- FindStat

Identifier

Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
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].