Identifier
Mp00283:
Perfect matchings
—non-nesting-exceedence permutation⟶
Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00066: Permutations —inverse⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Images
=>
Cc0012;cc-rep-0Cc0002;cc-rep-3
[(1,2)]=>[2,1]=>[2,1]=>[2]
[(1,2),(3,4)]=>[2,1,4,3]=>[2,1,4,3]=>[2,2]
[(1,3),(2,4)]=>[3,4,1,2]=>[3,4,1,2]=>[2,1,1]
[(1,4),(2,3)]=>[3,4,2,1]=>[4,3,1,2]=>[3,1]
[(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5]=>[2,2,2]
[(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[3,4,1,2,6,5]=>[2,2,1,1]
[(1,4),(2,3),(5,6)]=>[3,4,2,1,6,5]=>[4,3,1,2,6,5]=>[3,2,1]
[(1,5),(2,3),(4,6)]=>[3,5,2,6,1,4]=>[5,3,1,6,2,4]=>[3,2,1]
[(1,6),(2,3),(4,5)]=>[3,5,2,6,4,1]=>[6,3,1,5,2,4]=>[3,2,1]
[(1,6),(2,4),(3,5)]=>[4,5,6,2,3,1]=>[6,4,5,1,2,3]=>[3,1,1,1]
[(1,5),(2,4),(3,6)]=>[4,5,6,2,1,3]=>[5,4,6,1,2,3]=>[3,1,1,1]
[(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[4,5,6,1,2,3]=>[2,1,1,1,1]
[(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[3,5,1,6,2,4]=>[2,2,1,1]
[(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[2,1,5,6,3,4]=>[2,2,1,1]
[(1,2),(3,6),(4,5)]=>[2,1,5,6,4,3]=>[2,1,6,5,3,4]=>[3,2,1]
[(1,3),(2,6),(4,5)]=>[3,5,1,6,4,2]=>[3,6,1,5,2,4]=>[3,1,1,1]
[(1,4),(2,6),(3,5)]=>[4,5,6,1,3,2]=>[4,6,5,1,2,3]=>[3,1,1,1]
[(1,5),(2,6),(3,4)]=>[4,5,6,3,1,2]=>[5,6,4,1,2,3]=>[3,1,1,1]
[(1,6),(2,5),(3,4)]=>[4,5,6,3,2,1]=>[6,5,4,1,2,3]=>[4,1,1]
[(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[2,1,4,3,6,5,8,7]=>[2,2,2,2]
[(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[3,4,1,2,6,5,8,7]=>[2,2,2,1,1]
[(1,4),(2,3),(5,6),(7,8)]=>[3,4,2,1,6,5,8,7]=>[4,3,1,2,6,5,8,7]=>[3,2,2,1]
[(1,5),(2,3),(4,6),(7,8)]=>[3,5,2,6,1,4,8,7]=>[5,3,1,6,2,4,8,7]=>[3,2,2,1]
[(1,6),(2,3),(4,5),(7,8)]=>[3,5,2,6,4,1,8,7]=>[6,3,1,5,2,4,8,7]=>[3,2,2,1]
[(1,7),(2,3),(4,5),(6,8)]=>[3,5,2,7,4,8,1,6]=>[7,3,1,5,2,8,4,6]=>[3,2,2,1]
[(1,8),(2,3),(4,5),(6,7)]=>[3,5,2,7,4,8,6,1]=>[8,3,1,5,2,7,4,6]=>[3,2,2,1]
[(1,8),(2,4),(3,5),(6,7)]=>[4,5,7,2,3,8,6,1]=>[8,4,5,1,2,7,3,6]=>[3,2,1,1,1]
[(1,7),(2,4),(3,5),(6,8)]=>[4,5,7,2,3,8,1,6]=>[7,4,5,1,2,8,3,6]=>[3,2,1,1,1]
[(1,6),(2,4),(3,5),(7,8)]=>[4,5,6,2,3,1,8,7]=>[6,4,5,1,2,3,8,7]=>[3,2,1,1,1]
