Identifier
Mp00283:
Perfect matchings
—non-nesting-exceedence permutation⟶
Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Images
=>
Cc0012;cc-rep-0Cc0020;cc-rep-3Cc0002;cc-rep-4
[(1,2)]=>[2,1]=>[2,1]=>([(0,1)],2)=>[2]
[(1,2),(3,4)]=>[2,1,4,3]=>[2,4,1,3]=>([(0,3),(1,2),(2,3)],4)=>[4]
[(1,3),(2,4)]=>[3,4,1,2]=>[3,1,4,2]=>([(0,3),(1,2),(2,3)],4)=>[4]
[(1,4),(2,3)]=>[3,4,2,1]=>[3,4,2,1]=>([(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>[4]
[(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[2,4,6,1,3,5]=>([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)=>[6]
[(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[3,1,4,6,2,5]=>([(0,5),(1,4),(2,3),(3,5),(4,5)],6)=>[6]
[(1,4),(2,3),(5,6)]=>[3,4,2,1,6,5]=>[3,4,2,6,1,5]=>([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)=>[6]
[(1,5),(2,3),(4,6)]=>[3,5,2,6,1,4]=>[5,3,6,2,1,4]=>([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>[6]
[(1,6),(2,3),(4,5)]=>[3,5,2,6,4,1]=>[5,3,6,2,4,1]=>([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[6]
[(1,6),(2,4),(3,5)]=>[4,5,6,2,3,1]=>[4,5,2,6,3,1]=>([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[6]
[(1,5),(2,4),(3,6)]=>[4,5,6,2,1,3]=>[4,5,2,1,6,3]=>([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)=>[6]
[(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[4,1,5,2,6,3]=>([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)=>[6]
[(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[5,3,1,6,2,4]=>([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[6]
[(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[5,6,2,1,3,4]=>([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>[6]
[(1,2),(3,6),(4,5)]=>[2,1,5,6,4,3]=>[5,6,2,4,1,3]=>([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>[6]
[(1,3),(2,6),(4,5)]=>[3,5,1,6,4,2]=>[3,5,6,1,4,2]=>([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)=>[6]
[(1,4),(2,6),(3,5)]=>[4,5,6,1,3,2]=>[4,1,5,6,3,2]=>([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[6]
[(1,5),(2,6),(3,4)]=>[4,5,6,3,1,2]=>[4,5,1,6,3,2]=>([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[6]
[(1,6),(2,5),(3,4)]=>[4,5,6,3,2,1]=>[4,5,6,3,2,1]=>([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[6]
[(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[2,4,6,8,1,3,5,7]=>([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7),(6,7)],8)=>[8]
[(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[3,1,4,6,8,2,5,7]=>([(0,7),(1,6),(2,3),(3,7),(4,5),(4,7),(5,6),(6,7)],8)=>[8]
[(1,4),(2,3),(5,6),(7,8)]=>[3,4,2,1,6,5,8,7]=>[3,4,2,6,8,1,5,7]=>([(0,6),(1,2),(1,6),(2,7),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)=>[8]
[(1,5),(2,3),(4,6),(7,8)]=>[3,5,2,6,1,4,8,7]=>[5,3,6,2,8,1,4,7]=>([(0,4),(1,2),(1,4),(1,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,6),(2,3),(4,5),(7,8)]=>[3,5,2,6,4,1,8,7]=>[5,3,6,2,4,8,1,7]=>([(0,1),(1,7),(2,3),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,7),(2,3),(4,5),(6,8)]=>[3,5,2,7,4,8,1,6]=>[5,7,3,8,2,4,1,6]=>([(0,4),(0,6),(1,2),(1,4),(1,6),(1,7),(2,3),(2,5),(2,7),(3,5),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,8),(2,3),(4,5),(6,7)]=>[3,5,2,7,4,8,6,1]=>[5