Identifier
Mp00058:
Perfect matchings
—to permutation⟶
Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ 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]=>[2,1,4,3]
[(1,3),(2,4)]=>[3,4,1,2]=>[4,3,2,1]=>[2,3,4,1]
[(1,4),(2,3)]=>[4,3,2,1]=>[4,3,2,1]=>[2,3,4,1]
[(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5]
[(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[4,3,2,1,6,5]=>[2,3,4,1,6,5]
[(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[4,3,2,1,6,5]=>[2,3,4,1,6,5]
[(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>[6,5,3,4,2,1]=>[2,4,5,3,6,1]
[(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[6,5,3,4,2,1]=>[2,4,5,3,6,1]
[(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>[6,5,4,3,2,1]=>[2,3,4,5,6,1]
[(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>[6,5,4,3,2,1]=>[2,3,4,5,6,1]
[(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[6,5,4,3,2,1]=>[2,3,4,5,6,1]
[(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[5,6,3,4,1,2]=>[4,1,6,3,5,2]
[(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[2,1,6,5,4,3]=>[2,1,4,5,6,3]
[(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[2,1,6,5,4,3]=>[2,1,4,5,6,3]
[(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>[6,5,3,4,2,1]=>[2,4,5,3,6,1]
[(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>[6,5,4,3,2,1]=>[2,3,4,5,6,1]
[(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>[6,5,4,3,2,1]=>[2,3,4,5,6,1]
[(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[6,5,4,3,2,1]=>[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]=>[2,1,4,3,6,5,8,7]
[(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[4,3,2,1,6,5,8,7]=>[2,3,4,1,6,5,8,7]
[(1,4),(2,3),(5,6),(7,8)]=>[4,3,2,1,6,5,8,7]=>[4,3,2,1,6,5,8,7]=>[2,3,4,1,6,5,8,7]
[(1,5),(2,3),(4,6),(7,8)]=>[5,3,2,6,1,4,8,7]=>[6,5,3,4,2,1,8,7]=>[2,4,5,3,6,1,8,7]
[(1,6),(2,3),(4,5),(7,8)]=>[6,3,2,5,4,1,8,7]=>[6,5,3,4,2,1,8,7]=>[2,4,5,3,6,1,8,7]
[(1,7),(2,3),(4,5),(6,8)]=>[7,3,2,5,4,8,1,6]=>[8,7,3,5,4,6,2,1]=>[2,6,5,7,3,4,8,1]
[(1,8),(2,3),(4,5),(6,7)]=>[8,3,2,5,4,7,6,1]=>[8,7,3,5,4,6,2,1]=>[2,6,5,7,3,4,8,1]
[(1,8),(2,4),(3,5),(6,7)]=>[8,4,5,2,3,7,6,1]=>[8,7,5,4,3,6,2,1]=>[2,6,4,5,7,3,8,1]
[(1,7),(2,4),(3,5),(6,8)]=>[7,4,5,2,3,8,1,6]=>[8,7,5,4,3,6,2,1]=>[2,6,4,5,7,3,8,1]
[(1,6),(2,4),(3,5),(7,8)]=>[6,4,5,2,3,1,8,7]=>[6,5,4,3,2,1,8,7]=>[2,3,4,5,6,1,8,7]
[(1,5),(2,4),(3,6),(7,8)]=>[5,4,6,2,1,3,8,7]=>[6,5,4,3,2,1,8,7]=>[2,3,4,5,6,1,8,7]
