Identifier
Mp00116:
Perfect matchings
—Kasraoui-Zeng⟶
Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Images
=>
Cc0012;cc-rep-0Cc0012;cc-rep-1
[(1,2)]=>[(1,2)]=>[2,1]=>[2,1]
[(1,2),(3,4)]=>[(1,2),(3,4)]=>[2,1,4,3]=>[2,1,4,3]
[(1,3),(2,4)]=>[(1,4),(2,3)]=>[4,3,2,1]=>[3,2,4,1]
[(1,4),(2,3)]=>[(1,3),(2,4)]=>[3,4,1,2]=>[3,1,4,2]
[(1,2),(3,4),(5,6)]=>[(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5]
[(1,3),(2,4),(5,6)]=>[(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[3,2,4,1,6,5]
[(1,4),(2,3),(5,6)]=>[(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[3,1,4,2,6,5]
[(1,5),(2,3),(4,6)]=>[(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>[3,1,5,4,6,2]
[(1,6),(2,3),(4,5)]=>[(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[3,1,5,2,6,4]
[(1,6),(2,4),(3,5)]=>[(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>[4,2,5,1,6,3]
[(1,5),(2,4),(3,6)]=>[(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>[4,2,5,3,6,1]
[(1,4),(2,5),(3,6)]=>[(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[4,3,5,2,6,1]
[(1,3),(2,5),(4,6)]=>[(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[3,2,5,4,6,1]
[(1,2),(3,5),(4,6)]=>[(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[2,1,5,4,6,3]
[(1,2),(3,6),(4,5)]=>[(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[2,1,5,3,6,4]
[(1,3),(2,6),(4,5)]=>[(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>[3,2,5,1,6,4]
[(1,4),(2,6),(3,5)]=>[(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>[4,3,5,1,6,2]
[(1,5),(2,6),(3,4)]=>[(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>[4,1,5,3,6,2]
[(1,6),(2,5),(3,4)]=>[(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[4,1,5,2,6,3]
[(1,2),(3,4),(5,6),(7,8)]=>[(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)]=>[(1,4),(2,3),(5,6),(7,8)]=>[4,3,2,1,6,5,8,7]=>[3,2,4,1,6,5,8,7]
[(1,4),(2,3),(5,6),(7,8)]=>[(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[3,1,4,2,6,5,8,7]
[(1,5),(2,3),(4,6),(7,8)]=>[(1,3),(2,6),(4,5),(7,8)]=>[3,6,1,5,4,2,8,7]=>[3,1,5,4,6,2,8,7]
[(1,6),(2,3),(4,5),(7,8)]=>[(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[3,1,5,2,6,4,8,7]
