Identifier
Mp00283:
Perfect matchings
—non-nesting-exceedence permutation⟶
Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Images
=>
Cc0012;cc-rep-0
[(1,2)]=>[2,1]=>[2]=>[1,1]
[(1,2),(3,4)]=>[2,1,4,3]=>[2,2]=>[1,2,1]
[(1,3),(2,4)]=>[3,4,1,2]=>[4]=>[1,1,1,1]
[(1,4),(2,3)]=>[3,4,2,1]=>[3,1]=>[1,1,2]
[(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[2,2,2]=>[1,2,2,1]
[(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[4,2]=>[1,1,1,2,1]
[(1,4),(2,3),(5,6)]=>[3,4,2,1,6,5]=>[3,1,2]=>[1,1,3,1]
[(1,5),(2,3),(4,6)]=>[3,5,2,6,1,4]=>[3,3]=>[1,1,2,1,1]
[(1,6),(2,3),(4,5)]=>[3,5,2,6,4,1]=>[3,2,1]=>[1,1,2,2]
[(1,6),(2,4),(3,5)]=>[4,5,6,2,3,1]=>[5,1]=>[1,1,1,1,2]
[(1,5),(2,4),(3,6)]=>[4,5,6,2,1,3]=>[4,2]=>[1,1,1,2,1]
[(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[6]=>[1,1,1,1,1,1]
[(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[3,3]=>[1,1,2,1,1]
[(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[2,4]=>[1,2,1,1,1]
[(1,2),(3,6),(4,5)]=>[2,1,5,6,4,3]=>[2,3,1]=>[1,2,1,2]
[(1,3),(2,6),(4,5)]=>[3,5,1,6,4,2]=>[3,2,1]=>[1,1,2,2]
[(1,4),(2,6),(3,5)]=>[4,5,6,1,3,2]=>[5,1]=>[1,1,1,1,2]
[(1,5),(2,6),(3,4)]=>[4,5,6,3,1,2]=>[4,2]=>[1,1,1,2,1]
[(1,6),(2,5),(3,4)]=>[4,5,6,3,2,1]=>[4,1,1]=>[1,1,1,3]
[(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[2,2,2,2]=>[1,2,2,2,1]
[(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[4,2,2]=>[1,1,1,2,2,1]
[(1,4),(2,3),(5,6),(7,8)]=>[3,4,2,1,6,5,8,7]=>[3,1,2,2]=>[1,1,3,2,1]
[(1,5),(2,3),(4,6),(7,8)]=>[3,5,2,6,1,4,8,7]=>[3,3,2]=>[1,1,2,1,2,1]
[(1,6),(2,3),(4,5),(7,8)]=>[3,5,2,6,4,1,8,7]=>[3,2,1,2]=>[1,1,2,3,1]
[(1,7),(2,3),(4,5),(6,8)]=>[3,5,2,7,4,8,1,6]=>[3,2,3]=>[1,1,2,2,1,1]
[(1,8),(2,3),(4,5),(6,7)]=>[3,5,2,7,4,8,6,1]=>[3,2,2,1]=>[1,1,2,2,2]
[(1,8),(2,4),(3,5),(6,7)]=>[4,5,7,2,3,8,6,1]=>[5,2,1]=>[1,1,1,1,2,2]
[(1,7),(2,4),(3,5),(6,8)]=>[4,5,7,2,3,8,1,6]=>[5,3]=>[1,1,1,1,2,1,1]
[(1,6),(2,4),(3,5),(7,8)]=>[4,5,6,2,3,1,8,7]=>[5,1,2]=>[1,1,1,1,3,1]
[(1,5),(2,4),(3,6),(7,8)]=>[4,5,6,2,1,3,8,7]=>[4,2,2]=>[1,1,1,2,2,1]
[(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[6,2]=>[1,1,1,1,1,2,1]
[(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[3,3,2]=>[1,1,2,1,2,1]
[(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[2,4,2]=>[1,2,1,1,2,1]
[(1,2),(3,6),(4,5),(7,8)]=>[2,1,5,6,4,3,8,7]=>[2,3,1,2]=>[1,2,1,3,1]
[(1,3),(2,6),(4,5),(7,8)]=>[3,5,1,6,4,2,8,7]=>[3,2,1,2]=>[1,1,2,3,1]
[(1,4),(2,6),(3,5),(7,8)]=>[4,5,6,1,3,2,8,7]=>[5,1,2]=>[1,1,1,1,3,1]
[(1,5),(2,6),(3,4),(7,8)]=>[4,5,6,3,1,2,8,7]=>[4,2,2]=>[1,1,1,2,2,1]
[(1,6),(2,5),(3,4),(7,8)]=>[4,5,6,3,2,1,8,7]=>[4,1,1,2]=>[1,1,1,4,1]
