Identifier
Mp00052: Parking functions —to non-decreasing parking function⟶ Parking functions
Mp00319: Parking functions —to composition⟶ Integer compositions
Mp00314: Integer compositions —Foata bijection⟶ Integer compositions
Images
=>
[1]=>[1]=>[1]=>[1] [1,1]=>[1,1]=>[1,1]=>[1,1] [1,2]=>[1,2]=>[1,2]=>[1,2] [2,1]=>[1,2]=>[1,2]=>[1,2] [1,1,1]=>[1,1,1]=>[1,1,1]=>[1,1,1] [1,1,2]=>[1,1,2]=>[1,1,2]=>[1,1,2] [1,2,1]=>[1,1,2]=>[1,1,2]=>[1,1,2] [2,1,1]=>[1,1,2]=>[1,1,2]=>[1,1,2] [1,1,3]=>[1,1,3]=>[1,1,3]=>[1,1,3] [1,3,1]=>[1,1,3]=>[1,1,3]=>[1,1,3] [3,1,1]=>[1,1,3]=>[1,1,3]=>[1,1,3] [1,2,2]=>[1,2,2]=>[1,2,2]=>[1,2,2] [2,1,2]=>[1,2,2]=>[1,2,2]=>[1,2,2] [2,2,1]=>[1,2,2]=>[1,2,2]=>[1,2,2] [1,2,3]=>[1,2,3]=>[1,2,3]=>[1,2,3] [1,3,2]=>[1,2,3]=>[1,2,3]=>[1,2,3] [2,1,3]=>[1,2,3]=>[1,2,3]=>[1,2,3] [2,3,1]=>[1,2,3]=>[1,2,3]=>[1,2,3] [3,1,2]=>[1,2,3]=>[1,2,3]=>[1,2,3] [3,2,1]=>[1,2,3]=>[1,2,3]=>[1,2,3] [1,1,1,1]=>[1,1,1,1]=>[1,1,1,1]=>[1,1,1,1] [1,1,1,2]=>[1,1,1,2]=>[1,1,1,2]=>[1,1,1,2] [1,1,2,1]=>[1,1,1,2]=>[1,1,1,2]=>[1,1,1,2] [1,2,1,1]=>[1,1,1,2]=>[1,1,1,2]=>[1,1,1,2] [2,1,1,1]=>[1,1,1,2]=>[1,1,1,2]=>[1,1,1,2] [1,1,1,3]=>[1,1,1,3]=>[1,1,1,3]=>[1,1,1,3] [1,1,3,1]=>[1,1,1,3]=>[1,1,1,3]=>[1,1,1,3] [1,3,1,1]=>[1,1,1,3]=>[1,1,1,3]=>[1,1,1,3] [3,1,1,1]=>[1,1,1,3]=>[1,1,1,3]=>[1,1,1,3] [1,1,1,4]=>[1,1,1,4]=>[1,1,1,4]=>[1,1,1,4] [1,1,4,1]=>[1,1,1,4]=>[1,1,1,4]=>[1,1,1,4] [1,4,1,1]=>[1,1,1,4]=>[1,1,1,4]=>[1,1,1,4] [4,1,1,1]=>[1,1,1,4]=>[1,1,1,4]=>[1,1,1,4] [1,1,2,2]=>[1,1,2,2]=>[1,1,2,2]=>[1,1,2,2] [1,2,1,2]=>[1,1,2,2]=>[1,1,2,2]=>[1,1,2,2] [1,2,2,1]=>[1,1,2,2]=>[1,1,2,2]=>[1,1,2,2] [2,1,1,2]=>[1,1,2,2]=>[1,1,2,2]=>[1,1,2,2] [2,1,2,1]=>[1,1,2,2]=>[1,1,2,2]=>[1,1,2,2] [2,2,1,1]=>[1,1,2,2]=>[1,1,2,2]=>[1,1,2,2] [1,1,2,3]=>[1,1,2,3]=>[1,1,2,3]=>[1,1,2,3] [1,1,3,2]=>[1,1,2,3]=>[1,1,2,3]=>[1,1,2,3] [1,2,1,3]=>[1,1,2,3]=>[1,1,2,3]=>[1,1,2,3] [1,2,3,1]=>[1,1,2,3]=>[1,1,2,3]=>[1,1,2,3] [1,3,1,2]=>[1,1,2,3]=>[1,1,2,3]=>[1,1,2,3] [1,3,2,1]=>[1,1,2,3]=>[1,1,2,3]=>[1,1,2,3] [2,1,1,3]=>[1,1,2,3]=>[1,1,2,3]=>[1,1,2,3] [2,1,3,1]=>[1,1,2,3]=>[1,1,2,3]=>[1,1,2,3] [2,3,1,1]=>[1,1,2,3]=>[1,1,2,3]=>[1,1,2,3] [3,1,1,2]=>[1,1,2,3]=>[1,1,2,3]=>[1,1,2,3] [3,1,2,1]=>[1,1,2,3]=>[1,1,2,3]=>[1,1,2,3] [3,2,1,1]=>[1,1,2,3]=>[1,1,2,3]=>[1,1,2,3] [1,1,2,4]=>[1,1,2,4]=>[1,1,2,4]=>[1,1,2,4] [1,1,4,2]=>[1,1,2,4]=>[1,1,2,4]=>[1,1,2,4] [1,2,1,4]=>[1,1,2,4]=>[1,1,2,4]=>[1,1,2,4] [1,2,4,1]=>[1,1,2,4]=>[1,1,2,4]=>[1,1,2,4] [1,4,1,2]=>[1,1,2,4]=>[1,1,2,4]=>[1,1,2,4] [1,4,2,1]=>[1,1,2,4]=>[1,1,2,4]=>[1,1,2,4] [2,1,1,4]=>[1,1,2,4]=>[1,1,2,4]=>[1,1,2,4] [2,1,4,1]=>[1,1,2,4]=>[1,1,2,4]=>[1,1,2,4] [2,4,1,1]=>[1,1,2,4]=>[1,1,2,4]=>[1,1,2,4] [4,1,1,2]=>[1,1,2,4]=>[1,1,2,4]=>[1,1,2,4] [4,1,2,1]=>[1,1,2,4]=>[1,1,2,4]=>[1,1,2,4] [4,2,