Identifier
Mp00058: Perfect matchings to permutationPermutations
Mp00061: Permutations to increasing tree Binary trees
Images
=>
Cc0012;cc-rep-0Cc0010;cc-rep-2
[(1,2)]=>[2,1]=>[[.,.],.] [(1,2),(3,4)]=>[2,1,4,3]=>[[.,.],[[.,.],.]] [(1,3),(2,4)]=>[3,4,1,2]=>[[.,[.,.]],[.,.]] [(1,4),(2,3)]=>[4,3,2,1]=>[[[[.,.],.],.],.] [(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[[.,.],[[.,.],[[.,.],.]]] [(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[[.,[.,.]],[.,[[.,.],.]]] [(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[[[[.,.],.],.],[[.,.],.]] [(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>[[[[.,.],.],[.,.]],[.,.]] [(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[[[[.,.],.],[[.,.],.]],.] [(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>[[[[.,.],[.,.]],[.,.]],.] [(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>[[[[.,.],[.,.]],.],[.,.]] [(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[[.,[.,[.,.]]],[.,[.,.]]] [(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[[.,[.,.]],[[.,.],[.,.]]] [(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[[.,.],[[.,[.,.]],[.,.]]] [(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[[.,.],[[[[.,.],.],.],.]] [(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>[[.,[.,.]],[[[.,.],.],.]] [(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>[[.,[[.,.],.]],[[.,.],.]] [(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>[[[[.,[.,.]],.],.],[.,.]] [(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[[[[[[.,.],.],.],.],.],.] [(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[[.,.],[[.,.],[[.,.],[[.,.],.]]]] [(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[[.,[.,.]],[.,[[.,.],[[.,.],.]]]] [(1,4),(2,3),(5,6),(7,8)]=>[4,3,2,1,6,5,8,7]=>[[[[.,.],.],.],[[.,.],[[.,.],.]]] [(1,5),(2,3),(4,6),(7,8)]=>[5,3,2,6,1,4,8,7]=>[[[[.,.],.],[.,.]],[.,[[.,.],.]]] [(1,6),(2,3),(4,5),(7,8)]=>[6,3,2,5,4,1,8,7]=>[[[[.,.],.],[[.,.],.]],[[.,.],.]] [(1,7),(2,3),(4,5),(6,8)]=>[7,3,2,5,4,8,1,6]=>[[[[.,.],.],[[.,.],[.,.]]],[.,.]] [(1,8),(2,3),(4,5),(6,7)]=>[8,3,2,5,4,7,6,1]=>[[[[.,.],.],[[.,.],[[.,.],.]]],.] [(1,8),(2,4),(3,5),(6,7)]=>[8,4,5,2,3,7,6,1]=>[[[[.,.],[.,.]],[.,[[.,.],.]]],.] [(1,7),(2,4),(3,5),(6,8)]=>[7,4,5,2,3,8,1,6]=>[[[[.,.],[.,.]],[.,[.,.]]],[.,.]] [(1,6),(2,4),(3,5),(7,8)]=>[6,4,5,2,3,1,8,7]=>[[[[.,.],[.,.]],[.,.]],[[.,.],.]] [(1,5),(2,4),(3,6),(7,8)]=>[5,4,6,2,1,3,8,7]=>[[[[.,.],[.,.]],.],[.,[[.,.],.]]] [(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[[.,[.,[.,.]]],[.,[.,[[.,.],.]]]] [(