Identifier
Mp00058:
Perfect matchings
—to permutation⟶
Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00069: Permutations —complement⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Images
[(1,2)] => [2,1] => [1,2] => [.,[.,.]]
[(1,2),(3,4)] => [2,1,4,3] => [3,4,1,2] => [[.,[.,.]],[.,.]]
[(1,3),(2,4)] => [3,4,1,2] => [2,1,4,3] => [[.,.],[[.,.],.]]
[(1,4),(2,3)] => [4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [5,6,3,4,1,2] => [[[.,[.,.]],[.,.]],[.,.]]
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [4,3,6,5,1,2] => [[[.,.],[[.,.],.]],[.,.]]
[(1,4),(2,3),(5,6)] => [4,3,2,1,6,5] => [3,4,5,6,1,2] => [[.,[.,[.,[.,.]]]],[.,.]]
[(1,5),(2,3),(4,6)] => [5,3,2,6,1,4] => [2,4,5,1,6,3] => [[.,[.,[.,.]]],[[.,.],.]]
[(1,6),(2,3),(4,5)] => [6,3,2,5,4,1] => [1,4,5,2,3,6] => [.,[[.,[.,.]],[.,[.,.]]]]
[(1,6),(2,4),(3,5)] => [6,4,5,2,3,1] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
[(1,5),(2,4),(3,6)] => [5,4,6,2,1,3] => [2,3,1,5,6,4] => [[.,[.,.]],[[.,[.,.]],.]]
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [3,2,1,6,5,4] => [[[.,.],.],[[[.,.],.],.]]
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [4,2,6,1,5,3] => [[[.,.],[.,.]],[[.,.],.]]
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [5,6,2,1,4,3] => [[[.,[.,.]],.],[[.,.],.]]
[(1,2),(3,6),(4,5)] => [2,1,6,5,4,3] => [5,6,1,2,3,4] => [[.,[.,.]],[.,[.,[.,.]]]]
[(1,3),(2,6),(4,5)] => [3,6,1,5,4,2] => [4,1,6,2,3,5] => [[.,.],[[.,.],[.,[.,.]]]]
[(1,4),(2,6),(3,5)] => [4,6,5,1,3,2] => [3,1,2,6,4,5] => [[.,.],[.,[[.,.],[.,.]]]]
[(1,5),(2,6),(3,4)] => [5,6,4,3,1,2] => [2,1,3,4,6,5] => [[.,.],[.,[.,[[.,.],.]]]]
[(1,6),(2,5),(3,4)] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
[(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [7,8,5,6,3,4,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
[(1,3),(2,4),(5,6),(7,8)] => [3,4,1,2,6,5,8,7] => [6,5,8,7,3,4,1,2] => [[[[.,.],[[.,.],.]],[.,.]],[.,.]]
[(1,4),(2,3),(5,6),(7,8)] => [4,3,2,1,6,5,8,7] => [5,6,7,8,3,4,1,2] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
[(1,5),(2,3),(4,6),(7,8)] => [5,3,2,6,1,4,8,7] => [4,6,7,3,8,5,1,2] => [[[.,[.,[.,.]]],[[.,.],.]],[.,.]]
[(1,6),(2,3),(4,5),(7,8)] => [6,3,2,5,4,1,8,7] => [3,6,7,4,5,8,1,2] => [[.,[[.,[.,.]],[.,[.,.]]]],[.,.]]
[(1,7),(2,3),(4,5),(6,8)] => [7,3,2,5,4,8,1,6] => [2,6,7,4,5,1,8,3] => [[.,[[.,[.,.]],[.,.]]],[[.,.],.]]
[(1,8),(2,3),(4,5),(6,7)] => [8,3,2,5,4,7,6,1] => [1,6,7,4,5,2,3,8] => [.,[[[.,[.,.]],[.,.]],[.,[.,.]]]]
[(1,8),(2,4),(3,5),(6,7)] => [8,4,5,2,3,7,6,1] => [1,5,4,7,6,2,3,8] => [.,[[[.,.],[[.,.],.]],[.,[.,.]]]]
