Identifier
Mp00058:
Perfect matchings
—to permutation⟶
Permutations
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Images
=>
Cc0012;cc-rep-0
[(1,2)]=>[2,1]=>[2,1]=>[2,1]
[(1,2),(3,4)]=>[2,1,4,3]=>[2,1,4,3]=>[4,2,1,3]
[(1,3),(2,4)]=>[3,4,1,2]=>[4,1,3,2]=>[4,1,3,2]
[(1,4),(2,3)]=>[4,3,2,1]=>[2,3,4,1]=>[2,3,4,1]
[(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5]=>[6,4,2,1,3,5]
[(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[4,1,3,2,6,5]=>[6,4,1,3,2,5]
[(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[2,3,4,1,6,5]=>[6,2,3,4,1,5]
[(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>[6,3,5,1,2,4]=>[1,3,2,6,5,4]
[(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[4,3,5,6,1,2]=>[1,4,3,5,6,2]
[(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>[3,5,2,6,4,1]=>[5,3,6,2,4,1]
[(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>[2,6,1,5,4,3]=>[6,5,2,1,4,3]
[(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[6,1,2,4,5,3]=>[4,1,2,6,5,3]
[(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[5,6,3,2,1,4]=>[3,5,6,2,1,4]
[(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[2,1,6,3,5,4]=>[6,2,5,1,3,4]
[(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[2,1,4,5,6,3]=>[4,2,1,5,6,3]
[(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>[6,4,3,5,1,2]=>[1,4,6,3,5,2]
[(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>[5,3,1,4,6,2]=>[3,5,4,1,6,2]
[(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>[3,1,4,6,5,2]=>[6,3,4,1,5,2]
[(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]
[(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[2,1,4,3,6,5,8,7]=>[8,6,4,2,1,3,5,7]
[(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[4,1,3,2,6,5,8,7]=>[8,6,4,1,3,2,5,7]
[(1,4),(2,3),(5,6),(7,8)]=>[4,3,2,1,6,5,8,7]=>[2,3,4,1,6,5,8,7]=>[8,6,2,3,4,1,5,7]
[(1,5),(2,3),(4,6),(7,8)]=>[5,3,2,6,1,4,8,7]=>[6,3,5,1,2,4,8,7]=>[8,1,3,2,6,5,4,7]
[(1,6),(2,3),(4,5),(7,8)]=>[6,3,2,5,4,1,8,7]=>[4,3,5,6,1,2,8,7]=>[8,1,4,3,5,6,2,7]
[(1,7),(2,3),(4,5),(6,8)]=>[7,3,2,5,4,8,1,6]=>[8,3,7,5,2,1,4,6]=>[3,2,8,5,7,1,4,6]