1,1]=>[1,1,2,4]=>[1,1,2,4]=>[1,1,2,4] [1,1,3,3]=>[1,1,3,3]=>[1,1,3,3]=>[1,1,3,3] [1,3,1,3]=>[1,1,3,3]=>[1,1,3,3]=>[1,1,3,3] [1,3,3,1]=>[1,1,3,3]=>[1,1,3,3]=>[1,1,3,3] [3,1,1,3]=>[1,1,3,3]=>[1,1,3,3]=>[1,1,3,3] [3,1,3,1]=>[1,1,3,3]=>[1,1,3,3]=>[1,1,3,3] [3,3,1,1]=>[1,1,3,3]=>[1,1,3,3]=>[1,1,3,3] [1,1,3,4]=>[1,1,3,4]=>[1,1,3,4]=>[1,1,3,4] [1,1,4,3]=>[1,1,3,4]=>[1,1,3,4]=>[1,1,3,4] [1,3,1,4]=>[1,1,3,4]=>[1,1,3,4]=>[1,1,3,4] [1,3,4,1]=>[1,1,3,4]=>[1,1,3,4]=>[1,1,3,4] [1,4,1,3]=>[1,1,3,4]=>[1,1,3,4]=>[1,1,3,4] [1,4,3,1]=>[1,1,3,4]=>[1,1,3,4]=>[1,1,3,4] [3,1,1,4]=>[1,1,3,4]=>[1,1,3,4]=>[1,1,3,4] [3,1,4,1]=>[1,1,3,4]=>[1,1,3,4]=>[1,1,3,4] [3,4,1,1]=>[1,1,3,4]=>[1,1,3,4]=>[1,1,3,4] [4,1,1,3]=>[1,1,3,4]=>[1,1,3,4]=>[1,1,3,4] [4,1,3,1]=>[1,1,3,4]=>[1,1,3,4]=>[1,1,3,4] [4,3,1,1]=>[1,1,3,4]=>[1,1,3,4]=>[1,1,3,4] [1,2,2,2]=>[1,2,2,2]=>[1,2,2,2]=>[1,2,2,2] [2,1,2,2]=>[1,2,2,2]=>[1,2,2,2]=>[1,2,2,2] [2,2,1,2]=>[1,2,2,2]=>[1,2,2,2]=>[1,2,2,2] [2,2,2,1]=>[1,2,2,2]=>[1,2,2,2]=>[1,2,2,2] [1,2,2,3]=>[1,2,2,3]=>[1,2,2,3]=>[1,2,2,3] [1,2,3,2]=>[1,2,2,3]=>[1,2,2,3]=>[1,2,2,3] [1,3,2,2]=>[1,2,2,3]=>[1,2,2,3]=>[1,2,2,3] [2,1,2,3]=>[1,2,2,3]=>[1,2,2,3]=>[1,2,2,3] [2,1,3,2]=>[1,2,2,3]=>[1,2,2,3]=>[1,2,2,3] [2,2,1,3]=>[1,2,2,3]=>[1,2,2,3]=>[1,2,2,3] [2,2,3,1]=>[1,2,2,3]=>[1,2,2,3]=>[1,2,2,3] [2,3,1,2]=>[1,2,2,3]=>[1,2,2,3]=>[1,2,2,3] [2,3,2,1]=>[1,2,2,3]=>[1,2,2,3]=>[1,2,2,3] [3,1,2,2]=>[1,2,2,3]=>[1,2,2,3]=>[1,2,2,3] [3,2,1,2]=>[1,2,2,3]=>[1,2,2,3]=>[1,2,2,3] [3,2,2,1]=>[1,2,2,3]=>[1,2,2,3]=>[1,2,2,3] [1,2,2,4]=>[1,2,2,4]=>[1,2,2,4]=>[1,2,2,4] [1,2,4,2]=>[1,2,2,4]=>[1,2,2,4]=>[1,2,2,4] [1,4,2,2]=>[1,2,2,4]=>[1,2,2,4]=>[1,2,2,4] [2,1,2,4]=>[1,2,2,4]=>[1,2,2,4]=>[1,2,2,4] [2,1,4,2]=>[1,2,2,4]=>[1,2,2,4]=>[1,2,2,4] [2,2,1,4]=>[1,2,2,4]=>[1,2,2,4]=>[1,2,2,4] [2,2,4,1]=>[1,2,2,4]=>[1,2,2,4]=>[1,2,2,4] [2,4,1,2]=>[1,2,2,4]=>[1,2,2,4]=>[1,2,2,4] [2,4,2,1]=>[1,2,2,4]=>[1,2,2,4]=>[1,2,2,4] [4,1,2,2]=>[1,2,2,4]=>[1,2,2,4]=>[1,2,2,4] [4,2,1,2]=>[1,2,2,4]=>[1,2,2,4]=>[1,2,2,4] [4,2,2,1]=>[1,2,2,4]=>[1,2,2,4]=>[1,2,2,4] [1,2,3,3]=>[1,2,3,3]=>[1,2,3,3]=>[1,2,3,3] [1,3,2,3]=>[1,2,3,3]=>[1,2,3,3]=>[1,2,3,3] [1,3,3,2]=>[1,2,3,3]=>[1,2,3,3]=>[1,2,3,3] [2,1,3,3]=>[1,2,3,3]=>[1,2,3,3]=>[1,2,3,3] [2,3,1,3]=>[1,2,3,3]=>[1,2,3,3]=>[1,2,3,3] [2,3,3,1]=>[1,2,3,3]=>[1,2,3,3]=>[1,2,3,3] [3,1,2,3]=>[1,2,3,3]=>[1,2,3,3]=>[1,2,3,3] [3,1,3,2]=>[1,2,3,3]=>[1,2,3,3]=>[1,2,3,3] [3,2,1,3]=>[1,2,3,3]=>[1,2,3,3]=>[1,2,3,3] [3,2,3,1]=>[1,2,3,3]=>[1,2,3,3]=>[1,2,3,3] [3,3,1,2]=>[1,2,3,3]=>[1,2,3,3]=>[1,2,3,3] [3,3,2,1]=>[1,2,3,3]=>[1,2,3,3]=>[1,2,3,3] [1,2,3,4]=>[1,2,3,4]=>[1,2,3,4]=>[1,2,3,4] [1,2,4,3]=>[1,2,3,4]=>[1,2,3,4]=>[1,2,3,4] [1,3,2,4]=>[1,2,3,4]=>[1,2,3,4]=>[1,2,3,4] [1,3,4,2]=>[1,2,3,4]=>[1,2,3,4]=>[1,2,3,4] [1,4