[(1,7),(2,5),(3,4),(6,8)]=>[4,5,7,3,2,8,1,6]=>[4,1,3]=>[1,1,1,3,1,1]
[(1,8),(2,5),(3,4),(6,7)]=>[4,5,7,3,2,8,6,1]=>[4,1,2,1]=>[1,1,1,3,2]
[(1,8),(2,6),(3,4),(5,7)]=>[4,6,7,3,8,2,5,1]=>[4,3,1]=>[1,1,1,2,1,2]
[(1,7),(2,6),(3,4),(5,8)]=>[4,6,7,3,8,2,1,5]=>[4,2,2]=>[1,1,1,2,2,1]
[(1,6),(2,7),(3,4),(5,8)]=>[4,6,7,3,8,1,2,5]=>[4,4]=>[1,1,1,2,1,1,1]
[(1,5),(2,7),(3,4),(6,8)]=>[4,5,7,3,1,8,2,6]=>[4,1,3]=>[1,1,1,3,1,1]
[(1,4),(2,7),(3,5),(6,8)]=>[4,5,7,1,3,8,2,6]=>[5,3]=>[1,1,1,1,2,1,1]
[(1,3),(2,7),(4,5),(6,8)]=>[3,5,1,7,4,8,2,6]=>[3,2,3]=>[1,1,2,2,1,1]
[(1,2),(3,7),(4,5),(6,8)]=>[2,1,5,7,4,8,3,6]=>[2,3,3]=>[1,2,1,2,1,1]
[(1,2),(3,8),(4,5),(6,7)]=>[2,1,5,7,4,8,6,3]=>[2,3,2,1]=>[1,2,1,2,2]
[(1,3),(2,8),(4,5),(6,7)]=>[3,5,1,7,4,8,6,2]=>[3,2,2,1]=>[1,1,2,2,2]
[(1,4),(2,8),(3,5),(6,7)]=>[4,5,7,1,3,8,6,2]=>[5,2,1]=>[1,1,1,1,2,2]
[(1,5),(2,8),(3,4),(6,7)]=>[4,5,7,3,1,8,6,2]=>[4,1,2,1]=>[1,1,1,3,2]
[(1,6),(2,8),(3,4),(5,7)]=>[4,6,7,3,8,1,5,2]=>[4,3,1]=>[1,1,1,2,1,2]
[(1,7),(2,8),(3,4),(5,6)]=>[4,6,7,3,8,5,1,2]=>[4,2,2]=>[1,1,1,2,2,1]
[(1,8),(2,7),(3,4),(5,6)]=>[4,6,7,3,8,5,2,1]=>[4,2,1,1]=>[1,1,1,2,3]
[(1,8),(2,7),(3,5),(4,6)]=>[5,6,7,8,3,4,2,1]=>[6,1,1]=>[1,1,1,1,1,3]
[(1,7),(2,8),(3,5),(4,6)]=>[5,6,7,8,3,4,1,2]=>[6,2]=>[1,1,1,1,1,2,1]
[(1,6),(2,8),(3,5),(4,7)]=>[5,6,7,8,3,1,4,2]=>[5,2,1]=>[1,1,1,1,2,2]
[(1,5),(2,8),(3,6),(4,7)]=>[5,6,7,8,1,3,4,2]=>[7,1]=>[1,1,1,1,1,1,2]
[(1,4),(2,8),(3,6),(5,7)]=>[4,6,7,1,8,3,5,2]=>[4,3,1]=>[1,1,1,2,1,2]
[(1,3),(2,8),(4,6),(5,7)]=>[3,6,1,7,8,4,5,2]=>[3,4,1]=>[1,1,2,1,1,2]
[(1,2),(3,8),(4,6),(5,7)]=>[2,1,6,7,8,4,5,3]=>[2,5,1]=>[1,2,1,1,1,2]
[(1,2),(3,7),(4,6),(5,8)]=>[2,1,6,7,8,4,3,5]=>[2,4,2]=>[1,2,1,1,2,1]
[(1,3),(2,7),(4,6),(5,8)]=>[3,6,1,7,8,4,2,5]=>[3,3,2]=>[1,1,2,1,2,1]
[(1,4),(2,7),(3,6),(5,8)]=>[4,6,7,1,8,3,2,5]=>[4,2,2]=>[1,1,1,2,2,1]
[(1,5),(2,7),(3,6),(4,8)]=>[5,6,7,8,1,3,2,4]=>[6,2]=>[1,1,1,1,1,2,1]
[(1,6),(2,7),(3,5),(4,8)]=>[5,6,7,8,3,1,2,4]=>[5,3]=>[1,1,1,1,2,1,1]
[(1,7),(2,6),(3,5),(4,8)]=>[5,6,7,8,3,2,1,4]=>[5,1,2]=>[1,1,1,1,3,1]
[(1,8),(2,6),(3,5),(4,7)]=>[5,6,7,8,3,2,4,1]=>[5,2,1]=>[1,1,1,1,2,2]
[(1,8),(2,5),(3,6),(4,7)]=>[5,6,7,8,2,3,4,1]=>[7,1]=>[1,1,1,1,1,1,2]
[(1,7),(2,5),(3,6),(4,8)]=>[5,6,7,8,2,3,1,4]=>[6,2]=>[1,1,1,1,1,2,1]
[(1,6),(2,5),(3,7),(4,8)]=>[5,6,7,8,2,1,3,4]=>[5,3]=>[1,1,1,1,2,1,1]
[(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[8]=>[1,1,1,1,1,1,1,1]
[(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[4,4]=>[1,1,1,2,1,1,1]
[(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[3,5]=>[1,1,2,1,1,1,1]
[(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[2,6]=>[1,2,1,1,1,1,1]