[(1,8),(2,3),(4,5),(6,7)]=>[8,3,2,5,4,7,6,1]=>[6,3,7,5,2,8,1,4]=>[3,6,7,2,5,1,8,4]
[(1,8),(2,4),(3,5),(6,7)]=>[8,4,5,2,3,7,6,1]=>[6,5,2,7,4,8,1,3]=>[2,6,5,7,1,4,8,3]
[(1,7),(2,4),(3,5),(6,8)]=>[7,4,5,2,3,8,1,6]=>[8,5,2,7,4,1,3,6]=>[2,5,1,8,7,4,3,6]
[(1,6),(2,4),(3,5),(7,8)]=>[6,4,5,2,3,1,8,7]=>[3,5,2,6,4,1,8,7]=>[8,5,3,6,2,4,1,7]
[(1,5),(2,4),(3,6),(7,8)]=>[5,4,6,2,1,3,8,7]=>[2,6,1,5,4,3,8,7]=>[8,6,5,2,1,4,3,7]
[(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[6,1,2,4,5,3,8,7]=>[8,4,1,2,6,5,3,7]
[(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[5,6,3,2,1,4,8,7]=>[8,3,5,6,2,1,4,7]
[(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[2,1,6,3,5,4,8,7]=>[8,6,2,5,1,3,4,7]
[(1,2),(3,6),(4,5),(7,8)]=>[2,1,6,5,4,3,8,7]=>[2,1,4,5,6,3,8,7]=>[8,4,2,1,5,6,3,7]
[(1,3),(2,6),(4,5),(7,8)]=>[3,6,1,5,4,2,8,7]=>[6,4,3,5,1,2,8,7]=>[8,1,4,6,3,5,2,7]
[(1,4),(2,6),(3,5),(7,8)]=>[4,6,5,1,3,2,8,7]=>[5,3,1,4,6,2,8,7]=>[8,3,5,4,1,6,2,7]
[(1,5),(2,6),(3,4),(7,8)]=>[5,6,4,3,1,2,8,7]=>[3,1,4,6,5,2,8,7]=>[8,6,3,4,1,5,2,7]
[(1,6),(2,5),(3,4),(7,8)]=>[6,5,4,3,2,1,8,7]=>[2,3,4,5,6,1,8,7]=>[8,2,3,4,5,6,1,7]
[(1,7),(2,5),(3,4),(6,8)]=>[7,5,4,3,2,8,1,6]=>[8,3,4,5,7,1,2,6]=>[1,3,4,5,2,8,7,6]
[(1,8),(2,5),(3,4),(6,7)]=>[8,5,4,3,2,7,6,1]=>[6,3,4,5,7,8,1,2]=>[1,3,4,6,5,7,8,2]
[(1,8),(2,6),(3,4),(5,7)]=>[8,6,4,3,7,2,5,1]=>[5,7,4,6,2,8,3,1]=>[5,4,7,6,8,2,3,1]
[(1,7),(2,6),(3,4),(5,8)]=>[7,6,4,3,8,2,1,5]=>[2,8,4,6,1,7,3,5]=>[4,2,1,8,6,7,3,5]
[(1,6),(2,7),(3,4),(5,8)]=>[6,7,4,3,8,1,2,5]=>[8,1,4,7,2,6,3,5]=>[4,1,8,2,3,7,6,5]
[(1,5),(2,7),(3,4),(6,8)]=>[5,7,4,3,1,8,2,6]=>[3,4,7,8,5,2,1,6]=>[3,7,4,5,8,2,1,6]
[(1,2),(3,7),(4,5),(6,8)]=>[2,1,7,5,4,8,3,6]=>[2,1,8,5,7,3,4,6]=>[2,1,8,5,3,4,7,6]
[(1,2),(3,8),(4,5),(6,7)]=>[2,1,8,5,4,7,6,3]=>[2,1,6,5,7,8,3,4]=>[2,6,5,1,3,7,8,4]
[(1,3),(2,8),(4,5),(6,7)]=>[3,8,1,5,4,7,6,2]=>[8,6,3,5,1,7,4,2]=>[6,3,8,5,7,4,1,2]
[(1,4),(2,8),(3,5),(6,7)]=>[4,8,5,1,3,7,6,2]=>[5,8,1,4,6,7,2,3]=>[1,5,2,4,6,8,7,3]
[(1,5),(2,8),(3,4),(6,7)]=>[5,8,4,3,1,7,6,2]=>[3,4,8,6,5,7,1,2]=>[1,6,8,3,4,5,7,2]