,7,3,8,2,4,6,1]=>([(0,4),(0,6),(0,7),(1,2),(1,4),(1,6),(1,7),(2,3),(2,5),(2,7),(3,5),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,8),(2,4),(3,5),(6,7)]=>[4,5,7,2,3,8,6,1]=>[7,4,2,5,8,3,6,1]=>([(0,1),(0,6),(0,7),(1,5),(1,7),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,7),(2,4),(3,5),(6,8)]=>[4,5,7,2,3,8,1,6]=>[7,4,2,5,8,3,1,6]=>([(0,3),(0,7),(1,5),(1,6),(1,7),(2,4),(2,6),(2,7),(3,5),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,6),(2,4),(3,5),(7,8)]=>[4,5,6,2,3,1,8,7]=>[4,5,2,6,3,8,1,7]=>([(0,2),(1,6),(1,7),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,5),(2,4),(3,6),(7,8)]=>[4,5,6,2,1,3,8,7]=>[4,5,2,1,6,8,3,7]=>([(0,7),(1,2),(2,7),(3,4),(3,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)=>[8]
[(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[4,1,5,2,6,8,3,7]=>([(0,7),(1,6),(2,3),(3,7),(4,5),(4,7),(5,6),(6,7)],8)=>[8]
[(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[5,3,1,6,8,2,4,7]=>([(0,5),(1,3),(1,7),(2,4),(2,6),(3,5),(3,6),(4,6),(4,7),(5,7),(6,7)],8)=>[8]
[(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[5,6,2,8,1,3,4,7]=>([(0,7),(1,4),(1,5),(1,6),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)=>[8]
[(1,2),(3,6),(4,5),(7,8)]=>[2,1,5,6,4,3,8,7]=>[5,6,2,4,8,1,3,7]=>([(0,2),(1,5),(1,6),(1,7),(2,3),(2,7),(3,4),(3,5),(3,6),(4,5),(4,6),(4,7),(5,7),(6,7)],8)=>[8]
[(1,3),(2,6),(4,5),(7,8)]=>[3,5,1,6,4,2,8,7]=>[3,5,6,1,4,8,2,7]=>([(0,2),(1,6),(1,7),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7)],8)=>[8]
[(1,4),(2,6),(3,5),(7,8)]=>[4,5,6,1,3,2,8,7]=>[4,1,5,6,3,8,2,7]=>([(0,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,5),(2,6),(3,4),(7,8)]=>[4,5,6,3,1,2,8,7]=>[4,5,1,6,3,8,2,7]=>([(0,3),(1,6),(1,7),(2,4),(2,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,6),(2,5),(3,4),(7,8)]=>[4,5,6,3,2,1,8,7]=>[4,5,6,3,2,8,1,7]=>([(0,1),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,7),(2,5),(3,4),(6,8)]=>[4,5,7,3,2,8,1,6]=>[7,4,5,3,8,2,1,6]=>([(0,1),(0,7),(1,5),(1,6),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,8),(2,5),(3,4),(6,7)]=>[4,5,7,3,2,8,6,1]=>[7,4,5,3,8,2,6,1]=>([(0,1),(0,6),(0,7),(1,5),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,8),(2,6),(3,4),(5,7)]=>[4,6,7,3,8,2,5,1]=>[6,7,4,8,3,2,5,1]=>([(0,1),(0,3),(0,4),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,7),(2,6),(3,4),(5,8)]=>[4,6,7,3,8,2,1,5]=>[6,7,4,8,3,2,1,5]=>([(0,1),(0,3),(0,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,6),(2,7),(3,4),(5,8)]=>[4,6,7,3,8,1,2,5]=>[6,7,4,1,8,3,2,5]=>([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)=>[8]
[(1,5),(2,7),(3,4),(6,8)]=>[4,5,7,3,1,8,2,6]=>[7,4,5,3,1,8,2,6]=>([(0,1),(0,6),(1,7),(2,3),(2,4),(2,5),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,4),(2,7),(3,5),(6,8)]=>[4,5,7,1,3,8,2,6]=>[4,7,1,5,8,3,2,6]=>([(0,4),(0,7),(1,3),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(4,5),(4,6),(5,6),(5,7),(6,7)],8)=>[8]
[(1,3),(2,7),(4,5),(6,8)]=>[3,5,1,7,4,8,2,6]=>[5,3,7,8,1,4,2,6]=>([(0,5),(0,6),(1,3),(1,4),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(4,7),(5,7),(6,7)],8)=>[8]
[(1,2),(3,7),(4,5),(6,8)]=>[2,1,5,7,4,8,3,6]=>[7,5,8,2,4,1,3,6]=>([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7)],8)=>[8]