,2,3]=>[1,2,3,4]=>[1,2,3,4]=>[1,2,3,4] [1,4,3,2]=>[1,2,3,4]=>[1,2,3,4]=>[1,2,3,4] [2,1,3,4]=>[1,2,3,4]=>[1,2,3,4]=>[1,2,3,4] [2,1,4,3]=>[1,2,3,4]=>[1,2,3,4]=>[1,2,3,4] [2,3,1,4]=>[1,2,3,4]=>[1,2,3,4]=>[1,2,3,4] [2,3,4,1]=>[1,2,3,4]=>[1,2,3,4]=>[1,2,3,4] [2,4,1,3]=>[1,2,3,4]=>[1,2,3,4]=>[1,2,3,4] [2,4,3,1]=>[1,2,3,4]=>[1,2,3,4]=>[1,2,3,4] [3,1,2,4]=>[1,2,3,4]=>[1,2,3,4]=>[1,2,3,4] [3,1,4,2]=>[1,2,3,4]=>[1,2,3,4]=>[1,2,3,4] [3,2,1,4]=>[1,2,3,4]=>[1,2,3,4]=>[1,2,3,4] [3,2,4,1]=>[1,2,3,4]=>[1,2,3,4]=>[1,2,3,4] [3,4,1,2]=>[1,2,3,4]=>[1,2,3,4]=>[1,2,3,4] [3,4,2,1]=>[1,2,3,4]=>[1,2,3,4]=>[1,2,3,4] [4,1,2,3]=>[1,2,3,4]=>[1,2,3,4]=>[1,2,3,4] [4,1,3,2]=>[1,2,3,4]=>[1,2,3,4]=>[1,2,3,4] [4,2,1,3]=>[1,2,3,4]=>[1,2,3,4]=>[1,2,3,4] [4,2,3,1]=>[1,2,3,4]=>[1,2,3,4]=>[1,2,3,4] [4,3,1,2]=>[1,2,3,4]=>[1,2,3,4]=>[1,2,3,4] [4,3,2,1]=>[1,2,3,4]=>[1,2,3,4]=>[1,2,3,4] [1,1,1,1,1]=>[1,1,1,1,1]=>[1,1,1,1,1]=>[1,1,1,1,1] [1,1,1,1,2]=>[1,1,1,1,2]=>[1,1,1,1,2]=>[1,1,1,1,2] [1,1,1,2,1]=>[1,1,1,1,2]=>[1,1,1,1,2]=>[1,1,1,1,2] [1,1,2,1,1]=>[1,1,1,1,2]=>[1,1,1,1,2]=>[1,1,1,1,2] [1,2,1,1,1]=>[1,1,1,1,2]=>[1,1,1,1,2]=>[1,1,1,1,2] [2,1,1,1,1]=>[1,1,1,1,2]=>[1,1,1,1,2]=>[1,1,1,1,2] [1,1,1,1,3]=>[1,1,1,1,3]=>[1,1,1,1,3]=>[1,1,1,1,3] [1,1,1,3,1]=>[1,1,1,1,3]=>[1,1,1,1,3]=>[1,1,1,1,3] [1,1,3,1,1]=>[1,1,1,1,3]=>[1,1,1,1,3]=>[1,1,1,1,3] [1,3,1,1,1]=>[1,1,1,1,3]=>[1,1,1,1,3]=>[1,1,1,1,3] [3,1,1,1,1]=>[1,1,1,1,3]=>[1,1,1,1,3]=>[1,1,1,1,3] [1,1,1,1,4]=>[1,1,1,1,4]=>[1,1,1,1,4]=>[1,1,1,1,4] [1,1,1,4,1]=>[1,1,1,1,4]=>[1,1,1,1,4]=>[1,1,1,1,4] [1,1,4,1,1]=>[1,1,1,1,4]=>[1,1,1,1,4]=>[1,1,1,1,4] [1,4,1,1,1]=>[1,1,1,1,4]=>[1,1,1,1,4]=>[1,1,1,1,4] [4,1,1,1,1]=>[1,1,1,1,4]=>[1,1,1,1,4]=>[1,1,1,1,4] [1,1,1,1,5]=>[1,1,1,1,5]=>[1,1,1,1,5]=>[1,1,1,1,5] [1,1,1,5,1]=>[1,1,1,1,5]=>[1,1,1,1,5]=>[1,1,1,1,5] [1,1,5,1,1]=>[1,1,1,1,5]=>[1,1,1,1,5]=>[1,1,1,1,5] [1,5,1,1,1]=>[1,1,1,1,5]=>[1,1,1,1,5]=>[1,1,1,1,5] [5,1,1,1,1]=>[1,1,1,1,5]=>[1,1,1,1,5]=>[1,1,1,1,5] [1,1,1,2,2]=>[1,1,1,2,2]=>[1,1,1,2,2]=>[1,1,1,2,2] [1,1,2,1,2]=>[1,1,1,2,2]=>[1,1,1,2,2]=>[1,1,1,2,2] [1,1,2,2,1]=>[1,1,1,2,2]=>[1,1,1,2,2]=>[1,1,1,2,2] [1,2,1,1,2]=>[1,1,1,2,2]=>[1,1,1,2,2]=>[1,1,1,2,2] [1,2,1,2,1]=>[1,1,1,2,2]=>[1,1,1,2,2]=>[1,1,1,2,2] [1,2,2,1,1]=>[1,1,1,2,2]=>[1,1,1,2,2]=>[1,1,1,2,2] [2,1,1,1,2]=>[1,1,1,2,2]=>[1,1,1,2,2]=>[1,1,1,2,2] [2,1,1,2,1]=>[1,1,1,2,2]=>[1,1,1,2,2]=>[1,1,1,2,2] [2,1,2,1,1]=>[1,1,1,2,2]=>[1,1,1,2,2]=>[1,1,1,2,2] [2,2,1,1,1]=>[1,1,1,2,2]=>[1,1,1,2,2]=>[1,1,1,2,2] [1,1,1,2,3]=>[1,1,1,2,3]=>[1,1,1,2,3]=>[1,1,1,2,3] [1,1,1,3,2]=>[1,1,1,2,3]=>[1,1,1,2,3]=>[1,1,1,2,3] [1,1,2,1,3]=>[1,1,1,2,3]=>[1,1,1,2,3]=>[1,1,1,2,3] [1,1,2,3,1]=>[1,1,1,2,3]=>[1,1,1,2,3]=>[1,1,1,2,3] [1,1,3,1,2]=>[1,1,1,2,3]=>[1,1,1,2,3]=>[1,1,1,2,3] [1,1,3,2,1]=>[1,