[(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[6,5,4,3,2,1,8,7]=>[2,3,4,5,6,1,8,7]
[(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[5,6,3,4,1,2,8,7]=>[4,1,6,3,5,2,8,7]
[(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[2,1,6,5,4,3,8,7]=>[2,1,4,5,6,3,8,7]
[(1,2),(3,6),(4,5),(7,8)]=>[2,1,6,5,4,3,8,7]=>[2,1,6,5,4,3,8,7]=>[2,1,4,5,6,3,8,7]
[(1,3),(2,6),(4,5),(7,8)]=>[3,6,1,5,4,2,8,7]=>[6,5,3,4,2,1,8,7]=>[2,4,5,3,6,1,8,7]
[(1,4),(2,6),(3,5),(7,8)]=>[4,6,5,1,3,2,8,7]=>[6,5,4,3,2,1,8,7]=>[2,3,4,5,6,1,8,7]
[(1,5),(2,6),(3,4),(7,8)]=>[5,6,4,3,1,2,8,7]=>[6,5,4,3,2,1,8,7]=>[2,3,4,5,6,1,8,7]
[(1,6),(2,5),(3,4),(7,8)]=>[6,5,4,3,2,1,8,7]=>[6,5,4,3,2,1,8,7]=>[2,3,4,5,6,1,8,7]
[(1,7),(2,5),(3,4),(6,8)]=>[7,5,4,3,2,8,1,6]=>[8,7,5,4,3,6,2,1]=>[2,6,4,5,7,3,8,1]
[(1,8),(2,5),(3,4),(6,7)]=>[8,5,4,3,2,7,6,1]=>[8,7,5,4,3,6,2,1]=>[2,6,4,5,7,3,8,1]
[(1,8),(2,6),(3,4),(5,7)]=>[8,6,4,3,7,2,5,1]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[(1,7),(2,6),(3,4),(5,8)]=>[7,6,4,3,8,2,1,5]=>[8,7,6,4,5,3,2,1]=>[2,3,5,6,4,7,8,1]
[(1,6),(2,7),(3,4),(5,8)]=>[6,7,4,3,8,1,2,5]=>[8,7,6,4,5,3,2,1]=>[2,3,5,6,4,7,8,1]
[(1,5),(2,7),(3,4),(6,8)]=>[5,7,4,3,1,8,2,6]=>[7,8,5,4,3,6,1,2]=>[6,1,4,5,8,3,7,2]
[(1,4),(2,7),(3,5),(6,8)]=>[4,7,5,1,3,8,2,6]=>[7,8,5,4,3,6,1,2]=>[6,1,4,5,8,3,7,2]
[(1,3),(2,7),(4,5),(6,8)]=>[3,7,1,5,4,8,2,6]=>[7,8,3,5,4,6,1,2]=>[6,1,5,8,3,4,7,2]
[(1,2),(3,7),(4,5),(6,8)]=>[2,1,7,5,4,8,3,6]=>[2,1,8,7,5,6,4,3]=>[2,1,4,6,7,5,8,3]
[(1,2),(3,8),(4,5),(6,7)]=>[2,1,8,5,4,7,6,3]=>[2,1,8,7,5,6,4,3]=>[2,1,4,6,7,5,8,3]
[(1,3),(2,8),(4,5),(6,7)]=>[3,8,1,5,4,7,6,2]=>[8,7,3,5,4,6,2,1]=>[2,6,5,7,3,4,8,1]
[(1,4),(2,8),(3,5),(6,7)]=>[4,8,5,1,3,7,6,2]=>[8,7,5,4,3,6,2,1]=>[2,6,4,5,7,3,8,1]
[(1,5),(2,8),(3,4),(6,7)]=>[5,8,4,3,1,7,6,2]=>[8,7,5,4,3,6,2,1]=>[2,6,4,5,7,3,8,1]
[(1,6),(2,8),(3,4),(5,7)]=>[6,8,4,3,7,1,5,2]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[(1,7),(2,8),(3,4),(5,6)]=>[7,8,4,3,6,5,1,2]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[(1,8),(2,7),(3,4),(5,6)]=>[8,7,4,3,6,5,2,1]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[(1,8),(2,7),(3,5),(4,6)]=>[8,7,5,6,3,4,2,1]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[(1,7),(2,8),(3,5),(4,6)]=>[7,8,5,6,3,4,1,2]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[(1,6),(2,8),(3,5),(4,7)]=>[6,8,5,7,3,1,4,2]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[(1,5),(2,8),(3,6),(4,7)]=>[5,8,6,7,1,3,4,2]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[(1,4),(2,8),(3,6),(5,7)]=>[4,8,6,1,7,3,5,2]=>[8,7,6,4,5,3,2,1]=>[2,3,5,6,4,7,8,1]