[(1,5),(2,4),(3,6),(7,8)]=>[4,5,6,2,1,3,8,7]=>[5,4,6,1,2,3,8,7]=>[3,2,1,1,1]
[(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[4,5,6,1,2,3,8,7]=>[2,2,1,1,1,1]
[(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[3,5,1,6,2,4,8,7]=>[2,2,2,1,1]
[(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[2,1,5,6,3,4,8,7]=>[2,2,2,1,1]
[(1,2),(3,6),(4,5),(7,8)]=>[2,1,5,6,4,3,8,7]=>[2,1,6,5,3,4,8,7]=>[3,2,2,1]
[(1,3),(2,6),(4,5),(7,8)]=>[3,5,1,6,4,2,8,7]=>[3,6,1,5,2,4,8,7]=>[3,2,1,1,1]
[(1,4),(2,6),(3,5),(7,8)]=>[4,5,6,1,3,2,8,7]=>[4,6,5,1,2,3,8,7]=>[3,2,1,1,1]
[(1,5),(2,6),(3,4),(7,8)]=>[4,5,6,3,1,2,8,7]=>[5,6,4,1,2,3,8,7]=>[3,2,1,1,1]
[(1,6),(2,5),(3,4),(7,8)]=>[4,5,6,3,2,1,8,7]=>[6,5,4,1,2,3,8,7]=>[4,2,1,1]
[(1,7),(2,5),(3,4),(6,8)]=>[4,5,7,3,2,8,1,6]=>[7,5,4,1,2,8,3,6]=>[4,2,1,1]
[(1,8),(2,5),(3,4),(6,7)]=>[4,5,7,3,2,8,6,1]=>[8,5,4,1,2,7,3,6]=>[4,2,1,1]
[(1,8),(2,6),(3,4),(5,7)]=>[4,6,7,3,8,2,5,1]=>[8,6,4,1,7,2,3,5]=>[4,2,1,1]
[(1,7),(2,6),(3,4),(5,8)]=>[4,6,7,3,8,2,1,5]=>[7,6,4,1,8,2,3,5]=>[4,2,1,1]
[(1,6),(2,7),(3,4),(5,8)]=>[4,6,7,3,8,1,2,5]=>[6,7,4,1,8,2,3,5]=>[3,2,1,1,1]
[(1,5),(2,7),(3,4),(6,8)]=>[4,5,7,3,1,8,2,6]=>[5,7,4,1,2,8,3,6]=>[3,2,1,1,1]
[(1,4),(2,7),(3,5),(6,8)]=>[4,5,7,1,3,8,2,6]=>[4,7,5,1,2,8,3,6]=>[3,2,1,1,1]
[(1,3),(2,7),(4,5),(6,8)]=>[3,5,1,7,4,8,2,6]=>[3,7,1,5,2,8,4,6]=>[3,2,1,1,1]
[(1,2),(3,7),(4,5),(6,8)]=>[2,1,5,7,4,8,3,6]=>[2,1,7,5,3,8,4,6]=>[3,2,2,1]
[(1,2),(3,8),(4,5),(6,7)]=>[2,1,5,7,4,8,6,3]=>[2,1,8,5,3,7,4,6]=>[3,2,2,1]
[(1,3),(2,8),(4,5),(6,7)]=>[3,5,1,7,4,8,6,2]=>[3,8,1,5,2,7,4,6]=>[3,2,1,1,1]
[(1,4),(2,8),(3,5),(6,7)]=>[4,5,7,1,3,8,6,2]=>[4,8,5,1,2,7,3,6]=>[3,2,1,1,1]
[(1,5),(2,8),(3,4),(6,7)]=>[4,5,7,3,1,8,6,2]=>[5,8,4,1,2,7,3,6]=>[3,2,1,1,1]
[(1,6),(2,8),(3,4),(5,7)]=>[4,6,7,3,8,1,5,2]=>[6,8,4,1,7,2,3,5]=>[3,2,1,1,1]
[(1,7),(2,8),(3,4),(5,6)]=>[4,6,7,3,8,5,1,2]=>[7,8,4,1,6,2,3,5]=>[3,2,1,1,1]
[(1,8),(2,7),(3,4),(5,6)]=>[4,6,7,3,8,5,2,1]=>[8,7,4,1,6,2,3,5]=>[4,2,1,1]
[(1,8),(2,7),(3,5),(4,6)]=>[5,6,7,8,3,4,2,1]=>[8,7,5,6,1,2,3,4]=>[4,1,1,1,1]
[(1,7),(2,8),(3,5),(4,6)]=>[5,6,7,8,3,4,1,2]=>[7,8,5,6,1,2,3,4]=>[3,1,1,1,1,1]
[(1,6),(2,8),(3,5),(4,7)]=>[5,6,7,8,3,1,4,2]=>[6,8,5,7,1,2,3,4]=>[3,1,1,1,1,1]