1,1,2,3]=>[1,1,1,2,3]=>[1,1,1,2,3] [1,2,1,1,3]=>[1,1,1,2,3]=>[1,1,1,2,3]=>[1,1,1,2,3] [1,2,1,3,1]=>[1,1,1,2,3]=>[1,1,1,2,3]=>[1,1,1,2,3] [1,2,3,1,1]=>[1,1,1,2,3]=>[1,1,1,2,3]=>[1,1,1,2,3] [1,3,1,1,2]=>[1,1,1,2,3]=>[1,1,1,2,3]=>[1,1,1,2,3] [1,3,1,2,1]=>[1,1,1,2,3]=>[1,1,1,2,3]=>[1,1,1,2,3] [1,3,2,1,1]=>[1,1,1,2,3]=>[1,1,1,2,3]=>[1,1,1,2,3] [2,1,1,1,3]=>[1,1,1,2,3]=>[1,1,1,2,3]=>[1,1,1,2,3] [2,1,1,3,1]=>[1,1,1,2,3]=>[1,1,1,2,3]=>[1,1,1,2,3] [2,1,3,1,1]=>[1,1,1,2,3]=>[1,1,1,2,3]=>[1,1,1,2,3] [2,3,1,1,1]=>[1,1,1,2,3]=>[1,1,1,2,3]=>[1,1,1,2,3] [3,1,1,1,2]=>[1,1,1,2,3]=>[1,1,1,2,3]=>[1,1,1,2,3] [3,1,1,2,1]=>[1,1,1,2,3]=>[1,1,1,2,3]=>[1,1,1,2,3] [3,1,2,1,1]=>[1,1,1,2,3]=>[1,1,1,2,3]=>[1,1,1,2,3] [3,2,1,1,1]=>[1,1,1,2,3]=>[1,1,1,2,3]=>[1,1,1,2,3] [1,1,1,2,4]=>[1,1,1,2,4]=>[1,1,1,2,4]=>[1,1,1,2,4] [1,1,1,4,2]=>[1,1,1,2,4]=>[1,1,1,2,4]=>[1,1,1,2,4] [1,1,2,1,4]=>[1,1,1,2,4]=>[1,1,1,2,4]=>[1,1,1,2,4] [1,1,2,4,1]=>[1,1,1,2,4]=>[1,1,1,2,4]=>[1,1,1,2,4] [1,1,4,1,2]=>[1,1,1,2,4]=>[1,1,1,2,4]=>[1,1,1,2,4] [1,1,4,2,1]=>[1,1,1,2,4]=>[1,1,1,2,4]=>[1,1,1,2,4] [1,2,1,1,4]=>[1,1,1,2,4]=>[1,1,1,2,4]=>[1,1,1,2,4] [1,2,1,4,1]=>[1,1,1,2,4]=>[1,1,1,2,4]=>[1,1,1,2,4] [1,2,4,1,1]=>[1,1,1,2,4]=>[1,1,1,2,4]=>[1,1,1,2,4] [1,4,1,1,2]=>[1,1,1,2,4]=>[1,1,1,2,4]=>[1,1,1,2,4] [1,4,1,2,1]=>[1,1,1,2,4]=>[1,1,1,2,4]=>[1,1,1,2,4] [1,4,2,1,1]=>[1,1,1,2,4]=>[1,1,1,2,4]=>[1,1,1,2,4] [2,1,1,1,4]=>[1,1,1,2,4]=>[1,1,1,2,4]=>[1,1,1,2,4] [2,1,1,4,1]=>[1,1,1,2,4]=>[1,1,1,2,4]=>[1,1,1,2,4] [2,1,4,1,1]=>[1,1,1,2,4]=>[1,1,1,2,4]=>[1,1,1,2,4] [2,4,1,1,1]=>[1,1,1,2,4]=>[1,1,1,2,4]=>[1,1,1,2,4] [4,1,1,1,2]=>[1,1,1,2,4]=>[1,1,1,2,4]=>[1,1,1,2,4] [4,1,1,2,1]=>[1,1,1,2,4]=>[1,1,1,2,4]=>[1,1,1,2,4] [4,1,2,1,1]=>[1,1,1,2,4]=>[1,1,1,2,4]=>[1,1,1,2,4] [4,2,1,1,1]=>[1,1,1,2,4]=>[1,1,1,2,4]=>[1,1,1,2,4] [1,1,1,2,5]=>[1,1,1,2,5]=>[1,1,1,2,5]=>[1,1,1,2,5] [1,1,1,5,2]=>[1,1,1,2,5]=>[1,1,1,2,5]=>[1,1,1,2,5] [1,1,2,1,5]=>[1,1,1,2,5]=>[1,1,1,2,5]=>[1,1,1,2,5] [1,1,2,5,1]=>[1,1,1,2,5]=>[1,1,1,2,5]=>[1,1,1,2,5] [1,1,5,1,2]=>[1,1,1,2,5]=>[1,1,1,2,5]=>[1,1,1,2,5] [1,1,5,2,1]=>[1,1,1,2,5]=>[1,1,1,2,5]=>[1,1,1,2,5] [1,2,1,1,5]=>[1,1,1,2,5]=>[1,1,1,2,5]=>[1,1,1,2,5] [1,2,1,5,1]=>[1,1,1,2,5]=>[1,1,1,2,5]=>[1,1,1,2,5] [1,2,5,1,1]=>[1,1,1,2,5]=>[1,1,1,2,5]=>[1,1,1,2,5] [1,5,1,1,2]=>[1,1,1,2,5]=>[1,1,1,2,5]=>[1,1,1,2,5] [1,5,1,2,1]=>[1,1,1,2,5]=>[1,1,1,2,5]=>[1,1,1,2,5] [1,5,2,1,1]=>[1,1,1,2,5]=>[1,1,1,2,5]=>[1,1,1,2,5] [2,1,1,1,5]=>[1,1,1,2,5]=>[1,1,1,2,5]=>[1,1,1,2,5] [2,1,1,5,1]=>[1,1,1,2,5]=>[1,1,1,2,5]=>[1,1,1,2,5] [2,1,5,1,1]=>[1,1,1,2,5]=>[1,1,1,2,5]=>[1,1,1,2,5] [2,5,1,1,1]=>[1,1,1,2,5]=>[1,1,1,2,5]=>[1,1,1,2,5] [5,1,1,1,2]=>[1,1,1,2,5]=>[1,1,1,2,5]=>[1,1,1,2,5] [5,1,1,2,1]=>[1,1,1,2,5]=>[1,1,1,2,5]=>[1,1,1,2,5] [5,1,2,1,1]=>[1,1,1,2