Identifier
Mp00283:
Perfect matchings
—non-nesting-exceedence permutation⟶
Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Images
=>
Cc0012;cc-rep-0Cc0005;cc-rep-2Cc0005;cc-rep-3
[(1,2)]=>[2,1]=>[1,1,0,0]=>[1,0,1,0]
[(1,2),(3,4)]=>[2,1,4,3]=>[1,1,0,0,1,1,0,0]=>[1,0,1,1,0,0,1,0]
[(1,3),(2,4)]=>[3,4,1,2]=>[1,1,1,0,1,0,0,0]=>[1,1,0,1,0,1,0,0]
[(1,4),(2,3)]=>[3,4,2,1]=>[1,1,1,0,1,0,0,0]=>[1,1,0,1,0,1,0,0]
[(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[1,1,0,0,1,1,0,0,1,1,0,0]=>[1,0,1,1,0,0,1,1,0,0,1,0]
[(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[1,1,1,0,1,0,0,0,1,1,0,0]=>[1,1,0,1,0,1,1,0,0,0,1,0]
[(1,4),(2,3),(5,6)]=>[3,4,2,1,6,5]=>[1,1,1,0,1,0,0,0,1,1,0,0]=>[1,1,0,1,0,1,1,0,0,0,1,0]
[(1,5),(2,3),(4,6)]=>[3,5,2,6,1,4]=>[1,1,1,0,1,1,0,0,1,0,0,0]=>[1,1,0,1,1,0,1,0,0,1,0,0]
[(1,6),(2,3),(4,5)]=>[3,5,2,6,4,1]=>[1,1,1,0,1,1,0,0,1,0,0,0]=>[1,1,0,1,1,0,1,0,0,1,0,0]
[(1,6),(2,4),(3,5)]=>[4,5,6,2,3,1]=>[1,1,1,1,0,1,0,1,0,0,0,0]=>[1,1,1,0,1,0,1,0,1,0,0,0]
[(1,5),(2,4),(3,6)]=>[4,5,6,2,1,3]=>[1,1,1,1,0,1,0,1,0,0,0,0]=>[1,1,1,0,1,0,1,0,1,0,0,0]
[(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[1,1,1,1,0,1,0,1,0,0,0,0]=>[1,1,1,0,1,0,1,0,1,0,0,0]
[(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[1,1,1,0,1,1,0,0,1,0,0,0]=>[1,1,0,1,1,0,1,0,0,1,0,0]
[(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[1,1,0,0,1,1,1,0,1,0,0,0]=>[1,0,1,1,1,0,0,1,0,1,0,0]
[(1,2),(3,6),(4,5)]=>[2,1,5,6,4,3]=>[1,1,0,0,1,1,1,0,1,0,0,0]=>[1,0,1,1,1,0,0,1,0,1,0,0]
[(1,3),(2,6),(4,5)]=>[3,5,1,6,4,2]=>[1,1,1,0,1,1,0,0,1,0,0,0]=>[1,1,0,1,1,0,1,0,0,1,0,0]
[(1,4),(2,6),(3,5)]=>[4,5,6,1,3,2]=>[1,1,1,1,0,1,0,1,0,0,0,0]=>[1,1,1,0,1,0,1,0,1,0,0,0]
[(1,5),(2,6),(3,4)]=>[4,5,6,3,1,2]=>[1,1,1,1,0,1,0,1,0,0,0,0]=>[1,1,1,0,1,0,1,0,1,0,0,0]
[(1,6),(2,5),(3,4)]=>[4,5,6,3,2,1]=>[1,1,1,1,0,1,0,1,0,0,0,0]=>[1,1,1,0,1,0,1,0,1,0,0,0]
[(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
[(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]=>[1,1,0,1,0,1,1,0,0,0,1,1,0,0,1,0]
[(1,4),(2,3),(5,6),(7,8)]=>[3,4,2,1,6,5,8,7]=>[1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]=>[1,1,0,1,0,1,1,0,0,0,1,1,0,0,1,0]
[(1,5),(2,3),(4,6),(7,8)]=>[3,5,2,6,1,4,8,7]=>[1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]=>[1,1,0,1,1,0,1,0,0,1,1,0,0,0,1,0]
[(1,6),(2,3),(4,5),(7,8)]=>[3,5,2,6,4,1,8,7]=>[1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]=>[1,1,0,1,1,0,1,0,0,1,1,0,0,0,1,0]
[(1,7),(2,3),(4,5),(6,8)]=>[3,5,2,7,4,8,1,6]=>[1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]=>[1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
[(1,8),(2,3),(4,5),(6,7)]=>[3,5,2,7,4,8,6,1]=>[1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]=>[1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
[(1,8),(2,4),(3,5),(6,7)]=>[4,5,7,2,3,8,6,1]=>[1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]=>[1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
[(1,7),(2,4),(3,5),(6,8)]=>[4,5,7,2,3,8,1,6]=>[1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]=>[1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
[(1,6),(2,4),(3,5),(7,8)]=>[4,5,6,2,3,1,8,7]=>[1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0]=>[1,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0]
[(1,5),(2,4),(3,6),(7,8)]=>[4,5,6,2,1,3,8,7]=>[1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0]=>[1,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0]
[(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0]=>[1,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0]
[(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]=>[1,1,0,1,1,0,1,0,0,1,1,0,0,0,1,0]
[(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0]=>[1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
[(1,2),(3,6),(4,5),(7,8)]=>[2,1,5,6,4,3,8,7]=>[1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0]=>[1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
[(1,3),(2,6),(4,5),(7,8)]=>[3,5,1,6,4,2,8,7]=>[1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]=>[1,1,0,1,1,0,1,0,0,1,1,0,0,0,1,0]
[(1,4),(2,6),(3,5),(7,8)]=>[4,5,6,1,3,2,8,7]=>[1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0]=>[1,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0]