[(1,2),(3,8),(4,5),(6,7)]=>[2,1,5,7,4,8,6,3]=>[7,5,8,2,4,6,1,3]=>([(0,3),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(4,6),(4,7),(5,7),(6,7)],8)=>[8]
[(1,3),(2,8),(4,5),(6,7)]=>[3,5,1,7,4,8,6,2]=>[7,3,5,8,1,4,6,2]=>([(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)=>[8]
[(1,4),(2,8),(3,5),(6,7)]=>[4,5,7,1,3,8,6,2]=>[4,5,1,7,8,3,6,2]=>([(0,4),(0,5),(1,2),(1,3),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,5),(2,8),(3,4),(6,7)]=>[4,5,7,3,1,8,6,2]=>[7,4,1,5,8,3,6,2]=>([(0,4),(0,7),(1,2),(1,6),(1,7),(2,5),(2,6),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,6),(2,8),(3,4),(5,7)]=>[4,6,7,3,8,1,5,2]=>[6,7,4,3,8,1,5,2]=>([(0,1),(0,6),(0,7),(1,4),(1,5),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)=>[8]
[(1,7),(2,8),(3,4),(5,6)]=>[4,6,7,3,8,5,1,2]=>[6,7,4,8,3,1,5,2]=>([(0,1),(0,4),(0,5),(0,7),(1,3),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,8),(2,7),(3,4),(5,6)]=>[4,6,7,3,8,5,2,1]=>[6,7,4,8,3,5,2,1]=>([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,8),(2,7),(3,5),(4,6)]=>[5,6,7,8,3,4,2,1]=>[5,6,7,3,8,4,2,1]=>([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,7),(2,8),(3,5),(4,6)]=>[5,6,7,8,3,4,1,2]=>[5,6,7,3,1,8,4,2]=>([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)=>[8]
[(1,6),(2,8),(3,5),(4,7)]=>[5,6,7,8,3,1,4,2]=>[5,6,1,7,3,8,4,2]=>([(0,4),(0,5),(1,6),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,5),(2,8),(3,6),(4,7)]=>[5,6,7,8,1,3,4,2]=>[5,1,6,7,3,8,4,2]=>([(0,5),(1,6),(1,7),(2,4),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,4),(2,8),(3,6),(5,7)]=>[4,6,7,1,8,3,5,2]=>[6,4,7,1,8,3,5,2]=>([(0,2),(0,4),(0,5),(1,3),(1,6),(1,7),(2,3),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,3),(2,8),(4,6),(5,7)]=>[3,6,1,7,8,4,5,2]=>[6,7,8,3,1,4,5,2]=>([(0,3),(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,7),(6,7)],8)=>[8]
[(1,2),(3,8),(4,6),(5,7)]=>[2,1,6,7,8,4,5,3]=>[6,7,2,8,4,5,1,3]=>([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,7),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)=>[8]
[(1,2),(3,7),(4,6),(5,8)]=>[2,1,6,7,8,4,3,5]=>[6,7,2,8,4,1,3,5]=>([(0,5),(0,6),(0,7),(1,3),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(3,4),(3,5),(4,5),(4,6),(4,7),(5,6),(5,7)],8)=>[8]
[(1,3),(2,7),(4,6),(5,8)]=>[3,6,1,7,8,4,2,5]=>[3,6,1,7,8,4,2,5]=>([(0,1),(0,6),(1,7),(2,3),(2,4),(2,6),(3,5),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,4),(2,7),(3,6),(5,8)]=>[4,6,7,1,8,3,2,5]=>[6,1,7,4,8,3,2,5]=>([(0,7),(1,2),(1,5),(1,6),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(5,6),(5,7),(6,7)],8)=>[8]
[(1,5),(2,7),(3,6),(4,8)]=>[5,6,7,8,1,3,2,4]=>[5,1,6,7,3,2,8,4]=>([(0,7),(1,6),(2,4),(2,5),(2,7),(3,4),(3,5),(3,7),(4,5),(4,6),(5,6),(6,7)],8)=>[8]
[(1,6),(2,7),(3,5),(4,8)]=>[5,6,7,8,3,1,2,4]=>[5,6,1,7,3,2,8,4]=>([(0,7),(1,5),(1,6),(2,3),(2,4),(2,7),(3,4),(3,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)=>[8]
[(1,7),(2,6),(3,5),(4,8)]=>[5,6,7,8,3,2,1,4]=>[5,6,7,3,2,1,8,4]=>([(0,4),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,8),(2,6),(3,5),(4,7)]=>[5,6,7,8,3,2,4,1]=>[5,6,7,3,2,8,4,1]=>([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)=>[8]