,5]=>[1,1,1,2,5]=>[1,1,1,2,5] [5,2,1,1,1]=>[1,1,1,2,5]=>[1,1,1,2,5]=>[1,1,1,2,5] [1,1,1,3,3]=>[1,1,1,3,3]=>[1,1,1,3,3]=>[1,1,1,3,3] [1,1,3,1,3]=>[1,1,1,3,3]=>[1,1,1,3,3]=>[1,1,1,3,3] [1,1,3,3,1]=>[1,1,1,3,3]=>[1,1,1,3,3]=>[1,1,1,3,3] [1,3,1,1,3]=>[1,1,1,3,3]=>[1,1,1,3,3]=>[1,1,1,3,3] [1,3,1,3,1]=>[1,1,1,3,3]=>[1,1,1,3,3]=>[1,1,1,3,3] [1,3,3,1,1]=>[1,1,1,3,3]=>[1,1,1,3,3]=>[1,1,1,3,3] [3,1,1,1,3]=>[1,1,1,3,3]=>[1,1,1,3,3]=>[1,1,1,3,3] [3,1,1,3,1]=>[1,1,1,3,3]=>[1,1,1,3,3]=>[1,1,1,3,3] [3,1,3,1,1]=>[1,1,1,3,3]=>[1,1,1,3,3]=>[1,1,1,3,3] [3,3,1,1,1]=>[1,1,1,3,3]=>[1,1,1,3,3]=>[1,1,1,3,3] [1,1,1,3,4]=>[1,1,1,3,4]=>[1,1,1,3,4]=>[1,1,1,3,4] [1,1,1,4,3]=>[1,1,1,3,4]=>[1,1,1,3,4]=>[1,1,1,3,4] [1,1,3,1,4]=>[1,1,1,3,4]=>[1,1,1,3,4]=>[1,1,1,3,4] [1,1,3,4,1]=>[1,1,1,3,4]=>[1,1,1,3,4]=>[1,1,1,3,4] [1,1,4,1,3]=>[1,1,1,3,4]=>[1,1,1,3,4]=>[1,1,1,3,4] [1,1,4,3,1]=>[1,1,1,3,4]=>[1,1,1,3,4]=>[1,1,1,3,4] [1,3,1,1,4]=>[1,1,1,3,4]=>[1,1,1,3,4]=>[1,1,1,3,4] [1,3,1,4,1]=>[1,1,1,3,4]=>[1,1,1,3,4]=>[1,1,1,3,4] [1,3,4,1,1]=>[1,1,1,3,4]=>[1,1,1,3,4]=>[1,1,1,3,4] [1,4,1,1,3]=>[1,1,1,3,4]=>[1,1,1,3,4]=>[1,1,1,3,4] [1,4,1,3,1]=>[1,1,1,3,4]=>[1,1,1,3,4]=>[1,1,1,3,4] [1,4,3,1,1]=>[1,1,1,3,4]=>[1,1,1,3,4]=>[1,1,1,3,4] [3,1,1,1,4]=>[1,1,1,3,4]=>[1,1,1,3,4]=>[1,1,1,3,4] [3,1,1,4,1]=>[1,1,1,3,4]=>[1,1,1,3,4]=>[1,1,1,3,4] [3,1,4,1,1]=>[1,1,1,3,4]=>[1,1,1,3,4]=>[1,1,1,3,4] [3,4,1,1,1]=>[1,1,1,3,4]=>[1,1,1,3,4]=>[1,1,1,3,4] [4,1,1,1,3]=>[1,1,1,3,4]=>[1,1,1,3,4]=>[1,1,1,3,4] [4,1,1,3,1]=>[1,1,1,3,4]=>[1,1,1,3,4]=>[1,1,1,3,4] [4,1,3,1,1]=>[1,1,1,3,4]=>[1,1,1,3,4]=>[1,1,1,3,4] [4,3,1,1,1]=>[1,1,1,3,4]=>[1,1,1,3,4]=>[1,1,1,3,4] [1,1,2,2,2]=>[1,1,2,2,2]=>[1,1,2,2,2]=>[1,1,2,2,2] [1,2,1,2,2]=>[1,1,2,2,2]=>[1,1,2,2,2]=>[1,1,2,2,2] [1,2,2,1,2]=>[1,1,2,2,2]=>[1,1,2,2,2]=>[1,1,2,2,2] [1,2,2,2,1]=>[1,1,2,2,2]=>[1,1,2,2,2]=>[1,1,2,2,2] [2,1,1,2,2]=>[1,1,2,2,2]=>[1,1,2,2,2]=>[1,1,2,2,2] [2,1,2,1,2]=>[1,1,2,2,2]=>[1,1,2,2,2]=>[1,1,2,2,2] [2,1,2,2,1]=>[1,1,2,2,2]=>[1,1,2,2,2]=>[1,1,2,2,2] [2,2,1,1,2]=>[1,1,2,2,2]=>[1,1,2,2,2]=>[1,1,2,2,2] [2,2,1,2,1]=>[1,1,2,2,2]=>[1,1,2,2,2]=>[1,1,2,2,2] [2,2,2,1,1]=>[1,1,2,2,2]=>[1,1,2,2,2]=>[1,1,2,2,2] [1,1,2,2,3]=>[1,1,2,2,3]=>[1,1,2,2,3]=>[1,1,2,2,3] [1,1,2,3,2]=>[1,1,2,2,3]=>[1,1,2,2,3]=>[1,1,2,2,3] [1,1,3,2,2]=>[1,1,2,2,3]=>[1,1,2,2,3]=>[1,1,2,2,3] [1,2,1,2,3]=>[1,1,2,2,3]=>[1,1,2,2,3]=>[1,1,2,2,3] [1,2,1,3,2]=>[1,1,2,2,3]=>[1,1,2,2,3]=>[1,1,2,2,3] [1,2,2,1,3]=>[1,1,2,2,3]=>[1,1,2,2,3]=>[1,1,2,2,3] [1,2,2,3,1]=>[1,1,2,2,3]=>[1,1,2,2,3]=>[1,1,2,2,3] [1,2,3,1,2]=>[1,1,2,2,3]=>[1,1,2,2,3]=>[1,1,2,2,3] [1,2,3,2,1]=>[1,1,2,2,3]=>[1,1,2,2,3]=>[1,1,2,2,3] [1,3,1,2,2]=>[1,1,2,2,3]=>[1,1,2,2,3]=>[1,1,2,2,3] [1,3,2,1,2]=>[1,1,2,2,3]=>[1,1,2,2,3]=>[1,1,2,2,3] [1,3,2,2,1]=>[1,1,2,2,3]=>