[(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[2,3,3]=>[1,2,1,2,1,1]
[(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[3,2,3]=>[1,1,2,2,1,1]
[(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[5,3]=>[1,1,1,1,2,1,1]
[(1,5),(2,4),(3,7),(6,8)]=>[4,5,7,2,1,8,3,6]=>[4,1,3]=>[1,1,1,3,1,1]
[(1,6),(2,4),(3,7),(5,8)]=>[4,6,7,2,8,1,3,5]=>[4,4]=>[1,1,1,2,1,1,1]
[(1,7),(2,4),(3,6),(5,8)]=>[4,6,7,2,8,3,1,5]=>[4,2,2]=>[1,1,1,2,2,1]
[(1,8),(2,4),(3,6),(5,7)]=>[4,6,7,2,8,3,5,1]=>[4,3,1]=>[1,1,1,2,1,2]
[(1,8),(2,3),(4,6),(5,7)]=>[3,6,2,7,8,4,5,1]=>[3,4,1]=>[1,1,2,1,1,2]
[(1,7),(2,3),(4,6),(5,8)]=>[3,6,2,7,8,4,1,5]=>[3,3,2]=>[1,1,2,1,2,1]
[(1,6),(2,3),(4,7),(5,8)]=>[3,6,2,7,8,1,4,5]=>[3,5]=>[1,1,2,1,1,1,1]
[(1,5),(2,3),(4,7),(6,8)]=>[3,5,2,7,1,8,4,6]=>[3,2,3]=>[1,1,2,2,1,1]
[(1,4),(2,3),(5,7),(6,8)]=>[3,4,2,1,7,8,5,6]=>[3,1,4]=>[1,1,3,1,1,1]
[(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[4,4]=>[1,1,1,2,1,1,1]
[(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[2,2,4]=>[1,2,2,1,1,1]
[(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,7,8,6,5]=>[2,2,3,1]=>[1,2,2,1,2]
[(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,7,8,6,5]=>[4,3,1]=>[1,1,1,2,1,2]
[(1,4),(2,3),(5,8),(6,7)]=>[3,4,2,1,7,8,6,5]=>[3,1,3,1]=>[1,1,3,1,2]
[(1,5),(2,3),(4,8),(6,7)]=>[3,5,2,7,1,8,6,4]=>[3,2,2,1]=>[1,1,2,2,2]
[(1,6),(2,3),(4,8),(5,7)]=>[3,6,2,7,8,1,5,4]=>[3,4,1]=>[1,1,2,1,1,2]
[(1,7),(2,3),(4,8),(5,6)]=>[3,6,2,7,8,5,1,4]=>[3,3,2]=>[1,1,2,1,2,1]
[(1,8),(2,3),(4,7),(5,6)]=>[3,6,2,7,8,5,4,1]=>[3,3,1,1]=>[1,1,2,1,3]
[(1,8),(2,4),(3,7),(5,6)]=>[4,6,7,2,8,5,3,1]=>[4,2,1,1]=>[1,1,1,2,3]
[(1,7),(2,4),(3,8),(5,6)]=>[4,6,7,2,8,5,1,3]=>[4,2,2]=>[1,1,1,2,2,1]
[(1,6),(2,4),(3,8),(5,7)]=>[4,6,7,2,8,1,5,3]=>[4,3,1]=>[1,1,1,2,1,2]
[(1,5),(2,4),(3,8),(6,7)]=>[4,5,7,2,1,8,6,3]=>[4,1,2,1]=>[1,1,1,3,2]
[(1,4),(2,5),(3,8),(6,7)]=>[4,5,7,1,2,8,6,3]=>[5,2,1]=>[1,1,1,1,2,2]
[(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,7,2,8,6,4]=>[3,2,2,1]=>[1,1,2,2,2]
[(1,2),(3,5),(4,8),(6,7)]=>[2,1,5,7,3,8,6,4]=>[2,3,2,1]=>[1,2,1,2,2]
[(1,2),(3,6),(4,8),(5,7)]=>[2,1,6,7,8,3,5,4]=>[2,5,1]=>[1,2,1,1,1,2]
[(1,3),(2,6),(4,8),(5,7)]=>[3,6,1,7,8,2,5,4]=>[3,4,1]=>[1,1,2,1,1,2]
[(1,4),(2,6),(3,8),(5,7)]=>[4,6,7,1,8,2,5,3]=>[4,3,1]=>[1,1,1,2,1,2]
[(1,5),(2,6),(3,8),(4,7)]=>[5,6,7,8,1,2,4,3]=>[7,1]=>[1,1,1,1,1,1,2]
[(1,6),(2,5),(3,8),(4,7)]=>[5,6,7,8,2,1,4,3]=>[5,2,1]=>[1,1,1,1,2,2]
[(1,7),(2,5),(3,8),(4,6)]=>[5,6,7,8,2,4,1,3]=>[6,2]=>[1,1,1,1,1,2,1]
[(1,8),(2,5),(3,7),(4,6)]=>[5,6,7,8,2,4,3,1]=>[6,1,1]=>[1,1,1,1,1,3]