[(1,7),(2,4),(3,5),(6,8)] => [7,4,5,2,3,8,1,6] => [2,5,4,7,6,1,8,3] => [[.,[[.,.],[[.,.],.]]],[[.,.],.]]
[(1,6),(2,4),(3,5),(7,8)] => [6,4,5,2,3,1,8,7] => [3,5,4,7,6,8,1,2] => [[.,[[.,.],[[.,.],[.,.]]]],[.,.]]
[(1,5),(2,4),(3,6),(7,8)] => [5,4,6,2,1,3,8,7] => [4,5,3,7,8,6,1,2] => [[[.,[.,.]],[[.,[.,.]],.]],[.,.]]
[(1,4),(2,5),(3,6),(7,8)] => [4,5,6,1,2,3,8,7] => [5,4,3,8,7,6,1,2] => [[[[.,.],.],[[[.,.],.],.]],[.,.]]
[(1,3),(2,5),(4,6),(7,8)] => [3,5,1,6,2,4,8,7] => [6,4,8,3,7,5,1,2] => [[[[.,.],[.,.]],[[.,.],.]],[.,.]]
[(1,2),(3,5),(4,6),(7,8)] => [2,1,5,6,3,4,8,7] => [7,8,4,3,6,5,1,2] => [[[[.,[.,.]],.],[[.,.],.]],[.,.]]
[(1,2),(3,6),(4,5),(7,8)] => [2,1,6,5,4,3,8,7] => [7,8,3,4,5,6,1,2] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
[(1,3),(2,6),(4,5),(7,8)] => [3,6,1,5,4,2,8,7] => [6,3,8,4,5,7,1,2] => [[[.,.],[[.,.],[.,[.,.]]]],[.,.]]
[(1,4),(2,6),(3,5),(7,8)] => [4,6,5,1,3,2,8,7] => [5,3,4,8,6,7,1,2] => [[[.,.],[.,[[.,.],[.,.]]]],[.,.]]
[(1,5),(2,6),(3,4),(7,8)] => [5,6,4,3,1,2,8,7] => [4,3,5,6,8,7,1,2] => [[[.,.],[.,[.,[[.,.],.]]]],[.,.]]
[(1,6),(2,5),(3,4),(7,8)] => [6,5,4,3,2,1,8,7] => [3,4,5,6,7,8,1,2] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
[(1,7),(2,5),(3,4),(6,8)] => [7,5,4,3,2,8,1,6] => [2,4,5,6,7,1,8,3] => [[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]
[(1,8),(2,5),(3,4),(6,7)] => [8,5,4,3,2,7,6,1] => [1,4,5,6,7,2,3,8] => [.,[[.,[.,[.,[.,.]]]],[.,[.,.]]]]
[(1,8),(2,6),(3,4),(5,7)] => [8,6,4,3,7,2,5,1] => [1,3,5,6,2,7,4,8] => [.,[[.,[.,[.,.]]],[[.,.],[.,.]]]]
[(1,7),(2,6),(3,4),(5,8)] => [7,6,4,3,8,2,1,5] => [2,3,5,6,1,7,8,4] => [[.,[.,[.,[.,.]]]],[[.,[.,.]],.]]
[(1,6),(2,7),(3,4),(5,8)] => [6,7,4,3,8,1,2,5] => [3,2,5,6,1,8,7,4] => [[[.,.],[.,[.,.]]],[[[.,.],.],.]]
[(1,5),(2,7),(3,4),(6,8)] => [5,7,4,3,1,8,2,6] => [4,2,5,6,8,1,7,3] => [[[.,.],[.,[.,[.,.]]]],[[.,.],.]]
[(1,4),(2,7),(3,5),(6,8)] => [4,7,5,1,3,8,2,6] => [5,2,4,8,6,1,7,3] => [[[.,.],[.,[[.,.],.]]],[[.,.],.]]
[(1,3),(2,7),(4,5),(6,8)] => [3,7,1,5,4,8,2,6] => [6,2,8,4,5,1,7,3] => [[[.,.],[[.,.],[.,.]]],[[.,.],.]]
[(1,2),(3,7),(4,5),(6,8)] => [2,1,7,5,4,8,3,6] => [7,8,2,4,5,1,6,3] => [[[.,[.,.]],[.,[.,.]]],[[.,.],.]]
