Identifier
Mp00283: non-nesting-exceedence permutationPermutations
Mp00239: Permutations CorteelPermutations
Mp00087: Permutations inverse first fundamental transformationPermutations
Mp00109: Permutations descent wordBinary words
Images
=>
Cc0012;cc-rep-0
[(1,2)]=>[2,1]=>[2,1]=>[2,1]=>1 [(1,2),(3,4)]=>[2,1,4,3]=>[2,1,4,3]=>[2,1,4,3]=>101 [(1,3),(2,4)]=>[3,4,1,2]=>[4,3,2,1]=>[3,2,4,1]=>101 [(1,4),(2,3)]=>[3,4,2,1]=>[4,3,1,2]=>[4,2,3,1]=>101 [(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5]=>10101 [(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[4,3,2,1,6,5]=>[3,2,4,1,6,5]=>10101 [(1,4),(2,3),(5,6)]=>[3,4,2,1,6,5]=>[4,3,1,2,6,5]=>[4,2,3,1,6,5]=>10101 [(1,5),(2,3),(4,6)]=>[3,5,2,6,1,4]=>[6,3,1,5,4,2]=>[5,4,6,2,3,1]=>10101 [(1,6),(2,3),(4,5)]=>[3,5,2,6,4,1]=>[6,3,1,5,2,4]=>[6,4,5,2,3,1]=>10101 [(1,6),(2,4),(3,5)]=>[4,5,6,2,3,1]=>[6,5,4,2,1,3]=>[6,3,4,2,5,1]=>10101 [(1,5),(2,4),(3,6)]=>[4,5,6,2,1,3]=>[6,5,4,2,3,1]=>[5,3,4,2,6,1]=>10101 [(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[6,5,4,3,2,1]=>[4,3,5,2,6,1]=>10101 [(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[6,3,2,5,4,1]=>[3,2,5,4,6,1]=>10101 [(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[2,1,6,5,4,3]=>[2,1,5,4,6,3]=>10101 [(1,2),(3,6),(4,5)]=>[2,1,5,6,4,3]=>[2,1,6,5,3,4]=>[2,1,6,4,5,3]=>10101 [(1,3),(2,6),(4,5)]=>[3,5,1,6,4,2]=>[6,3,2,5,1,4]=>[3,2,6,4,5,1]=>10101 [(1,4),(2,6),(3,5)]=>[4,5,6,1,3,2]=>[6,5,4,3,1,2]=>[4,3,6,2,5,1]=>10101 [(1,5),(2,6),(3,4)]=>[4,5,6,3,1,2]=>[6,5,4,1,3,2]=>[6,2,5,3,4,1]=>10101 [(1,6),(2,5),(3,4)]=>[4,5,6,3,2,1]=>[6,5,4,1,2,3]=>[5,2,6,3,4,1]=>10101 [(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[2,1,4,3,6,5,8,7]=>[2,1,4,3,6,5,8,7]=>1010101 [(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[4,3,2,1,6,5,8,7]=>[3,2,4,1,6,5,8,7]=>1010101 [(1,4),(2,3),(5,6),(7,8)]=>[3,4,2,1,6,5,8,7]=>[4,3,1,2,6,5,8,7]=>[4,2,3,1,6,5,8,7]=>1010101 [(1,5),(2,3),(4,6),(7,8)]=>[3,5,2,6,1,4,8,7]=>[6,3,1,5,4,2,8,7]=>[5,4,6,2,3,1,8,7]=>1010101 [(1,6),(2,3),(4,5),(7,8)]=>[3,5,2,6,4,1,8,7]=>[6,3,1,5,2,4,8,7]=>[6,4,5,2,3,1,8,7]=>1010101 [(1,7),(2,3),(4,5),(6,8)]=>[3,5,2,7,4,8,1,6]=>[8,3,1,5,2,7,6,4]=>[7,6,8,4,5,2,3,1]=>1010101 [(1,8),(2,3),(4,5),(6,7)]=>[3,5,2,7,4,8,6,1]=>[8,3,1,5,2,7,4,6]=>[8,6,7,4,5,2,3,1]=>1010101 [(1,8),(2,4),(3,5),(6,7)]=>[4,5,7,2,3,8,6,1]=>[8,5,4,2,1,7,3,6]=>[8,6,7,3,4,2,5,1]=>1010101 [(1,7),(2,4),(3,5),(6,8)]=>[4,5,7,2,3,8,1,6]=>[8,5,4,2,1,7,6,3]=>[7,6,8,3,4,2,5,1]=>1010101 [(1,6),(2,4),(3,5),(7,8)]=>[4,5,6,2,3,1,8,7]=>[6,5,4,2,1,3,8,7]=>[6,3,4,2,5,1,8,7]=>1010101 [(1,5),(2,4),(3,6),(7,8)]=>[4,5,6,2,1,3,8,7]=>[6,5,4,2,3,1,8,7]=>[5,3,4,2,6,1,8,7]=>1010101 [(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[6,5,4,3,2,1,8,7]=>[4,3,5,2,6,1,8,7]=>1010101 [(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[6,3,2,5,4,1,8,7]=>[3,2,5,4,6,1,8,7]=>1010101 [(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[2,1,6,5,4,3,8,7]=>[2,1,5,4,6,3,8,7]=>1010101 [(1,2),(3,6),(4,5),(7,8)]=>[2,1,5,6,4,3,8,7]=>[2,1,6,5,3,4,8,7]=>[2,1,6,4,5,3,8,7]=>1010101 [(1,3),(2,6),(4,5),(7,8)]=>[3,5,1,6,4,2,8,7]=>[6,3,2,5,1,4,8,7]=>[3,2,6,4,5,1,8,7]=>1010101 [(1,4),(2,6),(3,5),(7,8)]=>[4,5,6,1,3,2,8,7]=>[6,5,4,3,1,2,8,7]=>[4,3,6,2,5,1,8,7]=>1010101 [(1,5),(2,6),(3,4),(7,8)]=>[4,5,6,3,1,2,8,7]=>[6,5,4,1,3,2,8,7]=>[6,2,5,3,4,1,8,7]=>1010101 [(1,6),(2,5),(3,4),(7,8)]=>[4,5,6,3,2,1,8,7]=>[6,5,4,1,2,3,8,7]=>[5,2,6,3,4,1,8,7]=>1010101 [(1,7),(2,5),(3,4),(6,8)]=>[4,5,7,3,2,8,1,6]=>[8,5,4,1,2,7,6,3]=>[5,2,7,6,8,3,4,1]=>1010101 [(1,8),(2,5),(3,4),(6,7)]=>[4,5,7,3,2,8,6,1]=>[8,5,4,1,2,7,3,6]=>[5,2,8,6,7,3,4,1]=>1010101 [(1,8),(2,6),(3,4),(5,7)]=>[4,6,7,3,8,2,5,1]=>[8,7,4,1,6,3,2,5]=>[7,2,8,5,6,3,4,1]=>1010101 [(1,7),(2,6),(3,4),(5,8)]=>[4,6,7,3,8,2,1,5]=>[8,7,4,1,6,3,5,2]=>[8,2,7,5,6,3,4,1]=>1010101 [(1,6),(2,7),(3,4),(5,8)]=>[4,6,7,3,8,1,2,5]=>[8,7,4,1,6,5,3,2]=>[6,5,8,2,7,3,4,1]=>1010101 [(1,5),(2,7),(3,4),(6,8)]=>[4,5,7,3,1,8,2,6]=>[8,5,4,1,3,7,6,2]=>[7,6,8,2,5,3,4,1]=>1010101 [(1,4),(2,7),(3,5),(6,8)]=>[4,5,7,1,3,8,2,6]=>[8,5,4,3,1,7,6,2]=>[4,3,7,6,8,2,5,1]=>1010101 [(1,3),(2,7),(4,5),(6,8)]=>[3,5,1,7,4,8,2,6]=>[8,3,2,5,1,7,6,4]=>[3,2,7,6,8,4,5,1]=>1010101 [(1,2),(3,7),(4,5),(6,8)]=>[2,1,5,7,4,8,3,6]=>[2,1,8,5,3,7,6,4]=>[2,1,7,6,8,4,5,3]=>1010101 [(1,2),(3,8),(4,5),(6,7)]=>[2,1,5,7,4,8,6,3]=>[2,1,8,5,3,7,4,6]=>[2,1,8,6,7,4,5,3]=>1010101 [(1,3),(2,8),(4,5),(6,7)]=>[3,5,1,7,4,8,6,2]=>[8,3,2,5,1,7,4,6]=>[3,2,8,6,7,4,5,1]=>1010101 [(1,4),(2,8),(3,5),(6,7)]=>[4,5,7,1,3,8,6,2]=>[8,5,4,3,1,7,2,6]=>[4,3,8,6,7,2,5,1]=>1010101 [(1,5),(2,8),(3,4),(6,7)]=>[4,5,7,3,1,8,6,2]=>[8,5,4,1,3,7,2,6]=>[8,6,7,2,5,3,4,1]=>1010101 [(1,6),(2,8),(3,4),(5,7)]=>[4,6,7,3,8,1,5,2]=>[8,7,4,1,6,5,2,3]=>[6,5,7,2,8,3,4,1]=>1010101 [(1,7),(2,8),(3,4),(5,6)]=>[4,6,7,3,8,5,1,2]=>[8,7,4,1,6,2,5,3]=>[7,5,6,2,8,3,4,1]=>1010101 [(1,8),(2,7),(3,4),(5,6)]=>[4,6,7,3,8,5,2,1]=>[8,7,4,1,6,2,3,5]=>[8,5,6,2,7,3,4,1]=>1010101 [(1,8),(2,7),(3,5),(4,6)]=>[5,6,7,8,3,4,2,1]=>[8,7,6,5,2,1,3,4]=>[8,4,5,2,7,3,6,1]=>1010101 [(1,7),(2,8),(3,5),(4,6)]=>[5,6,7,8,3,4,1,2]=>[8,7,6,5,2,1,4,3]=>[7,4,5,2,8,3,6,1]=>1010101 [(1,6),(2,8),(3,5),(4,7)]=>[5,6,7,8,3,1,4,2]=>[8,7,6,5,2,4,1,3]=>[8,3,6,4,5,2,7,1]=>1010101 [(1,5),(2,8),(3,6),(4,7)]=>[5,6,7,8,1,3,4,2]=>[8,7,6,5,4,2,1,3]=>[5,4,8,3,6,2,7,1]=>1010101 [(1,4),(2,8),(3,6),(5,7)]=>[4,6,7,1,8,3,5,2]=>[8,7,4,3,6,2,1,5]=>[4,3,8,5,6,2,7,1]=>1010101 [(1,3),(2,8),(4,6),(5,7)]=>[3,6,1,7,8,4,5,2]=>[8,3,2,7,6,4,1,5]=>[3,2,8,5,6,4,7,1]=>1010101 [(1,2),(3,8),(4,6),(5,7)]=>[2,1,6,7,8,4,5,3]=>[2,1,8,7,6,4,3,5]=>[2,1,8,5,6,4,7,3]=>1010101 [(1,2),(3,7),(4,6),(5,8)]=>[2,1,6,7,8,4,3,5]=>[2,1,8,7,6,4,5,3]=>[2,1,7,5,6,4,8,3]=>1010101 [(1,3),(2,7),(4,6),(5,8)]=>[3,6,1,7,8,4,2,5]=>[8,3,2,7,6,4,5,1]=>[3,2,7,5,6,4,8,1]=>1010101 [(1,4),(2,7),(3,6),(5,8)]=>[4,6,7,1,8,3,2,5]=>[8,7,4,3,6,2,5,1]=>[4,3,7,5,6,2,8,1]=>1010101 [(1,5),(2,7),(3,6),(4,8)]=>[5,6,7,8,1,3,2,4]=>[8,7,6,5,4,2,3,1]=>[5,4,7,3,6,2,8,1]=>1010101 [(1,6),(2,7),(3,5),(4,8)]=>[5,6,7,8,3,1,2,4]=>[8,7,6,5,2,4,3,1]=>[7,3,6,4,5,2,8,1]=>1010101 [(1,7),(2,6),(3,5),(4,8)]=>[5,6,7,8,3,2,1,4]=>[8,7,6,5,2,3,4,1]=>[6,3,7,4,5,2,8,1]=>1010101 [(1,8),(2,6),(3,5),(4,7)]=>[5,6,7,8,3,2,4,1]=>[8,7,6,5,2,3,1,4]=>[6,3,8,4,5,2,7,1]=>1010101 [(1,8),(2,5),(3,6),(4,7)]=>[5,6,7,8,2,3,4,1]=>[8,7,6,5,3,2,1,4]=>[8,4,5,3,6,2,7,1]=>1010101 [(1,7),(2,5),(3,6),(4,8)]=>[5,6,7,8,2,3,1,4]=>[8,7,6,5,3,2,4,1]=>[7,4,5,3,6,2,8,1]=>1010101 [(1,6),(2,5),(3,7),(4,8)]=>[5,6,7,8,2,1,3,4]=>[8,7,6,5,3,4,2,1]=>[6,4,5,3,7,2,8,1]=>1010101 [(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[8,7,6,5,4,3,2,1]=>[5,4,6,3,7,2,8,1]=>1010101 [(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[8,7,4,3,6,5,2,1]=>[4,3,6,5,7,2,8,1]=>1010101 [(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[8,3,2,7,6,5,4,1]=>[3,2,6,5,7,4,8,1]=>1010101 [(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[2,1,8,7,6,5,4,3]=>[2,1,6,5,7,4,8,3]=>1010101 [(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[2,1,8,5,4,7,6,3]=>[2,1,5,4,7,6,8,3]=>1010101 [(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[8,3,2,5,4,7,6,1]=>[3,2,5,4,7,6,8,1]=>1010101 [(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[8,5,4,3,2,7,6,1]=>[4,3,5,2,7,6,8,1]=>1010101 [(1,5),(2,4),(3,7),(6,8)]=>[4,5,7,2,1,8,3,6]=>[8,5,4,2,3,7,6,1]=>[5,3,4,2,7,6,8,1]=>1010101 [(1,6),(2,4),(3,7),(5,8)]=>[4,6,7,2,8,1,3,5]=>[8,7,4,2,6,5,3,1]=>[6,5,7,3,4,2,8,1]=>1010101 [(1,7),(2,4),(3,6),(5,8)]=>[4,6,7,2,8,3,1,5]=>[8,7,4,2,6,3,5,1]=>[7,5,6,3,4,2,8,1]=>1010101 [(1,8),(2,4),(3,6),(5,7)]=>[4,6,7,2,8,3,5,1]=>[8,7,4,2,6,3,1,5]=>[8,5,6,3,4,2,7,1]=>1010101 [(1,8),(2,3),(4,6),(5,7)]=>[3,6,2,7,8,4,5,1]=>[8,3,1,7,6,4,2,5]=>[8,5,6,4,7,2,3,1]=>1010101 [(1,7),(2,3),(4,6),(5,8)]=>[3,6,2,7,8,4,1,5]=>[8,3,1,7,6,4,5,2]=>[7,5,6,4,8,2,3,1]=>1010101 [(1,6),(2,3),(4,7),(5,8)]=>[3,6,2,7,8,1,4,5]=>[8,3,1,7,6,5,4,2]=>[6,5,7,4,8,2,3,1]=>1010101 [(1,5),(2,3),(4,7),(6,8)]=>[3,5,2,7,1,8,4,6]=>[8,3,1,5,4,7,6,2]=>[5,4,7,6,8,2,3,1]=>1010101 [(1,4),(2,3),(5,7),(6,8)]=>[3,4,2,1,7,8,5,6]=>[4,3,1,2,8,7,6,5]=>[4,2,3,1,7,6,8,5]=>1010101 [(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[4,3,2,1,8,7,6,5]=>[3,2,4,1,7,6,8,5]=>1010101 [(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[2,1,4,3,8,7,6,5]=>[2,1,4,3,7,6,8,5]=>1010101 [(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,7,8,6,5]=>[2,1,4,3,8,7,5,6]=>[2,1,4,3,8,6,7,5]=>1010101 [(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,7,8,6,5]=>[4,3,2,1,8,7,5,6]=>[3,2,4,1,8,6,7,5]=>1010101 [(1,4),(2,3),(5,8),(6,7)]=>[3,4,2,1,7,8,6,5]=>[4,3,1,2,8,7,5,6]=>[4,2,3,1,8,6,7,5]=>1010101 [(1,5),(2,3),(4,8),(6,7)]=>[3,5,2,7,1,8,6,4]=>[8,3,1,5,4,7,2,6]=>[5,4,8,6,7,2,3,1]=>1010101 [(1,6),(2,3),(4,8),(5,7)]=>[3,6,2,7,8,1,5,4]=>[8,3,1,7,6,5,2,4]=>[6,5,8,4,7,2,3,1]=>1010101 [(1,7),(2,3),(4,8),(5,6)]=>[3,6,2,7,8,5,1,4]=>[8,3,1,7,6,2,5,4]=>[8,4,7,5,6,2,3,1]=>1010101 [(1,8),(2,3),(4,7),(5,6)]=>[3,6,2,7,8,5,4,1]=>[8,3,1,7,6,2,4,5]=>[7,4,8,5,6,2,3,1]=>1010101 [(1,8),(2,4),(3,7),(5,6)]=>[4,6,7,2,8,5,3,1]=>[8,7,4,2,6,1,3,5]=>[7,3,4,2,8,5,6,1]=>1010101 [(1,7),(2,4),(3,8),(5,6)]=>[4,6,7,2,8,5,1,3]=>[8,7,4,2,6,1,5,3]=>[8,3,4,2,7,5,6,1]=>1010101 [(1,6),(2,4),(3,8),(5,7)]=>[4,6,7,2,8,1,5,3]=>[8,7,4,2,6,5,1,3]=>[6,5,8,3,4,2,7,1]=>1010101 [(1,5),(2,4),(3,8),(6,7)]=>[4,5,7,2,1,8,6,3]=>[8,5,4,2,3,7,1,6]=>[5,3,4,2,8,6,7,1]=>1010101 [(1,4),(2,5),(3,8),(6,7)]=>[4,5,7,1,2,8,6,3]=>[8,5,4,3,2,7,1,6]=>[4,3,5,2,8,6,7,1]=>1010101 [(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,7,2,8,6,4]=>[8,3,2,5,4,7,1,6]=>[3,2,5,4,8,6,7,1]=>1010101 [(1,2),(3,5),(4,8),(6,7)]=>[2,1,5,7,3,8,6,4]=>[2,1,8,5,4,7,3,6]=>[2,1,5,4,8,6,7,3]=>1010101 [(1,2),(3,6),(4,8),(5,7)]=>[2,1,6,7,8,3,5,4]=>[2,1,8,7,6,5,3,4]=>[2,1,6,5,8,4,7,3]=>1010101 [(1,3),(2,6),(4,8),(5,7)]=>[3,6,1,7,8,2,5,4]=>[8,3,2,7,6,5,1,4]=>[3,2,6,5,8,4,7,1]=>1010101 [(1,4),(2,6),(3,8),(5,7)]=>[4,6,7,1,8,2,5,3]=>[8,7,4,3,6,5,1,2]=>[4,3,6,5,8,2,7,1]=>1010101 [(1,5),(2,6),(3,8),(4,7)]=>[5,6,7,8,1,2,4,3]=>[8,7,6,5,4,3,1,2]=>[5,4,6,3,8,2,7,1]=>1010101 [(1,6),(2,5),(3,8),(4,7)]=>[5,6,7,8,2,1,4,3]=>[8,7,6,5,3,4,1,2]=>[6,4,5,3,8,2,7,1]=>1010101 [(1,7),(2,5),(3,8),(4,6)]=>[5,6,7,8,2,4,1,3]=>[8,7,6,5,3,1,4,2]=>[8,2,7,4,5,3,6,1]=>1010101 [(1,8),(2,5),(3,7),(4,6)]=>[5,6,7,8,2,4,3,1]=>[8,7,6,5,3,1,2,4]=>[7,2,8,4,5,3,6,1]=>1010101 [(1,8),(2,6),(3,7),(4,5)]=>[5,6,7,8,4,2,3,1]=>[8,7,6,5,1,3,2,4]=>[6,3,7,2,8,4,5,1]=>1010101 [(1,7),(2,6),(3,8),(4,5)]=>[5,6,7,8,4,2,1,3]=>[8,7,6,5,1,3,4,2]=>[6,3,8,2,7,4,5,1]=>1010101 [(1,6),(2,7),(3,8),(4,5)]=>[5,6,7,8,4,1,2,3]=>[8,7,6,5,1,4,3,2]=>[8,2,7,3,6,4,5,1]=>1010101 [(1,5),(2,7),(3,8),(4,6)]=>[5,6,7,8,1,4,2,3]=>[8,7,6,5,4,1,3,2]=>[5,4,8,2,7,3,6,1]=>1010101 [(1,4),(2,7),(3,8),(5,6)]=>[4,6,7,1,8,5,2,3]=>[8,7,4,3,6,1,5,2]=>[4,3,8,2,7,5,6,1]=>1010101 [(1,3),(2,7),(4,8),(5,6)]=>[3,6,1,7,8,5,2,4]=>[8,3,2,7,6,1,5,4]=>[3,2,8,4,7,5,6,1]=>1010101 [(1,2),(3,7),(4,8),(5,6)]=>[2,1,6,7,8,5,3,4]=>[2,1,8,7,6,3,5,4]=>[2,1,8,4,7,5,6,3]=>1010101 [(1,2),(3,8),(4,7),(5,6)]=>[2,1,6,7,8,5,4,3]=>[2,1,8,7,6,3,4,5]=>[2,1,7,4,8,5,6,3]=>1010101 [(1,3),(2,8),(4,7),(5,6)]=>[3,6,1,7,8,5,4,2]=>[8,3,2,7,6,1,4,5]=>[3,2,7,4,8,5,6,1]=>1010101 [(1,4),(2,8),(3,7),(5,6)]=>[4,6,7,1,8,5,3,2]=>[8,7,4,3,6,1,2,5]=>[4,3,7,2,8,5,6,1]=>1010101 [(1,5),(2,8),(3,7),(4,6)]=>[5,6,7,8,1,4,3,2]=>[8,7,6,5,4,1,2,3]=>[5,4,7,2,8,3,6,1]=>1010101 [(1,6),(2,8),(3,7),(4,5)]=>[5,6,7,8,4,1,3,2]=>[8,7,6,5,1,4,2,3]=>[7,2,8,3,6,4,5,1]=>1010101 [(1,7),(2,8),(3,6),(4,5)]=>[5,6,7,8,4,3,1,2]=>[8,7,6,5,1,2,4,3]=>[8,3,6,2,7,4,5,1]=>1010101 [(1,8),(2,7),(3,6),(4,5)]=>[5,6,7,8,4,3,2,1]=>[8,7,6,5,1,2,3,4]=>[7,3,6,2,8,4,5,1]=>1010101 [(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]=>[2,1,4,3,6,5,8,7,10,9]=>101010101 [(1,3),(2,4),(5,6),(7,8),(9,10)]=>[3,4,1,2,6,5,8,7,10,9]=>[4,3,2,1,6,5,8,7,10,9]=>[3,2,4,1,6,5,8,7,10,9]=>101010101 [(1,10),(2,3),(4,5),(6,7),(8,9)]=>[3,5,2,7,4,9,6,10,8,1]=>[10,3,1,5,2,7,4,9,6,8]=>[10,8,9,6,7,4,5,2,3,1]=>101010101 [(1,6),(2,7),(3,8),(4,9),(5,10)]=>[6,7,8,9,10,1,2,3,4,5]=>[10,9,8,7,6,5,4,3,2,1]=>[6,5,7,4,8,3,9,2,10,1]=>101010101 [(1,3),(2,5),(4,7),(6,9),(8,10)]=>[3,5,1,7,2,9,4,10,6,8]=>[10,3,2,5,4,7,6,9,8,1]=>[3,2,5,4,7,6,9,8,10,1]=>101010101 [(1,6),(2,10),(3,9),(4,8),(5,7)]=>[6,7,8,9,10,1,5,4,3,2]=>[10,9,8,7,6,5,1,2,3,4]=>[6,5,9,3,8,2,10,4,7,1]=>101010101 [(1,10),(2,9),(3,8),(4,7),(5,6)]=>[6,7,8,9,10,5,4,3,2,1]=>[10,9,8,7,6,1,2,3,4,5]=>[8,3,9,4,7,2,10,5,6,1]=>101010101 [(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]=>[2,1,4,3,6,5,8,7,10,9,12,11]=>10101010101 [(1,3),(2,5),(4,7),(6,9),(8,11),(10,12)]=>[3,5,1,7,2,9,4,11,6,12,8,10]=>[12,3,2,5,4,7,6,9,8,11,10,1]=>[3,2,5,4,7,6,9,8,11,10,12,1]=>10101010101
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.
Map
Corteel
Description
Corteel's map interchanging the number of crossings and the number of nestings of a permutation.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.
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.
Map
descent word
Description
The descent positions of a permutation as a binary word.
For a permutation $\pi$ of $n$ letters and each $1\leq i\leq n-1$ such that $\pi(i) > \pi(i+1)$ we set $w_i=1$, otherwise $w_i=0$.
Thus, the length of the word is one less the size of the permutation. In particular, the descent word is undefined for the empty permutation.