Identifier
Mp00283:
Perfect matchings
—non-nesting-exceedence permutation⟶
Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Images
=>
Cc0012;cc-rep-0
[(1,2)]=>[2,1]=>[1,2]=>[1,2]
[(1,2),(3,4)]=>[2,1,4,3]=>[3,4,1,2]=>[3,4,1,2]
[(1,3),(2,4)]=>[3,4,1,2]=>[2,1,4,3]=>[2,1,4,3]
[(1,4),(2,3)]=>[3,4,2,1]=>[1,2,4,3]=>[1,4,3,2]
[(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[5,6,3,4,1,2]=>[5,6,3,4,1,2]
[(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[5,6,2,1,4,3]=>[5,6,2,1,4,3]
[(1,4),(2,3),(5,6)]=>[3,4,2,1,6,5]=>[5,6,1,2,4,3]=>[5,6,1,4,3,2]
[(1,5),(2,3),(4,6)]=>[3,5,2,6,1,4]=>[4,1,6,2,5,3]=>[4,1,6,5,3,2]
[(1,6),(2,3),(4,5)]=>[3,5,2,6,4,1]=>[1,4,6,2,5,3]=>[1,6,5,4,3,2]
[(1,6),(2,4),(3,5)]=>[4,5,6,2,3,1]=>[1,3,2,6,5,4]=>[1,6,5,4,3,2]
[(1,5),(2,4),(3,6)]=>[4,5,6,2,1,3]=>[3,1,2,6,5,4]=>[3,1,6,5,4,2]
[(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[3,2,1,6,5,4]=>[3,2,1,6,5,4]
[(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[4,2,6,1,5,3]=>[4,2,6,1,5,3]
[(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[4,3,6,5,1,2]=>[4,3,6,5,1,2]
[(1,2),(3,6),(4,5)]=>[2,1,5,6,4,3]=>[3,4,6,5,1,2]=>[3,6,5,4,1,2]
[(1,3),(2,6),(4,5)]=>[3,5,1,6,4,2]=>[2,4,6,1,5,3]=>[2,6,5,1,4,3]
[(1,4),(2,6),(3,5)]=>[4,5,6,1,3,2]=>[2,3,1,6,5,4]=>[2,6,1,5,4,3]
[(1,5),(2,6),(3,4)]=>[4,5,6,3,1,2]=>[2,1,3,6,5,4]=>[2,1,6,5,4,3]
[(1,6),(2,5),(3,4)]=>[4,5,6,3,2,1]=>[1,2,3,6,5,4]=>[1,6,5,4,3,2]
[(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[7,8,5,6,3,4,1,2]=>[7,8,5,6,3,4,1,2]
[(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[7,8,5,6,2,1,4,3]=>[7,8,5,6,2,1,4,3]
[(1,4),(2,3),(5,6),(7,8)]=>[3,4,2,1,6,5,8,7]=>[7,8,5,6,1,2,4,3]=>[7,8,5,6,1,4,3,2]
[(1,5),(2,3),(4,6),(7,8)]=>[3,5,2,6,1,4,8,7]=>[7,8,4,1,6,2,5,3]=>[7,8,4,1,6,5,3,2]
[(1,6),(2,3),(4,5),(7,8)]=>[3,5,2,6,4,1,8,7]=>[7,8,1,4,6,2,5,3]=>[7,8,1,6,5,4,3,2]
[(1,7),(2,3),(4,5),(6,8)]=>[3,5,2,7,4,8,1,6]=>[6,1,8,4,7,2,5,3]=>[6,1,8,7,5,4,3,2]
[(1,8),(2,3),(4,5),(6,7)]=>[3,5,2,7,4,8,6,1]=>[1,6,8,4,7,2,5,3]=>[1,8,7,6,5,4,3,2]
[(1,8),(2,4),(3,5),(6,7)]=>[4,5,7,2,3,8,6,1]=>[1,6,8,3,2,7,5,4]=>[1,8,7,6,5,4,3,2]
[(1,7),(2,4),(3,5),(6,8)]=>[4,5,7,2,3,8,1,6]=>[6,1,8,3,2,7,5,4]=>[6,1,8,7,5,4,3,2]
