Identifier
Mp00058:
Perfect matchings
—to permutation⟶
Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Images
=>
Cc0012;cc-rep-0
[(1,2)]=>[2,1]=>[2,1]=>[2,1]=>[1,1]
[(1,2),(3,4)]=>[2,1,4,3]=>[2,1,4,3]=>[2,1,4,3]=>[1,2,1]
[(1,3),(2,4)]=>[3,4,1,2]=>[3,4,1,2]=>[3,1,4,2]=>[1,2,1]
[(1,4),(2,3)]=>[4,3,2,1]=>[4,3,2,1]=>[3,2,4,1]=>[1,2,1]
[(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[2,1,6,5,4,3]=>[2,1,5,4,6,3]=>[1,2,2,1]
[(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[3,6,1,5,4,2]=>[3,1,5,4,6,2]=>[1,2,2,1]
[(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[4,3,2,1,6,5]=>[3,2,4,1,6,5]=>[1,2,2,1]
[(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>[5,3,2,6,1,4]=>[3,2,5,1,6,4]=>[1,2,2,1]
[(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[6,3,2,5,4,1]=>[3,2,5,4,6,1]=>[1,2,2,1]
[(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>[6,4,5,2,3,1]=>[4,2,5,3,6,1]=>[1,2,2,1]
[(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>[5,4,6,2,1,3]=>[4,2,5,1,6,3]=>[1,2,2,1]
[(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[4,6,5,1,3,2]=>[4,1,5,3,6,2]=>[1,2,2,1]
[(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[3,6,1,5,4,2]=>[3,1,5,4,6,2]=>[1,2,2,1]
[(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[2,1,6,5,4,3]=>[2,1,5,4,6,3]=>[1,2,2,1]
[(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[2,1,6,5,4,3]=>[2,1,5,4,6,3]=>[1,2,2,1]
[(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>[3,6,1,5,4,2]=>[3,1,5,4,6,2]=>[1,2,2,1]
[(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>[4,6,5,1,3,2]=>[4,1,5,3,6,2]=>[1,2,2,1]
[(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>[5,6,4,3,1,2]=>[4,3,5,1,6,2]=>[1,2,2,1]
[(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[6,5,4,3,2,1]=>[4,3,5,2,6,1]=>[1,2,2,1]
[(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[2,1,8,7,6,5,4,3]=>[2,1,6,5,7,4,8,3]=>[1,2,2,2,1]
[(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[3,8,1,7,6,5,4,2]=>[3,1,6,5,7,4,8,2]=>[1,2,2,2,1]
[(1,4),(2,3),(5,6),(7,8)]=>[4,3,2,1,6,5,8,7]=>[4,3,2,1,8,7,6,5]=>[3,2,4,1,7,6,8,5]=>[1,2,2,2,1]
[(1,5),(2,3),(4,6),(7,8)]=>[5,3,2,6,1,4,8,7]=>[5,3,2,8,1,7,6,4]=>[3,2,5,1,7,6,8,4]=>[1,2,2,2,1]