[(1,8),(2,6),(3,7),(4,5)]=>[5,6,7,8,4,2,3,1]=>[5,2,1]=>[1,1,1,1,2,2]
[(1,7),(2,6),(3,8),(4,5)]=>[5,6,7,8,4,2,1,3]=>[5,1,2]=>[1,1,1,1,3,1]
[(1,6),(2,7),(3,8),(4,5)]=>[5,6,7,8,4,1,2,3]=>[5,3]=>[1,1,1,1,2,1,1]
[(1,5),(2,7),(3,8),(4,6)]=>[5,6,7,8,1,4,2,3]=>[6,2]=>[1,1,1,1,1,2,1]
[(1,4),(2,7),(3,8),(5,6)]=>[4,6,7,1,8,5,2,3]=>[4,2,2]=>[1,1,1,2,2,1]
[(1,3),(2,7),(4,8),(5,6)]=>[3,6,1,7,8,5,2,4]=>[3,3,2]=>[1,1,2,1,2,1]
[(1,2),(3,7),(4,8),(5,6)]=>[2,1,6,7,8,5,3,4]=>[2,4,2]=>[1,2,1,1,2,1]
[(1,2),(3,8),(4,7),(5,6)]=>[2,1,6,7,8,5,4,3]=>[2,4,1,1]=>[1,2,1,1,3]
[(1,3),(2,8),(4,7),(5,6)]=>[3,6,1,7,8,5,4,2]=>[3,3,1,1]=>[1,1,2,1,3]
[(1,4),(2,8),(3,7),(5,6)]=>[4,6,7,1,8,5,3,2]=>[4,2,1,1]=>[1,1,1,2,3]
[(1,5),(2,8),(3,7),(4,6)]=>[5,6,7,8,1,4,3,2]=>[6,1,1]=>[1,1,1,1,1,3]
[(1,6),(2,8),(3,7),(4,5)]=>[5,6,7,8,4,1,3,2]=>[5,2,1]=>[1,1,1,1,2,2]
[(1,7),(2,8),(3,6),(4,5)]=>[5,6,7,8,4,3,1,2]=>[5,1,2]=>[1,1,1,1,3,1]
[(1,8),(2,7),(3,6),(4,5)]=>[5,6,7,8,4,3,2,1]=>[5,1,1,1]=>[1,1,1,1,4]
[(1,2),(3,4),(5,6),(7,8),(9,10)]=>[2,1,4,3,6,5,8,7,10,9]=>[2,2,2,2,2]=>[1,2,2,2,2,1]
[(1,4),(2,3),(5,6),(7,8),(9,10)]=>[3,4,2,1,6,5,8,7,10,9]=>[3,1,2,2,2]=>[1,1,3,2,2,1]
[(1,6),(2,3),(4,5),(7,8),(9,10)]=>[3,5,2,6,4,1,8,7,10,9]=>[3,2,1,2,2]=>[1,1,2,3,2,1]
[(1,8),(2,3),(4,5),(6,7),(9,10)]=>[3,5,2,7,4,8,6,1,10,9]=>[3,2,2,1,2]=>[1,1,2,2,3,1]
[(1,10),(2,3),(4,5),(6,7),(8,9)]=>[3,5,2,7,4,9,6,10,8,1]=>[3,2,2,2,1]=>[1,1,2,2,2,2]
[(1,2),(3,6),(4,5),(7,8),(9,10)]=>[2,1,5,6,4,3,8,7,10,9]=>[2,3,1,2,2]=>[1,2,1,3,2,1]
[(1,6),(2,5),(3,4),(7,8),(9,10)]=>[4,5,6,3,2,1,8,7,10,9]=>[4,1,1,2,2]=>[1,1,1,4,2,1]
[(1,8),(2,5),(3,4),(6,7),(9,10)]=>[4,5,7,3,2,8,6,1,10,9]=>[4,1,2,1,2]=>[1,1,1,3,3,1]
[(1,10),(2,5),(3,4),(6,7),(8,9)]=>[4,5,7,3,2,9,6,10,8,1]=>[4,1,2,2,1]=>[1,1,1,3,2,2]
[(1,2),(3,8),(4,5),(6,7),(9,10)]=>[2,1,5,7,4,8,6,3,10,9]=>[2,3,2,1,2]=>[1,2,1,2,3,1]
[(1,8),(2,7),(3,4),(5,6),(9,10)]=>[4,6,7,3,8,5,2,1,10,9]=>[4,2,1,1,2]=>[1,1,1,2,4,1]
[(1,10),(2,7),(3,4),(5,6),(8,9)]=>[4,6,7,3,9,5,2,10,8,1]=>[4,2,1,2,1]=>[1,1,1,2,3,2]
[(1,2),(3,10),(4,5),(6,7),(8,9)]=>[2,1,5,7,4,9,6,10,8,3]=>[2,3,2,2,1]=>[1,2,1,2,2,2]
[(1,10),(2,9),(3,4),(5,6),(7,8)]=>[4,6,8,3,9,5,10,7,2,1]=>[4,2,2,1,1]=>[1,1,1,2,2,3]
[(1,2),(3,4),(5,8),(6,7),(9,10)]=>[2,1,4,3,7,8,6,5,10,9]=>[2,2,3,1,2]=>[1,2,2,1,3,1]
[(1,4),(2,3),(5,8),(6,7),(9,10)]=>[3,4,2,1,7,8,6,5,10,9]=>[3,1,3,1,2]=>[1,1,3,1,3,1]
[(1,8),(2,3),(4,7),(5,6),(9,10)]=>[3,6,2,7,8,5,4,1,10,9]=>[3,3,1,1,2]=>[1,1,2,1,4,1]
[(1,10),(2,3),(4,7),(5,6),(8,9)]=>[3,6,2,7,9,5,4,10,8,1]=>[3,3,1,2,1]=>[1,1,2,1,3,2]
[(1,2),(3,8),(4,7),(5,6),(9,10)]=>[2,1,6,7,8,5,4,3,10,9]=>[2,4,1,1,2]=>[1,2,1,1,4,1]