[(1,2),(3,8),(4,5),(6,7)] => [2,1,8,5,4,7,6,3] => [7,8,1,4,5,2,3,6] => [[.,[.,.]],[[.,[.,.]],[.,[.,.]]]]
[(1,3),(2,8),(4,5),(6,7)] => [3,8,1,5,4,7,6,2] => [6,1,8,4,5,2,3,7] => [[.,.],[[[.,.],[.,.]],[.,[.,.]]]]
[(1,4),(2,8),(3,5),(6,7)] => [4,8,5,1,3,7,6,2] => [5,1,4,8,6,2,3,7] => [[.,.],[[.,[[.,.],.]],[.,[.,.]]]]
[(1,5),(2,8),(3,4),(6,7)] => [5,8,4,3,1,7,6,2] => [4,1,5,6,8,2,3,7] => [[.,.],[[.,[.,[.,.]]],[.,[.,.]]]]
[(1,6),(2,8),(3,4),(5,7)] => [6,8,4,3,7,1,5,2] => [3,1,5,6,2,8,4,7] => [[.,.],[[.,[.,.]],[[.,.],[.,.]]]]
[(1,7),(2,8),(3,4),(5,6)] => [7,8,4,3,6,5,1,2] => [2,1,5,6,3,4,8,7] => [[.,.],[[.,[.,.]],[.,[[.,.],.]]]]
[(1,8),(2,7),(3,4),(5,6)] => [8,7,4,3,6,5,2,1] => [1,2,5,6,3,4,7,8] => [.,[.,[[.,[.,.]],[.,[.,[.,.]]]]]]
[(1,8),(2,7),(3,5),(4,6)] => [8,7,5,6,3,4,2,1] => [1,2,4,3,6,5,7,8] => [.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
[(1,7),(2,8),(3,5),(4,6)] => [7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
[(1,6),(2,8),(3,5),(4,7)] => [6,8,5,7,3,1,4,2] => [3,1,4,2,6,8,5,7] => [[.,.],[[.,.],[[.,[.,.]],[.,.]]]]
[(1,5),(2,8),(3,6),(4,7)] => [5,8,6,7,1,3,4,2] => [4,1,3,2,8,6,5,7] => [[.,.],[[.,.],[[[.,.],.],[.,.]]]]
[(1,4),(2,8),(3,6),(5,7)] => [4,8,6,1,7,3,5,2] => [5,1,3,8,2,6,4,7] => [[.,.],[[.,[.,.]],[[.,.],[.,.]]]]
[(1,3),(2,8),(4,6),(5,7)] => [3,8,1,6,7,4,5,2] => [6,1,8,3,2,5,4,7] => [[.,.],[[[.,.],.],[[.,.],[.,.]]]]
[(1,2),(3,8),(4,6),(5,7)] => [2,1,8,6,7,4,5,3] => [7,8,1,3,2,5,4,6] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
[(1,2),(3,7),(4,6),(5,8)] => [2,1,7,6,8,4,3,5] => [7,8,2,3,1,5,6,4] => [[[.,[.,.]],[.,.]],[[.,[.,.]],.]]
[(1,3),(2,7),(4,6),(5,8)] => [3,7,1,6,8,4,2,5] => [6,2,8,3,1,5,7,4] => [[[.,.],[[.,.],.]],[[.,[.,.]],.]]
[(1,4),(2,7),(3,6),(5,8)] => [4,7,6,1,8,3,2,5] => [5,2,3,8,1,6,7,4] => [[[.,.],[.,[.,.]]],[[.,[.,.]],.]]
[(1,5),(2,7),(3,6),(4,8)] => [5,7,6,8,1,3,2,4] => [4,2,3,1,8,6,7,5] => [[[.,.],[.,.]],[[[.,.],[.,.]],.]]
[(1,6),(2,7),(3,5),(4,8)] => [6,7,5,8,3,1,2,4] => [3,2,4,1,6,8,7,5] => [[[.,.],[.,.]],[[.,[[.,.],.]],.]]
