Identifier
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Images
=>
Cc0012;cc-rep-0Cc0010;cc-rep-3
[(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]=>[3,1,4,2]=>[[.,[.,.]],[.,.]] [(1,4),(2,3)]=>[4,3,2,1]=>[3,2,4,1]=>[[[.,.],.],[.,.]] [(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5]=>[[.,.],[[.,.],[[.,.],.]]] [(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[3,1,4,2,6,5]=>[[.,[.,.]],[.,[[.,.],.]]] [(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[3,2,4,1,6,5]=>[[[.,.],.],[.,[[.,.],.]]] [(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>[3,2,5,1,6,4]=>[[[.,.],.],[[.,.],[.,.]]] [(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[3,2,5,4,6,1]=>[[[.,.],.],[[.,.],[.,.]]] [(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>[4,2,5,3,6,1]=>[[[.,.],[.,.]],[.,[.,.]]] [(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>[4,2,5,1,6,3]=>[[[.,.],[.,.]],[.,[.,.]]] [(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[4,1,5,2,6,3]=>[[.,[.,[.,.]]],[.,[.,.]]] [(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[3,1,5,2,6,4]=>[[.,[.,.]],[[.,.],[.,.]]] [(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[2,1,5,3,6,4]=>[[.,.],[[.,[.,.]],[.,.]]] [(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[2,1,5,4,6,3]=>[[.,.],[[[.,.],.],[.,.]]] [(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>[3,1,5,4,6,2]=>[[.,[.,.]],[[.,.],[.,.]]] [(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>[4,1,5,3,6,2]=>[[.,[[.,.],.]],[.,[.,.]]] [(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>[4,3,5,1,6,2]=>[[[.,[.,.]],.],[.,[.,.]]] [(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[4,3,5,2,6,1]=>[[[[.,.],.],.],[.,[.,.]]] [(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[2,1,4,3,6,5,8,7]=>[[.,.],[[.,.],[[.,.],[[.,.],.]]]] [(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[3,1,4,2,6,5,8,7]=>[[.,[.,.]],[.,[[.,.],[[.,.],.]]]] [(1,4),(2,3),(5,6),(7,8)]=>[4,3,2,1,6,5,8,7]=>[3,2,4,1,6,5,8,7]=>[[[.,.],.],[.,[[.,.],[[.,.],.]]]] [(1,5),(2,3),(4,6),(7,8)]=>[5,3,2,6,1,4,8,7]=>[3,2,5,1,6,4,8,7]=>[[[.,.],.],[[.,.],[.,[[.,.],.]]]] [(1,6),(2,3),(4,5),(7,8)]=>[6,3,2,5,4,1,8,7]=>[3,