[(1,5),(2,6),(3,4),(7,8)]=>[4,5,6,3,1,2,8,7]=>[1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0]=>[1,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0]
[(1,6),(2,5),(3,4),(7,8)]=>[4,5,6,3,2,1,8,7]=>[1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0]=>[1,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0]
[(1,7),(2,5),(3,4),(6,8)]=>[4,5,7,3,2,8,1,6]=>[1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]=>[1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
[(1,8),(2,5),(3,4),(6,7)]=>[4,5,7,3,2,8,6,1]=>[1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]=>[1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
[(1,8),(2,6),(3,4),(5,7)]=>[4,6,7,3,8,2,5,1]=>[1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]=>[1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
[(1,7),(2,6),(3,4),(5,8)]=>[4,6,7,3,8,2,1,5]=>[1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]=>[1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
[(1,6),(2,7),(3,4),(5,8)]=>[4,6,7,3,8,1,2,5]=>[1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]=>[1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
[(1,5),(2,7),(3,4),(6,8)]=>[4,5,7,3,1,8,2,6]=>[1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]=>[1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
[(1,4),(2,7),(3,5),(6,8)]=>[4,5,7,1,3,8,2,6]=>[1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]=>[1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
[(1,3),(2,7),(4,5),(6,8)]=>[3,5,1,7,4,8,2,6]=>[1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]=>[1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
[(1,2),(3,7),(4,5),(6,8)]=>[2,1,5,7,4,8,3,6]=>[1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]=>[1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
[(1,2),(3,8),(4,5),(6,7)]=>[2,1,5,7,4,8,6,3]=>[1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]=>[1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
[(1,3),(2,8),(4,5),(6,7)]=>[3,5,1,7,4,8,6,2]=>[1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]=>[1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
[(1,4),(2,8),(3,5),(6,7)]=>[4,5,7,1,3,8,6,2]=>[1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]=>[1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
[(1,5),(2,8),(3,4),(6,7)]=>[4,5,7,3,1,8,6,2]=>[1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]=>[1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
[(1,6),(2,8),(3,4),(5,7)]=>[4,6,7,3,8,1,5,2]=>[1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]=>[1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
[(1,7),(2,8),(3,4),(5,6)]=>[4,6,7,3,8,5,1,2]=>[1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]=>[1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
[(1,8),(2,7),(3,4),(5,6)]=>[4,6,7,3,8,5,2,1]=>[1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]=>[1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
[(1,8),(2,7),(3,5),(4,6)]=>[5,6,7,8,3,4,2,1]=>[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
[(1,7),(2,8),(3,5),(4,6)]=>[5,6,7,8,3,4,1,2]=>[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
[(1,6),(2,8),(3,5),(4,7)]=>[5,6,7,8,3,1,4,2]=>[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
[(1,5),(2,8),(3,6),(4,7)]=>[5,6,7,8,1,3,4,2]=>[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
[(1,4),(2,8),(3,6),(5,7)]=>[4,6,7,1,8,3,5,2]=>[1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]=>[1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
[(1,3),(2,8),(4,6),(5,7)]=>[3,6,1,7,8,4,5,2]=>[1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]=>[1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
[(1,2),(3,8),(4,6),(5,7)]=>[2,1,6,7,8,4,5,3]=>[1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]=>[1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
[(1,2),(3,7),(4,6),(5,8)]=>[2,1,6,7,8,4,3,5]=>[1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]=>[1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
[(1,3),(2,7),(4,6),(5,8)]=>[3,6,1,7,8,4,2,5]=>[1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]=>[1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
