Identifier
Mp00231:
Integer compositions
—bounce path⟶
Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Images
=>
Cc0005;cc-rep-1
[1]=>[1,0]=>[2,1]=>[2,1]
[1,1]=>[1,0,1,0]=>[3,1,2]=>[3,1,2]
[2]=>[1,1,0,0]=>[2,3,1]=>[3,2,1]
[1,1,1]=>[1,0,1,0,1,0]=>[4,1,2,3]=>[4,1,2,3]
[1,2]=>[1,0,1,1,0,0]=>[3,1,4,2]=>[3,4,1,2]
[2,1]=>[1,1,0,0,1,0]=>[2,4,1,3]=>[4,2,1,3]
[3]=>[1,1,1,0,0,0]=>[2,3,4,1]=>[4,2,3,1]
[1,1,1,1]=>[1,0,1,0,1,0,1,0]=>[5,1,2,3,4]=>[5,1,2,3,4]
[1,1,2]=>[1,0,1,0,1,1,0,0]=>[4,1,2,5,3]=>[4,1,5,2,3]
[1,2,1]=>[1,0,1,1,0,0,1,0]=>[3,1,5,2,4]=>[3,5,1,2,4]
[1,3]=>[1,0,1,1,1,0,0,0]=>[3,1,4,5,2]=>[3,5,1,4,2]
[2,1,1]=>[1,1,0,0,1,0,1,0]=>[2,5,1,3,4]=>[5,2,1,3,4]
[2,2]=>[1,1,0,0,1,1,0,0]=>[2,4,1,5,3]=>[4,2,5,1,3]
[3,1]=>[1,1,1,0,0,0,1,0]=>[2,3,5,1,4]=>[5,2,3,1,4]
[4]=>[1,1,1,1,0,0,0,0]=>[2,3,4,5,1]=>[5,2,3,4,1]
[1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0]=>[6,1,2,3,4,5]=>[6,1,2,3,4,5]
[1,1,1,2]=>[1,0,1,0,1,0,1,1,0,0]=>[5,1,2,3,6,4]=>[5,1,2,6,3,4]
[1,1,2,1]=>[1,0,1,0,1,1,0,0,1,0]=>[4,1,2,6,3,5]=>[4,1,6,2,3,5]
[1,1,3]=>[1,0,1,0,1,1,1,0,0,0]=>[4,1,2,5,6,3]=>[4,1,6,2,5,3]
[1,2,1,1]=>[1,0,1,1,0,0,1,0,1,0]=>[3,1,6,2,4,5]=>[3,6,1,2,4,5]
[1,2,2]=>[1,0,1,1,0,0,1,1,0,0]=>[3,1,5,2,6,4]=>[3,5,1,6,2,4]
[1,3,1]=>[1,0,1,1,1,0,0,0,1,0]=>[3,1,4,6,2,5]=>[3,6,1,4,2,5]
[1,4]=>[1,0,1,1,1,1,0,0,0,0]=>[3,1,4,5,6,2]=>[3,6,1,4,5,2]
[2,1,1,1]=>[1,1,0,0,1,0,1,0,1,0]=>[2,6,1,3,4,5]=>[6,2,1,3,4,5]
[2,1,2]=>[1,1,0,0,1,0,1,1,0,0]=>[2,5,1,3,6,4]=>[5,2,1,6,3,4]
[2,2,1]=>[1,1,0,0,1,1,0,0,1,0]=>[2,4,1,6,3,5]=>[4,2,6,1,3,5]
[2,3]=>[1,1,0,0,1,1,1,0,0,0]=>[2,4,1,5,6,3]=>[4,2,6,1,5,3]
[3,1,1]=>[1,1,1,0,0,0,1,0,1,0]=>[2,3,6,1,4,5]=>[6,2,3,1,4,5]
[3,2]=>[1,1,1,0,0,0,1,1,0,0]=>[2,3,5,1,6,4]=>[5,2,3,6,1,4]
[4,1]=>[1,1,1,1,0,0,0,0,1,0]=>[2,3,4,6,1,5]=>[6,2,3,4,1,5]
[5]=>[1,1,1,1,1,0,0,0,0,0]=>[2,3,4,5,6,1]=>[6,2,3,4,5,1]
[1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0]=>[7,1,2,3,4,5,6]=>[7,1,2,3,4,5,6]
[1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,1,0,0]=>[6,1,2,3,4,7,5]=>[6,1,2,3,7,4,5]
[1,1,1,2,1]=>[1,0,1,0,1,0,1,1,0,0,1,0]=>[5,1,2,3,7,4,6]=>[5,1,2,7,3,4,6]
[1,1,1,3]=>[1,0,1,0,1,0,1,1,1,0,0,0]=>[5,1,2,3,6,7,4]=>[5,1,2,7,3,6,4]
[1,1,2,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0]=>[4,1,2,7,3,5,6]=>[4,1,7,2,3,5,6]
[1,1,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0]=>[4,1,2,6,3,7,5]=>[4,1,6,2,7,3,5]
[1,1,3,1]=>[1,0,1,0,1,1,1,0,0,0,1,0]=>[4,1,2,5,7,3,6]=>[4,1,7,2,5,3,6]
[1,1,4]=>[1,0,1,0,1,1,1,1,0,0,0,0]=>[4,1,2,5,6,7,3]=>[4,1,7,2,5,6,3]
[1,2,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0]=>[3,1,7,2,4,5,6]=>[3,7,1,2,4,5,6]