[(1,8),(2,7),(3,6),(4,5),(9,10)]=>[5,6,7,8,4,3,2,1,10,9]=>[5,1,1,1,2]=>[1,1,1,1,5,1]
[(1,10),(2,7),(3,6),(4,5),(8,9)]=>[5,6,7,9,4,3,2,10,8,1]=>[5,1,1,2,1]=>[1,1,1,1,4,2]
[(1,2),(3,10),(4,7),(5,6),(8,9)]=>[2,1,6,7,9,5,4,10,8,3]=>[2,4,1,2,1]=>[1,2,1,1,3,2]
[(1,10),(2,9),(3,6),(4,5),(7,8)]=>[5,6,8,9,4,3,10,7,2,1]=>[5,1,2,1,1]=>[1,1,1,1,3,3]
[(1,2),(3,4),(5,10),(6,7),(8,9)]=>[2,1,4,3,7,9,6,10,8,5]=>[2,2,3,2,1]=>[1,2,2,1,2,2]
[(1,4),(2,3),(5,10),(6,7),(8,9)]=>[3,4,2,1,7,9,6,10,8,5]=>[3,1,3,2,1]=>[1,1,3,1,2,2]
[(1,10),(2,3),(4,9),(5,6),(7,8)]=>[3,6,2,8,9,5,10,7,4,1]=>[3,3,2,1,1]=>[1,1,2,1,2,3]
[(1,2),(3,10),(4,9),(5,6),(7,8)]=>[2,1,6,8,9,5,10,7,4,3]=>[2,4,2,1,1]=>[1,2,1,1,2,3]
[(1,10),(2,9),(3,8),(4,5),(6,7)]=>[5,7,8,9,4,10,6,3,2,1]=>[5,2,1,1,1]=>[1,1,1,1,2,4]
[(1,6),(2,7),(3,8),(4,9),(5,10)]=>[6,7,8,9,10,1,2,3,4,5]=>[10]=>[1,1,1,1,1,1,1,1,1,1]
[(1,2),(3,4),(5,6),(7,10),(8,9)]=>[2,1,4,3,6,5,9,10,8,7]=>[2,2,2,3,1]=>[1,2,2,2,1,2]
[(1,4),(2,3),(5,6),(7,10),(8,9)]=>[3,4,2,1,6,5,9,10,8,7]=>[3,1,2,3,1]=>[1,1,3,2,1,2]
[(1,6),(2,3),(4,5),(7,10),(8,9)]=>[3,5,2,6,4,1,9,10,8,7]=>[3,2,1,3,1]=>[1,1,2,3,1,2]
[(1,10),(2,3),(4,5),(6,9),(7,8)]=>[3,5,2,8,4,9,10,7,6,1]=>[3,2,3,1,1]=>[1,1,2,2,1,3]
[(1,2),(3,6),(4,5),(7,10),(8,9)]=>[2,1,5,6,4,3,9,10,8,7]=>[2,3,1,3,1]=>[1,2,1,3,1,2]
[(1,6),(2,5),(3,4),(7,10),(8,9)]=>[4,5,6,3,2,1,9,10,8,7]=>[4,1,1,3,1]=>[1,1,1,4,1,2]
[(1,10),(2,5),(3,4),(6,9),(7,8)]=>[4,5,8,3,2,9,10,7,6,1]=>[4,1,3,1,1]=>[1,1,1,3,1,3]
[(1,2),(3,10),(4,5),(6,9),(7,8)]=>[2,1,5,8,4,9,10,7,6,3]=>[2,3,3,1,1]=>[1,2,1,2,1,3]
[(1,10),(2,9),(3,4),(5,8),(6,7)]=>[4,7,8,3,9,10,6,5,2,1]=>[4,3,1,1,1]=>[1,1,1,2,1,4]
[(1,2),(3,4),(5,10),(6,9),(7,8)]=>[2,1,4,3,8,9,10,7,6,5]=>[2,2,4,1,1]=>[1,2,2,1,1,3]
[(1,4),(2,3),(5,10),(6,9),(7,8)]=>[3,4,2,1,8,9,10,7,6,5]=>[3,1,4,1,1]=>[1,1,3,1,1,3]
[(1,10),(2,3),(4,9),(5,8),(6,7)]=>[3,7,2,8,9,10,6,5,4,1]=>[3,4,1,1,1]=>[1,1,2,1,1,4]
[(1,2),(3,10),(4,9),(5,8),(6,7)]=>[2,1,7,8,9,10,6,5,4,3]=>[2,5,1,1,1]=>[1,2,1,1,1,4]
[(1,10),(2,9),(3,8),(4,7),(5,6)]=>[6,7,8,9,10,5,4,3,2,1]=>[6,1,1,1,1]=>[1,1,1,1,1,5]
[(1,12),(2,3),(4,9),(5,8),(6,7),(10,11)]=>[3,7,2,8,9,11,6,5,4,12,10,1]=>[3,4,1,1,2,1]=>[1,1,2,1,1,4,2]
[(1,12),(2,11),(3,4),(5,8),(6,7),(9,10)]=>[4,7,8,3,10,11,6,5,12,9,2,1]=>[4,3,1,2,1,1]=>[1,1,1,2,1,3,3]
[(1,10),(2,9),(3,4),(5,8),(6,7),(11,12)]=>[4,7,8,3,9,10,6,5,2,1,12,11]=>[4,3,1,1,1,2]=>[1,1,1,2,1,5,1]
[(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)]=>[2,1,5,7,4,9,6,10,8,3,12,11]=>[2,3,2,2,1,2]=>[1,2,1,2,2,3,1]
[(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]=>[3,5,2,7,4,9,6,11,8,12,10,1]=>[3,2,2,2,2,1]=>[1,1,2,2,2,2,2]