[(1,4),(2,7),(3,6),(5,8)]=>[4,6,7,1,8,3,2,5]=>[1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]=>[1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
[(1,5),(2,7),(3,6),(4,8)]=>[5,6,7,8,1,3,2,4]=>[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
[(1,6),(2,7),(3,5),(4,8)]=>[5,6,7,8,3,1,2,4]=>[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
[(1,7),(2,6),(3,5),(4,8)]=>[5,6,7,8,3,2,1,4]=>[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
[(1,8),(2,6),(3,5),(4,7)]=>[5,6,7,8,3,2,4,1]=>[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
[(1,8),(2,5),(3,6),(4,7)]=>[5,6,7,8,2,3,4,1]=>[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
[(1,7),(2,5),(3,6),(4,8)]=>[5,6,7,8,2,3,1,4]=>[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
[(1,6),(2,5),(3,7),(4,8)]=>[5,6,7,8,2,1,3,4]=>[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
[(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
[(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]=>[1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
[(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]=>[1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
[(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]=>[1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
[(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]=>[1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
[(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]=>[1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
[(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]=>[1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
[(1,5),(2,4),(3,7),(6,8)]=>[4,5,7,2,1,8,3,6]=>[1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]=>[1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
[(1,6),(2,4),(3,7),(5,8)]=>[4,6,7,2,8,1,3,5]=>[1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]=>[1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
[(1,7),(2,4),(3,6),(5,8)]=>[4,6,7,2,8,3,1,5]=>[1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]=>[1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
[(1,8),(2,4),(3,6),(5,7)]=>[4,6,7,2,8,3,5,1]=>[1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]=>[1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
[(1,8),(2,3),(4,6),(5,7)]=>[3,6,2,7,8,4,5,1]=>[1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]=>[1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
[(1,7),(2,3),(4,6),(5,8)]=>[3,6,2,7,8,4,1,5]=>[1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]=>[1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
[(1,6),(2,3),(4,7),(5,8)]=>[3,6,2,7,8,1,4,5]=>[1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]=>[1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
[(1,5),(2,3),(4,7),(6,8)]=>[3,5,2,7,1,8,4,6]=>[1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]=>[1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
[(1,4),(2,3),(5,7),(6,8)]=>[3,4,2,1,7,8,5,6]=>[1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]=>[1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
[(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]=>[1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
[(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0]=>[1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
[(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,7,8,6,5]=>[1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0]=>[1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
[(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,7,8,6,5]=>[1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]=>[1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
[(1,4),(2,3),(5,8),(6,7)]=>[3,4,2,1,7,8,6,5]=>[1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]=>[1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
[(1,5),(2,3),(4,8),(6,7)]=>[3,5,2,7,1,8,6,4]=>[1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]=>[1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
[(1,6),(2,3),(4,8),(5,7)]=>[3,6,2,7,8,1,5,4]=>[1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]=>[1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
