,7,1,3,4,2] => [5,8,7,6,1,4,3,2] => [[.,[[[.,.],.],.]],[[[.,.],.],.]]
[(1,4),(2,8),(3,6),(5,7)] => [4,8,6,1,7,3,5,2] => [4,8,7,1,6,5,3,2] => [[.,[[.,.],.]],[[[[.,.],.],.],.]]
[(1,3),(2,8),(4,6),(5,7)] => [3,8,1,6,7,4,5,2] => [3,8,1,7,6,5,4,2] => [[.,[.,.]],[[[[[.,.],.],.],.],.]]
[(1,2),(3,8),(4,6),(5,7)] => [2,1,8,6,7,4,5,3] => [2,1,8,7,6,5,4,3] => [[.,.],[[[[[[.,.],.],.],.],.],.]]
[(1,2),(3,7),(4,6),(5,8)] => [2,1,7,6,8,4,3,5] => [2,1,8,7,6,5,4,3] => [[.,.],[[[[[[.,.],.],.],.],.],.]]
[(1,3),(2,7),(4,6),(5,8)] => [3,7,1,6,8,4,2,5] => [3,8,1,7,6,5,4,2] => [[.,[.,.]],[[[[[.,.],.],.],.],.]]
[(1,4),(2,7),(3,6),(5,8)] => [4,7,6,1,8,3,2,5] => [4,8,7,1,6,5,3,2] => [[.,[[.,.],.]],[[[[.,.],.],.],.]]
[(1,5),(2,7),(3,6),(4,8)] => [5,7,6,8,1,3,2,4] => [5,8,7,6,1,4,3,2] => [[.,[[[.,.],.],.]],[[[.,.],.],.]]
[(1,6),(2,7),(3,5),(4,8)] => [6,7,5,8,3,1,2,4] => [6,8,5,7,3,1,4,2] => [[[[.,[.,.]],[.,.]],.],[[.,.],.]]
[(1,7),(2,6),(3,5),(4,8)] => [7,6,5,8,3,2,1,4] => [7,6,5,8,3,2,1,4] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
[(1,8),(2,6),(3,5),(4,7)] => [8,6,5,7,3,2,4,1] => [8,6,5,7,3,2,4,1] => [[[[[[.,.],.],[.,.]],.],[.,.]],.]
[(1,8),(2,5),(3,6),(4,7)] => [8,5,6,7,2,3,4,1] => [8,5,7,6,2,4,3,1] => [[[[.,.],[[.,.],.]],[[.,.],.]],.]
[(1,7),(2,5),(3,6),(4,8)] => [7,5,6,8,2,3,1,4] => [7,5,8,6,2,4,1,3] => [[[[.,.],[[.,.],.]],[.,.]],[.,.]]
[(1,6),(2,5),(3,7),(4,8)] => [6,5,7,8,2,1,3,4] => [6,5,8,7,2,1,4,3] => [[[[.,.],[[.,.],.]],.],[[.,.],.]]
[(1,5),(2,6),(3,7),(4,8)] => [5,6,7,8,1,2,3,4] => [5,8,7,6,1,4,3,2] => [[.,[[[.,.],.],.]],[[[.,.],.],.]]
[(1,4),(2,6),(3,7),(5,8)] => [4,6,7,1,8,2,3,5] => [4,8,7,1,6,5,3,2] => [[.,[[.,.],.]],[[[[.,.],.],.],.]]
[(1,3),(2,6),(4,7),(5,8)] => [3,6,1,7,8,2,4,5] => [3,8,1,7,6,5,4,2] => [[.,[.,.]],[[[[[.,.],.],.],.],.]]
[(1,2),(3,6),(4,7),(5,8)] => [2,1,6,7,8,3,4,5] => [2,1,8,7,6,5,4,3] => [[.,.],[[[[[[.,.],.],.],.],.],.]]
[(1,2),(3,5),(4,7),(6,8)] => [2,1,5,7,3,8,4,6] => [2,1,8,7,6,5,4,3] => [[.,.],[[[[[[.,.],.],.],.],.],.]]
[(1,3),(2,5),(4,7),(6,8)] => [3,5,1,7,2,8,4,6] => [3,8,1,7,6,5,4,2] => [[.,[.,.]],[[[[[.,.],.],.],.],.]]
[(1,4),(2,5),(3,7),(6,8)] => [4,5,7,1,2,8,3,6] => [4,8,7,1,6,5,3,2] => [[.,[[.,.],.]],[[[[.,.],.],.],.]]
[(1,5),(2,4),(3,7),(6,8)] => [5,4,7,2,1,8,3,6] => [5,4,8,2,1,7,6,3] => [[[[.,.],[.,.]],.],[[[.,.],.],.]]
[(1,6),(2,4),(3,7),(5,8)] => [6,4,7,2,8,1,3,5] => [6,4,8,2,7,1,5,3] => [[[[.,.],[.,.]],[.,.]],[[.,.],.]]