[(1,8),(2,5),(3,6),(4,7)]=>[5,6,7,8,2,3,4,1]=>[5,6,2,7,3,8,4,1]=>([(0,6),(0,7),(1,4),(1,5),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,7),(2,5),(3,6),(4,8)]=>[5,6,7,8,2,3,1,4]=>[5,6,2,7,3,1,8,4]=>([(0,6),(1,4),(1,5),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7)],8)=>[8]
[(1,6),(2,5),(3,7),(4,8)]=>[5,6,7,8,2,1,3,4]=>[5,6,2,1,7,3,8,4]=>([(0,7),(1,4),(1,7),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)=>[8]
[(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[5,1,6,2,7,3,8,4]=>([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7),(6,7)],8)=>[8]
[(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[6,1,7,4,2,8,3,5]=>([(0,7),(1,3),(1,6),(2,4),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6),(5,7),(6,7)],8)=>[8]
[(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[3,6,1,7,2,8,4,5]=>([(0,4),(0,5),(1,2),(1,6),(2,7),(3,4),(3,5),(3,6),(4,7),(5,7),(6,7)],8)=>[8]
[(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[6,7,2,1,8,3,4,5]=>([(0,3),(0,4),(0,5),(1,2),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)=>[8]
[(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[7,5,8,2,1,3,4,6]=>([(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7)],8)=>[8]
[(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[5,7,3,1,8,2,4,6]=>([(0,3),(0,7),(1,3),(1,4),(1,7),(2,4),(2,5),(2,7),(3,6),(4,5),(4,6),(5,6),(5,7),(6,7)],8)=>[8]
[(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[4,7,1,5,2,8,3,6]=>([(0,1),(0,7),(1,6),(2,3),(2,7),(3,5),(3,6),(4,5),(4,6),(4,7),(5,7),(6,7)],8)=>[8]
[(1,5),(2,4),(3,7),(6,8)]=>[4,5,7,2,1,8,3,6]=>[7,4,2,5,1,8,3,6]=>([(0,1),(0,6),(1,7),(2,4),(2,5),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)=>[8]
[(1,6),(2,4),(3,7),(5,8)]=>[4,6,7,2,8,1,3,5]=>[6,7,4,2,1,8,3,5]=>([(0,1),(0,4),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)=>[8]
[(1,7),(2,4),(3,6),(5,8)]=>[4,6,7,2,8,3,1,5]=>[6,7,4,2,8,3,1,5]=>([(0,1),(0,5),(0,6),(1,4),(1,7),(2,3),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)=>[8]
[(1,8),(2,4),(3,6),(5,7)]=>[4,6,7,2,8,3,5,1]=>[6,7,4,2,8,3,5,1]=>([(0,2),(0,6),(0,7),(1,3),(1,4),(1,5),(1,7),(2,4),(2,5),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,8),(2,3),(4,6),(5,7)]=>[3,6,2,7,8,4,5,1]=>[3,6,7,2,8,4,5,1]=>([(0,6),(0,7),(1,4),(1,5),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,7),(6,7)],8)=>[8]
[(1,7),(2,3),(4,6),(5,8)]=>[3,6,2,7,8,4,1,5]=>[3,6,7,2,8,4,1,5]=>([(0,6),(0,7),(1,2),(1,4),(1,5),(2,3),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,6),(2,3),(4,7),(5,8)]=>[3,6,2,7,8,1,4,5]=>[3,6,7,2,1,8,4,5]=>([(0,6),(0,7),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,5),(2,3),(4,7),(6,8)]=>[3,5,2,7,1,8,4,6]=>[5,7,3,8,2,1,4,6]=>([(0,4),(0,7),(1,3),(1,4),(1,7),(2,3),(2,5),(2,6),(2,7),(3,5),(3,6),(4,5),(4,6),(5,6),(5,7),(6,7)],8)=>[8]
[(1,4),(2,3),(5,7),(6,8)]=>[3,4,2,1,7,8,5,6]=>[3,4,7,8,2,1,5,6]=>([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[3,4,7,1,8,2,5,6]=>([(0,5),(0,7),(1,5),(1,7),(2,4),(2,6),(3,4),(3,6),(4,7),(5,6),(6,7)],8)=>[8]
[(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[7,8,2,4,1,3,5,6]=>([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)=>[8]