[(1,6),(2,4),(3,5),(7,8)]=>[4,5,6,2,3,1,8,7]=>[7,8,1,3,2,6,5,4]=>[7,8,1,6,5,4,3,2]
[(1,5),(2,4),(3,6),(7,8)]=>[4,5,6,2,1,3,8,7]=>[7,8,3,1,2,6,5,4]=>[7,8,3,1,6,5,4,2]
[(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[7,8,3,2,1,6,5,4]=>[7,8,3,2,1,6,5,4]
[(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[7,8,4,2,6,1,5,3]=>[7,8,4,2,6,1,5,3]
[(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[7,8,4,3,6,5,1,2]=>[7,8,4,3,6,5,1,2]
[(1,2),(3,6),(4,5),(7,8)]=>[2,1,5,6,4,3,8,7]=>[7,8,3,4,6,5,1,2]=>[7,8,3,6,5,4,1,2]
[(1,3),(2,6),(4,5),(7,8)]=>[3,5,1,6,4,2,8,7]=>[7,8,2,4,6,1,5,3]=>[7,8,2,6,5,1,4,3]
[(1,4),(2,6),(3,5),(7,8)]=>[4,5,6,1,3,2,8,7]=>[7,8,2,3,1,6,5,4]=>[7,8,2,6,1,5,4,3]
[(1,5),(2,6),(3,4),(7,8)]=>[4,5,6,3,1,2,8,7]=>[7,8,2,1,3,6,5,4]=>[7,8,2,1,6,5,4,3]
[(1,6),(2,5),(3,4),(7,8)]=>[4,5,6,3,2,1,8,7]=>[7,8,1,2,3,6,5,4]=>[7,8,1,6,5,4,3,2]
[(1,7),(2,5),(3,4),(6,8)]=>[4,5,7,3,2,8,1,6]=>[6,1,8,2,3,7,5,4]=>[6,1,8,7,5,4,3,2]
[(1,8),(2,5),(3,4),(6,7)]=>[4,5,7,3,2,8,6,1]=>[1,6,8,2,3,7,5,4]=>[1,8,7,6,5,4,3,2]
[(1,8),(2,6),(3,4),(5,7)]=>[4,6,7,3,8,2,5,1]=>[1,5,2,8,3,7,6,4]=>[1,8,7,6,5,4,3,2]
[(1,7),(2,6),(3,4),(5,8)]=>[4,6,7,3,8,2,1,5]=>[5,1,2,8,3,7,6,4]=>[5,1,8,7,6,4,3,2]
[(1,6),(2,7),(3,4),(5,8)]=>[4,6,7,3,8,1,2,5]=>[5,2,1,8,3,7,6,4]=>[5,2,1,8,7,6,4,3]
[(1,5),(2,7),(3,4),(6,8)]=>[4,5,7,3,1,8,2,6]=>[6,2,8,1,3,7,5,4]=>[6,2,8,1,7,5,4,3]
[(1,4),(2,7),(3,5),(6,8)]=>[4,5,7,1,3,8,2,6]=>[6,2,8,3,1,7,5,4]=>[6,2,8,7,1,5,4,3]
[(1,3),(2,7),(4,5),(6,8)]=>[3,5,1,7,4,8,2,6]=>[6,2,8,4,7,1,5,3]=>[6,2,8,7,5,1,4,3]
[(1,2),(3,7),(4,5),(6,8)]=>[2,1,5,7,4,8,3,6]=>[6,3,8,4,7,5,1,2]=>[6,3,8,7,5,4,1,2]
[(1,2),(3,8),(4,5),(6,7)]=>[2,1,5,7,4,8,6,3]=>[3,6,8,4,7,5,1,2]=>[3,8,7,6,5,4,1,2]
[(1,3),(2,8),(4,5),(6,7)]=>[3,5,1,7,4,8,6,2]=>[2,6,8,4,7,1,5,3]=>[2,8,7,6,5,1,4,3]
[(1,4),(2,8),(3,5),(6,7)]=>[4,5,7,1,3,8,6,2]=>[2,6,8,3,1,7,5,4]=>[2,8,7,6,1,5,4,3]
[(1,5),(2,8),(3,4),(6,7)]=>[4,5,7,3,1,8,6,2]=>[2,6,8,1,3,7,5,4]=>[2,8,7,1,6,5,4,3]
[(1,6),(2,8),(3,4),(5,7)]=>[4,6,7,3,8,1,5,2]=>[2,5,1,8,3,7,6,4]=>[2,8,1,7,6,5,4,3]
[(1,7),(2,8),(3,4),(5,6)]=>[4,6,7,3,8,5,1,2]=>[2,1,5,8,3,7,6,4]=>[2,1,8,7,6,5,4,3]
[(1,8),(2,7),(3,4),(5,6)]=>[4,6,7,3,8,5,2,1]=>[1,2,5,8,3,7,6,4]=>[1,8,7,6,5,4,3,2]