[(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]=>[2,1,5,7,4,9,6,11,8,12,10,3]=>[2,3,2,2,2,1]=>[1,2,1,2,2,2,2]
[(1,10),(2,3),(4,5),(6,7),(8,9),(11,12)]=>[3,5,2,7,4,9,6,10,8,1,12,11]=>[3,2,2,2,1,2]=>[1,1,2,2,2,3,1]
[(1,2),(3,4),(5,10),(6,7),(8,9),(11,12)]=>[2,1,4,3,7,9,6,10,8,5,12,11]=>[2,2,3,2,1,2]=>[1,2,2,1,2,3,1]
[(1,12),(2,3),(4,11),(5,10),(6,7),(8,9)]=>[3,7,2,9,10,11,6,12,8,5,4,1]=>[3,4,2,1,1,1]=>[1,1,2,1,1,2,4]
[(1,12),(2,5),(3,4),(6,7),(8,9),(10,11)]=>[4,5,7,3,2,9,6,11,8,12,10,1]=>[4,1,2,2,2,1]=>[1,1,1,3,2,2,2]
[(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]=>[2,1,4,3,7,9,6,11,8,12,10,5]=>[2,2,3,2,2,1]=>[1,2,2,1,2,2,2]
[(1,10),(2,5),(3,4),(6,7),(8,9),(11,12)]=>[4,5,7,3,2,9,6,10,8,1,12,11]=>[4,1,2,2,1,2]=>[1,1,1,3,2,3,1]
[(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]=>[2,1,5,7,4,8,6,3,10,9,12,11]=>[2,3,2,1,2,2]=>[1,2,1,2,3,2,1]
[(1,12),(2,3),(4,5),(6,7),(8,11),(9,10)]=>[3,5,2,7,4,10,6,11,12,9,8,1]=>[3,2,2,3,1,1]=>[1,1,2,2,2,1,3]
[(1,2),(3,12),(4,5),(6,7),(8,11),(9,10)]=>[2,1,5,7,4,10,6,11,12,9,8,3]=>[2,3,2,3,1,1]=>[1,2,1,2,2,1,3]
[(1,8),(2,3),(4,5),(6,7),(9,10),(11,12)]=>[3,5,2,7,4,8,6,1,10,9,12,11]=>[3,2,2,1,2,2]=>[1,1,2,2,3,2,1]
[(1,2),(3,4),(5,12),(6,7),(8,11),(9,10)]=>[2,1,4,3,7,10,6,11,12,9,8,5]=>[2,2,3,3,1,1]=>[1,2,2,1,2,1,3]
[(1,8),(2,5),(3,4),(6,7),(9,10),(11,12)]=>[4,5,7,3,2,8,6,1,10,9,12,11]=>[4,1,2,1,2,2]=>[1,1,1,3,3,2,1]
[(1,2),(3,12),(4,9),(5,6),(7,8),(10,11)]=>[2,1,6,8,9,5,11,7,4,12,10,3]=>[2,4,2,1,2,1]=>[1,2,1,1,2,3,2]
[(1,10),(2,3),(4,9),(5,6),(7,8),(11,12)]=>[3,6,2,8,9,5,10,7,4,1,12,11]=>[3,3,2,1,1,2]=>[1,1,2,1,2,4,1]
[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]=>[2,1,4,3,6,5,8,7,10,9,12,11]=>[2,2,2,2,2,2]=>[1,2,2,2,2,2,1]
[(1,12),(2,3),(4,11),(5,6),(7,8),(9,10)]=>[3,6,2,8,10,5,11,7,12,9,4,1]=>[3,3,2,2,1,1]=>[1,1,2,1,2,2,3]
[(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)]=>[3,4,2,1,6,5,8,7,10,9,12,11]=>[3,1,2,2,2,2]=>[1,1,3,2,2,2,1]
[(1,12),(2,9),(3,4),(5,6),(7,8),(10,11)]=>[4,6,8,3,9,5,11,7,2,12,10,1]=>[4,2,2,1,2,1]=>[1,1,1,2,2,3,2]
[(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]=>[2,1,4,3,6,5,8,7,11,12,10,9]=>[2,2,2,2,3,1]=>[1,2,2,2,2,1,2]
[(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)]=>[2,1,5,8,4,9,10,7,6,3,12,11]=>[2,3,3,1,1,2]=>[1,2,1,2,1,4,1]
[(1,12),(2,11),(3,4),(5,10),(6,9),(7,8)]=>[4,8,9,3,10,11,12,7,6,5,2,1]=>[4,4,1,1,1,1]=>[1,1,1,2,1,1,5]
[(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]=>[2,1,5,6,4,3,8,7,10,9,12,11]=>[2,3,1,2,2,2]=>[1,2,1,3,2,2,1]
[(1,12),(2,3),(4,5),(6,11),(7,8),(9,10)]=>[3,5,2,8,4,10,11,7,12,9,6,1]=>[3,2,3,2,1,1]=>[1,1,2,2,1,2,3]