[(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,7,8,6,5]=>[7,8,2,4,6,1,3,5]=>([(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)=>[8]
[(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,7,8,6,5]=>[3,7,8,1,4,6,2,5]=>([(0,3),(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(4,7),(5,7),(6,7)],8)=>[8]
[(1,4),(2,3),(5,8),(6,7)]=>[3,4,2,1,7,8,6,5]=>[3,4,7,8,2,6,1,5]=>([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,5),(2,3),(4,8),(6,7)]=>[3,5,2,7,1,8,6,4]=>[5,7,3,8,2,6,1,4]=>([(0,1),(0,2),(0,3),(0,6),(1,2),(1,5),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,6),(2,3),(4,8),(5,7)]=>[3,6,2,7,8,1,5,4]=>[3,6,7,2,8,5,1,4]=>([(0,5),(0,7),(1,4),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(5,7),(6,7)],8)=>[8]
[(1,7),(2,3),(4,8),(5,6)]=>[3,6,2,7,8,5,1,4]=>[3,6,7,8,5,2,1,4]=>([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,8),(2,3),(4,7),(5,6)]=>[3,6,2,7,8,5,4,1]=>[3,6,7,8,5,2,4,1]=>([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,8),(2,4),(3,7),(5,6)]=>[4,6,7,2,8,5,3,1]=>[6,4,7,8,2,5,3,1]=>([(0,3),(0,4),(0,6),(0,7),(1,3),(1,4),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,7),(2,4),(3,8),(5,6)]=>[4,6,7,2,8,5,1,3]=>[6,4,7,8,2,1,5,3]=>([(0,1),(0,2),(0,4),(0,7),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,6),(2,4),(3,8),(5,7)]=>[4,6,7,2,8,1,5,3]=>[6,4,7,2,8,1,5,3]=>([(0,4),(0,6),(0,7),(1,2),(1,3),(1,5),(1,6),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7)],8)=>[8]
[(1,5),(2,4),(3,8),(6,7)]=>[4,5,7,2,1,8,6,3]=>[4,5,2,7,8,1,6,3]=>([(0,4),(0,5),(0,7),(1,2),(1,3),(1,6),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)=>[8]
[(1,4),(2,5),(3,8),(6,7)]=>[4,5,7,1,2,8,6,3]=>[4,1,5,7,8,2,6,3]=>([(0,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)=>[8]
[(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,7,2,8,6,4]=>[5,3,7,8,1,6,2,4]=>([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,6),(2,3),(2,4),(2,7),(3,5),(3,7),(4,5),(4,7),(5,6),(6,7)],8)=>[8]
[(1,2),(3,5),(4,8),(6,7)]=>[2,1,5,7,3,8,6,4]=>[7,5,8,2,6,1,3,4]=>([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(6,7)],8)=>[8]
[(1,2),(3,6),(4,8),(5,7)]=>[2,1,6,7,8,3,5,4]=>[6,7,2,8,5,1,3,4]=>([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(1,7),(2,4),(2,5),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,3),(2,6),(4,8),(5,7)]=>[3,6,1,7,8,2,5,4]=>[3,6,1,7,8,5,2,4]=>([(0,1),(0,6),(1,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)=>[8]
[(1,4),(2,6),(3,8),(5,7)]=>[4,6,7,1,8,2,5,3]=>[6,4,1,7,8,2,5,3]=>([(0,5),(0,7),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,5),(2,6),(3,8),(4,7)]=>[5,6,7,8,1,2,4,3]=>[5,1,6,2,7,8,4,3]=>([(0,5),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,6),(2,5),(3,8),(4,7)]=>[5,6,7,8,2,1,4,3]=>[5,6,2,1,7,8,4,3]=>([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,7),(2,5),(3,8),(4,6)]=>[5,6,7,8,2,4,1,3]=>[5,6,2,7,1,8,4,3]=>([(0,6),(0,7),(1,3),(1,4),(1,5),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(6,7)],8)=>[8]