[(1,8),(2,7),(3,5),(4,6)]=>[5,6,7,8,3,4,2,1]=>[1,2,4,3,8,7,6,5]=>[1,8,7,6,5,4,3,2]
[(1,7),(2,8),(3,5),(4,6)]=>[5,6,7,8,3,4,1,2]=>[2,1,4,3,8,7,6,5]=>[2,1,8,7,6,5,4,3]
[(1,6),(2,8),(3,5),(4,7)]=>[5,6,7,8,3,1,4,2]=>[2,4,1,3,8,7,6,5]=>[2,8,1,7,6,5,4,3]
[(1,5),(2,8),(3,6),(4,7)]=>[5,6,7,8,1,3,4,2]=>[2,4,3,1,8,7,6,5]=>[2,8,7,1,6,5,4,3]
[(1,4),(2,8),(3,6),(5,7)]=>[4,6,7,1,8,3,5,2]=>[2,5,3,8,1,7,6,4]=>[2,8,7,6,1,5,4,3]
[(1,3),(2,8),(4,6),(5,7)]=>[3,6,1,7,8,4,5,2]=>[2,5,4,8,7,1,6,3]=>[2,8,7,6,5,1,4,3]
[(1,2),(3,8),(4,6),(5,7)]=>[2,1,6,7,8,4,5,3]=>[3,5,4,8,7,6,1,2]=>[3,8,7,6,5,4,1,2]
[(1,2),(3,7),(4,6),(5,8)]=>[2,1,6,7,8,4,3,5]=>[5,3,4,8,7,6,1,2]=>[5,3,8,7,6,4,1,2]
[(1,3),(2,7),(4,6),(5,8)]=>[3,6,1,7,8,4,2,5]=>[5,2,4,8,7,1,6,3]=>[5,2,8,7,6,1,4,3]
[(1,4),(2,7),(3,6),(5,8)]=>[4,6,7,1,8,3,2,5]=>[5,2,3,8,1,7,6,4]=>[5,2,8,7,1,6,4,3]
[(1,5),(2,7),(3,6),(4,8)]=>[5,6,7,8,1,3,2,4]=>[4,2,3,1,8,7,6,5]=>[4,2,8,1,7,6,5,3]
[(1,6),(2,7),(3,5),(4,8)]=>[5,6,7,8,3,1,2,4]=>[4,2,1,3,8,7,6,5]=>[4,2,1,8,7,6,5,3]
[(1,7),(2,6),(3,5),(4,8)]=>[5,6,7,8,3,2,1,4]=>[4,1,2,3,8,7,6,5]=>[4,1,8,7,6,5,3,2]
[(1,8),(2,6),(3,5),(4,7)]=>[5,6,7,8,3,2,4,1]=>[1,4,2,3,8,7,6,5]=>[1,8,7,6,5,4,3,2]
[(1,8),(2,5),(3,6),(4,7)]=>[5,6,7,8,2,3,4,1]=>[1,4,3,2,8,7,6,5]=>[1,8,7,6,5,4,3,2]
[(1,7),(2,5),(3,6),(4,8)]=>[5,6,7,8,2,3,1,4]=>[4,1,3,2,8,7,6,5]=>[4,1,8,7,6,5,3,2]
[(1,6),(2,5),(3,7),(4,8)]=>[5,6,7,8,2,1,3,4]=>[4,3,1,2,8,7,6,5]=>[4,3,1,8,7,6,5,2]
[(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[4,3,2,1,8,7,6,5]=>[4,3,2,1,8,7,6,5]
[(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[5,3,2,8,1,7,6,4]=>[5,3,2,8,1,7,6,4]
[(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[5,4,2,8,7,1,6,3]=>[5,4,2,8,7,1,6,3]
[(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[5,4,3,8,7,6,1,2]=>[5,4,3,8,7,6,1,2]
[(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[6,4,8,3,7,5,1,2]=>[6,4,8,3,7,5,1,2]
[(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[6,4,8,2,7,1,5,3]=>[6,4,8,2,7,1,5,3]
[(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[6,3,8,2,1,7,5,4]=>[6,3,8,2,1,7,5,4]
[(1,5),(2,4),(3,7),(6,8)]=>[4,5,7,2,1,8,3,6]=>[6,3,8,1,2,7,5,4]=>[6,3,8,1,7,5,4,2]