[(1,3),(2,8),(4,6),(5,7)]=>[3,8,1,6,7,4,5,2]=>[8,7,3,6,5,4,2,1]=>[2,4,5,6,7,3,8,1]
[(1,2),(3,8),(4,6),(5,7)]=>[2,1,8,6,7,4,5,3]=>[2,1,8,7,6,5,4,3]=>[2,1,4,5,6,7,8,3]
[(1,2),(3,7),(4,6),(5,8)]=>[2,1,7,6,8,4,3,5]=>[2,1,8,7,6,5,4,3]=>[2,1,4,5,6,7,8,3]
[(1,3),(2,7),(4,6),(5,8)]=>[3,7,1,6,8,4,2,5]=>[7,8,3,6,5,4,1,2]=>[4,1,5,6,8,3,7,2]
[(1,4),(2,7),(3,6),(5,8)]=>[4,7,6,1,8,3,2,5]=>[7,8,6,4,5,3,1,2]=>[3,1,5,6,4,8,7,2]
[(1,5),(2,7),(3,6),(4,8)]=>[5,7,6,8,1,3,2,4]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[(1,6),(2,7),(3,5),(4,8)]=>[6,7,5,8,3,1,2,4]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[(1,7),(2,6),(3,5),(4,8)]=>[7,6,5,8,3,2,1,4]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[(1,8),(2,6),(3,5),(4,7)]=>[8,6,5,7,3,2,4,1]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[(1,8),(2,5),(3,6),(4,7)]=>[8,5,6,7,2,3,4,1]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[(1,7),(2,5),(3,6),(4,8)]=>[7,5,6,8,2,3,1,4]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[(1,6),(2,5),(3,7),(4,8)]=>[6,5,7,8,2,1,3,4]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[7,8,6,4,5,3,1,2]=>[3,1,5,6,4,8,7,2]
[(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[6,8,3,7,5,1,4,2]=>[4,5,7,3,8,6,1,2]
[(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[2,1,8,7,6,5,4,3]=>[2,1,4,5,6,7,8,3]
[(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[2,1,7,8,5,6,3,4]=>[2,1,6,3,8,5,7,4]
[(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[5,7,3,8,1,6,2,4]=>[6,8,7,3,5,1,2,4]
[(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[7,5,8,4,2,6,1,3]=>[6,4,2,8,7,5,1,3]
[(1,5),(2,4),(3,7),(6,8)]=>[5,4,7,2,1,8,3,6]=>[7,5,8,4,2,6,1,3]=>[6,4,2,8,7,5,1,3]
[(1,6),(2,4),(3,7),(5,8)]=>[6,4,7,2,8,1,3,5]=>[8,7,6,4,5,3,2,1]=>[2,3,5,6,4,7,8,1]
[(1,7),(2,4),(3,6),(5,8)]=>[7,4,6,2,8,3,1,5]=>[8,7,6,4,5,3,2,1]=>[2,3,5,6,4,7,8,1]
[(1,8),(2,4),(3,6),(5,7)]=>[8,4,6,2,7,3,5,1]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[(1,8),(2,3),(4,6),(5,7)]=>[8,3,2,6,7,4,5,1]=>[8,7,3,6,5,4,2,1]=>[2,4,5,6,7,3,8,1]
[(1,7),(2,3),(4,6),(5,8)]=>[7,3,2,6,8,4,1,5]=>[8,7,3,6,5,4,2,1]=>[2,4,5,6,7,3,8,1]
[(1,6),(2,3),(4,7),(5,8)]=>[6,3,2,7,8,1,4,5]=>[8,6,3,7,5,2,4,1]=>[4,5,7,3,6,8,2,1]
[(1,5),(2,3),(4,7),(6,8)]=>[5,3,2,7,1,8,4,6]=>[7,5,3,8,2,6,1,4]=>[6,8,5,3,7,2,1,4]
[(1,4),(2,3),(5,7),(6,8)]=>[4,3,2,1,7,8,5,6]=>[4,3,2,1,8,7,6,5]=>[2,3,4,1,6,7,8,5]