[(1,6),(2,3),(4,5),(7,8)]=>[6,3,2,5,4,1,8,7]=>[6,3,2,8,7,1,5,4]=>[3,2,6,1,7,5,8,4]=>[1,2,2,2,1]
[(1,7),(2,3),(4,5),(6,8)]=>[7,3,2,5,4,8,1,6]=>[7,3,2,8,6,5,1,4]=>[3,2,6,5,7,1,8,4]=>[1,2,2,2,1]
[(1,8),(2,3),(4,5),(6,7)]=>[8,3,2,5,4,7,6,1]=>[8,3,2,7,6,5,4,1]=>[3,2,6,5,7,4,8,1]=>[1,2,2,2,1]
[(1,8),(2,4),(3,5),(6,7)]=>[8,4,5,2,3,7,6,1]=>[8,4,7,2,6,5,3,1]=>[4,2,6,5,7,3,8,1]=>[1,2,2,2,1]
[(1,7),(2,4),(3,5),(6,8)]=>[7,4,5,2,3,8,1,6]=>[7,4,8,2,6,5,1,3]=>[4,2,6,5,7,1,8,3]=>[1,2,2,2,1]
[(1,6),(2,4),(3,5),(7,8)]=>[6,4,5,2,3,1,8,7]=>[6,4,8,2,7,1,5,3]=>[4,2,6,1,7,5,8,3]=>[1,2,2,2,1]
[(1,5),(2,4),(3,6),(7,8)]=>[5,4,6,2,1,3,8,7]=>[5,4,8,2,1,7,6,3]=>[4,2,5,1,7,6,8,3]=>[1,2,2,2,1]
[(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[4,8,7,1,6,5,3,2]=>[4,1,6,5,7,3,8,2]=>[1,2,2,2,1]
[(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[3,8,1,7,6,5,4,2]=>[3,1,6,5,7,4,8,2]=>[1,2,2,2,1]
[(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[2,1,8,7,6,5,4,3]=>[2,1,6,5,7,4,8,3]=>[1,2,2,2,1]
[(1,2),(3,6),(4,5),(7,8)]=>[2,1,6,5,4,3,8,7]=>[2,1,8,7,6,5,4,3]=>[2,1,6,5,7,4,8,3]=>[1,2,2,2,1]
[(1,3),(2,6),(4,5),(7,8)]=>[3,6,1,5,4,2,8,7]=>[3,8,1,7,6,5,4,2]=>[3,1,6,5,7,4,8,2]=>[1,2,2,2,1]
[(1,4),(2,6),(3,5),(7,8)]=>[4,6,5,1,3,2,8,7]=>[4,8,7,1,6,5,3,2]=>[4,1,6,5,7,3,8,2]=>[1,2,2,2,1]
[(1,5),(2,6),(3,4),(7,8)]=>[5,6,4,3,1,2,8,7]=>[5,8,4,3,1,7,6,2]=>[4,3,5,1,7,6,8,2]=>[1,2,2,2,1]
[(1,6),(2,5),(3,4),(7,8)]=>[6,5,4,3,2,1,8,7]=>[6,5,4,3,2,1,8,7]=>[4,3,5,2,6,1,8,7]=>[1,2,2,2,1]
[(1,7),(2,5),(3,4),(6,8)]=>[7,5,4,3,2,8,1,6]=>[7,5,4,3,2,8,1,6]=>[4,3,5,2,7,1,8,6]=>[1,2,2,2,1]
[(1,8),(2,5),(3,4),(6,7)]=>[8,5,4,3,2,7,6,1]=>[8,5,4,3,2,7,6,1]=>[4,3,5,2,7,6,8,1]=>[1,2,2,2,1]
[(1,8),(2,6),(3,4),(5,7)]=>[8,6,4,3,7,2,5,1]=>[8,6,4,3,7,2,5,1]=>[4,3,6,2,7,5,8,1]=>[1,2,2,2,1]
[(1,7),(2,6),(3,4),(5,8)]=>[7,6,4,3,8,2,1,5]=>[7,6,4,3,8,2,1,5]=>[4,3,6,2,7,1,8,5]=>[1,2,2,2,1]
[(1,6),(2,7),(3,4),(5,8)]=>[6,7,4,3,8,1,2,5]=>[6,8,4,3,7,1,5,2]=>[4,3,6,1,7,5,8,2]=>[1,2,2,2,1]
[(1,5),(2,7),(3,4),(6,8)]=>[5,7,4,3,1,8,2,6]=>[5,8,4,3,1,7,6,2]=>[4,3,5,1,7,6,8,2]=>[1,2,2,2,1]
[(1,4),(2,7),(3,5),(6,8)]=>[4,7,5,1,3,8,2,6]=>[4,8,7,1,6,5,3,2]=>[4,1,6,5,7,3,8,2]=>[1,2,2,2,1]
[(1,3),(2,7),(4,5),(6,8)]=>[3,7,1,5,4,8,2,6]=>[3,8,1,7,6,5,4,2]=>[3,1,6,5,7,4,8,2]=>[1,2,2,2,1]