2,5,4,6,1,8,7]=>[[[.,.],.],[[.,.],[.,[[.,.],.]]]] [(1,7),(2,3),(4,5),(6,8)]=>[7,3,2,5,4,8,1,6]=>[3,2,5,4,7,1,8,6]=>[[[.,.],.],[[.,.],[[.,.],[.,.]]]] [(1,8),(2,3),(4,5),(6,7)]=>[8,3,2,5,4,7,6,1]=>[3,2,5,4,7,6,8,1]=>[[[.,.],.],[[.,.],[[.,.],[.,.]]]] [(1,8),(2,4),(3,5),(6,7)]=>[8,4,5,2,3,7,6,1]=>[4,2,5,3,7,6,8,1]=>[[[.,.],[.,.]],[.,[[.,.],[.,.]]]] [(1,7),(2,4),(3,5),(6,8)]=>[7,4,5,2,3,8,1,6]=>[4,2,5,3,7,1,8,6]=>[[[.,.],[.,.]],[.,[[.,.],[.,.]]]] [(1,6),(2,4),(3,5),(7,8)]=>[6,4,5,2,3,1,8,7]=>[4,2,5,3,6,1,8,7]=>[[[.,.],[.,.]],[.,[.,[[.,.],.]]]] [(1,5),(2,4),(3,6),(7,8)]=>[5,4,6,2,1,3,8,7]=>[4,2,5,1,6,3,8,7]=>[[[.,.],[.,.]],[.,[.,[[.,.],.]]]] [(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[4,1,5,2,6,3,8,7]=>[[.,[.,[.,.]]],[.,[.,[[.,.],.]]]] [(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[3,1,5,2,6,4,8,7]=>[[.,[.,.]],[[.,.],[.,[[.,.],.]]]] [(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[2,1,5,3,6,4,8,7]=>[[.,.],[[.,[.,.]],[.,[[.,.],.]]]] [(1,2),(3,6),(4,5),(7,8)]=>[2,1,6,5,4,3,8,7]=>[2,1,5,4,6,3,8,7]=>[[.,.],[[[.,.],.],[.,[[.,.],.]]]] [(1,3),(2,6),(4,5),(7,8)]=>[3,6,1,5,4,2,8,7]=>[3,1,5,4,6,2,8,7]=>[[.,[.,.]],[[.,.],[.,[[.,.],.]]]] [(1,4),(2,6),(3,5),(7,8)]=>[4,6,5,1,3,2,8,7]=>[4,1,5,3,6,2,8,7]=>[[.,[[.,.],.]],[.,[.,[[.,.],.]]]] [(1,6),(2,5),(3,4),(7,8)]=>[6,5,4,3,2,1,8,7]=>[4,3,5,2,6,1,8,7]=>[[[[.,.],.],.],[.,[.,[[.,.],.]]]] [(1,8),(2,5),(3,4),(6,7)]=>[8,5,4,3,2,7,6,1]=>[4,3,5,2,7,6,8,1]=>[[[[.,.],.],.],[.,[[.,.],[.,.]]]] [(1,4),(2,7),(3,5),(6,8)]=>[4,7,5,1,3,8,2,6]=>[4,1,5,3,7,2,8,6]=>[[.,[[.,.],.]],[.,[[.,.],[.,.]]]] [(1,3),(2,7),(4,5),(6,8)]=>[3,7,1,5,4,8,2,6]=>[3,1,5,4,7,2,8,6]=>[[.,[.,.]],[[.,.],[[.,.],[.,.]]]] [(1,2),(3,7),(4,5),(6,8)]=>[2,1,7,5,4,8,3,6]=>[2,1,5,4,7,3,8,6]=>[[.,.],[[[.,.],.],[[.,.],[.,.]]]] [(1,2),(3,8),(4,5),(6,7)]=>[2,1,8,5,4,7,6,3]=>[2,1,5,4,7,6,8,3]=>[[.,.],[[[.,.],.],[[.,.],[.,.]]]] [(1,3),(2,8),(4,5),(6,7)]=>[3,8,1,5,4,7,6,2]=>[3,1,5,4,7,6,8,2]=>[[.,[.,.]],[[.,.],[[.,.],[.,.]]]] [(1,4),(2,8),(3,5),(6,7)]=>[4,8,5,1,3,7,6,2]=>[4,1,5,3,7,6,8,2]=>[[.,[[.,.],.]],[.,[[.,.],[.,.]]]] [(1,7),(2,8),(3,4),(5,6)]=>[7,8,4,3,6,5,1,2]=>[4,3,6,5,7,1,8,2]=>[[[.,[.,.]],.],[[.,.],[