[(1,7),(2,6),(3,5),(4,8)] => [7,6,5,8,3,2,1,4] => [2,3,4,1,6,7,8,5] => [[.,[.,[.,.]]],[[.,[.,[.,.]]],.]]
[(1,8),(2,6),(3,5),(4,7)] => [8,6,5,7,3,2,4,1] => [1,3,4,2,6,7,5,8] => [.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
[(1,8),(2,5),(3,6),(4,7)] => [8,5,6,7,2,3,4,1] => [1,4,3,2,7,6,5,8] => [.,[[[.,.],.],[[[.,.],.],[.,.]]]]
[(1,7),(2,5),(3,6),(4,8)] => [7,5,6,8,2,3,1,4] => [2,4,3,1,7,6,8,5] => [[.,[[.,.],.]],[[[.,.],[.,.]],.]]
[(1,6),(2,5),(3,7),(4,8)] => [6,5,7,8,2,1,3,4] => [3,4,2,1,7,8,6,5] => [[[.,[.,.]],.],[[[.,[.,.]],.],.]]
[(1,5),(2,6),(3,7),(4,8)] => [5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => [[[[.,.],.],.],[[[[.,.],.],.],.]]
[(1,4),(2,6),(3,7),(5,8)] => [4,6,7,1,8,2,3,5] => [5,3,2,8,1,7,6,4] => [[[[.,.],.],[.,.]],[[[.,.],.],.]]
[(1,3),(2,6),(4,7),(5,8)] => [3,6,1,7,8,2,4,5] => [6,3,8,2,1,7,5,4] => [[[[.,.],[.,.]],.],[[[.,.],.],.]]
[(1,2),(3,6),(4,7),(5,8)] => [2,1,6,7,8,3,4,5] => [7,8,3,2,1,6,5,4] => [[[[.,[.,.]],.],.],[[[.,.],.],.]]
[(1,2),(3,5),(4,7),(6,8)] => [2,1,5,7,3,8,4,6] => [7,8,4,2,6,1,5,3] => [[[[.,[.,.]],.],[.,.]],[[.,.],.]]
[(1,3),(2,5),(4,7),(6,8)] => [3,5,1,7,2,8,4,6] => [6,4,8,2,7,1,5,3] => [[[[.,.],[.,.]],[.,.]],[[.,.],.]]
[(1,4),(2,5),(3,7),(6,8)] => [4,5,7,1,2,8,3,6] => [5,4,2,8,7,1,6,3] => [[[[.,.],.],[[.,.],.]],[[.,.],.]]
[(1,5),(2,4),(3,7),(6,8)] => [5,4,7,2,1,8,3,6] => [4,5,2,7,8,1,6,3] => [[[.,[.,.]],[.,[.,.]]],[[.,.],.]]
[(1,6),(2,4),(3,7),(5,8)] => [6,4,7,2,8,1,3,5] => [3,5,2,7,1,8,6,4] => [[[.,[.,.]],[.,.]],[[[.,.],.],.]]
[(1,7),(2,4),(3,6),(5,8)] => [7,4,6,2,8,3,1,5] => [2,5,3,7,1,6,8,4] => [[.,[[.,.],[.,.]]],[[.,[.,.]],.]]
[(1,8),(2,4),(3,6),(5,7)] => [8,4,6,2,7,3,5,1] => [1,5,3,7,2,6,4,8] => [.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
[(1,8),(2,3),(4,6),(5,7)] => [8,3,2,6,7,4,5,1] => [1,6,7,3,2,5,4,8] => [.,[[[.,[.,.]],.],[[.,.],[.,.]]]]
[(1,7),(2,3),(4,6),(5,8)] => [7,3,2,6,8,4,1,5] => [2,6,7,3,1,5,8,4] => [[.,[[.,[.,.]],.]],[[.,[.,.]],.]]
[(1,6),(2,3),(4,7),(5,8)] => [6,3,2,7,8,1,4,5] => [3,6,7,2,1,8,5,4] => [[[.,[.,[.,.]]],.],[[[.,.],.],.]]
[(1,5),(2,3),(4,7),(6,8)] => [5,3,2,7,1,8,4,6] => [4,6,7,2,8,1,5,3] => [[[.,[.,[.,.]]],[.,.]],[[.,.],.]]