[(1,2),(3,7),(4,5),(6,8)]=>[2,1,7,5,4,8,3,6]=>[2,1,8,7,6,5,4,3]=>[2,1,6,5,7,4,8,3]=>[1,2,2,2,1]
[(1,2),(3,8),(4,5),(6,7)]=>[2,1,8,5,4,7,6,3]=>[2,1,8,7,6,5,4,3]=>[2,1,6,5,7,4,8,3]=>[1,2,2,2,1]
[(1,3),(2,8),(4,5),(6,7)]=>[3,8,1,5,4,7,6,2]=>[3,8,1,7,6,5,4,2]=>[3,1,6,5,7,4,8,2]=>[1,2,2,2,1]
[(1,4),(2,8),(3,5),(6,7)]=>[4,8,5,1,3,7,6,2]=>[4,8,7,1,6,5,3,2]=>[4,1,6,5,7,3,8,2]=>[1,2,2,2,1]
[(1,5),(2,8),(3,4),(6,7)]=>[5,8,4,3,1,7,6,2]=>[5,8,4,3,1,7,6,2]=>[4,3,5,1,7,6,8,2]=>[1,2,2,2,1]
[(1,6),(2,8),(3,4),(5,7)]=>[6,8,4,3,7,1,5,2]=>[6,8,4,3,7,1,5,2]=>[4,3,6,1,7,5,8,2]=>[1,2,2,2,1]
[(1,7),(2,8),(3,4),(5,6)]=>[7,8,4,3,6,5,1,2]=>[7,8,4,3,6,5,1,2]=>[4,3,6,5,7,1,8,2]=>[1,2,2,2,1]
[(1,8),(2,7),(3,4),(5,6)]=>[8,7,4,3,6,5,2,1]=>[8,7,4,3,6,5,2,1]=>[4,3,6,5,7,2,8,1]=>[1,2,2,2,1]
[(1,8),(2,7),(3,5),(4,6)]=>[8,7,5,6,3,4,2,1]=>[8,7,5,6,3,4,2,1]=>[5,3,6,4,7,2,8,1]=>[1,2,2,2,1]
[(1,7),(2,8),(3,5),(4,6)]=>[7,8,5,6,3,4,1,2]=>[7,8,5,6,3,4,1,2]=>[5,3,6,4,7,1,8,2]=>[1,2,2,2,1]
[(1,6),(2,8),(3,5),(4,7)]=>[6,8,5,7,3,1,4,2]=>[6,8,5,7,3,1,4,2]=>[5,3,6,1,7,4,8,2]=>[1,2,2,2,1]
[(1,5),(2,8),(3,6),(4,7)]=>[5,8,6,7,1,3,4,2]=>[5,8,7,6,1,4,3,2]=>[5,1,6,4,7,3,8,2]=>[1,2,2,2,1]
[(1,4),(2,8),(3,6),(5,7)]=>[4,8,6,1,7,3,5,2]=>[4,8,7,1,6,5,3,2]=>[4,1,6,5,7,3,8,2]=>[1,2,2,2,1]
[(1,3),(2,8),(4,6),(5,7)]=>[3,8,1,6,7,4,5,2]=>[3,8,1,7,6,5,4,2]=>[3,1,6,5,7,4,8,2]=>[1,2,2,2,1]
[(1,2),(3,8),(4,6),(5,7)]=>[2,1,8,6,7,4,5,3]=>[2,1,8,7,6,5,4,3]=>[2,1,6,5,7,4,8,3]=>[1,2,2,2,1]
[(1,2),(3,7),(4,6),(5,8)]=>[2,1,7,6,8,4,3,5]=>[2,1,8,7,6,5,4,3]=>[2,1,6,5,7,4,8,3]=>[1,2,2,2,1]
[(1,3),(2,7),(4,6),(5,8)]=>[3,7,1,6,8,4,2,5]=>[3,8,1,7,6,5,4,2]=>[3,1,6,5,7,4,8,2]=>[1,2,2,2,1]
[(1,4),(2,7),(3,6),(5,8)]=>[4,7,6,1,8,3,2,5]=>[4,8,7,1,6,5,3,2]=>[4,1,6,5,7,3,8,2]=>[1,2,2,2,1]
[(1,5),(2,7),(3,6),(4,8)]=>[5,7,6,8,1,3,2,4]=>[5,8,7,6,1,4,3,2]=>[5,1,6,4,7,3,8,2]=>[1,2,2,2,1]
[(1,6),(2,7),(3,5),(4,8)]=>[6,7,5,8,3,1,2,4]=>[6,8,5,7,3,1,4,2]=>[5,3,6,1,7,4,8,2]=>[1,2,2,2,1]
[(1,7),(2,6),(3,5),(4,8)]=>[7,6,5,8,3,2,1,4]=>[7,6,5,8,3,2,1,4]=>[5,3,6,2,7,1,8,4]=>[1,2,2,2,1]
[(1,8),(2,6),(3,5),(4,7)]=>[8,6,5,7,3,2,4,1]=>[8,6,5,7,3,2,4,1]=>[5,3,6,2,7,4,8,1]=>[1,2,2,2,1]