[(1,7),(2,3),(4,8),(5,6)]=>[3,6,2,7,8,5,1,4]=>[1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]=>[1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
[(1,8),(2,3),(4,7),(5,6)]=>[3,6,2,7,8,5,4,1]=>[1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]=>[1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
[(1,8),(2,4),(3,7),(5,6)]=>[4,6,7,2,8,5,3,1]=>[1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]=>[1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
[(1,7),(2,4),(3,8),(5,6)]=>[4,6,7,2,8,5,1,3]=>[1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]=>[1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
[(1,6),(2,4),(3,8),(5,7)]=>[4,6,7,2,8,1,5,3]=>[1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]=>[1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
[(1,5),(2,4),(3,8),(6,7)]=>[4,5,7,2,1,8,6,3]=>[1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]=>[1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
[(1,4),(2,5),(3,8),(6,7)]=>[4,5,7,1,2,8,6,3]=>[1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]=>[1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
[(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,7,2,8,6,4]=>[1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]=>[1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
[(1,2),(3,5),(4,8),(6,7)]=>[2,1,5,7,3,8,6,4]=>[1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]=>[1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
[(1,2),(3,6),(4,8),(5,7)]=>[2,1,6,7,8,3,5,4]=>[1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]=>[1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
[(1,3),(2,6),(4,8),(5,7)]=>[3,6,1,7,8,2,5,4]=>[1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]=>[1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
[(1,4),(2,6),(3,8),(5,7)]=>[4,6,7,1,8,2,5,3]=>[1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]=>[1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
[(1,5),(2,6),(3,8),(4,7)]=>[5,6,7,8,1,2,4,3]=>[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
[(1,6),(2,5),(3,8),(4,7)]=>[5,6,7,8,2,1,4,3]=>[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
[(1,7),(2,5),(3,8),(4,6)]=>[5,6,7,8,2,4,1,3]=>[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
[(1,8),(2,5),(3,7),(4,6)]=>[5,6,7,8,2,4,3,1]=>[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
[(1,8),(2,6),(3,7),(4,5)]=>[5,6,7,8,4,2,3,1]=>[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
[(1,7),(2,6),(3,8),(4,5)]=>[5,6,7,8,4,2,1,3]=>[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
[(1,6),(2,7),(3,8),(4,5)]=>[5,6,7,8,4,1,2,3]=>[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
[(1,5),(2,7),(3,8),(4,6)]=>[5,6,7,8,1,4,2,3]=>[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
[(1,4),(2,7),(3,8),(5,6)]=>[4,6,7,1,8,5,2,3]=>[1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]=>[1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
[(1,3),(2,7),(4,8),(5,6)]=>[3,6,1,7,8,5,2,4]=>[1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]=>[1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
[(1,2),(3,7),(4,8),(5,6)]=>[2,1,6,7,8,5,3,4]=>[1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]=>[1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
[(1,2),(3,8),(4,7),(5,6)]=>[2,1,6,7,8,5,4,3]=>[1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]=>[1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
[(1,3),(2,8),(4,7),(5,6)]=>[3,6,1,7,8,5,4,2]=>[1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]=>[1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
[(1,4),(2,8),(3,7),(5,6)]=>[4,6,7,1,8,5,3,2]=>[1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]=>[1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
[(1,5),(2,8),(3,7),(4,6)]=>[5,6,7,8,1,4,3,2]=>[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
[(1,6),(2,8),(3,7),(4,5)]=>[5,6,7,8,4,1,3,2]=>[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
[(1,7),(2,8),(3,6),(4,5)]=>[5,6,7,8,4,3,1,2]=>[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