.,[.,.]]]] [(1,8),(2,7),(3,4),(5,6)]=>[8,7,4,3,6,5,2,1]=>[4,3,6,5,7,2,8,1]=>[[[[.,.],.],.],[[.,.],[.,[.,.]]]] [(1,7),(2,8),(3,5),(4,6)]=>[7,8,5,6,3,4,1,2]=>[5,3,6,4,7,1,8,2]=>[[[.,[.,.]],[.,.]],[.,[.,[.,.]]]] [(1,5),(2,8),(3,6),(4,7)]=>[5,8,6,7,1,3,4,2]=>[5,1,6,3,7,4,8,2]=>[[.,[[.,.],[.,.]]],[.,[.,[.,.]]]] [(1,4),(2,8),(3,6),(5,7)]=>[4,8,6,1,7,3,5,2]=>[4,1,6,3,7,5,8,2]=>[[.,[[.,.],.]],[[.,.],[.,[.,.]]]] [(1,3),(2,8),(4,6),(5,7)]=>[3,8,1,6,7,4,5,2]=>[3,1,6,4,7,5,8,2]=>[[.,[.,.]],[[.,[.,.]],[.,[.,.]]]] [(1,2),(3,8),(4,6),(5,7)]=>[2,1,8,6,7,4,5,3]=>[2,1,6,4,7,5,8,3]=>[[.,.],[[[.,.],[.,.]],[.,[.,.]]]] [(1,2),(3,7),(4,6),(5,8)]=>[2,1,7,6,8,4,3,5]=>[2,1,6,4,7,3,8,5]=>[[.,.],[[[.,.],[.,.]],[.,[.,.]]]] [(1,3),(2,7),(4,6),(5,8)]=>[3,7,1,6,8,4,2,5]=>[3,1,6,4,7,2,8,5]=>[[.,[.,.]],[[.,[.,.]],[.,[.,.]]]] [(1,4),(2,7),(3,6),(5,8)]=>[4,7,6,1,8,3,2,5]=>[4,1,6,3,7,2,8,5]=>[[.,[[.,.],.]],[[.,.],[.,[.,.]]]] [(1,5),(2,7),(3,6),(4,8)]=>[5,7,6,8,1,3,2,4]=>[5,1,6,3,7,2,8,4]=>[[.,[[.,.],[.,.]]],[.,[.,[.,.]]]] [(1,8),(2,6),(3,5),(4,7)]=>[8,6,5,7,3,2,4,1]=>[5,3,6,2,7,4,8,1]=>[[[[.,.],.],[.,.]],[.,[.,[.,.]]]] [(1,8),(2,5),(3,6),(4,7)]=>[8,5,6,7,2,3,4,1]=>[5,2,6,3,7,4,8,1]=>[[[.,.],[.,[.,.]]],[.,[.,[.,.]]]] [(1,7),(2,5),(3,6),(4,8)]=>[7,5,6,8,2,3,1,4]=>[5,2,6,3,7,1,8,4]=>[[[.,.],[.,[.,.]]],[.,[.,[.,.]]]] [(1,6),(2,5),(3,7),(4,8)]=>[6,5,7,8,2,1,3,4]=>[5,2,6,1,7,3,8,4]=>[[[.,.],[.,[.,.]]],[.,[.,[.,.]]]] [(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[5,1,6,2,7,3,8,4]=>[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]] [(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[4,1,6,2,7,3,8,5]=>[[.,[.,[.,.]]],[[.,.],[.,[.,.]]]] [(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[3,1,6,2,7,4,8,5]=>[[.,[.,.]],[[.,[.,.]],[.,[.,.]]]] [(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[2,1,6,3,7,4,8,5]=>[[.,.],[[.,[.,[.,.]]],[.,[.,.]]]] [(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[2,1,5,3,7,4,8,6]=>[[.,.],[[.,[.,.]],[[.,.],[.,.]]]] [(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[3,1,5,2,7,4,8,6]=>[[.,[.,.]],[[.,.],[[.,.],[.,.]]]] [(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[4,1,5,2,7,3,8,6]=>[[