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[[.,[.,.]],[[.,.],[.,[[.,.],.]]]] [(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[[.,.],[[.,[.,.]],[.,[[.,.],.]]]] [(1,2),(3,6),(4,5),(7,8)]=>[2,1,6,5,4,3,8,7]=>[[.,.],[[[[.,.],.],.],[[.,.],.]]] [(1,3),(2,6),(4,5),(7,8)]=>[3,6,1,5,4,2,8,7]=>[[.,[.,.]],[[[.,.],.],[[.,.],.]]] [(1,4),(2,6),(3,5),(7,8)]=>[4,6,5,1,3,2,8,7]=>[[.,[[.,.],.]],[[.,.],[[.,.],.]]] [(1,5),(2,6),(3,4),(7,8)]=>[5,6,4,3,1,2,8,7]=>[[[[.,[.,.]],.],.],[.,[[.,.],.]]] [(1,6),(2,5),(3,4),(7,8)]=>[6,5,4,3,2,1,8,7]=>[[[[[[.,.],.],.],.],.],[[.,.],.]] [(1,7),(2,5),(3,4),(6,8)]=>[7,5,4,3,2,8,1,6]=>[[[[[[.,.],.],.],.],[.,.]],[.,.]] [(1,8),(2,5),(3,4),(6,7)]=>[8,5,4,3,2,7,6,1]=>[[[[[[.,.],.],.],.],[[.,.],.]],.] [(1,8),(2,6),(3,4),(5,7)]=>[8,6,4,3,7,2,5,1]=>[[[[[[.,.],.],.],[.,.]],[.,.]],.] [(1,7),(2,6),(3,4),(5,8)]=>[7,6,4,3,8,2,1,5]=>[[[[[[.,.],.],.],[.,.]],.],[.,.]] [(1,6),(2,7),(3,4),(5,8)]=>[6,7,4,3,8,1,2,5]=>[[[[.,[.,.]],.],[.,.]],[.,[.,.]]] [(1,5),(2,7),(3,4),(6,8)]=>[5,7,4,3,1,8,2,6]=>[[[[.,[.,.]],.],.],[[.,.],[.,.]]] [(1,4),(2,7),(3,5),(6,8)]=>[4,7,5,1,3,8,2,6]=>[[.,[[.,.],.]],[[.,[.,.]],[.,.]]] [(1,3),(2,7),(4,5),(6,8)]=>[3,7,1,5,4,8,2,6]=>[[.,[.,.]],[[[.,.],[.,.]],[.,.]]] [(1,2),(3,7),(4,5),(6,8)]=>[2,1,7,5,4,8,3,6]=>[[.,.],[[[[.,.],.],[.,.]],[.,.]]] [(1,2),(3,8),(4,5),(6,7)]=>[2,1,8,5,4,7,6,3]=>[[.,.],[[[[.,.],.],[[.,.],.]],.]] [(1,3),(2,8),(4,5),(6,7)]=>[3,8,1,5,4,7,6,2]=>[[.,[.,.]],[[[.,.],[[.,.],.]],.]] [(1,4),(2,8),(3,5),(6,7)]=>[4,8,5,1,3,7,6,2]=>[[.,[[.,.],.]],[[.,[[.,.],.]],.]] [(1,5),(2,8),(3,4),(6,7)]=>[5,8,4,3,1,7,6,2]=>[[[[.,[.,.]],.],.],[[[.,.],.],.]] [(1,6),(2,8),(3,4),(5,7)]=>[6,8,4,3,7,1,5,2]=>[[[[.,[.,.]],.],[.,.]],[[.,.],.]] [(1,7),(2,8),(3,4),(5,6)]=>[7,8,4,3,6,5,1,2]=>[[[[.,[.,.]],.],[[.,.],.]],[.,.]] [(1,8),(2,7),(3,4),(5,6)]=>[8,7,4,3,6,5,2,1]=>[[[[[[.,.],.],.],[[.,.],.]],.],.] [(1,8),(2,7),(3,5),(4,6)]=>[8,7,5,6,3,4,2,1]=>[[[[[[.,.],.],[.,.]],[.,.]],.],.] [(1,7),(2,8),(3,5),(4,6)]=>[7,8,5,6,3,4,1,2]=>[[[[.,[.,.]],[.,.]],[.,.]],[.,.]] [(1,6),(2,8),(3,5),(4,7)]=>[6,8,5,7,3,1,4,2]=>[[[[.,[.,.]],[.,.]],.],[[.,.],.]] [(1,5),(2,8),(3,6),(4,7)]=>[5,8,6,7,1,3,4,2]=>[[.,[[.,.],[.,.]]],[[.,[.,.]],.]] [(1,4),(2,8),(3,6),(5,7)]=>[4,8,6,1,7,3,5,2]=>[[.,[[.,.],.]],[[[.,.],[.,.]],.]] [(1,3),(2,8),(4,6),(5,7)]=>[3,8,1,6,7,4,5,2]=>[[.,[.,.]],[[[.,[.,.]],[.,.]],.]] [(1,2),(3,8),(4,6),(5,7)]=>[2,1,8,6,7,4,5,3]=>[[.,.],[[[[.,.],[.,.]],[.,.]],.]] [(1,2),(3,7),(4,6),(5,8)]=>[2,1,7,6,8,4,3,5]=>[[.,.],[[[[.,.],[.,.]],.],[.,.]]] [(1,3),(2,7),(4,6),(5,8)]=>[3,7,1,6,8,4,2,5]=>[[.,[.,.]],[[[.,[.,.]],.],[.,.]]