[(1,8),(2,7),(3,6),(4,5)]=>[5,6,7,8,4,3,2,1]=>[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
[(1,2),(3,4),(5,6),(7,8),(9,10)]=>[2,1,4,3,6,5,8,7,10,9]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
[(1,3),(2,4),(5,6),(7,8),(9,10)]=>[3,4,1,2,6,5,8,7,10,9]=>[1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[1,1,0,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
[(1,4),(2,3),(5,6),(7,8),(9,10)]=>[3,4,2,1,6,5,8,7,10,9]=>[1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[1,1,0,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
[(1,10),(2,3),(4,5),(6,7),(8,9)]=>[3,5,2,7,4,9,6,10,8,1]=>[1,1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,0]=>[1,1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
[(1,8),(2,5),(3,4),(6,7),(9,10)]=>[4,5,7,3,2,8,6,1,10,9]=>[1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0,1,1,0,0]=>[1,1,1,0,1,0,1,1,0,1,0,0,0,1,1,0,0,0,1,0]
[(1,10),(2,9),(3,4),(5,6),(7,8)]=>[4,6,8,3,9,5,10,7,2,1]=>[1,1,1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0,0,0]=>[1,1,1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0,0,0]
[(1,3),(2,6),(4,7),(5,8),(9,10)]=>[3,6,1,7,8,2,4,5,10,9]=>[1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0,1,1,0,0]=>[1,1,0,1,1,1,0,1,0,0,1,0,1,1,0,0,0,0,1,0]
[(1,4),(2,5),(3,7),(6,8),(9,10)]=>[4,5,7,1,2,8,3,6,10,9]=>[1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0,1,1,0,0]=>[1,1,1,0,1,0,1,1,0,1,0,0,0,1,1,0,0,0,1,0]
[(1,8),(2,3),(4,7),(5,6),(9,10)]=>[3,6,2,7,8,5,4,1,10,9]=>[1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0,1,1,0,0]=>[1,1,0,1,1,1,0,1,0,0,1,0,1,1,0,0,0,0,1,0]
[(1,2),(3,10),(4,7),(5,6),(8,9)]=>[2,1,6,7,9,5,4,10,8,3]=>[1,1,0,0,1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]=>[1,0,1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,1,0,0]
[(1,10),(2,9),(3,8),(4,6),(5,7)]=>[6,7,8,9,10,4,5,3,2,1]=>[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]=>[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
[(1,6),(2,10),(3,9),(4,7),(5,8)]=>[6,7,8,9,10,1,4,5,3,2]=>[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]=>[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
[(1,6),(2,9),(3,10),(4,7),(5,8)]=>[6,7,8,9,10,1,4,5,2,3]=>[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]=>[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
[(1,10),(2,8),(3,9),(4,6),(5,7)]=>[6,7,8,9,10,4,5,2,3,1]=>[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]=>[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
[(1,9),(2,8),(3,7),(4,6),(5,10)]=>[6,7,8,9,10,4,3,2,1,5]=>[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]=>[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
[(1,10),(2,9),(3,6),(4,7),(5,8)]=>[6,7,8,9,10,3,4,5,2,1]=>[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]=>[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
[(1,9),(2,10),(3,6),(4,7),(5,8)]=>[6,7,8,9,10,3,4,5,1,2]=>[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]=>[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
[(1,6),(2,10),(3,7),(4,8),(5,9)]=>[6,7,8,9,10,1,3,4,5,2]=>[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]=>[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
[(1,8),(2,9),(3,6),(4,7),(5,10)]=>[6,7,8,9,10,3,4,1,2,5]=>[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]=>[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
[(1,9),(2,8),(3,6),(4,7),(5,10)]=>[6,7,8,9,10,3,4,2,1,5]=>[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]=>[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
[(1,2),(3,7),(4,8),(5,9),(6,10)]=>[2,1,7,8,9,10,3,4,5,6]=>[1,1,0,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[1,0,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0,0]
[(1,6),(2,7),(3,8),(4,9),(5,10)]=>[6,7,8,9,10,1,2,3,4,5]=>[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]=>[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
[(1,9),(2,6),(3,7),(4,8),(5,10)]=>[6,7,8,9,10,2,3,4,1,5]=>[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]=>[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
[(1,10),(2,6),(3,7),(4,8),(5,9)]=>[6,7,8,9,10,2,3,4,5,1]=>[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]=>[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