.,[.,[.,.]]],[.,[[.,.],[.,.]]]] [(1,5),(2,4),(3,7),(6,8)]=>[5,4,7,2,1,8,3,6]=>[4,2,5,1,7,3,8,6]=>[[[.,.],[.,.]],[.,[[.,.],[.,.]]]] [(1,6),(2,4),(3,7),(5,8)]=>[6,4,7,2,8,1,3,5]=>[4,2,6,1,7,3,8,5]=>[[[.,.],[.,.]],[[.,.],[.,[.,.]]]] [(1,7),(2,4),(3,6),(5,8)]=>[7,4,6,2,8,3,1,5]=>[4,2,6,3,7,1,8,5]=>[[[.,.],[.,.]],[[.,.],[.,[.,.]]]] [(1,8),(2,4),(3,6),(5,7)]=>[8,4,6,2,7,3,5,1]=>[4,2,6,3,7,5,8,1]=>[[[.,.],[.,.]],[[.,.],[.,[.,.]]]] [(1,8),(2,3),(4,6),(5,7)]=>[8,3,2,6,7,4,5,1]=>[3,2,6,4,7,5,8,1]=>[[[.,.],.],[[.,[.,.]],[.,[.,.]]]] [(1,7),(2,3),(4,6),(5,8)]=>[7,3,2,6,8,4,1,5]=>[3,2,6,4,7,1,8,5]=>[[[.,.],.],[[.,[.,.]],[.,[.,.]]]] [(1,6),(2,3),(4,7),(5,8)]=>[6,3,2,7,8,1,4,5]=>[3,2,6,1,7,4,8,5]=>[[[.,.],.],[[.,[.,.]],[.,[.,.]]]] [(1,5),(2,3),(4,7),(6,8)]=>[5,3,2,7,1,8,4,6]=>[3,2,5,1,7,4,8,6]=>[[[.,.],.],[[.,.],[[.,.],[.,.]]]] [(1,4),(2,3),(5,7),(6,8)]=>[4,3,2,1,7,8,5,6]=>[3,2,4,1,7,5,8,6]=>[[[.,.],.],[.,[[.,[.,.]],[.,.]]]] [(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[3,1,4,2,7,5,8,6]=>[[.,[.,.]],[.,[[.,[.,.]],[.,.]]]] [(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[2,1,4,3,7,5,8,6]=>[[.,.],[[.,.],[[.,[.,.]],[.,.]]]] [(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,8,7,6,5]=>[2,1,4,3,7,6,8,5]=>[[.,.],[[.,.],[[[.,.],.],[.,.]]]] [(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,8,7,6,5]=>[3,1,4,2,7,6,8,5]=>[[.,[.,.]],[.,[[[.,.],.],[.,.]]]] [(1,4),(2,3),(5,8),(6,7)]=>[4,3,2,1,8,7,6,5]=>[3,2,4,1,7,6,8,5]=>[[[.,.],.],[.,[[[.,.],.],[.,.]]]] [(1,5),(2,3),(4,8),(6,7)]=>[5,3,2,8,1,7,6,4]=>[3,2,5,1,7,6,8,4]=>[[[.,.],.],[[.,.],[[.,.],[.,.]]]] [(1,6),(2,3),(4,8),(5,7)]=>[6,3,2,8,7,1,5,4]=>[3,2,6,1,7,5,8,4]=>[[[.,.],.],[[[.,.],.],[.,[.,.]]]] [(1,8),(2,3),(4,7),(5,6)]=>[8,3,2,7,6,5,4,1]=>[3,2,6,5,7,4,8,1]=>[[[.,.],.],[[[.,.],.],[.,[.,.]]]] [(1,6),(2,4),(3,8),(5,7)]=>[6,4,8,2,7,1,5,3]=>[4,2,6,1,7,5,8,3]=>[[[.,.],[.,.]],[[.,.],[.,[.,.]]]] [(1,5),(2,4),(3,8),(6,7)]=>[5,4,8,2,1,7,6,3]=>[4,2,5,1,7,6,8,3]=>[[[.,.],[.,.]],[.,[[.,.],[.,.]]]] [(1,4),(2,5),(3,8),(6,7)]=>[4,5,8,1,2,7,6,3]=>[4,1,5,2,7,6,8,3]=>[[.,[.,[.,.]]],[.,[[.,.],[.,.]]]] [(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,8,2,7,6,4]=>[3,1,5,