[1,2,1,2]=>[1,0,1,1,0,0,1,0,1,1,0,0]=>[3,1,6,2,4,7,5]=>[3,6,1,2,7,4,5]
[1,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,0]=>[3,1,5,2,7,4,6]=>[3,5,1,7,2,4,6]
[1,2,3]=>[1,0,1,1,0,0,1,1,1,0,0,0]=>[3,1,5,2,6,7,4]=>[3,5,1,7,2,6,4]
[1,3,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0]=>[3,1,4,7,2,5,6]=>[3,7,1,4,2,5,6]
[1,3,2]=>[1,0,1,1,1,0,0,0,1,1,0,0]=>[3,1,4,6,2,7,5]=>[3,6,1,4,7,2,5]
[1,4,1]=>[1,0,1,1,1,1,0,0,0,0,1,0]=>[3,1,4,5,7,2,6]=>[3,7,1,4,5,2,6]
[1,5]=>[1,0,1,1,1,1,1,0,0,0,0,0]=>[3,1,4,5,6,7,2]=>[3,7,1,4,5,6,2]
[2,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0]=>[2,7,1,3,4,5,6]=>[7,2,1,3,4,5,6]
[2,1,1,2]=>[1,1,0,0,1,0,1,0,1,1,0,0]=>[2,6,1,3,4,7,5]=>[6,2,1,3,7,4,5]
[2,1,2,1]=>[1,1,0,0,1,0,1,1,0,0,1,0]=>[2,5,1,3,7,4,6]=>[5,2,1,7,3,4,6]
[2,1,3]=>[1,1,0,0,1,0,1,1,1,0,0,0]=>[2,5,1,3,6,7,4]=>[5,2,1,7,3,6,4]
[2,2,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0]=>[2,4,1,7,3,5,6]=>[4,2,7,1,3,5,6]
[2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0]=>[2,4,1,6,3,7,5]=>[4,2,6,1,7,3,5]
[2,3,1]=>[1,1,0,0,1,1,1,0,0,0,1,0]=>[2,4,1,5,7,3,6]=>[4,2,7,1,5,3,6]
[2,4]=>[1,1,0,0,1,1,1,1,0,0,0,0]=>[2,4,1,5,6,7,3]=>[4,2,7,1,5,6,3]
[3,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0]=>[2,3,7,1,4,5,6]=>[7,2,3,1,4,5,6]
[3,1,2]=>[1,1,1,0,0,0,1,0,1,1,0,0]=>[2,3,6,1,4,7,5]=>[6,2,3,1,7,4,5]
[3,2,1]=>[1,1,1,0,0,0,1,1,0,0,1,0]=>[2,3,5,1,7,4,6]=>[5,2,3,7,1,4,6]
[3,3]=>[1,1,1,0,0,0,1,1,1,0,0,0]=>[2,3,5,1,6,7,4]=>[5,2,3,7,1,6,4]
[4,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0]=>[2,3,4,7,1,5,6]=>[7,2,3,4,1,5,6]
[4,2]=>[1,1,1,1,0,0,0,0,1,1,0,0]=>[2,3,4,6,1,7,5]=>[6,2,3,4,7,1,5]
[5,1]=>[1,1,1,1,1,0,0,0,0,0,1,0]=>[2,3,4,5,7,1,6]=>[7,2,3,4,5,1,6]
[6]=>[1,1,1,1,1,1,0,0,0,0,0,0]=>[2,3,4,5,6,7,1]=>[7,2,3,4,5,6,1]
[1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[8,1,2,3,4,5,6,7]=>[8,1,2,3,4,5,6,7]
[1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[7,1,2,3,4,5,8,6]=>[7,1,2,3,4,8,5,6]
[1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[6,1,2,3,4,8,5,7]=>[6,1,2,3,8,4,5,7]
[1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[6,1,2,3,4,7,8,5]=>[6,1,2,3,8,4,7,5]
[1,1,1,2,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[5,1,2,3,8,4,6,7]=>[5,1,2,8,3,4,6,7]
[1,1,1,2,2]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[5,1,2,3,7,4,8,6]=>[5,1,2,7,3,8,4,6]
[1,1,1,3,1]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[5,1,2,3,6,8,4,7]=>[5,1,2,8,3,6,4,7]
[1,1,1,4]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[5,1,2,3,6,7,8,4]=>[5,1,2,8,3,6,7,4]
[1,1,2,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[4,1,2,8,3,5,6,7]=>[4,1,8,2,3,5,6,7]