[(1,4),(2,6),(3,8),(5,9),(7,10)]=>[4,6,8,1,9,2,10,3,5,7]=>[1,1,1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0,0,0]=>[1,1,1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0,0,0]
[(1,2),(3,5),(4,8),(6,9),(7,10)]=>[2,1,5,8,3,9,10,4,6,7]=>[1,1,0,0,1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]=>[1,0,1,1,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
[(1,3),(2,5),(4,7),(6,9),(8,10)]=>[3,5,1,7,2,9,4,10,6,8]=>[1,1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,0]=>[1,1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
[(1,2),(3,6),(4,7),(5,9),(8,10)]=>[2,1,6,7,9,3,4,10,5,8]=>[1,1,0,0,1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]=>[1,0,1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,1,0,0]
[(1,3),(2,4),(5,6),(7,9),(8,10)]=>[3,4,1,2,6,5,9,10,7,8]=>[1,1,1,0,1,0,0,0,1,1,0,0,1,1,1,0,1,0,0,0]=>[1,1,0,1,0,1,1,0,0,0,1,1,1,0,0,1,0,1,0,0]
[(1,2),(3,4),(5,6),(7,9),(8,10)]=>[2,1,4,3,6,5,9,10,7,8]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0]=>[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
[(1,2),(3,4),(5,6),(7,10),(8,9)]=>[2,1,4,3,6,5,9,10,8,7]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0]=>[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
[(1,4),(2,3),(5,6),(7,10),(8,9)]=>[3,4,2,1,6,5,9,10,8,7]=>[1,1,1,0,1,0,0,0,1,1,0,0,1,1,1,0,1,0,0,0]=>[1,1,0,1,0,1,1,0,0,0,1,1,1,0,0,1,0,1,0,0]
[(1,2),(3,10),(4,5),(6,9),(7,8)]=>[2,1,5,8,4,9,10,7,6,3]=>[1,1,0,0,1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]=>[1,0,1,1,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
[(1,2),(3,10),(4,9),(5,8),(6,7)]=>[2,1,7,8,9,10,6,5,4,3]=>[1,1,0,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[1,0,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0,0]
[(1,6),(2,10),(3,9),(4,8),(5,7)]=>[6,7,8,9,10,1,5,4,3,2]=>[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]=>[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
[(1,10),(2,9),(3,8),(4,7),(5,6)]=>[6,7,8,9,10,5,4,3,2,1]=>[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]=>[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
[(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]=>[7,8,9,10,11,12,6,5,4,3,2,1]=>[1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0]=>[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
[(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]=>[3,5,2,7,4,9,6,11,8,12,10,1]=>[1,1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,0]=>[1,1,0,1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]=>[2,1,4,3,6,5,8,7,10,9,12,11]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
[(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]=>[2,1,4,3,6,5,8,7,11,12,10,9]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
[(1,2),(3,4),(5,6),(7,8),(9,11),(10,12)]=>[2,1,4,3,6,5,8,7,11,12,9,10]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
[(1,3),(2,5),(4,7),(6,9),(8,11),(10,12)]=>[3,5,1,7,2,9,4,11,6,12,8,10]=>[1,1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,0]=>[1,1,0,1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
[(1,7),(2,8),(3,9),(4,10),(5,11),(6,12)]=>[7,8,9,10,11,12,1,2,3,4,5,6]=>[1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0]=>[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
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
left-to-right-maxima to Dyck path
Description
The left-to-right maxima of a permutation as a Dyck path.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.
Map
peaks-to-valleys
Description
Return the path that has a valley wherever the original path has a peak of height at least one.
More precisely, the height of a valley in the image is the height of the corresponding peak minus $2$.
This is also (the inverse of) rowmotion on Dyck paths regarded as order ideals in the triangular poset.
More precisely, the height of a valley in the image is the height of the corresponding peak minus $2$.
This is also (the inverse of) rowmotion on Dyck paths regarded as order ideals in the triangular poset.
searching the database
Sorry, this map was not found in the database.