] [(1,4),(2,7),(3,6),(5,8)]=>[4,7,6,1,8,3,2,5]=>[[.,[[.,.],.]],[[[.,.],.],[.,.]]] [(1,5),(2,7),(3,6),(4,8)]=>[5,7,6,8,1,3,2,4]=>[[.,[[.,.],[.,.]]],[[.,.],[.,.]]] [(1,6),(2,7),(3,5),(4,8)]=>[6,7,5,8,3,1,2,4]=>[[[[.,[.,.]],[.,.]],.],[.,[.,.]]] [(1,7),(2,6),(3,5),(4,8)]=>[7,6,5,8,3,2,1,4]=>[[[[[[.,.],.],[.,.]],.],.],[.,.]] [(1,8),(2,6),(3,5),(4,7)]=>[8,6,5,7,3,2,4,1]=>[[[[[[.,.],.],[.,.]],.],[.,.]],.] [(1,8),(2,5),(3,6),(4,7)]=>[8,5,6,7,2,3,4,1]=>[[[[.,.],[.,[.,.]]],[.,[.,.]]],.] [(1,7),(2,5),(3,6),(4,8)]=>[7,5,6,8,2,3,1,4]=>[[[[.,.],[.,[.,.]]],[.,.]],[.,.]] [(1,6),(2,5),(3,7),(4,8)]=>[6,5,7,8,2,1,3,4]=>[[[[.,.],[.,[.,.]]],.],[.,[.,.]]] [(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]] [(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[[.,[.,[.,.]]],[[.,.],[.,[.,.]]]] [(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[[.,[.,.]],[[.,[.,.]],[.,[.,.]]]] [(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[[.,.],[[.,[.,[.,.]]],[.,[.,.]]]] [(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[[.,.],[[.,[.,.]],[[.,.],[.,.]]]] [(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[[.,[.,.]],[[.,.],[[.,.],[.,.]]]] [(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[[.,[.,[.,.]]],[.,[[.,.],[.,.]]]] [(1,5),(2,4),(3,7),(6,8)]=>[5,4,7,2,1,8,3,6]=>[[[[.,.],[.,.]],.],[[.,.],[.,.]]] [(1,6),(2,4),(3,7),(5,8)]=>[6,4,7,2,8,1,3,5]=>[[[[.,.],[.,.]],[.,.]],[.,[.,.]]] [(1,7),(2,4),(3,6),(5,8)]=>[7,4,6,2,8,3,1,5]=>[[[[.,.],[.,.]],[[.,.],.]],[.,.]] [(1,8),(2,4),(3,6),(5,7)]=>[8,4,6,2,7,3,5,1]=>[[[[.,.],[.,.]],[[.,.],[.,.]]],.] [(1,8),(2,3),(4,6),(5,7)]=>[8,3,2,6,7,4,5,1]=>[[[[.,.],.],[[.,[.,.]],[.,.]]],.] [(1,7),(2,3),(4,6),(5,8)]=>[7,3,2,6,8,4,1,5]=>[[[[.,.],.],[[.,[.,.]],.]],[.,.]] [(1,6),(2,3),(4,7),(5,8)]=>[6,3,2,7,8,1,4,5]=>[[[[.,.],.],[.,[.,.]]],[.,[.,.]]] [(1,5),(2,3),(4,7),(6,8)]=>[5,3,2,7,1,8,4,6]=>[[[[.,.],.],[.,.]],[[.,.],[.,.]]] [(1,4),(2,3),(5,7),(6,8)]=>[4,3,2,1,7,8,5,6]=>[[[[.,.],.],.],[[.,[.,.]],[.,.]]] [(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[[.,[.,.]],[.,[[.,[.,.]],[.,.]]]] [(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[[.,.],[[.,.],[[.,[.,.]],[.,.]]]] [(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,8,7,6,5]=>[[.,.],[[.,.],[[[[.,.],.],.],.]]] [(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,8,7,6,5]=>[[.,[.,.]],[.,[[[[.,.],.],.],.]]] [(1,4),(2,3),(5,8),(6,7)]=>[4,3,2,1,8,7,6,5]=>[[[[.,.],.],.],[[[[.,.],.],.],.]] [(1,5),(2,3),(4,8),(6,7)]=>[5,3,2,8,1,7,6,4]=>[[[[.,.],.],[.,.]],[[[.,.],.],.]] [(1,6),(2,3),(4,8),(5,7)]=>[6,3,2,8,7,1,5,4]=>[[[[.,.],.],[