[1,1,2,1,2]=>[1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[4,1,2,7,3,5,8,6]=>[4,1,7,2,3,8,5,6]
[1,1,2,2,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[4,1,2,6,3,8,5,7]=>[4,1,6,2,8,3,5,7]
[1,1,2,3]=>[1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[4,1,2,6,3,7,8,5]=>[4,1,6,2,8,3,7,5]
[1,1,3,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>[4,1,2,5,8,3,6,7]=>[4,1,8,2,5,3,6,7]
[1,1,3,2]=>[1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>[4,1,2,5,7,3,8,6]=>[4,1,7,2,5,8,3,6]
[1,1,4,1]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[4,1,2,5,6,8,3,7]=>[4,1,8,2,5,6,3,7]
[1,1,5]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[4,1,2,5,6,7,8,3]=>[4,1,8,2,5,6,7,3]
[1,2,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[3,1,8,2,4,5,6,7]=>[3,8,1,2,4,5,6,7]
[1,2,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[3,1,7,2,4,5,8,6]=>[3,7,1,2,4,8,5,6]
[1,2,1,2,1]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[3,1,6,2,4,8,5,7]=>[3,6,1,2,8,4,5,7]
[1,2,1,3]=>[1,0,1,1,0,0,1,0,1,1,1,0,0,0]=>[3,1,6,2,4,7,8,5]=>[3,6,1,2,8,4,7,5]
[1,2,2,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[3,1,5,2,8,4,6,7]=>[3,5,1,8,2,4,6,7]
[1,2,2,2]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[3,1,5,2,7,4,8,6]=>[3,5,1,7,2,8,4,6]
[1,2,3,1]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,0]=>[3,1,5,2,6,8,4,7]=>[3,5,1,8,2,6,4,7]
[1,2,4]=>[1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[3,1,5,2,6,7,8,4]=>[3,5,1,8,2,6,7,4]
[1,3,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[3,1,4,8,2,5,6,7]=>[3,8,1,4,2,5,6,7]
[1,3,1,2]=>[1,0,1,1,1,0,0,0,1,0,1,1,0,0]=>[3,1,4,7,2,5,8,6]=>[3,7,1,4,2,8,5,6]
[1,3,2,1]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,0]=>[3,1,4,6,2,8,5,7]=>[3,6,1,4,8,2,5,7]
[1,3,3]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[3,1,4,6,2,7,8,5]=>[3,6,1,4,8,2,7,5]
[1,4,1,1]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[3,1,4,5,8,2,6,7]=>[3,8,1,4,5,2,6,7]
[1,4,2]=>[1,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[3,1,4,5,7,2,8,6]=>[3,7,1,4,5,8,2,6]
[1,5,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[3,1,4,5,6,8,2,7]=>[3,8,1,4,5,6,2,7]
[1,6]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[3,1,4,5,6,7,8,2]=>[3,8,1,4,5,6,7,2]
[2,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[2,8,1,3,4,5,6,7]=>[8,2,1,3,4,5,6,7]
[2,1,1,3]=>[1,1,0,0,1,0,1,0,1,1,1,0,0,0]=>[2,6,1,3,4,7,8,5]=>[6,2,1,3,8,4,7,5]
[2,1,2,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>[2,5,1,3,8,4,6,7]=>[5,2,1,8,3,4,6,7]