[(1,6),(2,8),(3,4),(5,7)]=>[6,8,4,3,7,1,5,2]=>[7,5,4,8,1,6,3,2]=>[5,7,4,8,6,1,3,2]
[(1,7),(2,8),(3,4),(5,6)]=>[7,8,4,3,6,5,1,2]=>[5,1,4,6,8,2,7,3]=>[5,4,1,6,8,2,7,3]
[(1,8),(2,7),(3,4),(5,6)]=>[8,7,4,3,6,5,2,1]=>[2,5,4,6,7,1,8,3]=>[5,4,2,6,7,8,1,3]
[(1,8),(2,7),(3,5),(4,6)]=>[8,7,5,6,3,4,2,1]=>[2,4,6,3,7,5,8,1]=>[6,4,2,7,3,5,8,1]
[(1,7),(2,8),(3,5),(4,6)]=>[7,8,5,6,3,4,1,2]=>[4,1,6,3,8,5,7,2]=>[6,4,8,1,3,5,7,2]
[(1,6),(2,8),(3,5),(4,7)]=>[6,8,5,7,3,1,4,2]=>[3,7,4,2,8,6,5,1]=>[7,8,6,3,4,2,5,1]
[(1,5),(2,8),(3,6),(4,7)]=>[5,8,6,7,1,3,4,2]=>[7,4,1,3,5,8,6,2]=>[7,4,1,8,3,5,6,2]
[(1,3),(2,8),(4,6),(5,7)]=>[3,8,1,6,7,4,5,2]=>[8,5,3,7,4,1,6,2]=>[5,3,8,7,4,6,1,2]
[(1,2),(3,8),(4,6),(5,7)]=>[2,1,8,6,7,4,5,3]=>[2,1,5,7,4,8,6,3]=>[7,5,8,2,1,4,6,3]
[(1,2),(3,7),(4,6),(5,8)]=>[2,1,7,6,8,4,3,5]=>[2,1,4,8,3,7,6,5]=>[8,7,4,2,1,3,6,5]
[(1,3),(2,7),(4,6),(5,8)]=>[3,7,1,6,8,4,2,5]=>[7,4,3,8,2,1,6,5]=>[7,4,8,3,2,6,1,5]
[(1,4),(2,7),(3,6),(5,8)]=>[4,7,6,1,8,3,2,5]=>[6,3,7,4,2,8,5,1]=>[6,7,8,3,4,2,5,1]
[(1,5),(2,7),(3,6),(4,8)]=>[5,7,6,8,1,3,2,4]=>[8,3,1,2,5,7,6,4]=>[8,7,1,3,2,5,6,4]
[(1,6),(2,7),(3,5),(4,8)]=>[6,7,5,8,3,1,2,4]=>[3,1,8,2,7,6,5,4]=>[8,7,6,3,1,5,2,4]
[(1,7),(2,6),(3,5),(4,8)]=>[7,6,5,8,3,2,1,4]=>[2,3,8,1,6,7,5,4]=>[6,8,2,3,1,7,5,4]
[(1,8),(2,6),(3,5),(4,7)]=>[8,6,5,7,3,2,4,1]=>[4,3,7,2,6,8,5,1]=>[7,4,3,2,6,8,5,1]
[(1,8),(2,5),(3,6),(4,7)]=>[8,5,6,7,2,3,4,1]=>[4,7,2,3,8,5,6,1]=>[2,7,4,3,8,5,6,1]
[(1,7),(2,5),(3,6),(4,8)]=>[7,5,6,8,2,3,1,4]=>[3,8,2,1,7,5,6,4]=>[8,3,7,2,1,5,6,4]
[(1,6),(2,5),(3,7),(4,8)]=>[6,5,7,8,2,1,3,4]=>[2,8,1,3,6,5,7,4]=>[8,2,6,1,3,5,7,4]
[(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[8,1,2,3,5,6,7,4]=>[5,1,2,3,6,8,7,4]
[(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[7,8,2,4,3,6,1,5]=>[2,4,7,3,8,1,6,5]
[(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[6,8,3,2,4,1,7,5]=>[6,3,8,2,7,4,1,5]
[(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[2,1,8,3,4,6,7,5]=>[8,2,1,3,6,4,7,5]
[(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[2,1,7,8,5,4,3,6]=>[5,7,8,2,1,4,3,6]
[(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[5,7,3,8,1,4,2,6]=>[3,5,1,7,8,4,2,6]