[(1,5),(2,8),(3,6),(4,7)]=>[5,6,7,8,1,3,4,2]=>[5,8,6,7,1,2,3,4]=>[3,1,1,1,1,1]
[(1,4),(2,8),(3,6),(5,7)]=>[4,6,7,1,8,3,5,2]=>[4,8,6,1,7,2,3,5]=>[3,2,1,1,1]
[(1,3),(2,8),(4,6),(5,7)]=>[3,6,1,7,8,4,5,2]=>[3,8,1,6,7,2,4,5]=>[3,1,1,1,1,1]
[(1,2),(3,8),(4,6),(5,7)]=>[2,1,6,7,8,4,5,3]=>[2,1,8,6,7,3,4,5]=>[3,2,1,1,1]
[(1,2),(3,7),(4,6),(5,8)]=>[2,1,6,7,8,4,3,5]=>[2,1,7,6,8,3,4,5]=>[3,2,1,1,1]
[(1,3),(2,7),(4,6),(5,8)]=>[3,6,1,7,8,4,2,5]=>[3,7,1,6,8,2,4,5]=>[3,1,1,1,1,1]
[(1,4),(2,7),(3,6),(5,8)]=>[4,6,7,1,8,3,2,5]=>[4,7,6,1,8,2,3,5]=>[3,2,1,1,1]
[(1,5),(2,7),(3,6),(4,8)]=>[5,6,7,8,1,3,2,4]=>[5,7,6,8,1,2,3,4]=>[3,1,1,1,1,1]
[(1,6),(2,7),(3,5),(4,8)]=>[5,6,7,8,3,1,2,4]=>[6,7,5,8,1,2,3,4]=>[3,1,1,1,1,1]
[(1,7),(2,6),(3,5),(4,8)]=>[5,6,7,8,3,2,1,4]=>[7,6,5,8,1,2,3,4]=>[4,1,1,1,1]
[(1,8),(2,6),(3,5),(4,7)]=>[5,6,7,8,3,2,4,1]=>[8,6,5,7,1,2,3,4]=>[4,1,1,1,1]
[(1,8),(2,5),(3,6),(4,7)]=>[5,6,7,8,2,3,4,1]=>[8,5,6,7,1,2,3,4]=>[3,1,1,1,1,1]
[(1,7),(2,5),(3,6),(4,8)]=>[5,6,7,8,2,3,1,4]=>[7,5,6,8,1,2,3,4]=>[3,1,1,1,1,1]
[(1,6),(2,5),(3,7),(4,8)]=>[5,6,7,8,2,1,3,4]=>[6,5,7,8,1,2,3,4]=>[3,1,1,1,1,1]
[(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[5,6,7,8,1,2,3,4]=>[2,1,1,1,1,1,1]
[(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[4,6,7,1,8,2,3,5]=>[2,2,1,1,1,1]
[(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[3,6,1,7,8,2,4,5]=>[2,2,1,1,1,1]
[(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[2,1,6,7,8,3,4,5]=>[2,2,1,1,1,1]
[(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[2,1,5,7,3,8,4,6]=>[2,2,2,1,1]
[(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[3,5,1,7,2,8,4,6]=>[2,2,2,1,1]
[(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[4,5,7,1,2,8,3,6]=>[2,2,1,1,1,1]
[(1,5),(2,4),(3,7),(6,8)]=>[4,5,7,2,1,8,3,6]=>[5,4,7,1,2,8,3,6]=>[3,2,1,1,1]
[(1,6),(2,4),(3,7),(5,8)]=>[4,6,7,2,8,1,3,5]=>[6,4,7,1,8,2,3,5]=>[3,2,1,1,1]
[(1,7),(2,4),(3,6),(5,8)]=>[4,6,7,2,8,3,1,5]=>[7,4,6,1,8,2,3,5]=>[3,2,1,1,1]
[(1,8),(2,4),(3,6),(5,7)]=>[4,6,7,2,8,3,5,1]=>[8,4,6,1,7,2,3,5]=>[3,2,1,1,1]
[(1,8),(2,3),(4,6),(5,7)]=>[3,6,2,7,8,4,5,1]=>[8,3,1,6,7,2,4,5]=>[3,2,1,1,1]
[(1,7),(2,3),(4,6),(5,8)]=>[3,6,2,7,8,4,1,5]=>[7,3,1,6,8,2,4,5]=>[3,2,1,1,1]
