Identifier
Mp00283: non-nesting-exceedence permutationPermutations
Mp00087: Permutations inverse first fundamental transformationPermutations
Mp00064: Permutations reversePermutations
Mp00109: Permutations descent wordBinary words
Images
=>
Cc0012;cc-rep-0
[(1,2)]=>[2,1]=>[2,1]=>[1,2]=>0 [(1,2),(3,4)]=>[2,1,4,3]=>[2,1,4,3]=>[3,4,1,2]=>010 [(1,3),(2,4)]=>[3,4,1,2]=>[3,1,4,2]=>[2,4,1,3]=>010 [(1,4),(2,3)]=>[3,4,2,1]=>[4,1,3,2]=>[2,3,1,4]=>010 [(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5]=>[5,6,3,4,1,2]=>01010 [(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[3,1,4,2,6,5]=>[5,6,2,4,1,3]=>01010 [(1,4),(2,3),(5,6)]=>[3,4,2,1,6,5]=>[4,1,3,2,6,5]=>[5,6,2,3,1,4]=>01010 [(1,5),(2,3),(4,6)]=>[3,5,2,6,1,4]=>[5,1,3,2,6,4]=>[4,6,2,3,1,5]=>01010 [(1,6),(2,3),(4,5)]=>[3,5,2,6,4,1]=>[6,1,3,2,5,4]=>[4,5,2,3,1,6]=>01010 [(1,6),(2,4),(3,5)]=>[4,5,6,2,3,1]=>[6,1,4,2,5,3]=>[3,5,2,4,1,6]=>01010 [(1,5),(2,4),(3,6)]=>[4,5,6,2,1,3]=>[5,1,4,2,6,3]=>[3,6,2,4,1,5]=>01010 [(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[4,1,5,2,6,3]=>[3,6,2,5,1,4]=>01010 [(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[3,1,5,2,6,4]=>[4,6,2,5,1,3]=>01010 [(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[2,1,5,3,6,4]=>[4,6,3,5,1,2]=>01010 [(1,2),(3,6),(4,5)]=>[2,1,5,6,4,3]=>[2,1,6,3,5,4]=>[4,5,3,6,1,2]=>01010 [(1,3),(2,6),(4,5)]=>[3,5,1,6,4,2]=>[3,1,6,2,5,4]=>[4,5,2,6,1,3]=>01010 [(1,4),(2,6),(3,5)]=>[4,5,6,1,3,2]=>[4,1,6,2,5,3]=>[3,5,2,6,1,4]=>01010 [(1,5),(2,6),(3,4)]=>[4,5,6,3,1,2]=>[6,2,5,1,4,3]=>[3,4,1,5,2,6]=>01010 [(1,6),(2,5),(3,4)]=>[4,5,6,3,2,1]=>[5,2,6,1,4,3]=>[3,4,1,6,2,5]=>01010 [(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[2,1,4,3,6,5,8,7]=>[7,8,5,6,3,4,1,2]=>0101010 [(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[3,1,4,2,6,5,8,7]=>[7,8,5,6,2,4,1,3]=>0101010 [(1,4),(2,3),(5,6),(7,8)]=>[3,4,2,1,6,5,8,7]=>[4,1,3,2,6,5,8,7]=>[7,8,5,6,2,3,1,4]=>0101010 [(1,5),(2,3),(4,6),(7,8)]=>[3,5,2,6,1,4,8,7]=>[5,1,3,2,6,4,8,7]=>[7,8,4,6,2,3,1,5]=>0101010 [(1,6),(2,3),(4,5),(7,8)]=>[3,5,2,6,4,1,8,7]=>[6,1,3,2,5,4,8,7]=>[7,8,4,5,2,3,1,6]=>0101010 [(1,7),(2,3),(4,5),(6,8)]=>[3,5,2,7,4,8,1,6]=>[7,1,3,2,5,4,8,6]=>[6,8,4,5,2,3,1,7]=>0101010 [(1,8),(2,3),(4,5),(6,7)]=>[3,5,2,7,4,8,6,1]=>[8,1,3,2,5,4,7,6]=>[6,7,4,5,2,3,1,8]=>0101010 [(1,8),(2,4),(3,5),(6,7)]=>[4,5,7,2,3,8,6,1]=>[8,1,4,2,5,3,7,6]=>[6,7,3,5,2,4,1,8]=>0101010 [(1,7),(2,4),(3,5),(6,8)]=>[4,5,7,2,3,8,1,6]=>[7,1,4,2,5,3,8,6]=>[6,8,3,5,2,4,1,7]=>0101010 [(1,6),(2,4),(3,5),(7,8)]=>[4,5,6,2,3,1,8,7]=>[6,1,4,2,5,3,8,7]=>[7,8,3,5,2,4,1,6]=>0101010 [(1,5),(2,4),(3,6),(7,8)]=>[4,5,6,2,1,3,8,7]=>[5,1,4,2,6,3,8,7]=>[7,8,3,6,2,4,1,5]=>0101010 [(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[4,1,5,2,6,3,8,7]=>[7,8,3,6,2,5,1,4]=>0101010 [(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[3,1,5,2,6,4,8,7]=>[7,8,4,6,2,5,1,3]=>0101010 [(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[2,1,5,3,6,4,8,7]=>[7,8,4,6,3,5,1,2]=>0101010 [(1,2),(3,6),(4,5),(7,8)]=>[2,1,5,6,4,3,8,7]=>[2,1,6,3,5,4,8,7]=>[7,8,4,5,3,6,1,2]=>0101010 [(1,3),(2,6),(4,5),(7,8)]=>[3,5,1,6,4,2,8,7]=>[3,1,6,2,5,4,8,7]=>[7,8,4,5,2,6,1,3]=>0101010 [(1,4),(2,6),(3,5),(7,8)]=>[4,5,6,1,3,2,8,7]=>[4,1,6,2,5,3,8,7]=>[7,8,3,5,2,6,1,4]=>0101010 [(1,5),(2,6),(3,4),(7,8)]=>[4,5,6,3,1,2,8,7]=>[6,2,5,1,4,3,8,7]=>[7,8,3,4,1,5,2,6]=>0101010 [(1,6),(2,5),(3,4),(7,8)]=>[4,5,6,3,2,1,8,7]=>[5,2,6,1,4,3,8,7]=>[7,8,3,4,1,6,2,5]=>0101010 [(1,7),(2,5),(3,4),(6,8)]=>[4,5,7,3,2,8,1,6]=>[5,2,7,1,4,3,8,6]=>[6,8,3,4,1,7,2,5]=>0101010 [(1,8),(2,5),(3,4),(6,7)]=>[4,5,7,3,2,8,6,1]=>[5,2,8,1,4,3,7,6]=>[6,7,3,4,1,8,2,5]=>0101010 [(1,8),(2,6),(3,4),(5,7)]=>[4,6,7,3,8,2,5,1]=>[6,2,8,1,4,3,7,5]=>[5,7,3,4,1,8,2,6]=>0101010 [(1,7),(2,6),(3,4),(5,8)]=>[4,6,7,3,8,2,1,5]=>[6,2,7,1,4,3,8,5]=>[5,8,3,4,1,7,2,6]=>0101010 [(1,6),(2,7),(3,4),(5,8)]=>[4,6,7,3,8,1,2,5]=>[7,2,6,1,4,3,8,5]=>[5,8,3,4,1,6,2,7]=>0101010 [(1,5),(2,7),(3,4),(6,8)]=>[4,5,7,3,1,8,2,6]=>[7,2,5,1,4,3,8,6]=>[6,8,3,4,1,5,2,7]=>0101010 [(1,4),(2,7),(3,5),(6,8)]=>[4,5,7,1,3,8,2,6]=>[4,1,7,2,5,3,8,6]=>[6,8,3,5,2,7,1,4]=>0101010 [(1,3),(2,7),(4,5),(6,8)]=>[3,5,1,7,4,8,2,6]=>[3,1,7,2,5,4,8,6]=>[6,8,4,5,2,7,1,3]=>0101010 [(1,2),(3,7),(4,5),(6,8)]=>[2,1,5,7,4,8,3,6]=>[2,1,7,3,5,4,8,6]=>[6,8,4,5,3,7,1,2]=>0101010 [(1,2),(3,8),(4,5),(6,7)]=>[2,1,5,7,4,8,6,3]=>[2,1,8,3,5,4,7,6]=>[6,7,4,5,3,8,1,2]=>0101010 [(1,3),(2,8),(4,5),(6,7)]=>[3,5,1,7,4,8,6,2]=>[3,1,8,2,5,4,7,6]=>[6,7,4,5,2,8,1,3]=>0101010 [(1,4),(2,8),(3,5),(6,7)]=>[4,5,7,1,3,8,6,2]=>[4,1,8,2,5,3,7,6]=>[6,7,3,5,2,8,1,4]=>0101010 [(1,5),(2,8),(3,4),(6,7)]=>[4,5,7,3,1,8,6,2]=>[8,2,5,1,4,3,7,6]=>[6,7,3,4,1,5,2,8]=>0101010 [(1,6),(2,8),(3,4),(5,7)]=>[4,6,7,3,8,1,5,2]=>[8,2,6,1,4,3,7,5]=>[5,7,3,4,1,6,2,8]=>0101010 [(1,7),(2,8),(3,4),(5,6)]=>[4,6,7,3,8,5,1,2]=>[7,1,4,3,8,2,6,5]=>[5,6,2,8,3,4,1,7]=>0101010 [(1,8),(2,7),(3,4),(5,6)]=>[4,6,7,3,8,5,2,1]=>[8,1,4,3,7,2,6,5]=>[5,6,2,7,3,4,1,8]=>0101010 [(1,8),(2,7),(3,5),(4,6)]=>[5,6,7,8,3,4,2,1]=>[8,1,5,3,7,2,6,4]=>[4,6,2,7,3,5,1,8]=>0101010 [(1,7),(2,8),(3,5),(4,6)]=>[5,6,7,8,3,4,1,2]=>[7,1,5,3,8,2,6,4]=>[4,6,2,8,3,5,1,7]=>0101010 [(1,6),(2,8),(3,5),(4,7)]=>[5,6,7,8,3,1,4,2]=>[8,2,6,1,5,3,7,4]=>[4,7,3,5,1,6,2,8]=>0101010 [(1,5),(2,8),(3,6),(4,7)]=>[5,6,7,8,1,3,4,2]=>[5,1,8,2,6,3,7,4]=>[4,7,3,6,2,8,1,5]=>0101010 [(1,4),(2,8),(3,6),(5,7)]=>[4,6,7,1,8,3,5,2]=>[4,1,8,2,6,3,7,5]=>[5,7,3,6,2,8,1,4]=>0101010 [(1,3),(2,8),(4,6),(5,7)]=>[3,6,1,7,8,4,5,2]=>[3,1,8,2,6,4,7,5]=>[5,7,4,6,2,8,1,3]=>0101010 [(1,2),(3,8),(4,6),(5,7)]=>[2,1,6,7,8,4,5,3]=>[2,1,8,3,6,4,7,5]=>[5,7,4,6,3,8,1,2]=>0101010 [(1,2),(3,7),(4,6),(5,8)]=>[2,1,6,7,8,4,3,5]=>[2,1,7,3,6,4,8,5]=>[5,8,4,6,3,7,1,2]=>0101010 [(1,3),(2,7),(4,6),(5,8)]=>[3,6,1,7,8,4,2,5]=>[3,1,7,2,6,4,8,5]=>[5,8,4,6,2,7,1,3]=>0101010 [(1,4),(2,7),(3,6),(5,8)]=>[4,6,7,1,8,3,2,5]=>[4,1,7,2,6,3,8,5]=>[5,8,3,6,2,7,1,4]=>0101010 [(1,5),(2,7),(3,6),(4,8)]=>[5,6,7,8,1,3,2,4]=>[5,1,7,2,6,3,8,4]=>[4,8,3,6,2,7,1,5]=>0101010 [(1,6),(2,7),(3,5),(4,8)]=>[5,6,7,8,3,1,2,4]=>[7,2,6,1,5,3,8,4]=>[4,8,3,5,1,6,2,7]=>0101010 [(1,7),(2,6),(3,5),(4,8)]=>[5,6,7,8,3,2,1,4]=>[6,2,7,1,5,3,8,4]=>[4,8,3,5,1,7,2,6]=>0101010 [(1,8),(2,6),(3,5),(4,7)]=>[5,6,7,8,3,2,4,1]=>[6,2,8,1,5,3,7,4]=>[4,7,3,5,1,8,2,6]=>0101010 [(1,8),(2,5),(3,6),(4,7)]=>[5,6,7,8,2,3,4,1]=>[8,1,5,2,6,3,7,4]=>[4,7,3,6,2,5,1,8]=>0101010 [(1,7),(2,5),(3,6),(4,8)]=>[5,6,7,8,2,3,1,4]=>[7,1,5,2,6,3,8,4]=>[4,8,3,6,2,5,1,7]=>0101010 [(1,6),(2,5),(3,7),(4,8)]=>[5,6,7,8,2,1,3,4]=>[6,1,5,2,7,3,8,4]=>[4,8,3,7,2,5,1,6]=>0101010 [(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[5,1,6,2,7,3,8,4]=>[4,8,3,7,2,6,1,5]=>0101010 [(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[4,1,6,2,7,3,8,5]=>[5,8,3,7,2,6,1,4]=>0101010 [(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[3,1,6,2,7,4,8,5]=>[5,8,4,7,2,6,1,3]=>0101010 [(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[2,1,6,3,7,4,8,5]=>[5,8,4,7,3,6,1,2]=>0101010 [(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[2,1,5,3,7,4,8,6]=>[6,8,4,7,3,5,1,2]=>0101010 [(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[3,1,5,2,7,4,8,6]=>[6,8,4,7,2,5,1,3]=>0101010 [(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[4,1,5,2,7,3,8,6]=>[6,8,3,7,2,5,1,4]=>0101010 [(1,5),(2,4),(3,7),(6,8)]=>[4,5,7,2,1,8,3,6]=>[5,1,4,2,7,3,8,6]=>[6,8,3,7,2,4,1,5]=>0101010 [(1,6),(2,4),(3,7),(5,8)]=>[4,6,7,2,8,1,3,5]=>[6,1,4,2,7,3,8,5]=>[5,8,3,7,2,4,1,6]=>0101010 [(1,7),(2,4),(3,6),(5,8)]=>[4,6,7,2,8,3,1,5]=>[7,1,4,2,6,3,8,5]=>[5,8,3,6,2,4,1,7]=>0101010 [(1,8),(2,4),(3,6),(5,7)]=>[4,6,7,2,8,3,5,1]=>[8,1,4,2,6,3,7,5]=>[5,7,3,6,2,4,1,8]=>0101010 [(1,8),(2,3),(4,6),(5,7)]=>[3,6,2,7,8,4,5,1]=>[8,1,3,2,6,4,7,5]=>[5,7,4,6,2,3,1,8]=>0101010 [(1,7),(2,3),(4,6),(5,8)]=>[3,6,2,7,8,4,1,5]=>[7,1,3,2,6,4,8,5]=>[5,8,4,6,2,3,1,7]=>0101010 [(1,6),(2,3),(4,7),(5,8)]=>[3,6,2,7,8,1,4,5]=>[6,1,3,2,7,4,8,5]=>[5,8,4,7,2,3,1,6]=>0101010 [(1,5),(2,3),(4,7),(6,8)]=>[3,5,2,7,1,8,4,6]=>[5,1,3,2,7,4,8,6]=>[6,8,4,7,2,3,1,5]=>0101010 [(1,4),(2,3),(5,7),(6,8)]=>[3,4,2,1,7,8,5,6]=>[4,1,3,2,7,5,8,6]=>[6,8,5,7,2,3,1,4]=>0101010 [(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[3,1,4,2,7,5,8,6]=>[6,8,5,7,2,4,1,3]=>0101010 [(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[2,1,4,3,7,5,8,6]=>[6,8,5,7,3,4,1,2]=>0101010 [(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,7,8,6,5]=>[2,1,4,3,8,5,7,6]=>[6,7,5,8,3,4,1,2]=>0101010 [(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,7,8,6,5]=>[3,1,4,2,8,5,7,6]=>[6,7,5,8,2,4,1,3]=>0101010 [(1,4),(2,3),(5,8),(6,7)]=>[3,4,2,1,7,8,6,5]=>[4,1,3,2,8,5,7,6]=>[6,7,5,8,2,3,1,4]=>0101010 [(1,5),(2,3),(4,8),(6,7)]=>[3,5,2,7,1,8,6,4]=>[5,1,3,2,8,4,7,6]=>[6,7,4,8,2,3,1,5]=>0101010 [(1,6),(2,3),(4,8),(5,7)]=>[3,6,2,7,8,1,5,4]=>[6,1,3,2,8,4,7,5]=>[5,7,4,8,2,3,1,6]=>0101010 [(1,7),(2,3),(4,8),(5,6)]=>[3,6,2,7,8,5,1,4]=>[8,4,7,1,3,2,6,5]=>[5,6,2,3,1,7,4,8]=>0101010 [(1,8),(2,3),(4,7),(5,6)]=>[3,6,2,7,8,5,4,1]=>[7,4,8,1,3,2,6,5]=>[5,6,2,3,1,8,4,7]=>0101010 [(1,8),(2,4),(3,7),(5,6)]=>[4,6,7,2,8,5,3,1]=>[7,3,8,1,4,2,6,5]=>[5,6,2,4,1,8,3,7]=>0101010 [(1,7),(2,4),(3,8),(5,6)]=>[4,6,7,2,8,5,1,3]=>[8,3,7,1,4,2,6,5]=>[5,6,2,4,1,7,3,8]=>0101010 [(1,6),(2,4),(3,8),(5,7)]=>[4,6,7,2,8,1,5,3]=>[6,1,4,2,8,3,7,5]=>[5,7,3,8,2,4,1,6]=>0101010 [(1,5),(2,4),(3,8),(6,7)]=>[4,5,7,2,1,8,6,3]=>[5,1,4,2,8,3,7,6]=>[6,7,3,8,2,4,1,5]=>0101010 [(1,4),(2,5),(3,8),(6,7)]=>[4,5,7,1,2,8,6,3]=>[4,1,5,2,8,3,7,6]=>[6,7,3,8,2,5,1,4]=>0101010 [(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,7,2,8,6,4]=>[3,1,5,2,8,4,7,6]=>[6,7,4,8,2,5,1,3]=>0101010 [(1,2),(3,5),(4,8),(6,7)]=>[2,1,5,7,3,8,6,4]=>[2,1,5,3,8,4,7,6]=>[6,7,4,8,3,5,1,2]=>0101010 [(1,2),(3,6),(4,8),(5,7)]=>[2,1,6,7,8,3,5,4]=>[2,1,6,3,8,4,7,5]=>[5,7,4,8,3,6,1,2]=>0101010 [(1,3),(2,6),(4,8),(5,7)]=>[3,6,1,7,8,2,5,4]=>[3,1,6,2,8,4,7,5]=>[5,7,4,8,2,6,1,3]=>0101010 [(1,4),(2,6),(3,8),(5,7)]=>[4,6,7,1,8,2,5,3]=>[4,1,6,2,8,3,7,5]=>[5,7,3,8,2,6,1,4]=>0101010 [(1,5),(2,6),(3,8),(4,7)]=>[5,6,7,8,1,2,4,3]=>[5,1,6,2,8,3,7,4]=>[4,7,3,8,2,6,1,5]=>0101010 [(1,6),(2,5),(3,8),(4,7)]=>[5,6,7,8,2,1,4,3]=>[6,1,5,2,8,3,7,4]=>[4,7,3,8,2,5,1,6]=>0101010 [(1,7),(2,5),(3,8),(4,6)]=>[5,6,7,8,2,4,1,3]=>[8,3,7,1,5,2,6,4]=>[4,6,2,5,1,7,3,8]=>0101010 [(1,8),(2,5),(3,7),(4,6)]=>[5,6,7,8,2,4,3,1]=>[7,3,8,1,5,2,6,4]=>[4,6,2,5,1,8,3,7]=>0101010 [(1,8),(2,6),(3,7),(4,5)]=>[5,6,7,8,4,2,3,1]=>[6,2,7,3,8,1,5,4]=>[4,5,1,8,3,7,2,6]=>0101010 [(1,7),(2,6),(3,8),(4,5)]=>[5,6,7,8,4,2,1,3]=>[6,2,8,3,7,1,5,4]=>[4,5,1,7,3,8,2,6]=>0101010 [(1,6),(2,7),(3,8),(4,5)]=>[5,6,7,8,4,1,2,3]=>[8,3,7,2,6,1,5,4]=>[4,5,1,6,2,7,3,8]=>0101010 [(1,5),(2,7),(3,8),(4,6)]=>[5,6,7,8,1,4,2,3]=>[5,1,8,3,7,2,6,4]=>[4,6,2,7,3,8,1,5]=>0101010 [(1,4),(2,7),(3,8),(5,6)]=>[4,6,7,1,8,5,2,3]=>[4,1,8,3,7,2,6,5]=>[5,6,2,7,3,8,1,4]=>0101010 [(1,3),(2,7),(4,8),(5,6)]=>[3,6,1,7,8,5,2,4]=>[3,1,8,4,7,2,6,5]=>[5,6,2,7,4,8,1,3]=>0101010 [(1,2),(3,7),(4,8),(5,6)]=>[2,1,6,7,8,5,3,4]=>[2,1,8,4,7,3,6,5]=>[5,6,3,7,4,8,1,2]=>0101010 [(1,2),(3,8),(4,7),(5,6)]=>[2,1,6,7,8,5,4,3]=>[2,1,7,4,8,3,6,5]=>[5,6,3,8,4,7,1,2]=>0101010 [(1,3),(2,8),(4,7),(5,6)]=>[3,6,1,7,8,5,4,2]=>[3,1,7,4,8,2,6,5]=>[5,6,2,8,4,7,1,3]=>0101010 [(1,4),(2,8),(3,7),(5,6)]=>[4,6,7,1,8,5,3,2]=>[4,1,7,3,8,2,6,5]=>[5,6,2,8,3,7,1,4]=>0101010 [(1,5),(2,8),(3,7),(4,6)]=>[5,6,7,8,1,4,3,2]=>[5,1,7,3,8,2,6,4]=>[4,6,2,8,3,7,1,5]=>0101010 [(1,6),(2,8),(3,7),(4,5)]=>[5,6,7,8,4,1,3,2]=>[7,3,8,2,6,1,5,4]=>[4,5,1,6,2,8,3,7]=>0101010 [(1,7),(2,8),(3,6),(4,5)]=>[5,6,7,8,4,3,1,2]=>[8,2,6,3,7,1,5,4]=>[4,5,1,7,3,6,2,8]=>0101010 [(1,8),(2,7),(3,6),(4,5)]=>[5,6,7,8,4,3,2,1]=>[7,2,6,3,8,1,5,4]=>[4,5,1,8,3,6,2,7]=>0101010 [(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]=>[9,10,7,8,5,6,3,4,1,2]=>010101010 [(1,3),(2,4),(5,6),(7,8),(9,10)]=>[3,4,1,2,6,5,8,7,10,9]=>[3,1,4,2,6,5,8,7,10,9]=>[9,10,7,8,5,6,2,4,1,3]=>010101010 [(1,8),(2,3),(4,5),(6,7),(9,10)]=>[3,5,2,7,4,8,6,1,10,9]=>[8,1,3,2,5,4,7,6,10,9]=>[9,10,6,7,4,5,2,3,1,8]=>010101010 [(1,4),(2,5),(3,6),(7,8),(9,10)]=>[4,5,6,1,2,3,8,7,10,9]=>[4,1,5,2,6,3,8,7,10,9]=>[9,10,7,8,3,6,2,5,1,4]=>010101010 [(1,3),(2,5),(4,6),(7,8),(9,10)]=>[3,5,1,6,2,4,8,7,10,9]=>[3,1,5,2,6,4,8,7,10,9]=>[9,10,7,8,4,6,2,5,1,3]=>010101010 [(1,2),(3,5),(4,6),(7,8),(9,10)]=>[2,1,5,6,3,4,8,7,10,9]=>[2,1,5,3,6,4,8,7,10,9]=>[9,10,7,8,4,6,3,5,1,2]=>010101010 [(1,5),(2,6),(3,7),(4,8),(9,10)]=>[5,6,7,8,1,2,3,4,10,9]=>[5,1,6,2,7,3,8,4,10,9]=>[9,10,4,8,3,7,2,6,1,5]=>010101010 [(1,3),(2,5),(4,7),(6,8),(9,10)]=>[3,5,1,7,2,8,4,6,10,9]=>[3,1,5,2,7,4,8,6,10,9]=>[9,10,6,8,4,7,2,5,1,3]=>010101010 [(1,3),(2,4),(5,7),(6,8),(9,10)]=>[3,4,1,2,7,8,5,6,10,9]=>[3,1,4,2,7,5,8,6,10,9]=>[9,10,6,8,5,7,2,4,1,3]=>010101010 [(1,2),(3,4),(5,7),(6,8),(9,10)]=>[2,1,4,3,7,8,5,6,10,9]=>[2,1,4,3,7,5,8,6,10,9]=>[9,10,6,8,5,7,3,4,1,2]=>010101010 [(1,2),(3,7),(4,8),(5,9),(6,10)]=>[2,1,7,8,9,10,3,4,5,6]=>[2,1,7,3,8,4,9,5,10,6]=>[6,10,5,9,4,8,3,7,1,2]=>010101010 [(1,5),(2,7),(3,8),(4,9),(6,10)]=>[5,7,8,9,1,10,2,3,4,6]=>[5,1,7,2,8,3,9,4,10,6]=>[6,10,4,9,3,8,2,7,1,5]=>010101010 [(1,6),(2,7),(3,8),(4,9),(5,10)]=>[6,7,8,9,10,1,2,3,4,5]=>[6,1,7,2,8,3,9,4,10,5]=>[5,10,4,9,3,8,2,7,1,6]=>010101010 [(1,4),(2,6),(3,8),(5,9),(7,10)]=>[4,6,8,1,9,2,10,3,5,7]=>[4,1,6,2,8,3,9,5,10,7]=>[7,10,5,9,3,8,2,6,1,4]=>010101010 [(1,2),(3,4),(5,7),(6,9),(8,10)]=>[2,1,4,3,7,9,5,10,6,8]=>[2,1,4,3,7,5,9,6,10,8]=>[8,10,6,9,5,7,3,4,1,2]=>010101010 [(1,3),(2,5),(4,7),(6,9),(8,10)]=>[3,5,1,7,2,9,4,10,6,8]=>[3,1,5,2,7,4,9,6,10,8]=>[8,10,6,9,4,7,2,5,1,3]=>010101010 [(1,2),(3,5),(4,7),(6,9),(8,10)]=>[2,1,5,7,3,9,4,10,6,8]=>[2,1,5,3,7,4,9,6,10,8]=>[8,10,6,9,4,7,3,5,1,2]=>010101010 [(1,2),(3,5),(4,6),(7,9),(8,10)]=>[2,1,5,6,3,4,9,10,7,8]=>[2,1,5,3,6,4,9,7,10,8]=>[8,10,7,9,4,6,3,5,1,2]=>010101010 [(1,3),(2,4),(5,6),(7,9),(8,10)]=>[3,4,1,2,6,5,9,10,7,8]=>[3,1,4,2,6,5,9,7,10,8]=>[8,10,7,9,5,6,2,4,1,3]=>010101010 [(1,2),(3,4),(5,6),(7,9),(8,10)]=>[2,1,4,3,6,5,9,10,7,8]=>[2,1,4,3,6,5,9,7,10,8]=>[8,10,7,9,5,6,3,4,1,2]=>010101010 [(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]=>[11,12,9,10,7,8,5,6,3,4,1,2]=>01010101010 [(1,2),(3,4),(5,6),(7,8),(9,11),(10,12)]=>[2,1,4,3,6,5,8,7,11,12,9,10]=>[2,1,4,3,6,5,8,7,11,9,12,10]=>[10,12,9,11,7,8,5,6,3,4,1,2]=>01010101010 [(1,2),(3,4),(5,6),(7,9),(8,10),(11,12)]=>[2,1,4,3,6,5,9,10,7,8,12,11]=>[2,1,4,3,6,5,9,7,10,8,12,11]=>[11,12,8,10,7,9,5,6,3,4,1,2]=>01010101010 [(1,2),(3,4),(5,7),(6,8),(9,10),(11,12)]=>[2,1,4,3,7,8,5,6,10,9,12,11]=>[2,1,4,3,7,5,8,6,10,9,12,11]=>[11,12,9,10,6,8,5,7,3,4,1,2]=>01010101010 [(1,2),(3,4),(5,7),(6,8),(9,11),(10,12)]=>[2,1,4,3,7,8,5,6,11,12,9,10]=>[2,1,4,3,7,5,8,6,11,9,12,10]=>[10,12,9,11,6,8,5,7,3,4,1,2]=>01010101010 [(1,2),(3,5),(4,6),(7,8),(9,10),(11,12)]=>[2,1,5,6,3,4,8,7,10,9,12,11]=>[2,1,5,3,6,4,8,7,10,9,12,11]=>[11,12,9,10,7,8,4,6,3,5,1,2]=>01010101010 [(1,2),(3,5),(4,6),(7,8),(9,11),(10,12)]=>[2,1,5,6,3,4,8,7,11,12,9,10]=>[2,1,5,3,6,4,8,7,11,9,12,10]=>[10,12,9,11,7,8,4,6,3,5,1,2]=>01010101010 [(1,2),(3,5),(4,6),(7,9),(8,10),(11,12)]=>[2,1,5,6,3,4,9,10,7,8,12,11]=>[2,1,5,3,6,4,9,7,10,8,12,11]=>[11,12,8,10,7,9,4,6,3,5,1,2]=>01010101010 [(1,3),(2,4),(5,6),(7,8),(9,10),(11,12)]=>[3,4,1,2,6,5,8,7,10,9,12,11]=>[3,1,4,2,6,5,8,7,10,9,12,11]=>[11,12,9,10,7,8,5,6,2,4,1,3]=>01010101010 [(1,3),(2,4),(5,6),(7,8),(9,11),(10,12)]=>[3,4,1,2,6,5,8,7,11,12,9,10]=>[3,1,4,2,6,5,8,7,11,9,12,10]=>[10,12,9,11,7,8,5,6,2,4,1,3]=>01010101010 [(1,3),(2,4),(5,6),(7,9),(8,10),(11,12)]=>[3,4,1,2,6,5,9,10,7,8,12,11]=>[3,1,4,2,6,5,9,7,10,8,12,11]=>[11,12,8,10,7,9,5,6,2,4,1,3]=>01010101010 [(1,3),(2,4),(5,7),(6,8),(9,10),(11,12)]=>[3,4,1,2,7,8,5,6,10,9,12,11]=>[3,1,4,2,7,5,8,6,10,9,12,11]=>[11,12,9,10,6,8,5,7,2,4,1,3]=>01010101010 [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12)]=>[3,4,1,2,7,8,5,6,11,12,9,10]=>[3,1,4,2,7,5,8,6,11,9,12,10]=>[10,12,9,11,6,8,5,7,2,4,1,3]=>01010101010 [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12)]=>[7,8,9,10,11,12,1,2,3,4,5,6]=>[7,1,8,2,9,3,10,4,11,5,12,6]=>[6,12,5,11,4,10,3,9,2,8,1,7]=>01010101010
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
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
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
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.