[(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[4,3,2,1,8,7,6,5]=>[2,3,4,1,6,7,8,5]
[(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[2,1,4,3,8,7,6,5]=>[2,1,4,3,6,7,8,5]
[(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,8,7,6,5]=>[2,1,4,3,8,7,6,5]=>[2,1,4,3,6,7,8,5]
[(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,8,7,6,5]=>[4,3,2,1,8,7,6,5]=>[2,3,4,1,6,7,8,5]
[(1,4),(2,3),(5,8),(6,7)]=>[4,3,2,1,8,7,6,5]=>[4,3,2,1,8,7,6,5]=>[2,3,4,1,6,7,8,5]
[(1,5),(2,3),(4,8),(6,7)]=>[5,3,2,8,1,7,6,4]=>[8,5,3,7,2,6,4,1]=>[4,6,7,5,8,3,2,1]
[(1,6),(2,3),(4,8),(5,7)]=>[6,3,2,8,7,1,5,4]=>[8,6,3,7,5,2,4,1]=>[4,5,7,3,6,8,2,1]
[(1,7),(2,3),(4,8),(5,6)]=>[7,3,2,8,6,5,1,4]=>[8,7,3,6,5,4,2,1]=>[2,4,5,6,7,3,8,1]
[(1,8),(2,3),(4,7),(5,6)]=>[8,3,2,7,6,5,4,1]=>[8,7,3,6,5,4,2,1]=>[2,4,5,6,7,3,8,1]
[(1,8),(2,4),(3,7),(5,6)]=>[8,4,7,2,6,5,3,1]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[(1,7),(2,4),(3,8),(5,6)]=>[7,4,8,2,6,5,1,3]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[(1,6),(2,4),(3,8),(5,7)]=>[6,4,8,2,7,1,5,3]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[(1,5),(2,4),(3,8),(6,7)]=>[5,4,8,2,1,7,6,3]=>[8,5,7,4,2,6,3,1]=>[3,6,4,7,8,5,2,1]
[(1,4),(2,5),(3,8),(6,7)]=>[4,5,8,1,2,7,6,3]=>[8,5,7,4,2,6,3,1]=>[3,6,4,7,8,5,2,1]
[(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,8,2,7,6,4]=>[5,8,3,7,1,6,4,2]=>[4,6,7,8,5,3,1,2]
[(1,2),(3,5),(4,8),(6,7)]=>[2,1,5,8,3,7,6,4]=>[2,1,8,7,5,6,4,3]=>[2,1,4,6,7,5,8,3]
[(1,2),(3,6),(4,8),(5,7)]=>[2,1,6,8,7,3,5,4]=>[2,1,8,7,6,5,4,3]=>[2,1,4,5,6,7,8,3]
[(1,3),(2,6),(4,8),(5,7)]=>[3,6,1,8,7,2,5,4]=>[6,8,3,7,5,1,4,2]=>[4,5,7,3,8,6,1,2]
[(1,4),(2,6),(3,8),(5,7)]=>[4,6,8,1,7,2,5,3]=>[8,7,6,4,5,3,2,1]=>[2,3,5,6,4,7,8,1]
[(1,5),(2,6),(3,8),(4,7)]=>[5,6,8,7,1,2,4,3]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[(1,6),(2,5),(3,8),(4,7)]=>[6,5,8,7,2,1,4,3]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[(1,7),(2,5),(3,8),(4,6)]=>[7,5,8,6,2,4,1,3]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[(1,8),(2,5),(3,7),(4,6)]=>[8,5,7,6,2,4,3,1]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[(1,8),(2,6),(3,7),(4,5)]=>[8,6,7,5,4,2,3,1]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[(1,7),(2,6),(3,8),(4,5)]=>[7,6,8,5,4,2,1,3]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[(1,6),(2,7),(3,8),(4,5)]=>[6,7,8,5,4,1,2,3]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[(1,5),(2,7),(3,8),(4,6)]=>[5,7,8,6,1,4,2,3]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[(1,4),(2,7),(3,8),(5,6)]=>[4,7,8,1,6,5,2,3]=>[8,7,6,4,5,3,2,1]=>[2,3,5,6,4,7,8,1]