[(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[7,1,8,4,5,3,2,6]=>[1,7,8,4,5,3,2,6]
[(1,5),(2,4),(3,7),(6,8)]=>[5,4,7,2,1,8,3,6]=>[2,7,8,5,4,3,1,6]=>[5,7,8,2,4,3,1,6]
[(1,6),(2,4),(3,7),(5,8)]=>[6,4,7,2,8,1,3,5]=>[8,7,1,6,2,4,5,3]=>[8,1,2,4,7,6,5,3]
[(1,8),(2,4),(3,6),(5,7)]=>[8,4,6,2,7,3,5,1]=>[5,6,7,8,3,4,2,1]=>[3,5,6,7,8,4,2,1]
[(1,5),(2,3),(4,7),(6,8)]=>[5,3,2,7,1,8,4,6]=>[7,3,5,8,2,4,1,6]=>[3,2,5,7,4,8,1,6]
[(1,4),(2,3),(5,7),(6,8)]=>[4,3,2,1,7,8,5,6]=>[2,3,4,1,8,5,7,6]=>[8,2,7,3,4,1,5,6]
[(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[4,1,3,2,8,5,7,6]=>[8,4,7,1,3,2,5,6]
[(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[2,1,4,3,8,5,7,6]=>[8,4,7,2,1,3,5,6]
[(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,8,7,6,5]=>[2,1,4,3,6,7,8,5]=>[6,4,2,1,3,7,8,5]
[(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,8,7,6,5]=>[4,1,3,2,6,7,8,5]=>[6,4,1,3,2,7,8,5]
[(1,4),(2,3),(5,8),(6,7)]=>[4,3,2,1,8,7,6,5]=>[2,3,4,1,6,7,8,5]=>[6,2,3,4,1,7,8,5]
[(1,5),(2,3),(4,8),(6,7)]=>[5,3,2,8,1,7,6,4]=>[8,3,5,6,2,7,1,4]=>[3,2,5,6,8,1,7,4]
[(1,6),(2,3),(4,8),(5,7)]=>[6,3,2,8,7,1,5,4]=>[7,3,6,5,1,2,8,4]=>[7,1,6,3,5,8,2,4]
[(1,7),(2,3),(4,8),(5,6)]=>[7,3,2,8,6,5,1,4]=>[5,3,6,1,8,7,2,4]=>[3,8,5,1,6,7,2,4]
[(1,8),(2,3),(4,7),(5,6)]=>[8,3,2,7,6,5,4,1]=>[4,3,5,6,7,8,1,2]=>[1,4,3,5,6,7,8,2]
[(1,8),(2,4),(3,7),(5,6)]=>[8,4,7,2,6,5,3,1]=>[3,5,6,7,8,2,1,4]=>[3,2,5,6,7,8,1,4]
[(1,7),(2,4),(3,8),(5,6)]=>[7,4,8,2,6,5,1,3]=>[5,6,1,8,7,3,2,4]=>[3,8,5,6,7,1,2,4]
[(1,6),(2,4),(3,8),(5,7)]=>[6,4,8,2,7,1,5,3]=>[7,8,5,6,1,4,2,3]=>[1,5,2,7,8,6,4,3]
[(1,5),(2,4),(3,8),(6,7)]=>[5,4,8,2,1,7,6,3]=>[2,8,6,5,4,7,1,3]=>[2,6,8,5,1,4,7,3]
[(1,4),(2,5),(3,8),(6,7)]=>[4,5,8,1,2,7,6,3]=>[8,1,6,4,5,7,2,3]=>[1,4,2,8,6,5,7,3]
[(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,8,2,7,6,4]=>[5,8,3,6,1,7,2,4]=>[3,5,1,6,8,7,2,4]
[(1,2),(3,5),(4,8),(6,7)]=>[2,1,5,8,3,7,6,4]=>[2,1,8,6,5,7,3,4]=>[2,6,8,5,1,3,7,4]
[(1,2),(3,6),(4,8),(5,7)]=>[2,1,6,8,7,3,5,4]=>[2,1,7,5,3,6,8,4]=>[5,7,6,2,1,3,8,4]