[(1,8),(2,5),(3,6),(4,7)]=>[8,5,6,7,2,3,4,1]=>[8,5,7,6,2,4,3,1]=>[5,2,6,4,7,3,8,1]=>[1,2,2,2,1]
[(1,7),(2,5),(3,6),(4,8)]=>[7,5,6,8,2,3,1,4]=>[7,5,8,6,2,4,1,3]=>[5,2,6,4,7,1,8,3]=>[1,2,2,2,1]
[(1,6),(2,5),(3,7),(4,8)]=>[6,5,7,8,2,1,3,4]=>[6,5,8,7,2,1,4,3]=>[5,2,6,1,7,4,8,3]=>[1,2,2,2,1]
[(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[5,8,7,6,1,4,3,2]=>[5,1,6,4,7,3,8,2]=>[1,2,2,2,1]
[(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[4,8,7,1,6,5,3,2]=>[4,1,6,5,7,3,8,2]=>[1,2,2,2,1]
[(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[3,8,1,7,6,5,4,2]=>[3,1,6,5,7,4,8,2]=>[1,2,2,2,1]
[(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]=>[1,2,2,2,1]
[(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[2,1,8,7,6,5,4,3]=>[2,1,6,5,7,4,8,3]=>[1,2,2,2,1]
[(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[3,8,1,7,6,5,4,2]=>[3,1,6,5,7,4,8,2]=>[1,2,2,2,1]
[(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[4,8,7,1,6,5,3,2]=>[4,1,6,5,7,3,8,2]=>[1,2,2,2,1]
[(1,5),(2,4),(3,7),(6,8)]=>[5,4,7,2,1,8,3,6]=>[5,4,8,2,1,7,6,3]=>[4,2,5,1,7,6,8,3]=>[1,2,2,2,1]
[(1,6),(2,4),(3,7),(5,8)]=>[6,4,7,2,8,1,3,5]=>[6,4,8,2,7,1,5,3]=>[4,2,6,1,7,5,8,3]=>[1,2,2,2,1]
[(1,7),(2,4),(3,6),(5,8)]=>[7,4,6,2,8,3,1,5]=>[7,4,8,2,6,5,1,3]=>[4,2,6,5,7,1,8,3]=>[1,2,2,2,1]
[(1,8),(2,4),(3,6),(5,7)]=>[8,4,6,2,7,3,5,1]=>[8,4,7,2,6,5,3,1]=>[4,2,6,5,7,3,8,1]=>[1,2,2,2,1]
[(1,8),(2,3),(4,6),(5,7)]=>[8,3,2,6,7,4,5,1]=>[8,3,2,7,6,5,4,1]=>[3,2,6,5,7,4,8,1]=>[1,2,2,2,1]
[(1,7),(2,3),(4,6),(5,8)]=>[7,3,2,6,8,4,1,5]=>[7,3,2,8,6,5,1,4]=>[3,2,6,5,7,1,8,4]=>[1,2,2,2,1]
[(1,6),(2,3),(4,7),(5,8)]=>[6,3,2,7,8,1,4,5]=>[6,3,2,8,7,1,5,4]=>[3,2,6,1,7,5,8,4]=>[1,2,2,2,1]
[(1,5),(2,3),(4,7),(6,8)]=>[5,3,2,7,1,8,4,6]=>[5,3,2,8,1,7,6,4]=>[3,2,5,1,7,6,8,4]=>[1,2,2,2,1]
[(1,4),(2,3),(5,7),(6,8)]=>[4,3,2,1,7,8,5,6]=>[4,3,2,1,8,7,6,5]=>[3,2,4,1,7,6,8,5]=>[1,2,2,2,1]
[(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[3,8,1,7,6,5,4,2]=>[3,1,6,5,7,4,8,2]=>[1,2,2,2,1]
[(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[2,1,8,7,6,5,4,3]=>[2,1,6,5,7,4,8,3]=>[1,2,2,2,1]
[(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,8,7,6,5]=>[2,1,8,7,6,5,4,3]=>[2,1,6,5,7,4,8,3]=>[1,2,2,2,1]