[(1,6),(2,3),(4,7),(5,8)]=>[3,6,2,7,8,1,4,5]=>[6,3,1,7,8,2,4,5]=>[3,2,1,1,1]
[(1,5),(2,3),(4,7),(6,8)]=>[3,5,2,7,1,8,4,6]=>[5,3,1,7,2,8,4,6]=>[3,2,2,1]
[(1,4),(2,3),(5,7),(6,8)]=>[3,4,2,1,7,8,5,6]=>[4,3,1,2,7,8,5,6]=>[3,2,1,1,1]
[(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[3,4,1,2,7,8,5,6]=>[2,2,1,1,1,1]
[(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[2,1,4,3,7,8,5,6]=>[2,2,2,1,1]
[(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,7,8,6,5]=>[2,1,4,3,8,7,5,6]=>[3,2,2,1]
[(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,7,8,6,5]=>[3,4,1,2,8,7,5,6]=>[3,2,1,1,1]
[(1,4),(2,3),(5,8),(6,7)]=>[3,4,2,1,7,8,6,5]=>[4,3,1,2,8,7,5,6]=>[3,3,1,1]
[(1,5),(2,3),(4,8),(6,7)]=>[3,5,2,7,1,8,6,4]=>[5,3,1,8,2,7,4,6]=>[3,3,1,1]
[(1,6),(2,3),(4,8),(5,7)]=>[3,6,2,7,8,1,5,4]=>[6,3,1,8,7,2,4,5]=>[3,3,1,1]
[(1,7),(2,3),(4,8),(5,6)]=>[3,6,2,7,8,5,1,4]=>[7,3,1,8,6,2,4,5]=>[3,3,1,1]
[(1,8),(2,3),(4,7),(5,6)]=>[3,6,2,7,8,5,4,1]=>[8,3,1,7,6,2,4,5]=>[4,2,1,1]
[(1,8),(2,4),(3,7),(5,6)]=>[4,6,7,2,8,5,3,1]=>[8,4,7,1,6,2,3,5]=>[4,1,1,1,1]
[(1,7),(2,4),(3,8),(5,6)]=>[4,6,7,2,8,5,1,3]=>[7,4,8,1,6,2,3,5]=>[3,2,1,1,1]
[(1,6),(2,4),(3,8),(5,7)]=>[4,6,7,2,8,1,5,3]=>[6,4,8,1,7,2,3,5]=>[3,2,1,1,1]
[(1,5),(2,4),(3,8),(6,7)]=>[4,5,7,2,1,8,6,3]=>[5,4,8,1,2,7,3,6]=>[3,2,1,1,1]
[(1,4),(2,5),(3,8),(6,7)]=>[4,5,7,1,2,8,6,3]=>[4,5,8,1,2,7,3,6]=>[3,1,1,1,1,1]
[(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,7,2,8,6,4]=>[3,5,1,8,2,7,4,6]=>[3,2,1,1,1]
[(1,2),(3,5),(4,8),(6,7)]=>[2,1,5,7,3,8,6,4]=>[2,1,5,8,3,7,4,6]=>[3,2,1,1,1]
[(1,2),(3,6),(4,8),(5,7)]=>[2,1,6,7,8,3,5,4]=>[2,1,6,8,7,3,4,5]=>[3,2,1,1,1]
[(1,3),(2,6),(4,8),(5,7)]=>[3,6,1,7,8,2,5,4]=>[3,6,1,8,7,2,4,5]=>[3,2,1,1,1]
[(1,4),(2,6),(3,8),(5,7)]=>[4,6,7,1,8,2,5,3]=>[4,6,8,1,7,2,3,5]=>[3,1,1,1,1,1]
[(1,5),(2,6),(3,8),(4,7)]=>[5,6,7,8,1,2,4,3]=>[5,6,8,7,1,2,3,4]=>[3,1,1,1,1,1]
[(1,6),(2,5),(3,8),(4,7)]=>[5,6,7,8,2,1,4,3]=>[6,5,8,7,1,2,3,4]=>[3,2,1,1,1]
[(1,7),(2,5),(3,8),(4,6)]=>[5,6,7,8,2,4,1,3]=>[7,5,8,6,1,2,3,4]=>[3,2,1,1,1]
[(1,8),(2,5),(3,7),(4,6)]=>[5,6,7,8,2,4,3,1]=>[8,5,7,6,1,2,3,4]=>[4,1,1,1,1]
[(1,8),(2,6),(3,7),(4,5)]=>[5,6,7,8,4,2,3,1]=>[8,6,7,5,1,2,3,4]=>[4,1,1,1,1]