[1,1,2,2,3]=>[1,1,2,2,3] [2,1,1,2,3]=>[1,1,2,2,3]=>[1,1,2,2,3]=>[1,1,2,2,3] [2,1,1,3,2]=>[1,1,2,2,3]=>[1,1,2,2,3]=>[1,1,2,2,3] [2,1,2,1,3]=>[1,1,2,2,3]=>[1,1,2,2,3]=>[1,1,2,2,3] [2,1,2,3,1]=>[1,1,2,2,3]=>[1,1,2,2,3]=>[1,1,2,2,3] [2,1,3,1,2]=>[1,1,2,2,3]=>[1,1,2,2,3]=>[1,1,2,2,3] [2,1,3,2,1]=>[1,1,2,2,3]=>[1,1,2,2,3]=>[1,1,2,2,3] [2,2,1,1,3]=>[1,1,2,2,3]=>[1,1,2,2,3]=>[1,1,2,2,3] [2,2,1,3,1]=>[1,1,2,2,3]=>[1,1,2,2,3]=>[1,1,2,2,3] [2,2,3,1,1]=>[1,1,2,2,3]=>[1,1,2,2,3]=>[1,1,2,2,3] [2,3,1,1,2]=>[1,1,2,2,3]=>[1,1,2,2,3]=>[1,1,2,2,3] [2,3,1,2,1]=>[1,1,2,2,3]=>[1,1,2,2,3]=>[1,1,2,2,3] [2,3,2,1,1]=>[1,1,2,2,3]=>[1,1,2,2,3]=>[1,1,2,2,3] [3,1,1,2,2]=>[1,1,2,2,3]=>[1,1,2,2,3]=>[1,1,2,2,3] [3,1,2,1,2]=>[1,1,2,2,3]=>[1,1,2,2,3]=>[1,1,2,2,3] [3,1,2,2,1]=>[1,1,2,2,3]=>[1,1,2,2,3]=>[1,1,2,2,3] [3,2,1,1,2]=>[1,1,2,2,3]=>[1,1,2,2,3]=>[1,1,2,2,3] [3,2,1,2,1]=>[1,1,2,2,3]=>[1,1,2,2,3]=>[1,1,2,2,3] [3,2,2,1,1]=>[1,1,2,2,3]=>[1,1,2,2,3]=>[1,1,2,2,3] [1,1,2,2,4]=>[1,1,2,2,4]=>[1,1,2,2,4]=>[1,1,2,2,4] [1,1,2,4,2]=>[1,1,2,2,4]=>[1,1,2,2,4]=>[1,1,2,2,4] [1,1,4,2,2]=>[1,1,2,2,4]=>[1,1,2,2,4]=>[1,1,2,2,4] [1,2,1,2,4]=>[1,1,2,2,4]=>[1,1,2,2,4]=>[1,1,2,2,4] [1,2,1,4,2]=>[1,1,2,2,4]=>[1,1,2,2,4]=>[1,1,2,2,4] [1,2,2,1,4]=>[1,1,2,2,4]=>[1,1,2,2,4]=>[1,1,2,2,4] [1,2,2,4,1]=>[1,1,2,2,4]=>[1,1,2,2,4]=>[1,1,2,2,4] [1,2,4,1,2]=>[1,1,2,2,4]=>[1,1,2,2,4]=>[1,1,2,2,4] [1,2,4,2,1]=>[1,1,2,2,4]=>[1,1,2,2,4]=>[1,1,2,2,4] [1,4,1,2,2]=>[1,1,2,2,4]=>[1,1,2,2,4]=>[1,1,2,2,4] [1,4,2,1,2]=>[1,1,2,2,4]=>[1,1,2,2,4]=>[1,1,2,2,4] [1,4,2,2,1]=>[1,1,2,2,4]=>[1,1,2,2,4]=>[1,1,2,2,4] [2,1,1,2,4]=>[1,1,2,2,4]=>[1,1,2,2,4]=>[1,1,2,2,4] [2,1,1,4,2]=>[1,1,2,2,4]=>[1,1,2,2,4]=>[1,1,2,2,4] [2,1,2,1,4]=>[1,1,2,2,4]=>[1,1,2,2,4]=>[1,1,2,2,4] [2,1,2,4,1]=>[1,1,2,2,4]=>[1,1,2,2,4]=>[1,1,2,2,4] [2,1,4,1,2]=>[1,1,2,2,4]=>[1,1,2,2,4]=>[1,1,2,2,4] [2,1,4,2,1]=>[1,1,2,2,4]=>[1,1,2,2,4]=>[1,1,2,2,4] [2,2,1,1,4]=>[1,1,2,2,4]=>[1,1,2,2,4]=>[1,1,2,2,4] [2,2,1,4,1]=>[1,1,2,2,4]=>[1,1,2,2,4]=>[1,1,2,2,4] [2,2,4,1,1]=>[1,1,2,2,4]=>[1,1,2,2,4]=>[1,1,2,2,4] [2,4,1,1,2]=>[1,1,2,2,4]=>[1,1,2,2,4]=>[1,1,2,2,4] [2,4,1,2,1]=>[1,1,2,2,4]=>[1,1,2,2,4]=>[1,1,2,2,4] [2,4,2,1,1]=>[1,1,2,2,4]=>[1,1,2,2,4]=>[1,1,2,2,4] [4,1,1,2,2]=>[1,1,2,2,4]=>[1,1,2,2,4]=>[1,1,2,2,4] [4,1,2,1,2]=>[1,1,2,2,4]=>[1,1,2,2,4]=>[1,1,2,2,4] [4,1,2,2,1]=>[1,1,2,2,4]=>[1,1,2,2,4]=>[1,1,2,2,4] [4,2,1,1,2]=>[1,1,2,2,4]=>[1,1,2,2,4]=>[1,1,2,2,4] [4,2,1,2,1]=>[1,1,2,2,4]=>[1,1,2,2,4]=>[1,1,2,2,4] [4,2,2,1,1]=>[1,1,2,2,4]=>[1,1,2,2,4]=>[1,1,2,2,4] [1,1,2,3,3]=>[1,1,2,3,3]=>[1,1,2,3,3]=>[1,1,2,3,3] [1,1,3,2,3]=>[1,1,2,3,3]=>[1,1,2,3,3]=>[1,1,2,3,3] [1,1,3,3,2]=>[1,1,2,3,3]=>[1,1,2,3,3]=>[1,1,2,3,3] [1,2,1,3,3]=>[1,1,2,3,3]=>[1,1,2,3,3]=>[1,1,2,3,3] [1,2,3,1,3]=>[1,1,2,3,3]=>[1,1,