[2,1,2,2]=>[1,1,0,0,1,0,1,1,0,0,1,1,0,0]=>[2,5,1,3,7,4,8,6]=>[5,2,1,7,3,8,4,6]
[2,1,3,1]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,0]=>[2,5,1,3,6,8,4,7]=>[5,2,1,8,3,6,4,7]
[2,1,4]=>[1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>[2,5,1,3,6,7,8,4]=>[5,2,1,8,3,6,7,4]
[2,2,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[2,4,1,8,3,5,6,7]=>[4,2,8,1,3,5,6,7]
[2,2,1,2]=>[1,1,0,0,1,1,0,0,1,0,1,1,0,0]=>[2,4,1,7,3,5,8,6]=>[4,2,7,1,3,8,5,6]
[2,2,2,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[2,4,1,6,3,8,5,7]=>[4,2,6,1,8,3,5,7]
[2,2,3]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>[2,4,1,6,3,7,8,5]=>[4,2,6,1,8,3,7,5]
[2,3,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>[2,4,1,5,8,3,6,7]=>[4,2,8,1,5,3,6,7]
[2,3,2]=>[1,1,0,0,1,1,1,0,0,0,1,1,0,0]=>[2,4,1,5,7,3,8,6]=>[4,2,7,1,5,8,3,6]
[2,4,1]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,0]=>[2,4,1,5,6,8,3,7]=>[4,2,8,1,5,6,3,7]
[2,5]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[2,4,1,5,6,7,8,3]=>[4,2,8,1,5,6,7,3]
[3,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>[2,3,8,1,4,5,6,7]=>[8,2,3,1,4,5,6,7]
[3,1,1,2]=>[1,1,1,0,0,0,1,0,1,0,1,1,0,0]=>[2,3,7,1,4,5,8,6]=>[7,2,3,1,4,8,5,6]
[3,1,2,1]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,0]=>[2,3,6,1,4,8,5,7]=>[6,2,3,1,8,4,5,7]
[3,1,3]=>[1,1,1,0,0,0,1,0,1,1,1,0,0,0]=>[2,3,6,1,4,7,8,5]=>[6,2,3,1,8,4,7,5]
[3,2,1,1]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,0]=>[2,3,5,1,8,4,6,7]=>[5,2,3,8,1,4,6,7]
[3,2,2]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>[2,3,5,1,7,4,8,6]=>[5,2,3,7,1,8,4,6]
[3,3,1]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>[2,3,5,1,6,8,4,7]=>[5,2,3,8,1,6,4,7]
[3,4]=>[1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>[2,3,5,1,6,7,8,4]=>[5,2,3,8,1,6,7,4]
[4,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>[2,3,4,8,1,5,6,7]=>[8,2,3,4,1,5,6,7]
[4,1,2]=>[1,1,1,1,0,0,0,0,1,0,1,1,0,0]=>[2,3,4,7,1,5,8,6]=>[7,2,3,4,1,8,5,6]
[4,2,1]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,0]=>[2,3,4,6,1,8,5,7]=>[6,2,3,4,8,1,5,7]
[4,3]=>[1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>[2,3,4,6,1,7,8,5]=>[6,2,3,4,8,1,7,5]
[5,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>[2,3,4,5,8,1,6,7]=>[8,2,3,4,5,1,6,7]
[6,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[2,3,4,5,6,8,1,7]=>[8,2,3,4,5,6,1,7]
[7]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[2,3,4,5,6,7,8,1]=>[8,2,3,4,5,6,7,1]
[1,1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[9,1,2,3,4,5,6,7,8]=>[9,1,2,3,4,5,6,7,8]
[1,2,1,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[3,1,9,2,4,5,6,7,8]=>[3,9,1,2,4,5,6,7,8]