[(1,7),(2,3),(4,5),(6,8)]=>[(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,8,2,7,6,4]=>[3,1,5,2,7,6,8,4]
[(1,8),(2,3),(4,5),(6,7)]=>[(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[3,1,5,2,7,4,8,6]
[(1,8),(2,4),(3,5),(6,7)]=>[(1,5),(2,4),(3,7),(6,8)]=>[5,4,7,2,1,8,3,6]=>[4,2,5,1,7,3,8,6]
[(1,7),(2,4),(3,5),(6,8)]=>[(1,5),(2,4),(3,8),(6,7)]=>[5,4,8,2,1,7,6,3]=>[4,2,5,1,7,6,8,3]
[(1,6),(2,4),(3,5),(7,8)]=>[(1,5),(2,4),(3,6),(7,8)]=>[5,4,6,2,1,3,8,7]=>[4,2,5,1,6,3,8,7]
[(1,5),(2,4),(3,6),(7,8)]=>[(1,6),(2,4),(3,5),(7,8)]=>[6,4,5,2,3,1,8,7]=>[4,2,5,3,6,1,8,7]
[(1,4),(2,5),(3,6),(7,8)]=>[(1,6),(2,5),(3,4),(7,8)]=>[6,5,4,3,2,1,8,7]=>[4,3,5,2,6,1,8,7]
[(1,3),(2,5),(4,6),(7,8)]=>[(1,6),(2,3),(4,5),(7,8)]=>[6,3,2,5,4,1,8,7]=>[3,2,5,4,6,1,8,7]
[(1,2),(3,5),(4,6),(7,8)]=>[(1,2),(3,6),(4,5),(7,8)]=>[2,1,6,5,4,3,8,7]=>[2,1,5,4,6,3,8,7]
[(1,2),(3,6),(4,5),(7,8)]=>[(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[2,1,5,3,6,4,8,7]
[(1,3),(2,6),(4,5),(7,8)]=>[(1,5),(2,3),(4,6),(7,8)]=>[5,3,2,6,1,4,8,7]=>[3,2,5,1,6,4,8,7]
[(1,4),(2,6),(3,5),(7,8)]=>[(1,5),(2,6),(3,4),(7,8)]=>[5,6,4,3,1,2,8,7]=>[4,3,5,1,6,2,8,7]
[(1,5),(2,6),(3,4),(7,8)]=>[(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)]=>[(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[4,1,5,2,6,3,8,7]
[(1,7),(2,5),(3,4),(6,8)]=>[(1,4),(2,5),(3,8),(6,7)]=>[4,5,8,1,2,7,6,3]=>[4,1,5,2,7,6,8,3]
[(1,8),(2,5),(3,4),(6,7)]=>[(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[4,1,5,2,7,3,8,6]
[(1,8),(2,6),(3,4),(5,7)]=>[(1,4),(2,7),(3,6),(5,8)]=>[4,7,6,1,8,3,2,5]=>[4,1,6,3,7,2,8,5]
[(1,7),(2,6),(3,4),(5,8)]=>[(1,4),(2,8),(3,6),(5,7)]=>[4,8,6,1,7,3,5,2]=>[4,1,6,3,7,5,8,2]
[(1,6),(2,7),(3,4),(5,8)]=>[(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),(2,7),(3,4),(6,8)]=>[(1,4),(2,8),(3,5),(6,7)]=>[4,8,5,1,3,7,6,2]=>[4,1,5,3,7,6,8,2]
[(1,4),(2,7),(3,5),(6,8)]=>[(1,5),(2,8),(3,4),(6,7)]=>[5,8,4,3,1,7,6,2]=>[4,3,5,1,7,6,8,2]
[(1,3),(2,7),(4,5),(6,8)]=>[(1,5),(2,3),(4,8),(6,7)]=>[5,3,2,8,1,7,6,4]=>[3,2,5,1,7,6,8,4]