[(1,7),(2,4),(3,6),(5,8)] => [7,4,6,2,8,3,1,5] => [7,4,8,2,6,5,1,3] => [[[[.,.],[.,.]],[[.,.],.]],[.,.]]
[(1,8),(2,4),(3,6),(5,7)] => [8,4,6,2,7,3,5,1] => [8,4,7,2,6,5,3,1] => [[[[.,.],[.,.]],[[[.,.],.],.]],.]
[(1,8),(2,3),(4,6),(5,7)] => [8,3,2,6,7,4,5,1] => [8,3,2,7,6,5,4,1] => [[[[.,.],.],[[[[.,.],.],.],.]],.]
[(1,7),(2,3),(4,6),(5,8)] => [7,3,2,6,8,4,1,5] => [7,3,2,8,6,5,1,4] => [[[[.,.],.],[[[.,.],.],.]],[.,.]]
[(1,6),(2,3),(4,7),(5,8)] => [6,3,2,7,8,1,4,5] => [6,3,2,8,7,1,5,4] => [[[[.,.],.],[[.,.],.]],[[.,.],.]]
[(1,5),(2,3),(4,7),(6,8)] => [5,3,2,7,1,8,4,6] => [5,3,2,8,1,7,6,4] => [[[[.,.],.],[.,.]],[[[.,.],.],.]]
[(1,4),(2,3),(5,7),(6,8)] => [4,3,2,1,7,8,5,6] => [4,3,2,1,8,7,6,5] => [[[[.,.],.],.],[[[[.,.],.],.],.]]
[(1,3),(2,4),(5,7),(6,8)] => [3,4,1,2,7,8,5,6] => [3,8,1,7,6,5,4,2] => [[.,[.,.]],[[[[[.,.],.],.],.],.]]
[(1,2),(3,4),(5,7),(6,8)] => [2,1,4,3,7,8,5,6] => [2,1,8,7,6,5,4,3] => [[.,.],[[[[[[.,.],.],.],.],.],.]]
[(1,2),(3,4),(5,8),(6,7)] => [2,1,4,3,8,7,6,5] => [2,1,8,7,6,5,4,3] => [[.,.],[[[[[[.,.],.],.],.],.],.]]
[(1,3),(2,4),(5,8),(6,7)] => [3,4,1,2,8,7,6,5] => [3,8,1,7,6,5,4,2] => [[.,[.,.]],[[[[[.,.],.],.],.],.]]
[(1,4),(2,3),(5,8),(6,7)] => [4,3,2,1,8,7,6,5] => [4,3,2,1,8,7,6,5] => [[[[.,.],.],.],[[[[.,.],.],.],.]]
[(1,5),(2,3),(4,8),(6,7)] => [5,3,2,8,1,7,6,4] => [5,3,2,8,1,7,6,4] => [[[[.,.],.],[.,.]],[[[.,.],.],.]]
[(1,6),(2,3),(4,8),(5,7)] => [6,3,2,8,7,1,5,4] => [6,3,2,8,7,1,5,4] => [[[[.,.],.],[[.,.],.]],[[.,.],.]]
[(1,7),(2,3),(4,8),(5,6)] => [7,3,2,8,6,5,1,4] => [7,3,2,8,6,5,1,4] => [[[[.,.],.],[[[.,.],.],.]],[.,.]]
[(1,8),(2,3),(4,7),(5,6)] => [8,3,2,7,6,5,4,1] => [8,3,2,7,6,5,4,1] => [[[[.,.],.],[[[[.,.],.],.],.]],.]
[(1,8),(2,4),(3,7),(5,6)] => [8,4,7,2,6,5,3,1] => [8,4,7,2,6,5,3,1] => [[[[.,.],[.,.]],[[[.,.],.],.]],.]
[(1,7),(2,4),(3,8),(5,6)] => [7,4,8,2,6,5,1,3] => [7,4,8,2,6,5,1,3] => [[[[.,.],[.,.]],[[.,.],.]],[.,.]]
[(1,6),(2,4),(3,8),(5,7)] => [6,4,8,2,7,1,5,3] => [6,4,8,2,7,1,5,3] => [[[[.,.],[.,.]],[.,.]],[[.,.],.]]
[(1,5),(2,4),(3,8),(6,7)] => [5,4,8,2,1,7,6,3] => [5,4,8,2,1,7,6,3] => [[[[.,.],[.,.]],.],[[[.,.],.],.]]
[(1,4),(2,5),(3,8),(6,7)] => [4,5,8,1,2,7,6,3] => [4,8,7,1,6,5,3,2] => [[.,[[.,.],.]],[[[[.,.],.],.],.]]
>>> Load all 185 entries. <<<Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
Simion-Schmidt map
Description
The Simion-Schmidt map sends any permutation to a $123$-avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].
Map
binary search tree: left to right
Description
Return the shape of the binary search tree of the permutation as a non labelled binary tree.
searching the database
Sorry, this map was not found in the database.