2,3,3]=>[1,1,2,3,3] [1,2,3,3,1]=>[1,1,2,3,3]=>[1,1,2,3,3]=>[1,1,2,3,3] [1,3,1,2,3]=>[1,1,2,3,3]=>[1,1,2,3,3]=>[1,1,2,3,3] [1,3,1,3,2]=>[1,1,2,3,3]=>[1,1,2,3,3]=>[1,1,2,3,3] [1,3,2,1,3]=>[1,1,2,3,3]=>[1,1,2,3,3]=>[1,1,2,3,3] [1,3,2,3,1]=>[1,1,2,3,3]=>[1,1,2,3,3]=>[1,1,2,3,3] [1,3,3,1,2]=>[1,1,2,3,3]=>[1,1,2,3,3]=>[1,1,2,3,3] [1,3,3,2,1]=>[1,1,2,3,3]=>[1,1,2,3,3]=>[1,1,2,3,3] [2,1,1,3,3]=>[1,1,2,3,3]=>[1,1,2,3,3]=>[1,1,2,3,3] [2,1,3,1,3]=>[1,1,2,3,3]=>[1,1,2,3,3]=>[1,1,2,3,3] [2,1,3,3,1]=>[1,1,2,3,3]=>[1,1,2,3,3]=>[1,1,2,3,3] [2,3,1,1,3]=>[1,1,2,3,3]=>[1,1,2,3,3]=>[1,1,2,3,3] [2,3,1,3,1]=>[1,1,2,3,3]=>[1,1,2,3,3]=>[1,1,2,3,3] [2,3,3,1,1]=>[1,1,2,3,3]=>[1,1,2,3,3]=>[1,1,2,3,3] [3,1,1,2,3]=>[1,1,2,3,3]=>[1,1,2,3,3]=>[1,1,2,3,3] [3,1,1,3,2]=>[1,1,2,3,3]=>[1,1,2,3,3]=>[1,1,2,3,3] [3,1,2,1,3]=>[1,1,2,3,3]=>[1,1,2,3,3]=>[1,1,2,3,3] [3,1,2,3,1]=>[1,1,2,3,3]=>[1,1,2,3,3]=>[1,1,2,3,3] [3,1,3,1,2]=>[1,1,2,3,3]=>[1,1,2,3,3]=>[1,1,2,3,3] [3,1,3,2,1]=>[1,1,2,3,3]=>[1,1,2,3,3]=>[1,1,2,3,3] [3,2,1,1,3]=>[1,1,2,3,3]=>[1,1,2,3,3]=>[1,1,2,3,3] [3,2,1,3,1]=>[1,1,2,3,3]=>[1,1,2,3,3]=>[1,1,2,3,3] [3,2,3,1,1]=>[1,1,2,3,3]=>[1,1,2,3,3]=>[1,1,2,3,3] [3,3,1,1,2]=>[1,1,2,3,3]=>[1,1,2,3,3]=>[1,1,2,3,3] [3,3,1,2,1]=>[1,1,2,3,3]=>[1,1,2,3,3]=>[1,1,2,3,3] [3,3,2,1,1]=>[1,1,2,3,3]=>[1,1,2,3,3]=>[1,1,2,3,3] [1,2,2,2,2]=>[1,2,2,2,2]=>[1,2,2,2,2]=>[1,2,2,2,2] [2,1,2,2,2]=>[1,2,2,2,2]=>[1,2,2,2,2]=>[1,2,2,2,2] [2,2,1,2,2]=>[1,2,2,2,2]=>[1,2,2,2,2]=>[1,2,2,2,2] [2,2,2,1,2]=>[1,2,2,2,2]=>[1,2,2,2,2]=>[1,2,2,2,2] [2,2,2,2,1]=>[1,2,2,2,2]=>[1,2,2,2,2]=>[1,2,2,2,2] [1,2,2,2,3]=>[1,2,2,2,3]=>[1,2,2,2,3]=>[1,2,2,2,3] [1,2,2,3,2]=>[1,2,2,2,3]=>[1,2,2,2,3]=>[1,2,2,2,3] [1,2,3,2,2]=>[1,2,2,2,3]=>[1,2,2,2,3]=>[1,2,2,2,3] [1,3,2,2,2]=>[1,2,2,2,3]=>[1,2,2,2,3]=>[1,2,2,2,3] [2,1,2,2,3]=>[1,2,2,2,3]=>[1,2,2,2,3]=>[1,2,2,2,3] [2,1,2,3,2]=>[1,2,2,2,3]=>[1,2,2,2,3]=>[1,2,2,2,3] [2,1,3,2,2]=>[1,2,2,2,3]=>[1,2,2,2,3]=>[1,2,2,2,3] [2,2,1,2,3]=>[1,2,2,2,3]=>[1,2,2,2,3]=>[1,2,2,2,3] [2,2,1,3,2]=>[1,2,2,2,3]=>[1,2,2,2,3]=>[1,2,2,2,3] [2,2,2,1,3]=>[1,2,2,2,3]=>[1,2,2,2,3]=>[1,2,2,2,3] [2,2,2,3,1]=>[1,2,2,2,3]=>[1,2,2,2,3]=>[1,2,2,2,3] [2,2,3,1,2]=>[1,2,2,2,3]=>[1,2,2,2,3]=>[1,2,2,2,3] [2,2,3,2,1]=>[1,2,2,2,3]=>[1,2,2,2,3]=>[1,2,2,2,3] [2,3,1,2,2]=>[1,2,2,2,3]=>[1,2,2,2,3]=>[1,2,2,2,3] [2,3,2,1,2]=>[1,2,2,2,3]=>[1,2,2,2,3]=>[1,2,2,2,3] [2,3,2,2,1]=>[1,2,2,2,3]=>[1,2,2,2,3]=>[1,2,2,2,3] [3,1,2,2,2]=>[1,2,2,2,3]=>[1,2,2,2,3]=>[1,2,2,2,3] [3,2,1,2,2]=>[1,2,2,2,3]=>[1,2,2,2,3]=>[1,2,2,2,3] [3,2,2,1,2]=>[1,2,2,2,3]=>[1,2,2,2,3]=>[1,2,2,2,3] [3,2,2,2,1]=>[1,2,2,2,3]=>[1,2,2,2,3]=>[1,2,2,2,3] [1,1,1,1,1,1]=>[1,1,1,1,1,1]=>[1,1,1,1,1,1]=>[1,1,1,1,1,1] [1,1,1,1,1,2]=>[1,1,1,1,1,2]=>[1,1,1,1,1,2]=>[1,1,1,1,1,2] [1,1,1,1,2,1]=>[1,1,