[(1,6),(2,4),(3,7),(5,8)]=>[4,6,7,2,8,1,3,5]=>[5,3,1,8,2,7,6,4]=>[5,3,1,8,7,6,4,2]
[(1,7),(2,4),(3,6),(5,8)]=>[4,6,7,2,8,3,1,5]=>[5,1,3,8,2,7,6,4]=>[5,1,8,7,6,4,3,2]
[(1,8),(2,4),(3,6),(5,7)]=>[4,6,7,2,8,3,5,1]=>[1,5,3,8,2,7,6,4]=>[1,8,7,6,5,4,3,2]
[(1,8),(2,3),(4,6),(5,7)]=>[3,6,2,7,8,4,5,1]=>[1,5,4,8,7,2,6,3]=>[1,8,7,6,5,4,3,2]
[(1,7),(2,3),(4,6),(5,8)]=>[3,6,2,7,8,4,1,5]=>[5,1,4,8,7,2,6,3]=>[5,1,8,7,6,4,3,2]
[(1,6),(2,3),(4,7),(5,8)]=>[3,6,2,7,8,1,4,5]=>[5,4,1,8,7,2,6,3]=>[5,4,1,8,7,6,3,2]
[(1,5),(2,3),(4,7),(6,8)]=>[3,5,2,7,1,8,4,6]=>[6,4,8,1,7,2,5,3]=>[6,4,8,1,7,5,3,2]
[(1,4),(2,3),(5,7),(6,8)]=>[3,4,2,1,7,8,5,6]=>[6,5,8,7,1,2,4,3]=>[6,5,8,7,1,4,3,2]
[(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[6,5,8,7,2,1,4,3]=>[6,5,8,7,2,1,4,3]
[(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[6,5,8,7,3,4,1,2]=>[6,5,8,7,3,4,1,2]
[(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,7,8,6,5]=>[5,6,8,7,3,4,1,2]=>[5,8,7,6,3,4,1,2]
[(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,7,8,6,5]=>[5,6,8,7,2,1,4,3]=>[5,8,7,6,2,1,4,3]
[(1,4),(2,3),(5,8),(6,7)]=>[3,4,2,1,7,8,6,5]=>[5,6,8,7,1,2,4,3]=>[5,8,7,6,1,4,3,2]
[(1,5),(2,3),(4,8),(6,7)]=>[3,5,2,7,1,8,6,4]=>[4,6,8,1,7,2,5,3]=>[4,8,7,1,6,5,3,2]
[(1,6),(2,3),(4,8),(5,7)]=>[3,6,2,7,8,1,5,4]=>[4,5,1,8,7,2,6,3]=>[4,8,1,7,6,5,3,2]
[(1,7),(2,3),(4,8),(5,6)]=>[3,6,2,7,8,5,1,4]=>[4,1,5,8,7,2,6,3]=>[4,1,8,7,6,5,3,2]
[(1,8),(2,3),(4,7),(5,6)]=>[3,6,2,7,8,5,4,1]=>[1,4,5,8,7,2,6,3]=>[1,8,7,6,5,4,3,2]
[(1,8),(2,4),(3,7),(5,6)]=>[4,6,7,2,8,5,3,1]=>[1,3,5,8,2,7,6,4]=>[1,8,7,6,5,4,3,2]
[(1,7),(2,4),(3,8),(5,6)]=>[4,6,7,2,8,5,1,3]=>[3,1,5,8,2,7,6,4]=>[3,1,8,7,6,5,4,2]
[(1,6),(2,4),(3,8),(5,7)]=>[4,6,7,2,8,1,5,3]=>[3,5,1,8,2,7,6,4]=>[3,8,1,7,6,5,4,2]
[(1,5),(2,4),(3,8),(6,7)]=>[4,5,7,2,1,8,6,3]=>[3,6,8,1,2,7,5,4]=>[3,8,7,1,6,5,4,2]
[(1,4),(2,5),(3,8),(6,7)]=>[4,5,7,1,2,8,6,3]=>[3,6,8,2,1,7,5,4]=>[3,8,7,2,1,6,5,4]
[(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,7,2,8,6,4]=>[4,6,8,2,7,1,5,3]=>[4,8,7,2,6,1,5,3]
[(1,2),(3,5),(4,8),(6,7)]=>[2,1,5,7,3,8,6,4]=>[4,6,8,3,7,5,1,2]=>[4,8,7,3,6,5,1,2]
[(1,2),(3,6),(4,8),(5,7)]=>[2,1,6,7,8,3,5,4]=>[4,5,3,8,7,6,1,2]=>[4,8,3,7,6,5,1,2]
[(1,3),(2,6),(4,8),(5,7)]=>[3,6,1,7,8,2,5,4]=>[4,5,2,8,7,1,6,3]=>[4,8,2,7,6,1,5,3]