[(1,7),(2,6),(3,8),(4,5)]=>[5,6,7,8,4,2,1,3]=>[7,6,8,5,1,2,3,4]=>[4,1,1,1,1]
[(1,6),(2,7),(3,8),(4,5)]=>[5,6,7,8,4,1,2,3]=>[6,7,8,5,1,2,3,4]=>[3,1,1,1,1,1]
[(1,5),(2,7),(3,8),(4,6)]=>[5,6,7,8,1,4,2,3]=>[5,7,8,6,1,2,3,4]=>[3,1,1,1,1,1]
[(1,4),(2,7),(3,8),(5,6)]=>[4,6,7,1,8,5,2,3]=>[4,7,8,1,6,2,3,5]=>[3,1,1,1,1,1]
[(1,3),(2,7),(4,8),(5,6)]=>[3,6,1,7,8,5,2,4]=>[3,7,1,8,6,2,4,5]=>[3,2,1,1,1]
[(1,2),(3,7),(4,8),(5,6)]=>[2,1,6,7,8,5,3,4]=>[2,1,7,8,6,3,4,5]=>[3,2,1,1,1]
[(1,2),(3,8),(4,7),(5,6)]=>[2,1,6,7,8,5,4,3]=>[2,1,8,7,6,3,4,5]=>[4,2,1,1]
[(1,3),(2,8),(4,7),(5,6)]=>[3,6,1,7,8,5,4,2]=>[3,8,1,7,6,2,4,5]=>[4,1,1,1,1]
[(1,4),(2,8),(3,7),(5,6)]=>[4,6,7,1,8,5,3,2]=>[4,8,7,1,6,2,3,5]=>[4,1,1,1,1]
[(1,5),(2,8),(3,7),(4,6)]=>[5,6,7,8,1,4,3,2]=>[5,8,7,6,1,2,3,4]=>[4,1,1,1,1]
[(1,6),(2,8),(3,7),(4,5)]=>[5,6,7,8,4,1,3,2]=>[6,8,7,5,1,2,3,4]=>[4,1,1,1,1]
[(1,7),(2,8),(3,6),(4,5)]=>[5,6,7,8,4,3,1,2]=>[7,8,6,5,1,2,3,4]=>[4,1,1,1,1]
[(1,8),(2,7),(3,6),(4,5)]=>[5,6,7,8,4,3,2,1]=>[8,7,6,5,1,2,3,4]=>[5,1,1,1]
[(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]=>[2,2,2,2,2]
[(1,3),(2,4),(5,6),(7,8),(9,10)]=>[3,4,1,2,6,5,8,7,10,9]=>[3,4,1,2,6,5,8,7,10,9]=>[2,2,2,2,1,1]
[(1,4),(2,3),(5,6),(7,8),(9,10)]=>[3,4,2,1,6,5,8,7,10,9]=>[4,3,1,2,6,5,8,7,10,9]=>[3,2,2,2,1]
[(1,4),(2,5),(3,6),(7,8),(9,10)]=>[4,5,6,1,2,3,8,7,10,9]=>[4,5,6,1,2,3,8,7,10,9]=>[2,2,2,1,1,1,1]
[(1,5),(2,6),(3,7),(4,8),(9,10)]=>[5,6,7,8,1,2,3,4,10,9]=>[5,6,7,8,1,2,3,4,10,9]=>[2,2,1,1,1,1,1,1]
[(1,3),(2,4),(5,7),(6,8),(9,10)]=>[3,4,1,2,7,8,5,6,10,9]=>[3,4,1,2,7,8,5,6,10,9]=>[2,2,2,1,1,1,1]
[(1,2),(3,7),(4,8),(5,9),(6,10)]=>[2,1,7,8,9,10,3,4,5,6]=>[2,1,7,8,9,10,3,4,5,6]=>[2,2,1,1,1,1,1,1]
[(1,6),(2,7),(3,8),(4,9),(5,10)]=>[6,7,8,9,10,1,2,3,4,5]=>[6,7,8,9,10,1,2,3,4,5]=>[2,1,1,1,1,1,1,1,1]
[(1,10),(2,6),(3,7),(4,8),(5,9)]=>[6,7,8,9,10,2,3,4,5,1]=>[10,6,7,8,9,1,2,3,4,5]=>[3,1,1,1,1,1,1,1]
[(1,3),(2,4),(5,8),(6,9),(7,10)]=>[3,4,1,2,8,9,10,5,6,7]=>[3,4,1,2,8,9,10,5,6,7]=>[2,2,1,1,1,1,1,1]
[(1,2),(3,4),(5,8),(6,9),(7,10)]=>[2,1,4,3,8,9,10,5,6,7]=>[2,1,4,3,8,9,10,5,6,7]=>[2,2,2,1,1,1,1]
[(1,2),(3,5),(4,6),(7,9),(8,10)]=>[2,1,5,6,3,4,9,10,7,8]=>[2,1,5,6,3,4,9,10,7,8]=>[2,2,2,1,1,1,1]