[(1,3),(2,6),(4,8),(5,7)]=>[3,6,1,8,7,2,5,4]=>[6,7,3,5,2,1,8,4]=>[6,3,7,5,8,2,1,4]
[(1,4),(2,6),(3,8),(5,7)]=>[4,6,8,1,7,2,5,3]=>[8,7,5,4,2,6,1,3]=>[2,5,8,7,4,1,6,3]
[(1,5),(2,6),(3,8),(4,7)]=>[5,6,8,7,1,2,4,3]=>[7,1,4,2,5,6,8,3]=>[4,1,7,5,2,6,8,3]
[(1,6),(2,5),(3,8),(4,7)]=>[6,5,8,7,2,1,4,3]=>[2,7,4,1,6,5,8,3]=>[7,6,2,4,5,1,8,3]
[(1,7),(2,5),(3,8),(4,6)]=>[7,5,8,6,2,4,1,3]=>[4,6,1,2,8,7,3,5]=>[4,8,1,2,6,7,3,5]
[(1,8),(2,5),(3,7),(4,6)]=>[8,5,7,6,2,4,3,1]=>[3,4,6,1,7,8,2,5]=>[3,4,1,6,7,2,8,5]
[(1,8),(2,6),(3,7),(4,5)]=>[8,6,7,5,4,2,3,1]=>[3,4,2,5,7,8,6,1]=>[7,3,4,2,5,8,6,1]
[(1,7),(2,6),(3,8),(4,5)]=>[7,6,8,5,4,2,1,3]=>[2,4,1,5,8,7,6,3]=>[8,7,4,2,5,1,6,3]
[(1,6),(2,7),(3,8),(4,5)]=>[6,7,8,5,4,1,2,3]=>[4,1,2,5,8,6,7,3]=>[8,1,4,5,2,6,7,3]
[(1,5),(2,7),(3,8),(4,6)]=>[5,7,8,6,1,4,2,3]=>[6,4,2,1,5,8,7,3]=>[8,4,6,5,2,1,7,3]
[(1,4),(2,7),(3,8),(5,6)]=>[4,7,8,1,6,5,2,3]=>[8,5,2,4,6,1,7,3]=>[5,2,4,8,6,7,1,3]
[(1,3),(2,7),(4,8),(5,6)]=>[3,7,1,8,6,5,2,4]=>[7,5,3,2,6,8,1,4]=>[3,5,7,2,1,6,8,4]
[(1,2),(3,7),(4,8),(5,6)]=>[2,1,7,8,6,5,3,4]=>[2,1,5,3,6,8,7,4]=>[8,5,6,2,1,3,7,4]
[(1,2),(3,8),(4,7),(5,6)]=>[2,1,8,7,6,5,4,3]=>[2,1,4,5,6,7,8,3]=>[4,2,1,5,6,7,8,3]
[(1,3),(2,8),(4,7),(5,6)]=>[3,8,1,7,6,5,4,2]=>[8,4,3,5,6,7,1,2]=>[1,4,3,5,8,6,7,2]
[(1,4),(2,8),(3,7),(5,6)]=>[4,8,7,1,6,5,3,2]=>[7,3,5,4,6,1,8,2]=>[3,7,5,4,6,8,1,2]
[(1,5),(2,8),(3,7),(4,6)]=>[5,8,7,6,1,4,3,2]=>[6,3,4,1,5,7,8,2]=>[3,4,6,5,1,7,8,2]
[(1,6),(2,8),(3,7),(4,5)]=>[6,8,7,5,4,1,3,2]=>[4,3,1,5,7,6,8,2]=>[7,4,3,5,1,6,8,2]
[(1,7),(2,8),(3,6),(4,5)]=>[7,8,6,5,4,3,1,2]=>[3,1,4,5,6,8,7,2]=>[8,3,4,1,5,6,7,2]
[(1,8),(2,7),(3,6),(4,5)]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]=>[2,3,4,5,6,7,8,1]
[(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]=>[10,8,6,4,2,1,3,5,7,9]
[(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]=>[12,10,8,6,4,2,1,3,5,7,9,11]
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
descent views to invisible inversion bottoms
Description
Return a permutation whose multiset of invisible inversion bottoms is the multiset of descent views of the given permutation.
