Identifier
Mp00058: Perfect matchings to permutationPermutations
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00064: Permutations reversePermutations
Mp00066: Permutations inversePermutations
Mp00159: Permutations Demazure product with inversePermutations
Images
=>
Cc0012;cc-rep-0
[(1,2)]=>[2,1]=>[1,2]=>[2,1]=>[2,1]=>[2,1] [(1,2),(3,4)]=>[2,1,4,3]=>[1,2,3,4]=>[4,3,2,1]=>[4,3,2,1]=>[4,3,2,1] [(1,3),(2,4)]=>[3,4,1,2]=>[1,3,2,4]=>[4,2,3,1]=>[4,2,3,1]=>[4,3,2,1] [(1,4),(2,3)]=>[4,3,2,1]=>[1,4,2,3]=>[3,2,4,1]=>[4,2,1,3]=>[4,3,2,1] [(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[1,2,3,4,5,6]=>[6,5,4,3,2,1]=>[6,5,4,3,2,1]=>[6,5,4,3,2,1] [(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[1,3,2,4,5,6]=>[6,5,4,2,3,1]=>[6,4,5,3,2,1]=>[6,5,4,3,2,1] [(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[1,4,2,3,5,6]=>[6,5,3,2,4,1]=>[6,4,3,5,2,1]=>[6,5,4,3,2,1] [(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>[1,5,2,3,4,6]=>[6,4,3,2,5,1]=>[6,4,3,2,5,1]=>[6,5,4,3,2,1] [(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[1,6,2,3,4,5]=>[5,4,3,2,6,1]=>[6,4,3,2,1,5]=>[6,5,4,3,2,1] [(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>[1,6,2,4,3,5]=>[5,3,4,2,6,1]=>[6,4,2,3,1,5]=>[6,5,4,3,2,1] [(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>[1,5,2,4,3,6]=>[6,3,4,2,5,1]=>[6,4,2,3,5,1]=>[6,5,4,3,2,1] [(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[1,4,2,5,3,6]=>[6,3,5,2,4,1]=>[6,4,2,5,3,1]=>[6,5,4,3,2,1] [(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[1,3,2,5,4,6]=>[6,4,5,2,3,1]=>[6,4,5,2,3,1]=>[6,5,4,3,2,1] [(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[1,2,3,5,4,6]=>[6,4,5,3,2,1]=>[6,5,4,2,3,1]=>[6,5,4,3,2,1] [(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[1,2,3,6,4,5]=>[5,4,6,3,2,1]=>[6,5,4,2,1,3]=>[6,5,4,3,2,1] [(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>[1,3,2,6,4,5]=>[5,4,6,2,3,1]=>[6,4,5,2,1,3]=>[6,5,4,3,2,1] [(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>[1,4,2,6,3,5]=>[5,3,6,2,4,1]=>[6,4,2,5,1,3]=>[6,5,4,3,2,1] [(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>[1,5,2,6,3,4]=>[4,3,6,2,5,1]=>[6,4,2,1,5,3]=>[6,5,4,3,2,1] [(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[1,6,2,5,3,4]=>[4,3,5,2,6,1]=>[6,4,2,1,3,5]=>[6,5,4,3,2,1] [(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[1,2,3,4,5,6,7,8]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1] [(1,5),(2,7),(3,4),(6,8)]=>[5,7,4,3,1,8,2,6]=>[1,5,2,7,3,4,6,8]=>[8,6,4,3,7,2,5,1]=>[8,6,4,3,7,2,5,1]=>[8,7,6,5,4,3,2,1] [(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[1,3,2,4,5,7,6,8]=>[8,6,7,5,4,2,3,1]=>[8,6,7,5,4,2,3,1]=>[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
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.
Map
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
Map
inverse
Description
Sends a permutation to its inverse.
Map
Demazure product with inverse
Description
This map sends a permutation $\pi$ to $\pi^{-1} \star \pi$ where $\star$ denotes the Demazure product on permutations.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.