[(1,4),(2,3),(5,7),(6,8)] => [4,3,2,1,7,8,5,6] => [5,6,7,8,2,1,4,3] => [[[.,[.,[.,[.,.]]]],.],[[.,.],.]]
[(1,3),(2,4),(5,7),(6,8)] => [3,4,1,2,7,8,5,6] => [6,5,8,7,2,1,4,3] => [[[[.,.],[[.,.],.]],.],[[.,.],.]]
[(1,2),(3,4),(5,7),(6,8)] => [2,1,4,3,7,8,5,6] => [7,8,5,6,2,1,4,3] => [[[[.,[.,.]],[.,.]],.],[[.,.],.]]
[(1,2),(3,4),(5,8),(6,7)] => [2,1,4,3,8,7,6,5] => [7,8,5,6,1,2,3,4] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
[(1,3),(2,4),(5,8),(6,7)] => [3,4,1,2,8,7,6,5] => [6,5,8,7,1,2,3,4] => [[[.,.],[[.,.],.]],[.,[.,[.,.]]]]
[(1,4),(2,3),(5,8),(6,7)] => [4,3,2,1,8,7,6,5] => [5,6,7,8,1,2,3,4] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
[(1,5),(2,3),(4,8),(6,7)] => [5,3,2,8,1,7,6,4] => [4,6,7,1,8,2,3,5] => [[.,[.,[.,.]]],[[.,.],[.,[.,.]]]]
[(1,6),(2,3),(4,8),(5,7)] => [6,3,2,8,7,1,5,4] => [3,6,7,1,2,8,4,5] => [[.,[.,[.,.]]],[.,[[.,.],[.,.]]]]
[(1,7),(2,3),(4,8),(5,6)] => [7,3,2,8,6,5,1,4] => [2,6,7,1,3,4,8,5] => [[.,[.,[.,.]]],[.,[.,[[.,.],.]]]]
[(1,8),(2,3),(4,7),(5,6)] => [8,3,2,7,6,5,4,1] => [1,6,7,2,3,4,5,8] => [.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
[(1,8),(2,4),(3,7),(5,6)] => [8,4,7,2,6,5,3,1] => [1,5,2,7,3,4,6,8] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
[(1,7),(2,4),(3,8),(5,6)] => [7,4,8,2,6,5,1,3] => [2,5,1,7,3,4,8,6] => [[.,[.,.]],[[.,.],[.,[[.,.],.]]]]
[(1,6),(2,4),(3,8),(5,7)] => [6,4,8,2,7,1,5,3] => [3,5,1,7,2,8,4,6] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
[(1,5),(2,4),(3,8),(6,7)] => [5,4,8,2,1,7,6,3] => [4,5,1,7,8,2,3,6] => [[.,[.,.]],[[.,[.,.]],[.,[.,.]]]]
[(1,4),(2,5),(3,8),(6,7)] => [4,5,8,1,2,7,6,3] => [5,4,1,8,7,2,3,6] => [[[.,.],.],[[[.,.],.],[.,[.,.]]]]
>>> Load all 149 entries. <<<Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
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 $\tau(i) = n+1-\sigma(i)$
Map
to increasing tree
Description
Sends a permutation to its associated increasing tree.
This tree is recursively obtained by sending the unique permutation of length $0$ to the empty tree, and sending a permutation $\sigma$ of length $n \geq 1$ to a root node with two subtrees $L$ and $R$ by splitting $\sigma$ at the index $\sigma^{-1}(1)$, normalizing both sides again to permutations and sending the permutations on the left and on the right of $\sigma^{-1}(1)$ to the trees $L$ and $R$, respectively.
This tree is recursively obtained by sending the unique permutation of length $0$ to the empty tree, and sending a permutation $\sigma$ of length $n \geq 1$ to a root node with two subtrees $L$ and $R$ by splitting $\sigma$ at the index $\sigma^{-1}(1)$, normalizing both sides again to permutations and sending the permutations on the left and on the right of $\sigma^{-1}(1)$ to the trees $L$ and $R$, respectively.
searching the database
Sorry, this map was not found in the database.