[(1,2),(3,12),(4,5),(6,11),(7,8),(9,10)]=>[2,1,5,8,4,10,11,7,12,9,6,3]=>[2,3,3,2,1,1]=>[1,2,1,2,1,2,3]
[(1,6),(2,3),(4,5),(7,8),(9,10),(11,12)]=>[3,5,2,6,4,1,8,7,10,9,12,11]=>[3,2,1,2,2,2]=>[1,1,2,3,2,2,1]
[(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]=>[2,1,5,6,4,3,8,7,11,12,10,9]=>[2,3,1,2,3,1]=>[1,2,1,3,2,1,2]
[(1,6),(2,5),(3,4),(7,8),(9,10),(11,12)]=>[4,5,6,3,2,1,8,7,10,9,12,11]=>[4,1,1,2,2,2]=>[1,1,1,4,2,2,1]
[(1,12),(2,3),(4,7),(5,6),(8,9),(10,11)]=>[3,6,2,7,9,5,4,11,8,12,10,1]=>[3,3,1,2,2,1]=>[1,1,2,1,3,2,2]
[(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]=>[2,1,4,3,6,5,9,10,8,7,12,11]=>[2,2,2,3,1,2]=>[1,2,2,2,1,3,1]
[(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]=>[2,1,6,9,10,5,11,12,8,7,4,3]=>[2,4,3,1,1,1]=>[1,2,1,1,2,1,4]
[(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]=>[3,4,2,1,6,5,9,10,8,7,12,11]=>[3,1,2,3,1,2]=>[1,1,3,2,1,3,1]
[(1,12),(2,7),(3,4),(5,6),(8,9),(10,11)]=>[4,6,7,3,9,5,2,11,8,12,10,1]=>[4,2,1,2,2,1]=>[1,1,1,2,3,2,2]
[(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]=>[2,1,4,3,6,5,9,11,8,12,10,7]=>[2,2,2,3,2,1]=>[1,2,2,2,1,2,2]
[(1,10),(2,7),(3,4),(5,6),(8,9),(11,12)]=>[4,6,7,3,9,5,2,10,8,1,12,11]=>[4,2,1,2,1,2]=>[1,1,1,2,3,3,1]
[(1,4),(2,3),(5,6),(7,12),(8,9),(10,11)]=>[3,4,2,1,6,5,9,11,8,12,10,7]=>[3,1,2,3,2,1]=>[1,1,3,2,1,2,2]
[(1,8),(2,3),(4,7),(5,6),(9,10),(11,12)]=>[3,6,2,7,8,5,4,1,10,9,12,11]=>[3,3,1,1,2,2]=>[1,1,2,1,4,2,1]
[(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]=>[2,1,6,7,8,5,4,3,11,12,10,9]=>[2,4,1,1,3,1]=>[1,2,1,1,4,1,2]
[(1,12),(2,7),(3,4),(5,6),(8,11),(9,10)]=>[4,6,7,3,10,5,2,11,12,9,8,1]=>[4,2,1,3,1,1]=>[1,1,1,2,3,1,3]
[(1,8),(2,7),(3,4),(5,6),(9,10),(11,12)]=>[4,6,7,3,8,5,2,1,10,9,12,11]=>[4,2,1,1,2,2]=>[1,1,1,2,4,2,1]
[(1,4),(2,3),(5,6),(7,12),(8,11),(9,10)]=>[3,4,2,1,6,5,10,11,12,9,8,7]=>[3,1,2,4,1,1]=>[1,1,3,2,1,1,3]
[(1,6),(2,3),(4,5),(7,12),(8,9),(10,11)]=>[3,5,2,6,4,1,9,11,8,12,10,7]=>[3,2,1,3,2,1]=>[1,1,2,3,1,2,2]
[(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]=>[4,5,6,3,2,1,9,10,8,7,12,11]=>[4,1,1,3,1,2]=>[1,1,1,4,1,3,1]
[(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]=>[4,5,6,3,2,1,10,11,12,9,8,7]=>[4,1,1,4,1,1]=>[1,1,1,4,1,1,3]
[(1,7),(2,8),(3,9),(4,10),(5,11),(6,12)]=>[7,8,9,10,11,12,1,2,3,4,5,6]=>[12]=>[1,1,1,1,1,1,1,1,1,1,1,1]
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
DEX composition
Description
The DEX composition of a permutation.