[(1,2),(3,7),(4,5),(6,8)]=>[(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),(3,8),(4,5),(6,7)]=>[(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),(2,8),(4,5),(6,7)]=>[(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),(2,8),(3,5),(6,7)]=>[(1,5),(2,7),(3,4),(6,8)]=>[5,7,4,3,1,8,2,6]=>[4,3,5,1,7,2,8,6]
[(1,5),(2,8),(3,4),(6,7)]=>[(1,4),(2,7),(3,5),(6,8)]=>[4,7,5,1,3,8,2,6]=>[4,1,5,3,7,2,8,6]
[(1,6),(2,8),(3,4),(5,7)]=>[(1,4),(2,7),(3,8),(5,6)]=>[4,7,8,1,6,5,2,3]=>[4,1,6,5,7,2,8,3]
[(1,7),(2,8),(3,4),(5,6)]=>[(1,4),(2,6),(3,8),(5,7)]=>[4,6,8,1,7,2,5,3]=>[4,1,6,2,7,5,8,3]
[(1,8),(2,7),(3,4),(5,6)]=>[(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[4,1,6,2,7,3,8,5]
[(1,8),(2,7),(3,5),(4,6)]=>[(1,6),(2,5),(3,7),(4,8)]=>[6,5,7,8,2,1,3,4]=>[5,2,6,1,7,3,8,4]
[(1,7),(2,8),(3,5),(4,6)]=>[(1,6),(2,5),(3,8),(4,7)]=>[6,5,8,7,2,1,4,3]=>[5,2,6,1,7,4,8,3]
[(1,6),(2,8),(3,5),(4,7)]=>[(1,7),(2,5),(3,8),(4,6)]=>[7,5,8,6,2,4,1,3]=>[5,2,6,4,7,1,8,3]
[(1,5),(2,8),(3,6),(4,7)]=>[(1,7),(2,6),(3,8),(4,5)]=>[7,6,8,5,4,2,1,3]=>[5,4,6,2,7,1,8,3]
[(1,4),(2,8),(3,6),(5,7)]=>[(1,7),(2,6),(3,4),(5,8)]=>[7,6,4,3,8,2,1,5]=>[4,3,6,2,7,1,8,5]
[(1,3),(2,8),(4,6),(5,7)]=>[(1,7),(2,3),(4,6),(5,8)]=>[7,3,2,6,8,4,1,5]=>[3,2,6,4,7,1,8,5]
[(1,2),(3,8),(4,6),(5,7)]=>[(1,2),(3,7),(4,6),(5,8)]=>[2,1,7,6,8,4,3,5]=>[2,1,6,4,7,3,8,5]
[(1,2),(3,7),(4,6),(5,8)]=>[(1,2),(3,8),(4,6),(5,7)]=>[2,1,8,6,7,4,5,3]=>[2,1,6,4,7,5,8,3]
[(1,3),(2,7),(4,6),(5,8)]=>[(1,8),(2,3),(4,6),(5,7)]=>[8,3,2,6,7,4,5,1]=>[3,2,6,4,7,5,8,1]
[(1,4),(2,7),(3,6),(5,8)]=>[(1,8),(2,6),(3,4),(5,7)]=>[8,6,4,3,7,2,5,1]=>[4,3,6,2,7,5,8,1]
[(1,5),(2,7),(3,6),(4,8)]=>[(1,8),(2,6),(3,7),(4,5)]=>[8,6,7,5,4,2,3,1]=>[5,4,6,2,7,3,8,1]
[(1,6),(2,7),(3,5),(4,8)]=>[(1,8),(2,5),(3,7),(4,6)]=>[8,5,7,6,2,4,3,1]=>[5,2,6,4,7,3,8,1]
[(1,7),(2,6),(3,5),(4,8)]=>[(1,8),(2,5),(3,6),(4,7)]=>[8,5,6,7,2,3,4,1]=>[5,2,6,3,7,4,8,1]
[(1,8),(2,6),(3,5),(4,7)]=>[(1,7),(2,5),(3,6),(4,8)]=>[7,5,6,8,2,3,1,4]=>[5,2,6,3,7,1,8,4]
[(1,8),(2,5),(3,6),(4,7)]=>[(1,7),(2,6),(3,5),(4,8)]=>[7,6,5,8,3,2,1,4]=>[5,3,6,2,7,1,8,4]
[(1,7),(2,5),(3,6),(4,8)]=>[(1,8),(2,6),(3,5),(4,7)]=>[8,6,5,7,3,2,4,1]=>[5,3,6,2,7,4,8,1]
[(1,6),(2,5),(3,7),(4,8)]=>[(1,8),(2,7),(3,5),(4,6)]=>[8,7,5,6,3,4,2,1]=>[5,3,6,4,7,2,8,1]