[(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,8,7,6,5]=>[3,8,1,7,6,5,4,2]=>[3,1,6,5,7,4,8,2]=>[1,2,2,2,1]
[(1,4),(2,3),(5,8),(6,7)]=>[4,3,2,1,8,7,6,5]=>[4,3,2,1,8,7,6,5]=>[3,2,4,1,7,6,8,5]=>[1,2,2,2,1]
[(1,5),(2,3),(4,8),(6,7)]=>[5,3,2,8,1,7,6,4]=>[5,3,2,8,1,7,6,4]=>[3,2,5,1,7,6,8,4]=>[1,2,2,2,1]
[(1,6),(2,3),(4,8),(5,7)]=>[6,3,2,8,7,1,5,4]=>[6,3,2,8,7,1,5,4]=>[3,2,6,1,7,5,8,4]=>[1,2,2,2,1]
[(1,7),(2,3),(4,8),(5,6)]=>[7,3,2,8,6,5,1,4]=>[7,3,2,8,6,5,1,4]=>[3,2,6,5,7,1,8,4]=>[1,2,2,2,1]
[(1,8),(2,3),(4,7),(5,6)]=>[8,3,2,7,6,5,4,1]=>[8,3,2,7,6,5,4,1]=>[3,2,6,5,7,4,8,1]=>[1,2,2,2,1]
[(1,8),(2,4),(3,7),(5,6)]=>[8,4,7,2,6,5,3,1]=>[8,4,7,2,6,5,3,1]=>[4,2,6,5,7,3,8,1]=>[1,2,2,2,1]
[(1,7),(2,4),(3,8),(5,6)]=>[7,4,8,2,6,5,1,3]=>[7,4,8,2,6,5,1,3]=>[4,2,6,5,7,1,8,3]=>[1,2,2,2,1]
[(1,6),(2,4),(3,8),(5,7)]=>[6,4,8,2,7,1,5,3]=>[6,4,8,2,7,1,5,3]=>[4,2,6,1,7,5,8,3]=>[1,2,2,2,1]
[(1,5),(2,4),(3,8),(6,7)]=>[5,4,8,2,1,7,6,3]=>[5,4,8,2,1,7,6,3]=>[4,2,5,1,7,6,8,3]=>[1,2,2,2,1]
[(1,4),(2,5),(3,8),(6,7)]=>[4,5,8,1,2,7,6,3]=>[4,8,7,1,6,5,3,2]=>[4,1,6,5,7,3,8,2]=>[1,2,2,2,1]
[(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,8,2,7,6,4]=>[3,8,1,7,6,5,4,2]=>[3,1,6,5,7,4,8,2]=>[1,2,2,2,1]
[(1,2),(3,5),(4,8),(6,7)]=>[2,1,5,8,3,7,6,4]=>[2,1,8,7,6,5,4,3]=>[2,1,6,5,7,4,8,3]=>[1,2,2,2,1]
[(1,2),(3,6),(4,8),(5,7)]=>[2,1,6,8,7,3,5,4]=>[2,1,8,7,6,5,4,3]=>[2,1,6,5,7,4,8,3]=>[1,2,2,2,1]
[(1,3),(2,6),(4,8),(5,7)]=>[3,6,1,8,7,2,5,4]=>[3,8,1,7,6,5,4,2]=>[3,1,6,5,7,4,8,2]=>[1,2,2,2,1]
[(1,4),(2,6),(3,8),(5,7)]=>[4,6,8,1,7,2,5,3]=>[4,8,7,1,6,5,3,2]=>[4,1,6,5,7,3,8,2]=>[1,2,2,2,1]
[(1,5),(2,6),(3,8),(4,7)]=>[5,6,8,7,1,2,4,3]=>[5,8,7,6,1,4,3,2]=>[5,1,6,4,7,3,8,2]=>[1,2,2,2,1]
[(1,6),(2,5),(3,8),(4,7)]=>[6,5,8,7,2,1,4,3]=>[6,5,8,7,2,1,4,3]=>[5,2,6,1,7,4,8,3]=>[1,2,2,2,1]
[(1,7),(2,5),(3,8),(4,6)]=>[7,5,8,6,2,4,1,3]=>[7,5,8,6,2,4,1,3]=>[5,2,6,4,7,1,8,3]=>[1,2,2,2,1]
[(1,8),(2,5),(3,7),(4,6)]=>[8,5,7,6,2,4,3,1]=>[8,5,7,6,2,4,3,1]=>[5,2,6,4,7,3,8,1]=>[1,2,2,2,1]
[(1,8),(2,6),(3,7),(4,5)]=>[8,6,7,5,4,2,3,1]=>[8,6,7,5,4,2,3,1]=>[5,4,6,2,7,3,8,1]=>[1,2,2,2,1]
[(1,7),(2,6),(3,8),(4,5)]=>[7,6,8,5,4,2,1,3]=>[7,6,8,5,4,2,1,3]=>[5,4,6,2,7,1,8,3]=>[1,2,2,2,1]
[(1,6),(2,7),(3,8),(4,5)]=>[6,7,8,5,4,1,2,3]=>[6,8,7,5,4,1,3,2]=>[5,4,6,1,7,3,8,2]=>[1,2,2,2,1]