Let $\pi$ be a permutation in $\mathfrak S_n$. Let $\bar\pi$ be the word in the ordered set $\bar 1 < \dots < \bar n < 1 \dots < n$ obtained from $\pi$ by replacing every excedance $\pi(i) > i$ by $\overline{\pi(i)}$. Then the DEX set of $\pi$ is the set of indices $1 \leq i < n$ such that $\bar\pi(i) > \bar\pi(i+1)$. Finally, the DEX composition $c_1, \dots, c_k$ of $n$ corresponds to the DEX subset $\{c_1, c_1 + c_2, \dots, c_1 + \dots + c_{k-1}\}$.
The (quasi)symmetric function
$$ \sum_{\pi\in\mathfrak S_{\lambda, j}} F_{DEX(\pi)}, $$
where the sum is over the set of permutations of cycle type $\lambda$ with $j$ excedances, is the Eulerian quasisymmetric function.
Let $\pi$ be a permutation in $\mathfrak S_n$. Let $\bar\pi$ be the word in the ordered set $\bar 1 < \dots < \bar n < 1 \dots < n$ obtained from $\pi$ by replacing every excedance $\pi(i) > i$ by $\overline{\pi(i)}$. Then the DEX set of $\pi$ is the set of indices $1 \leq i < n$ such that $\bar\pi(i) > \bar\pi(i+1)$. Finally, the DEX composition $c_1, \dots, c_k$ of $n$ corresponds to the DEX subset $\{c_1, c_1 + c_2, \dots, c_1 + \dots + c_{k-1}\}$.
The (quasi)symmetric function
$$ \sum_{\pi\in\mathfrak S_{\lambda, j}} F_{DEX(\pi)}, $$
where the sum is over the set of permutations of cycle type $\lambda$ with $j$ excedances, is the Eulerian quasisymmetric function.
Map
complement
Description
The complement of a composition.
The complement of a composition $I$ is defined as follows:
If $I$ is the empty composition, then the complement is also the empty composition. Otherwise, let $S$ be the descent set corresponding to $I=(i_1,\dots,i_k)$, that is, the subset
$$\{ i_1, i_1 + i_2, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$$
of $\{ 1, 2, \ldots, |I|-1 \}$. Then, the complement of $I$ is the composition of the same size as $I$, whose descent set is $\{ 1, 2, \ldots, |I|-1 \} \setminus S$.
The complement of a composition $I$ coincides with the reversal (Mp00038reverse) of the composition conjugate (Mp00041conjugate) to $I$.
The complement of a composition $I$ is defined as follows:
If $I$ is the empty composition, then the complement is also the empty composition. Otherwise, let $S$ be the descent set corresponding to $I=(i_1,\dots,i_k)$, that is, the subset
$$\{ i_1, i_1 + i_2, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$$
of $\{ 1, 2, \ldots, |I|-1 \}$. Then, the complement of $I$ is the composition of the same size as $I$, whose descent set is $\{ 1, 2, \ldots, |I|-1 \} \setminus S$.
The complement of a composition $I$ coincides with the reversal (Mp00038reverse) of the composition conjugate (Mp00041conjugate) to $I$.
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