[(1,5),(2,6),(3,7),(4,8)]=>[(1,8),(2,7),(3,6),(4,5)]=>[8,7,6,5,4,3,2,1]=>[5,4,6,3,7,2,8,1]
[(1,4),(2,6),(3,7),(5,8)]=>[(1,8),(2,7),(3,4),(5,6)]=>[8,7,4,3,6,5,2,1]=>[4,3,6,5,7,2,8,1]
[(1,3),(2,6),(4,7),(5,8)]=>[(1,8),(2,3),(4,7),(5,6)]=>[8,3,2,7,6,5,4,1]=>[3,2,6,5,7,4,8,1]
[(1,2),(3,6),(4,7),(5,8)]=>[(1,2),(3,8),(4,7),(5,6)]=>[2,1,8,7,6,5,4,3]=>[2,1,6,5,7,4,8,3]
[(1,2),(3,5),(4,7),(6,8)]=>[(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),(2,5),(4,7),(6,8)]=>[(1,8),(2,3),(4,5),(6,7)]=>[8,3,2,5,4,7,6,1]=>[3,2,5,4,7,6,8,1]
[(1,4),(2,5),(3,7),(6,8)]=>[(1,8),(2,5),(3,4),(6,7)]=>[8,5,4,3,2,7,6,1]=>[4,3,5,2,7,6,8,1]
[(1,5),(2,4),(3,7),(6,8)]=>[(1,8),(2,4),(3,5),(6,7)]=>[8,4,5,2,3,7,6,1]=>[4,2,5,3,7,6,8,1]
[(1,6),(2,4),(3,7),(5,8)]=>[(1,8),(2,4),(3,7),(5,6)]=>[8,4,7,2,6,5,3,1]=>[4,2,6,5,7,3,8,1]
[(1,7),(2,4),(3,6),(5,8)]=>[(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),(2,4),(3,6),(5,7)]=>[(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),(2,3),(4,6),(5,7)]=>[(1,3),(2,7),(4,6),(5,8)]=>[3,7,1,6,8,4,2,5]=>[3,1,6,4,7,2,8,5]
[(1,7),(2,3),(4,6),(5,8)]=>[(1,3),(2,8),(4,6),(5,7)]=>[3,8,1,6,7,4,5,2]=>[3,1,6,4,7,5,8,2]
[(1,6),(2,3),(4,7),(5,8)]=>[(1,3),(2,8),(4,7),(5,6)]=>[3,8,1,7,6,5,4,2]=>[3,1,6,5,7,4,8,2]
[(1,5),(2,3),(4,7),(6,8)]=>[(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),(2,3),(5,7),(6,8)]=>[(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,8,7,6,5]=>[3,1,4,2,7,6,8,5]
[(1,3),(2,4),(5,7),(6,8)]=>[(1,4),(2,3),(5,8),(6,7)]=>[4,3,2,1,8,7,6,5]=>[3,2,4,1,7,6,8,5]
[(1,2),(3,4),(5,7),(6,8)]=>[(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,8,7,6,5]=>[2,1,4,3,7,6,8,5]
[(1,2),(3,4),(5,8),(6,7)]=>[(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[2,1,4,3,7,5,8,6]
[(1,3),(2,4),(5,8),(6,7)]=>[(1,4),(2,3),(5,7),(6,8)]=>[4,3,2,1,7,8,5,6]=>[3,2,4,1,7,5,8,6]
[(1,4),(2,3),(5,8),(6,7)]=>[(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[3,1,4,2,7,5,8,6]
[(1,5),(2,3),(4,8),(6,7)]=>[(1,3),(2,7),(4,5),(6,8)]=>[3,7,1,5,4,8,2,6]=>[3,1,5,4,7,2,8,6]
[(1,6),(2,3),(4,8),(5,7)]=>[(1,3),(2,7),(4,8),(5,6)]=>[3,7,1,8,6,5,2,4]=>[3,1,6,5,7,2,8,4]