This map is similar to Mp00235descent views to invisible inversion bottoms, but different beginning with permutations of six elements.
An invisible inversion of a permutation $\sigma$ is a pair $i < j$ such that $i < \sigma(j) < \sigma(i)$. The element $\sigma(j)$ is then an invisible inversion bottom.
A descent view in a permutation $\pi$ is an element $\pi(j)$ such that $\pi(i+1) < \pi(j) < \pi(i)$, and additionally the smallest element in the decreasing run containing $\pi(i)$ is smaller than the smallest element in the decreasing run containing $\pi(j)$.
This map is a bijection $\chi:\mathfrak S_n \to \mathfrak S_n$, such that
This map is similar to Mp00235descent views to invisible inversion bottoms, but different beginning with permutations of six elements.
An invisible inversion of a permutation $\sigma$ is a pair $i < j$ such that $i < \sigma(j) < \sigma(i)$. The element $\sigma(j)$ is then an invisible inversion bottom.
A descent view in a permutation $\pi$ is an element $\pi(j)$ such that $\pi(i+1) < \pi(j) < \pi(i)$, and additionally the smallest element in the decreasing run containing $\pi(i)$ is smaller than the smallest element in the decreasing run containing $\pi(j)$.
This map is a bijection $\chi:\mathfrak S_n \to \mathfrak S_n$, such that
- the multiset of descent views in $\pi$ is the multiset of invisible inversion bottoms in $\chi(\pi)$,
- the set of left-to-right maximima of $\pi$ is the set of maximal elements in the cycles of $\chi(\pi)$,
- the set of global ascent of $\pi$ is the set of global ascent of $\chi(\pi)$,
- the set of maximal elements in the decreasing runs of $\pi$ is the set of deficiency positions of $\chi(\pi)$, and
- the set of minimal elements in the decreasing runs of $\pi$ is the set of deficiency values of $\chi(\pi)$.
Map
Foata bijection
Description
Sends a permutation to its image under the Foata bijection.
The Foata bijection $\phi$ is a bijection on the set of words with no two equal letters. It can be defined by induction on the size of the word:
Given a word $w_1 w_2 ... w_n$, compute the image inductively by starting with $\phi(w_1) = w_1$.
At the $i$-th step, if $\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$, define $\phi(w_1 w_2 ... w_i w_{i+1})$ by placing $w_{i+1}$ on the end of the word $v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.
To compute $\phi([1,4,2,5,3])$, the sequence of words is
This bijection sends the major index (St000004The major index of a permutation.) to the number of inversions (St000018The number of inversions of a permutation.).
The Foata bijection $\phi$ is a bijection on the set of words with no two equal letters. It can be defined by induction on the size of the word:
Given a word $w_1 w_2 ... w_n$, compute the image inductively by starting with $\phi(w_1) = w_1$.
At the $i$-th step, if $\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$, define $\phi(w_1 w_2 ... w_i w_{i+1})$ by placing $w_{i+1}$ on the end of the word $v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.
- If $w_{i+1} \geq v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} \geq v_k$.
- If $w_{i+1} < v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} < v_k$.
To compute $\phi([1,4,2,5,3])$, the sequence of words is
- $1$
- $|1|4 \to 14$
- $|14|2 \to 412$
- $|4|1|2|5 \to 4125$
- $|4|125|3 \to 45123.$
This bijection sends the major index (St000004The major index of a permutation.) to the number of inversions (St000018The number of inversions of a permutation.).
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