[1,2,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[3,1,5,2,7,4,9,6,8]=>[3,5,1,7,2,9,4,6,8]
[1,2,2,3]=>[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>[3,1,5,2,7,4,8,9,6]=>[3,5,1,7,2,9,4,8,6]
[1,3,1,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>[3,1,4,9,2,5,6,7,8]=>[3,9,1,4,2,5,6,7,8]
[1,7]=>[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[3,1,4,5,6,7,8,9,2]=>[3,9,1,4,5,6,7,8,2]
[2,1,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[2,9,1,3,4,5,6,7,8]=>[9,2,1,3,4,5,6,7,8]
[2,2,1,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[2,4,1,9,3,5,6,7,8]=>[4,2,9,1,3,5,6,7,8]
[2,2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[2,4,1,6,3,8,5,9,7]=>[4,2,6,1,8,3,9,5,7]
[2,3,1,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[2,4,1,5,9,3,6,7,8]=>[4,2,9,1,5,3,6,7,8]
[2,6]=>[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[2,4,1,5,6,7,8,9,3]=>[4,2,9,1,5,6,7,8,3]
[8]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[2,3,4,5,6,7,8,9,1]=>[9,2,3,4,5,6,7,8,1]
[1,1,1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[10,1,2,3,4,5,6,7,8,9]=>[10,1,2,3,4,5,6,7,8,9]
[1,2,1,1,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[3,1,10,2,4,5,6,7,8,9]=>[3,10,1,2,4,5,6,7,8,9]
[1,8]=>[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[3,1,4,5,6,7,8,9,10,2]=>[3,10,1,4,5,6,7,8,9,2]
[2,1,1,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[2,10,1,3,4,5,6,7,8,9]=>[10,2,1,3,4,5,6,7,8,9]
[2,2,1,1,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[2,4,1,10,3,5,6,7,8,9]=>[4,2,10,1,3,5,6,7,8,9]
[2,7]=>[1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[2,4,1,5,6,7,8,9,10,3]=>[4,2,10,1,5,6,7,8,9,3]
[9]=>[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]=>[2,3,4,5,6,7,8,9,10,1]=>[10,2,3,4,5,6,7,8,9,1]
[1,1,1,1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[11,1,2,3,4,5,6,7,8,9,10]=>[11,1,2,3,4,5,6,7,8,9,10]
[2,1,1,1,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[2,11,1,3,4,5,6,7,8,9,10]=>[11,2,1,3,4,5,6,7,8,9,10]
[2,8]=>[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[2,4,1,5,6,7,8,9,10,11,3]=>[4,2,11,1,5,6,7,8,9,10,3]
[10]=>[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]=>[2,3,4,5,6,7,8,9,10,11,1]=>[11,2,3,4,5,6,7,8,9,10,1]
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
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.
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
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 $\chi(\pi)$.
searching the database
Sorry, this map was not found in the database.