[(1,7),(2,3),(4,8),(5,6)]=>[(1,3),(2,6),(4,8),(5,7)]=>[3,6,1,8,7,2,5,4]=>[3,1,6,2,7,5,8,4]
[(1,8),(2,3),(4,7),(5,6)]=>[(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[3,1,6,2,7,4,8,5]
[(1,8),(2,4),(3,7),(5,6)]=>[(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),(3,8),(5,6)]=>[(1,6),(2,4),(3,8),(5,7)]=>[6,4,8,2,7,1,5,3]=>[4,2,6,1,7,5,8,3]
[(1,6),(2,4),(3,8),(5,7)]=>[(1,7),(2,4),(3,8),(5,6)]=>[7,4,8,2,6,5,1,3]=>[4,2,6,5,7,1,8,3]
[(1,5),(2,4),(3,8),(6,7)]=>[(1,7),(2,4),(3,5),(6,8)]=>[7,4,5,2,3,8,1,6]=>[4,2,5,3,7,1,8,6]
[(1,4),(2,5),(3,8),(6,7)]=>[(1,7),(2,5),(3,4),(6,8)]=>[7,5,4,3,2,8,1,6]=>[4,3,5,2,7,1,8,6]
[(1,3),(2,5),(4,8),(6,7)]=>[(1,7),(2,3),(4,5),(6,8)]=>[7,3,2,5,4,8,1,6]=>[3,2,5,4,7,1,8,6]
[(1,2),(3,5),(4,8),(6,7)]=>[(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),(3,6),(4,8),(5,7)]=>[(1,2),(3,7),(4,8),(5,6)]=>[2,1,7,8,6,5,3,4]=>[2,1,6,5,7,3,8,4]
[(1,3),(2,6),(4,8),(5,7)]=>[(1,7),(2,3),(4,8),(5,6)]=>[7,3,2,8,6,5,1,4]=>[3,2,6,5,7,1,8,4]
[(1,4),(2,6),(3,8),(5,7)]=>[(1,7),(2,8),(3,4),(5,6)]=>[7,8,4,3,6,5,1,2]=>[4,3,6,5,7,1,8,2]
[(1,5),(2,6),(3,8),(4,7)]=>[(1,7),(2,8),(3,6),(4,5)]=>[7,8,6,5,4,3,1,2]=>[5,4,6,3,7,1,8,2]
[(1,6),(2,5),(3,8),(4,7)]=>[(1,7),(2,8),(3,5),(4,6)]=>[7,8,5,6,3,4,1,2]=>[5,3,6,4,7,1,8,2]
[(1,7),(2,5),(3,8),(4,6)]=>[(1,6),(2,8),(3,5),(4,7)]=>[6,8,5,7,3,1,4,2]=>[5,3,6,1,7,4,8,2]
[(1,8),(2,5),(3,7),(4,6)]=>[(1,6),(2,7),(3,5),(4,8)]=>[6,7,5,8,3,1,2,4]=>[5,3,6,1,7,2,8,4]
[(1,8),(2,6),(3,7),(4,5)]=>[(1,5),(2,7),(3,6),(4,8)]=>[5,7,6,8,1,3,2,4]=>[5,1,6,3,7,2,8,4]
[(1,7),(2,6),(3,8),(4,5)]=>[(1,5),(2,8),(3,6),(4,7)]=>[5,8,6,7,1,3,4,2]=>[5,1,6,3,7,4,8,2]
[(1,6),(2,7),(3,8),(4,5)]=>[(1,5),(2,8),(3,7),(4,6)]=>[5,8,7,6,1,4,3,2]=>[5,1,6,4,7,3,8,2]
[(1,5),(2,7),(3,8),(4,6)]=>[(1,6),(2,8),(3,7),(4,5)]=>[6,8,7,5,4,1,3,2]=>[5,4,6,1,7,3,8,2]
[(1,4),(2,7),(3,8),(5,6)]=>[(1,6),(2,8),(3,4),(5,7)]=>[6,8,4,3,7,1,5,2]=>[4,3,6,1,7,5,8,2]
[(1,3),(2,7),(4,8),(5,6)]=>[(1,6),(2,3),(4,8),(5,7)]=>[6,3,2,8,7,1,5,4]=>[3,2,6,1,7,5,8,4]
[(1,2),(3,7),(4,8),(5,6)]=>[(1,2),(3,6),(4,8),(5,7)]=>[2,1,6,8,7,3,5,4]=>[2,1,6,3,7,5,8,4]
[(1,2),(3,8),(4,7),(5,6)]=>[(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[2,1,6,3,7,4,8,5]