[.,.],.]],[[.,.],.]] [(1,7),(2,3),(4,8),(5,6)]=>[7,3,2,8,6,5,1,4]=>[[[[.,.],.],[[[.,.],.],.]],[.,.]] [(1,8),(2,3),(4,7),(5,6)]=>[8,3,2,7,6,5,4,1]=>[[[[.,.],.],[[[[.,.],.],.],.]],.] [(1,8),(2,4),(3,7),(5,6)]=>[8,4,7,2,6,5,3,1]=>[[[[.,.],[.,.]],[[[.,.],.],.]],.] [(1,7),(2,4),(3,8),(5,6)]=>[7,4,8,2,6,5,1,3]=>[[[[.,.],[.,.]],[[.,.],.]],[.,.]] [(1,6),(2,4),(3,8),(5,7)]=>[6,4,8,2,7,1,5,3]=>[[[[.,.],[.,.]],[.,.]],[[.,.],.]] [(1,5),(2,4),(3,8),(6,7)]=>[5,4,8,2,1,7,6,3]=>[[[[.,.],[.,.]],.],[[[.,.],.],.]] [(1,4),(2,5),(3,8),(6,7)]=>[4,5,8,1,2,7,6,3]=>[[.,[.,[.,.]]],[.,[[[.,.],.],.]]] [(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,8,2,7,6,4]=>[[.,[.,.]],[[.,.],[[[.,.],.],.]]] [(1,2),(3,5),(4,8),(6,7)]=>[2,1,5,8,3,7,6,4]=>[[.,.],[[.,[.,.]],[[[.,.],.],.]]] [(1,2),(3,6),(4,8),(5,7)]=>[2,1,6,8,7,3,5,4]=>[[.,.],[[.,[[.,.],.]],[[.,.],.]]] [(1,3),(2,6),(4,8),(5,7)]=>[3,6,1,8,7,2,5,4]=>[[.,[.,.]],[[[.,.],.],[[.,.],.]]] [(1,4),(2,6),(3,8),(5,7)]=>[4,6,8,1,7,2,5,3]=>[[.,[.,[.,.]]],[[.,.],[[.,.],.]]] [(1,5),(2,6),(3,8),(4,7)]=>[5,6,8,7,1,2,4,3]=>[[.,[.,[[.,.],.]]],[.,[[.,.],.]]] [(1,6),(2,5),(3,8),(4,7)]=>[6,5,8,7,2,1,4,3]=>[[[[.,.],[[.,.],.]],.],[[.,.],.]] [(1,7),(2,5),(3,8),(4,6)]=>[7,5,8,6,2,4,1,3]=>[[[[.,.],[[.,.],.]],[.,.]],[.,.]] [(1,8),(2,5),(3,7),(4,6)]=>[8,5,7,6,2,4,3,1]=>[[[[.,.],[[.,.],.]],[[.,.],.]],.] [(1,8),(2,6),(3,7),(4,5)]=>[8,6,7,5,4,2,3,1]=>[[[[[[.,.],[.,.]],.],.],[.,.]],.] [(1,7),(2,6),(3,8),(4,5)]=>[7,6,8,5,4,2,1,3]=>[[[[[[.,.],[.,.]],.],.],.],[.,.]] [(1,6),(2,7),(3,8),(4,5)]=>[6,7,8,5,4,1,2,3]=>[[[[.,[.,[.,.]]],.],.],[.,[.,.]]] [(1,5),(2,7),(3,8),(4,6)]=>[5,7,8,6,1,4,2,3]=>[[.,[[.,[.,.]],.]],[[.,.],[.,.]]] [(1,4),(2,7),(3,8),(5,6)]=>[4,7,8,1,6,5,2,3]=>[[.,[.,[.,.]]],[[[.,.],.],[.,.]]] [(1,3),(2,7),(4,8),(5,6)]=>[3,7,1,8,6,5,2,4]=>[[.,[.,.]],[[[[.,.],.],.],[.,.]]] [(1,2),(3,7),(4,8),(5,6)]=>[2,1,7,8,6,5,3,4]=>[[.,.],[[[[.,[.,.]],.],.],[.,.]]] [(1,2),(3,8),(4,7),(5,6)]=>[2,1,8,7,6,5,4,3]=>[[.,.],[[[[[[.,.],.],.],.],.],.]] [(1,3),(2,8),(4,7),(5,6)]=>[3,8,1,7,6,5,4,2]=>[[.,[.,.]],[[[[[.,.],.],.],.],.]] [(1,4),(2,8),(3,7),(5,6)]=>[4,8,7,1,6,5,3,2]=>[[.,[[.,.],.]],[[[[.,.],.],.],.]] [(1,5),(2,8),(3,7),(4,6)]=>[5,8,7,6,1,4,3,2]=>[[.,[[[.,.],.],.]],[[[.,.],.],.]] [(1,6),(2,8),(3,7),(4,5)]=>[6,8,7,5,4,1,3,2]=>[[[[.,[[.,.],.]],.],.],[[.,.],.]] [(1,7),(2,8),(3,6),(4,5)]=>[7,8,6,5,4,3,1,2]=>[[[[[[.,[.,.]],.],.],.],.],[.,.]] [(1,8),(2,7),(3,6),(4,5)]=>[8,7,6,5,4,3,2,1]=>[[[[[[[[.,.],.],.],.],.],.],.],.]
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
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.