[(1,4),(2,6),(3,8),(5,7)]=>[4,6,7,1,8,2,5,3]=>[3,5,2,8,1,7,6,4]=>[3,8,2,7,1,6,5,4]
[(1,5),(2,6),(3,8),(4,7)]=>[5,6,7,8,1,2,4,3]=>[3,4,2,1,8,7,6,5]=>[3,8,2,1,7,6,5,4]
[(1,6),(2,5),(3,8),(4,7)]=>[5,6,7,8,2,1,4,3]=>[3,4,1,2,8,7,6,5]=>[3,8,1,7,6,5,4,2]
[(1,7),(2,5),(3,8),(4,6)]=>[5,6,7,8,2,4,1,3]=>[3,1,4,2,8,7,6,5]=>[3,1,8,7,6,5,4,2]
[(1,8),(2,5),(3,7),(4,6)]=>[5,6,7,8,2,4,3,1]=>[1,3,4,2,8,7,6,5]=>[1,8,7,6,5,4,3,2]
[(1,8),(2,6),(3,7),(4,5)]=>[5,6,7,8,4,2,3,1]=>[1,3,2,4,8,7,6,5]=>[1,8,7,6,5,4,3,2]
[(1,7),(2,6),(3,8),(4,5)]=>[5,6,7,8,4,2,1,3]=>[3,1,2,4,8,7,6,5]=>[3,1,8,7,6,5,4,2]
[(1,6),(2,7),(3,8),(4,5)]=>[5,6,7,8,4,1,2,3]=>[3,2,1,4,8,7,6,5]=>[3,2,1,8,7,6,5,4]
[(1,5),(2,7),(3,8),(4,6)]=>[5,6,7,8,1,4,2,3]=>[3,2,4,1,8,7,6,5]=>[3,2,8,1,7,6,5,4]
[(1,4),(2,7),(3,8),(5,6)]=>[4,6,7,1,8,5,2,3]=>[3,2,5,8,1,7,6,4]=>[3,2,8,7,1,6,5,4]
[(1,3),(2,7),(4,8),(5,6)]=>[3,6,1,7,8,5,2,4]=>[4,2,5,8,7,1,6,3]=>[4,2,8,7,6,1,5,3]
[(1,2),(3,7),(4,8),(5,6)]=>[2,1,6,7,8,5,3,4]=>[4,3,5,8,7,6,1,2]=>[4,3,8,7,6,5,1,2]
[(1,2),(3,8),(4,7),(5,6)]=>[2,1,6,7,8,5,4,3]=>[3,4,5,8,7,6,1,2]=>[3,8,7,6,5,4,1,2]
[(1,3),(2,8),(4,7),(5,6)]=>[3,6,1,7,8,5,4,2]=>[2,4,5,8,7,1,6,3]=>[2,8,7,6,5,1,4,3]
[(1,4),(2,8),(3,7),(5,6)]=>[4,6,7,1,8,5,3,2]=>[2,3,5,8,1,7,6,4]=>[2,8,7,6,1,5,4,3]
[(1,5),(2,8),(3,7),(4,6)]=>[5,6,7,8,1,4,3,2]=>[2,3,4,1,8,7,6,5]=>[2,8,7,1,6,5,4,3]
[(1,6),(2,8),(3,7),(4,5)]=>[5,6,7,8,4,1,3,2]=>[2,3,1,4,8,7,6,5]=>[2,8,1,7,6,5,4,3]
[(1,7),(2,8),(3,6),(4,5)]=>[5,6,7,8,4,3,1,2]=>[2,1,3,4,8,7,6,5]=>[2,1,8,7,6,5,4,3]
[(1,8),(2,7),(3,6),(4,5)]=>[5,6,7,8,4,3,2,1]=>[1,2,3,4,8,7,6,5]=>[1,8,7,6,5,4,3,2]
[(1,2),(3,4),(5,6),(7,8),(9,10)]=>[2,1,4,3,6,5,8,7,10,9]=>[9,10,7,8,5,6,3,4,1,2]=>[9,10,7,8,5,6,3,4,1,2]
[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]=>[2,1,4,3,6,5,8,7,10,9,12,11]=>[11,12,9,10,7,8,5,6,3,4,1,2]=>[11,12,9,10,7,8,5,6,3,4,1,2]
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
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
Map
Simion-Schmidt map
Description
The Simion-Schmidt map sends any permutation to a $123$-avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].
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