[(1,3),(2,8),(4,7),(5,6)]=>[(1,6),(2,3),(4,7),(5,8)]=>[6,3,2,7,8,1,4,5]=>[3,2,6,1,7,4,8,5]
[(1,4),(2,8),(3,7),(5,6)]=>[(1,6),(2,7),(3,4),(5,8)]=>[6,7,4,3,8,1,2,5]=>[4,3,6,1,7,2,8,5]
[(1,5),(2,8),(3,7),(4,6)]=>[(1,6),(2,7),(3,8),(4,5)]=>[6,7,8,5,4,1,2,3]=>[5,4,6,1,7,2,8,3]
[(1,6),(2,8),(3,7),(4,5)]=>[(1,5),(2,7),(3,8),(4,6)]=>[5,7,8,6,1,4,2,3]=>[5,1,6,4,7,2,8,3]
[(1,7),(2,8),(3,6),(4,5)]=>[(1,5),(2,6),(3,8),(4,7)]=>[5,6,8,7,1,2,4,3]=>[5,1,6,2,7,4,8,3]
[(1,8),(2,7),(3,6),(4,5)]=>[(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[5,1,6,2,7,3,8,4]
[(1,2),(3,4),(5,6),(7,8),(9,10)]=>[(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]
[(1,3),(2,4),(5,6),(7,8),(9,10)]=>[(1,4),(2,3),(5,6),(7,8),(9,10)]=>[4,3,2,1,6,5,8,7,10,9]=>[3,2,4,1,6,5,8,7,10,9]
[(1,5),(2,4),(3,6),(7,8),(9,10)]=>[(1,6),(2,4),(3,5),(7,8),(9,10)]=>[6,4,5,2,3,1,8,7,10,9]=>[4,2,5,3,6,1,8,7,10,9]
[(1,4),(2,5),(3,6),(7,8),(9,10)]=>[(1,6),(2,5),(3,4),(7,8),(9,10)]=>[6,5,4,3,2,1,8,7,10,9]=>[4,3,5,2,6,1,8,7,10,9]
[(1,3),(2,5),(4,6),(7,8),(9,10)]=>[(1,6),(2,3),(4,5),(7,8),(9,10)]=>[6,3,2,5,4,1,8,7,10,9]=>[3,2,5,4,6,1,8,7,10,9]
[(1,2),(3,5),(4,6),(7,8),(9,10)]=>[(1,2),(3,6),(4,5),(7,8),(9,10)]=>[2,1,6,5,4,3,8,7,10,9]=>[2,1,5,4,6,3,8,7,10,9]
[(1,2),(3,7),(4,6),(5,8),(9,10)]=>[(1,2),(3,8),(4,6),(5,7),(9,10)]=>[2,1,8,6,7,4,5,3,10,9]=>[2,1,6,4,7,5,8,3,10,9]
[(1,5),(2,6),(3,7),(4,8),(9,10)]=>[(1,8),(2,7),(3,6),(4,5),(9,10)]=>[8,7,6,5,4,3,2,1,10,9]=>[5,4,6,3,7,2,8,1,10,9]
[(1,4),(2,6),(3,7),(5,8),(9,10)]=>[(1,8),(2,7),(3,4),(5,6),(9,10)]=>[8,7,4,3,6,5,2,1,10,9]=>[4,3,6,5,7,2,8,1,10,9]
[(1,3),(2,6),(4,7),(5,8),(9,10)]=>[(1,8),(2,3),(4,7),(5,6),(9,10)]=>[8,3,2,7,6,5,4,1,10,9]=>[3,2,6,5,7,4,8,1,10,9]
[(1,2),(3,6),(4,7),(5,8),(9,10)]=>[(1,2),(3,8),(4,7),(5,6),(9,10)]=>[2,1,8,7,6,5,4,3,10,9]=>[2,1,6,5,7,4,8,3,10,9]
[(1,2),(3,5),(4,7),(6,8),(9,10)]=>[(1,2),(3,8),(4,5),(6,7),(9,10)]=>[2,1,8,5,4,7,6,3,10,9]=>[2,1,5,4,7,6,8,3,10,9]
[(1,3),(2,5),(4,7),(6,8),(9,10)]=>[(1,8),(2,3),(4,5),(6,7),(9,10)]=>[8,3,2,5,4,7,6,1,10,9]=>[3,2,5,4,7,6,8,1,10,9]
[(1,4),(2,5),(3,7),(6,8),(9,10)]=>[(1,8),(2,5),(3,4),(6,7),(9,10)]=>[8,5,4,3,2,7,6,1,10,9]=>[4,3,5,2,7,6,8,1,10,9]