2,7,6,8,4]=>[[.,[.,.]],[[.,.],[[.,.],[.,.]]]] [(1,2),(3,5),(4,8),(6,7)]=>[2,1,5,8,3,7,6,4]=>[2,1,5,3,7,6,8,4]=>[[.,.],[[.,[.,.]],[[.,.],[.,.]]]] [(1,2),(3,6),(4,8),(5,7)]=>[2,1,6,8,7,3,5,4]=>[2,1,6,3,7,5,8,4]=>[[.,.],[[.,[[.,.],.]],[.,[.,.]]]] [(1,3),(2,6),(4,8),(5,7)]=>[3,6,1,8,7,2,5,4]=>[3,1,6,2,7,5,8,4]=>[[.,[.,.]],[[[.,.],.],[.,[.,.]]]] [(1,4),(2,6),(3,8),(5,7)]=>[4,6,8,1,7,2,5,3]=>[4,1,6,2,7,5,8,3]=>[[.,[.,[.,.]]],[[.,.],[.,[.,.]]]] [(1,5),(2,6),(3,8),(4,7)]=>[5,6,8,7,1,2,4,3]=>[5,1,6,2,7,4,8,3]=>[[.,[.,[[.,.],.]]],[.,[.,[.,.]]]] [(1,6),(2,5),(3,8),(4,7)]=>[6,5,8,7,2,1,4,3]=>[5,2,6,1,7,4,8,3]=>[[[.,.],[[.,.],.]],[.,[.,[.,.]]]] [(1,7),(2,6),(3,8),(4,5)]=>[7,6,8,5,4,2,1,3]=>[5,4,6,2,7,1,8,3]=>[[[[.,.],[.,.]],.],[.,[.,[.,.]]]] [(1,5),(2,7),(3,8),(4,6)]=>[5,7,8,6,1,4,2,3]=>[5,1,6,4,7,2,8,3]=>[[.,[[.,[.,.]],.]],[.,[.,[.,.]]]] [(1,3),(2,7),(4,8),(5,6)]=>[3,7,1,8,6,5,2,4]=>[3,1,6,5,7,2,8,4]=>[[.,[.,.]],[[[.,.],.],[.,[.,.]]]] [(1,2),(3,8),(4,7),(5,6)]=>[2,1,8,7,6,5,4,3]=>[2,1,6,5,7,4,8,3]=>[[.,.],[[[[.,.],.],.],[.,[.,.]]]] [(1,3),(2,8),(4,7),(5,6)]=>[3,8,1,7,6,5,4,2]=>[3,1,6,5,7,4,8,2]=>[[.,[.,.]],[[[.,.],.],[.,[.,.]]]] [(1,4),(2,8),(3,7),(5,6)]=>[4,8,7,1,6,5,3,2]=>[4,1,6,5,7,3,8,2]=>[[.,[[.,.],.]],[[.,.],[.,[.,.]]]] [(1,5),(2,8),(3,7),(4,6)]=>[5,8,7,6,1,4,3,2]=>[5,1,6,4,7,3,8,2]=>[[.,[[[.,.],.],.]],[.,[.,[.,.]]]] [(1,8),(2,7),(3,6),(4,5)]=>[8,7,6,5,4,3,2,1]=>[5,4,6,3,7,2,8,1]=>[[[[[.,.],.],.],.],[.,[.,[.,.]]]] [(1,2),(3,4),(5,6),(7,8),(9,10)]=>[2,1,4,3,6,5,8,7,10,9]=>[2,1,4,3,6,5,8,7,10,9]=>[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],.]]]]] [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]=>[2,1,4,3,6,5,8,7,10,9,12,11]=>[2,1,4,3,6,5,8,7,10,9,12,11]=>[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],.]]]]]]
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
inverse first fundamental transformation
Description
Let $\sigma = (i_{11}\cdots i_{1k_1})\cdots(i_{\ell 1}\cdots i_{\ell k_\ell})$ be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.
Maps $\sigma$ to the permutation $[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.
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.