[(1,3),(2,7),(4,8),(5,6)]=>[3,7,1,8,6,5,2,4]=>[7,8,3,6,5,4,1,2]=>[4,1,5,6,8,3,7,2]
[(1,2),(3,7),(4,8),(5,6)]=>[2,1,7,8,6,5,3,4]=>[2,1,8,7,6,5,4,3]=>[2,1,4,5,6,7,8,3]
[(1,2),(3,8),(4,7),(5,6)]=>[2,1,8,7,6,5,4,3]=>[2,1,8,7,6,5,4,3]=>[2,1,4,5,6,7,8,3]
[(1,3),(2,8),(4,7),(5,6)]=>[3,8,1,7,6,5,4,2]=>[8,7,3,6,5,4,2,1]=>[2,4,5,6,7,3,8,1]
[(1,4),(2,8),(3,7),(5,6)]=>[4,8,7,1,6,5,3,2]=>[8,7,6,4,5,3,2,1]=>[2,3,5,6,4,7,8,1]
[(1,5),(2,8),(3,7),(4,6)]=>[5,8,7,6,1,4,3,2]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[(1,6),(2,8),(3,7),(4,5)]=>[6,8,7,5,4,1,3,2]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[(1,7),(2,8),(3,6),(4,5)]=>[7,8,6,5,4,3,1,2]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[(1,8),(2,7),(3,6),(4,5)]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>[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]=>[2,1,4,3,6,5,8,7,10,9]
[(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]=>[2,1,4,3,6,5,8,7,10,9,12,11]
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
Demazure product with inverse
Description
This map sends a permutation $\pi$ to $\pi^{-1} \star \pi$ where $\star$ denotes the Demazure product on permutations.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.
Map
Clarke-Steingrimsson-Zeng
Description
The Clarke-Steingrimsson-Zeng map sending descents to excedances.
This is the map $\Phi$ in [1, sec.3]. In particular, it satisfies
$$ (des, Dbot, Ddif, Res)\pi = (exc, Ebot, Edif, Ine)\Phi(\pi), $$
where
This is the map $\Phi$ in [1, sec.3]. In particular, it satisfies
$$ (des, Dbot, Ddif, Res)\pi = (exc, Ebot, Edif, Ine)\Phi(\pi), $$
where
- $des$ is the number of descents, St000021The number of descents of a permutation.,
- $exc$ is the number of (strict) excedances, St000155The number of exceedances (also excedences) of a permutation.,
- $Dbot$ is the sum of the descent bottoms, St000154The sum of the descent bottoms of a permutation.,
- $Ebot$ is the sum of the excedance bottoms,
- $Ddif$ is the sum of the descent differences, St000030The sum of the descent differences of a permutations.,
- $Edif$ is the sum of the excedance differences (or depth), St000029The depth of a permutation.,
- $Res$ is the sum of the (right) embracing numbers,
- $Ine$ is the sum of the side numbers.
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