[(1,3),(2,4),(5,7),(6,8),(9,10)]=>[(1,4),(2,3),(5,8),(6,7),(9,10)]=>[4,3,2,1,8,7,6,5,10,9]=>[3,2,4,1,7,6,8,5,10,9]
[(1,2),(3,4),(5,7),(6,8),(9,10)]=>[(1,2),(3,4),(5,8),(6,7),(9,10)]=>[2,1,4,3,8,7,6,5,10,9]=>[2,1,4,3,7,6,8,5,10,9]
[(1,2),(3,4),(5,9),(6,8),(7,10)]=>[(1,2),(3,4),(5,10),(6,8),(7,9)]=>[2,1,4,3,10,8,9,6,7,5]=>[2,1,4,3,8,6,9,7,10,5]
[(1,7),(2,6),(3,9),(4,8),(5,10)]=>[(1,10),(2,8),(3,9),(4,6),(5,7)]=>[10,8,9,6,7,4,5,2,3,1]=>[6,4,7,5,8,2,9,3,10,1]
[(1,2),(3,7),(4,8),(5,9),(6,10)]=>[(1,2),(3,10),(4,9),(5,8),(6,7)]=>[2,1,10,9,8,7,6,5,4,3]=>[2,1,7,6,8,5,9,4,10,3]
[(1,3),(2,7),(4,8),(5,9),(6,10)]=>[(1,10),(2,3),(4,9),(5,8),(6,7)]=>[10,3,2,9,8,7,6,5,4,1]=>[3,2,7,6,8,5,9,4,10,1]
[(1,4),(2,7),(3,8),(5,9),(6,10)]=>[(1,10),(2,9),(3,4),(5,8),(6,7)]=>[10,9,4,3,8,7,6,5,2,1]=>[4,3,7,6,8,5,9,2,10,1]
[(1,5),(2,7),(3,8),(4,9),(6,10)]=>[(1,10),(2,9),(3,8),(4,5),(6,7)]=>[10,9,8,5,4,7,6,3,2,1]=>[5,4,7,6,8,3,9,2,10,1]
[(1,6),(2,7),(3,8),(4,9),(5,10)]=>[(1,10),(2,9),(3,8),(4,7),(5,6)]=>[10,9,8,7,6,5,4,3,2,1]=>[6,5,7,4,8,3,9,2,10,1]
[(1,5),(2,6),(3,8),(4,9),(7,10)]=>[(1,10),(2,9),(3,6),(4,5),(7,8)]=>[10,9,6,5,4,3,8,7,2,1]=>[5,4,6,3,8,7,9,2,10,1]
[(1,4),(2,6),(3,8),(5,9),(7,10)]=>[(1,10),(2,9),(3,4),(5,6),(7,8)]=>[10,9,4,3,6,5,8,7,2,1]=>[4,3,6,5,8,7,9,2,10,1]
[(1,3),(2,6),(4,8),(5,9),(7,10)]=>[(1,10),(2,3),(4,9),(5,6),(7,8)]=>[10,3,2,9,6,5,8,7,4,1]=>[3,2,6,5,8,7,9,4,10,1]
[(1,2),(3,6),(4,8),(5,9),(7,10)]=>[(1,2),(3,10),(4,9),(5,6),(7,8)]=>[2,1,10,9,6,5,8,7,4,3]=>[2,1,6,5,8,7,9,4,10,3]
[(1,2),(3,5),(4,8),(6,9),(7,10)]=>[(1,2),(3,10),(4,5),(6,9),(7,8)]=>[2,1,10,5,4,9,8,7,6,3]=>[2,1,5,4,8,7,9,6,10,3]
[(1,3),(2,5),(4,8),(6,9),(7,10)]=>[(1,10),(2,3),(4,5),(6,9),(7,8)]=>[10,3,2,5,4,9,8,7,6,1]=>[3,2,5,4,8,7,9,6,10,1]
[(1,4),(2,5),(3,8),(6,9),(7,10)]=>[(1,10),(2,5),(3,4),(6,9),(7,8)]=>[10,5,4,3,2,9,8,7,6,1]=>[4,3,5,2,8,7,9,6,10,1]
[(1,3),(2,4),(5,8),(6,9),(7,10)]=>[(1,4),(2,3),(5,10),(6,9),(7,8)]=>[4,3,2,1,10,9,8,7,6,5]=>[3,2,4,1,8,7,9,6,10,5]
[(1,2),(3,4),(5,8),(6,9),(7,10)]=>[(1,2),(3,4),(5,10),(6,9),(7,8)]=>[2,1,4,3,10,9,8,7,6,5]=>[2,1,4,3,8,7,9,6,10,5]