[(1,5),(2,7),(3,8),(4,6)]=>[5,7,8,6,1,4,2,3]=>[5,8,7,6,1,4,3,2]=>[5,1,6,4,7,3,8,2]=>[1,2,2,2,1]
[(1,4),(2,7),(3,8),(5,6)]=>[4,7,8,1,6,5,2,3]=>[4,8,7,1,6,5,3,2]=>[4,1,6,5,7,3,8,2]=>[1,2,2,2,1]
[(1,3),(2,7),(4,8),(5,6)]=>[3,7,1,8,6,5,2,4]=>[3,8,1,7,6,5,4,2]=>[3,1,6,5,7,4,8,2]=>[1,2,2,2,1]
[(1,2),(3,7),(4,8),(5,6)]=>[2,1,7,8,6,5,3,4]=>[2,1,8,7,6,5,4,3]=>[2,1,6,5,7,4,8,3]=>[1,2,2,2,1]
[(1,2),(3,8),(4,7),(5,6)]=>[2,1,8,7,6,5,4,3]=>[2,1,8,7,6,5,4,3]=>[2,1,6,5,7,4,8,3]=>[1,2,2,2,1]
[(1,3),(2,8),(4,7),(5,6)]=>[3,8,1,7,6,5,4,2]=>[3,8,1,7,6,5,4,2]=>[3,1,6,5,7,4,8,2]=>[1,2,2,2,1]
[(1,4),(2,8),(3,7),(5,6)]=>[4,8,7,1,6,5,3,2]=>[4,8,7,1,6,5,3,2]=>[4,1,6,5,7,3,8,2]=>[1,2,2,2,1]
[(1,5),(2,8),(3,7),(4,6)]=>[5,8,7,6,1,4,3,2]=>[5,8,7,6,1,4,3,2]=>[5,1,6,4,7,3,8,2]=>[1,2,2,2,1]
[(1,6),(2,8),(3,7),(4,5)]=>[6,8,7,5,4,1,3,2]=>[6,8,7,5,4,1,3,2]=>[5,4,6,1,7,3,8,2]=>[1,2,2,2,1]
[(1,7),(2,8),(3,6),(4,5)]=>[7,8,6,5,4,3,1,2]=>[7,8,6,5,4,3,1,2]=>[5,4,6,3,7,1,8,2]=>[1,2,2,2,1]
[(1,8),(2,7),(3,6),(4,5)]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>[5,4,6,3,7,2,8,1]=>[1,2,2,2,1]
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
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].
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.
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 composition
Description
The descent composition of a permutation.
The descent composition of a permutation $\pi$ of length $n$ is the integer composition of $n$ whose descent set equals the descent set of $\pi$. The descent set of a permutation $\pi$ is $\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$. The descent set of a composition $c = (i_1, i_2, \ldots, i_k)$ is the set $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.
The descent composition of a permutation $\pi$ of length $n$ is the integer composition of $n$ whose descent set equals the descent set of $\pi$. The descent set of a permutation $\pi$ is $\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$. The descent set of a composition $c = (i_1, i_2, \ldots, i_k)$ is the set $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.
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