[(1,8),(2,5),(3,7),(4,6)]=>[5,6,7,8,2,4,3,1]=>[5,6,2,7,8,4,3,1]=>([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,8),(2,6),(3,7),(4,5)]=>[5,6,7,8,4,2,3,1]=>[5,6,7,2,8,4,3,1]=>([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,7),(2,6),(3,8),(4,5)]=>[5,6,7,8,4,2,1,3]=>[5,6,7,2,1,8,4,3]=>([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,6),(2,7),(3,8),(4,5)]=>[5,6,7,8,4,1,2,3]=>[5,6,1,7,2,8,4,3]=>([(0,6),(0,7),(1,4),(1,5),(2,3),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,5),(2,7),(3,8),(4,6)]=>[5,6,7,8,1,4,2,3]=>[5,1,6,7,2,8,4,3]=>([(0,5),(1,6),(1,7),(2,4),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(5,6),(5,7),(6,7)],8)=>[8]
[(1,4),(2,7),(3,8),(5,6)]=>[4,6,7,1,8,5,2,3]=>[6,4,7,1,8,2,5,3]=>([(0,3),(0,5),(0,7),(1,2),(1,4),(1,6),(2,3),(2,5),(2,7),(3,4),(3,6),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,3),(2,7),(4,8),(5,6)]=>[3,6,1,7,8,5,2,4]=>[3,6,7,1,8,5,2,4]=>([(0,2),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(6,7)],8)=>[8]
[(1,2),(3,7),(4,8),(5,6)]=>[2,1,6,7,8,5,3,4]=>[6,7,8,5,2,1,3,4]=>([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,7),(6,7)],8)=>[8]
[(1,2),(3,8),(4,7),(5,6)]=>[2,1,6,7,8,5,4,3]=>[6,7,8,5,2,4,1,3]=>([(0,2),(0,3),(0,4),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,3),(2,8),(4,7),(5,6)]=>[3,6,1,7,8,5,4,2]=>[6,7,8,3,5,1,4,2]=>([(0,1),(0,3),(0,4),(0,5),(0,7),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,4),(2,8),(3,7),(5,6)]=>[4,6,7,1,8,5,3,2]=>[4,6,7,8,1,5,3,2]=>([(0,4),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,5),(2,8),(3,7),(4,6)]=>[5,6,7,8,1,4,3,2]=>[5,1,6,7,8,4,3,2]=>([(0,4),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,6),(2,8),(3,7),(4,5)]=>[5,6,7,8,4,1,3,2]=>[5,6,1,7,8,4,3,2]=>([(0,3),(0,4),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,7),(2,8),(3,6),(4,5)]=>[5,6,7,8,4,3,1,2]=>[5,6,7,1,8,4,3,2]=>([(0,2),(0,3),(0,4),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
[(1,8),(2,7),(3,6),(4,5)]=>[5,6,7,8,4,3,2,1]=>[5,6,7,8,4,3,2,1]=>([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>[8]
Map
non-nesting-exceedence permutation
Description
The fixed-point-free permutation with deficiencies given by the perfect matching, no alignments and no inversions between exceedences.
Put differently, the exceedences form the unique non-nesting perfect matching whose openers coincide with those of the given perfect matching.
Put differently, the exceedences form the unique non-nesting perfect matching whose openers coincide with those of the given perfect matching.
Map
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00067Foata bijection.
See Mp00067Foata bijection.
Map
graph of inversions
Description
The graph of inversions of a permutation.
For a permutation of $\{1,\dots,n\}$, this is the graph with vertices $\{1,\dots,n\}$, where $(i,j)$ is an edge if and only if it is an inversion of the permutation.
For a permutation of $\{1,\dots,n\}$, this is the graph with vertices $\{1,\dots,n\}$, where $(i,j)$ is an edge if and only if it is an inversion of the permutation.
Map
to partition of connected components
Description
Return the partition of the sizes of the connected components of the graph.
searching the database
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