[(1,2),(3,4),(5,7),(6,9),(8,10)]=>[(1,2),(3,4),(5,10),(6,7),(8,9)]=>[2,1,4,3,10,7,6,9,8,5]=>[2,1,4,3,7,6,9,8,10,5]
[(1,3),(2,4),(5,7),(6,9),(8,10)]=>[(1,4),(2,3),(5,10),(6,7),(8,9)]=>[4,3,2,1,10,7,6,9,8,5]=>[3,2,4,1,7,6,9,8,10,5]
[(1,4),(2,5),(3,7),(6,9),(8,10)]=>[(1,10),(2,5),(3,4),(6,7),(8,9)]=>[10,5,4,3,2,7,6,9,8,1]=>[4,3,5,2,7,6,9,8,10,1]
[(1,3),(2,5),(4,7),(6,9),(8,10)]=>[(1,10),(2,3),(4,5),(6,7),(8,9)]=>[10,3,2,5,4,7,6,9,8,1]=>[3,2,5,4,7,6,9,8,10,1]
[(1,2),(3,5),(4,7),(6,9),(8,10)]=>[(1,2),(3,10),(4,5),(6,7),(8,9)]=>[2,1,10,5,4,7,6,9,8,3]=>[2,1,5,4,7,6,9,8,10,3]
[(1,2),(3,6),(4,7),(5,9),(8,10)]=>[(1,2),(3,10),(4,7),(5,6),(8,9)]=>[2,1,10,7,6,5,4,9,8,3]=>[2,1,6,5,7,4,9,8,10,3]
[(1,3),(2,6),(4,7),(5,9),(8,10)]=>[(1,10),(2,3),(4,7),(5,6),(8,9)]=>[10,3,2,7,6,5,4,9,8,1]=>[3,2,6,5,7,4,9,8,10,1]
[(1,4),(2,6),(3,7),(5,9),(8,10)]=>[(1,10),(2,7),(3,4),(5,6),(8,9)]=>[10,7,4,3,6,5,2,9,8,1]=>[4,3,6,5,7,2,9,8,10,1]
[(1,5),(2,6),(3,7),(4,9),(8,10)]=>[(1,10),(2,7),(3,6),(4,5),(8,9)]=>[10,7,6,5,4,3,2,9,8,1]=>[5,4,6,3,7,2,9,8,10,1]
[(1,2),(3,5),(4,6),(7,9),(8,10)]=>[(1,2),(3,6),(4,5),(7,10),(8,9)]=>[2,1,6,5,4,3,10,9,8,7]=>[2,1,5,4,6,3,9,8,10,7]
[(1,3),(2,5),(4,6),(7,9),(8,10)]=>[(1,6),(2,3),(4,5),(7,10),(8,9)]=>[6,3,2,5,4,1,10,9,8,7]=>[3,2,5,4,6,1,9,8,10,7]
[(1,4),(2,5),(3,6),(7,9),(8,10)]=>[(1,6),(2,5),(3,4),(7,10),(8,9)]=>[6,5,4,3,2,1,10,9,8,7]=>[4,3,5,2,6,1,9,8,10,7]
[(1,3),(2,4),(5,6),(7,9),(8,10)]=>[(1,4),(2,3),(5,6),(7,10),(8,9)]=>[4,3,2,1,6,5,10,9,8,7]=>[3,2,4,1,6,5,9,8,10,7]
[(1,2),(3,4),(5,6),(7,9),(8,10)]=>[(1,2),(3,4),(5,6),(7,10),(8,9)]=>[2,1,4,3,6,5,10,9,8,7]=>[2,1,4,3,6,5,9,8,10,7]
[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]=>[(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
Kasraoui-Zeng
Description
The Kasraoui-Zeng involution for perfect matchings.
This yields the perfect matching with the number of nestings and crossings exchanged.
This yields the perfect matching with the number of nestings and crossings exchanged.
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.
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.
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