[(1,4),(2,5),(3,6),(7,9),(8,10)]=>[4,5,6,1,2,3,9,10,7,8]=>[4,5,6,1,2,3,9,10,7,8]=>[2,2,1,1,1,1,1,1]
[(1,2),(3,4),(5,6),(7,9),(8,10)]=>[2,1,4,3,6,5,9,10,7,8]=>[2,1,4,3,6,5,9,10,7,8]=>[2,2,2,2,1,1]
[(1,10),(2,9),(3,8),(4,7),(5,6)]=>[6,7,8,9,10,5,4,3,2,1]=>[10,9,8,7,6,1,2,3,4,5]=>[6,1,1,1,1]
[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]=>[2,1,4,3,6,5,8,7,10,9,12,11]=>[2,1,4,3,6,5,8,7,10,9,12,11]=>[2,2,2,2,2,2]
[(1,4),(2,5),(3,6),(7,10),(8,11),(9,12)]=>[4,5,6,1,2,3,10,11,12,7,8,9]=>[4,5,6,1,2,3,10,11,12,7,8,9]=>[2,2,1,1,1,1,1,1,1,1]
[(1,7),(2,8),(3,9),(4,10),(5,11),(6,12)]=>[7,8,9,10,11,12,1,2,3,4,5,6]=>[7,8,9,10,11,12,1,2,3,4,5,6]=>[2,1,1,1,1,1,1,1,1,1,1]
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
inverse
Description
Sends a permutation to its inverse.
Map
LLPS
Description
The Lewis-Lyu-Pylyavskyy-Sen shape of a permutation.
An ascent in a sequence $u = (u_1, u_2, \ldots)$ is an index $i$ such that $u_i < u_{i+1}$. Let $\mathrm{asc}(u)$ denote the number of ascents of $u$, and let
$$\mathrm{asc}^{*}(u) := \begin{cases} 0 &\textrm{if u is empty}, \\ 1 + \mathrm{asc}(u) &\textrm{otherwise}.\end{cases}$$
Given a permutation $w$ in the symmetric group $\mathfrak{S}_n$, define
$A'_k := \max_{u_1, \ldots, u_k} (\mathrm{asc}^{*}(u_1) + \cdots + \mathrm{asc}^{*}(u_k))$
where the maximum is taken over disjoint subsequences ${u_i}$ of $w$.
Then $A'_1, A'_2-A'_1, A'_3-A'_2,\dots$ is a partition of $n$. Its conjugate is the Lewis-Lyu-Pylyavskyy-Sen shape of a permutation.
An ascent in a sequence $u = (u_1, u_2, \ldots)$ is an index $i$ such that $u_i < u_{i+1}$. Let $\mathrm{asc}(u)$ denote the number of ascents of $u$, and let
$$\mathrm{asc}^{*}(u) := \begin{cases} 0 &\textrm{if u is empty}, \\ 1 + \mathrm{asc}(u) &\textrm{otherwise}.\end{cases}$$
Given a permutation $w$ in the symmetric group $\mathfrak{S}_n$, define
$A'_k := \max_{u_1, \ldots, u_k} (\mathrm{asc}^{*}(u_1) + \cdots + \mathrm{asc}^{*}(u_k))$
where the maximum is taken over disjoint subsequences ${u_i}$ of $w$.
Then $A'_1, A'_2-A'_1, A'_3-A'_2,\dots$ is a partition of $n$. Its conjugate is the Lewis-Lyu-Pylyavskyy-Sen shape of a permutation.
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