Identifier
Mp00058: Perfect matchings to permutationPermutations
Mp00086: Permutations first fundamental transformationPermutations
Mp00151: Permutations to cycle type Set partitions
Images
=>
Cc0012;cc-rep-0Cc0009;cc-rep-3
[(1,2)]=>[2,1]=>[2,1]=>{{1,2}} [(1,2),(3,4)]=>[2,1,4,3]=>[2,1,4,3]=>{{1,2},{3,4}} [(1,3),(2,4)]=>[3,4,1,2]=>[2,4,3,1]=>{{1,2,4},{3}} [(1,4),(2,3)]=>[4,3,2,1]=>[4,1,2,3]=>{{1,2,3,4}} [(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5]=>{{1,2},{3,4},{5,6}} [(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[2,4,3,1,6,5]=>{{1,2,4},{3},{5,6}} [(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[4,1,2,3,6,5]=>{{1,2,3,4},{5,6}} [(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>[4,5,2,6,3,1]=>{{1,4,6},{2,3,5}} [(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[6,5,2,1,4,3]=>{{1,2,3,4,5,6}} [(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>[6,3,1,5,2,4]=>{{1,2,3,4,5,6}} [(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>[3,1,6,5,4,2]=>{{1,2,3,6},{4,5}} [(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[2,3,6,4,5,1]=>{{1,2,3,6},{4},{5}} [(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[5,4,3,6,1,2]=>{{1,5},{2,4,6},{3}} [(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[2,1,4,6,5,3]=>{{1,2},{3,4,6},{5}} [(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[2,1,6,3,4,5]=>{{1,2},{3,4,5,6}} [(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>[5,6,3,2,4,1]=>{{1,2,4,5,6},{3}} [(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>[3,6,2,4,1,5]=>{{1,2,3,5,6},{4}} [(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>[2,6,1,3,5,4]=>{{1,2,3,4,6},{5}} [(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[6,1,2,3,4,5]=>{{1,2,3,4,5,6}} [(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[2,1,4,3,6,5,8,7]=>{{1,2},{3,4},{5,6},{7,8}} [(1,4),(2,3),(5,8),(6,7)]=>[4,3,2,1,8,7,6,5]=>[4,1,2,3,8,5,6,7]=>{{1,2,3,4},{5,6,7,8}}
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
first fundamental transformation
Description
Return the permutation whose cycles are the subsequences between successive left to right maxima.
Map
to cycle type
Description
Let $\pi=c_1\dots c_r$ a permutation of size $n$ decomposed in its cyclic parts. The associated set partition of $[n]$ then is $S=S_1\cup\dots\cup S_r$ such that $S_i$ is the set of integers in the cycle $c_i$.
A permutation is cyclic [1] if and only if its cycle type is a hook partition [2].