1,1,1,2]=>[1,1,1,1,1,2]=>[1,1,1,1,1,2] [1,1,1,2,1,1]=>[1,1,1,1,1,2]=>[1,1,1,1,1,2]=>[1,1,1,1,1,2] [1,1,2,1,1,1]=>[1,1,1,1,1,2]=>[1,1,1,1,1,2]=>[1,1,1,1,1,2] [1,2,1,1,1,1]=>[1,1,1,1,1,2]=>[1,1,1,1,1,2]=>[1,1,1,1,1,2] [2,1,1,1,1,1]=>[1,1,1,1,1,2]=>[1,1,1,1,1,2]=>[1,1,1,1,1,2] [1,1,1,1,1,3]=>[1,1,1,1,1,3]=>[1,1,1,1,1,3]=>[1,1,1,1,1,3] [1,1,1,1,3,1]=>[1,1,1,1,1,3]=>[1,1,1,1,1,3]=>[1,1,1,1,1,3] [1,1,1,3,1,1]=>[1,1,1,1,1,3]=>[1,1,1,1,1,3]=>[1,1,1,1,1,3] [1,1,3,1,1,1]=>[1,1,1,1,1,3]=>[1,1,1,1,1,3]=>[1,1,1,1,1,3] [1,3,1,1,1,1]=>[1,1,1,1,1,3]=>[1,1,1,1,1,3]=>[1,1,1,1,1,3] [3,1,1,1,1,1]=>[1,1,1,1,1,3]=>[1,1,1,1,1,3]=>[1,1,1,1,1,3] [1,1,1,1,1,4]=>[1,1,1,1,1,4]=>[1,1,1,1,1,4]=>[1,1,1,1,1,4] [1,1,1,1,4,1]=>[1,1,1,1,1,4]=>[1,1,1,1,1,4]=>[1,1,1,1,1,4] [1,1,1,4,1,1]=>[1,1,1,1,1,4]=>[1,1,1,1,1,4]=>[1,1,1,1,1,4] [1,1,4,1,1,1]=>[1,1,1,1,1,4]=>[1,1,1,1,1,4]=>[1,1,1,1,1,4] [1,4,1,1,1,1]=>[1,1,1,1,1,4]=>[1,1,1,1,1,4]=>[1,1,1,1,1,4] [4,1,1,1,1,1]=>[1,1,1,1,1,4]=>[1,1,1,1,1,4]=>[1,1,1,1,1,4] [1,1,1,1,1,5]=>[1,1,1,1,1,5]=>[1,1,1,1,1,5]=>[1,1,1,1,1,5] [1,1,1,1,5,1]=>[1,1,1,1,1,5]=>[1,1,1,1,1,5]=>[1,1,1,1,1,5] [1,1,1,5,1,1]=>[1,1,1,1,1,5]=>[1,1,1,1,1,5]=>[1,1,1,1,1,5] [1,1,5,1,1,1]=>[1,1,1,1,1,5]=>[1,1,1,1,1,5]=>[1,1,1,1,1,5] [1,5,1,1,1,1]=>[1,1,1,1,1,5]=>[1,1,1,1,1,5]=>[1,1,1,1,1,5] [5,1,1,1,1,1]=>[1,1,1,1,1,5]=>[1,1,1,1,1,5]=>[1,1,1,1,1,5] [1,1,1,1,2,2]=>[1,1,1,1,2,2]=>[1,1,1,1,2,2]=>[1,1,1,1,2,2] [1,1,1,2,1,2]=>[1,1,1,1,2,2]=>[1,1,1,1,2,2]=>[1,1,1,1,2,2] [1,1,1,2,2,1]=>[1,1,1,1,2,2]=>[1,1,1,1,2,2]=>[1,1,1,1,2,2] [1,1,2,1,1,2]=>[1,1,1,1,2,2]=>[1,1,1,1,2,2]=>[1,1,1,1,2,2] [1,1,2,1,2,1]=>[1,1,1,1,2,2]=>[1,1,1,1,2,2]=>[1,1,1,1,2,2] [1,1,2,2,1,1]=>[1,1,1,1,2,2]=>[1,1,1,1,2,2]=>[1,1,1,1,2,2] [1,2,1,1,1,2]=>[1,1,1,1,2,2]=>[1,1,1,1,2,2]=>[1,1,1,1,2,2] [1,2,1,1,2,1]=>[1,1,1,1,2,2]=>[1,1,1,1,2,2]=>[1,1,1,1,2,2] [1,2,1,2,1,1]=>[1,1,1,1,2,2]=>[1,1,1,1,2,2]=>[1,1,1,1,2,2] [1,2,2,1,1,1]=>[1,1,1,1,2,2]=>[1,1,1,1,2,2]=>[1,1,1,1,2,2] [2,1,1,1,1,2]=>[1,1,1,1,2,2]=>[1,1,1,1,2,2]=>[1,1,1,1,2,2] [2,1,1,1,2,1]=>[1,1,1,1,2,2]=>[1,1,1,1,2,2]=>[1,1,1,1,2,2] [2,1,1,2,1,1]=>[1,1,1,1,2,2]=>[1,1,1,1,2,2]=>[1,1,1,1,2,2] [2,1,2,1,1,1]=>[1,1,1,1,2,2]=>[1,1,1,1,2,2]=>[1,1,1,1,2,2] [2,2,1,1,1,1]=>[1,1,1,1,2,2]=>[1,1,1,1,2,2]=>[1,1,1,1,2,2] [1,1,1,1,2,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3] [1,1,1,1,3,2]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3] [1,1,1,2,1,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3] [1,1,1,2,3,1]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3] [1,1,1,3,1,2]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3] [1,1,1,3,2,1]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3] [1,1,2,1,1,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3] [1,1,2,1,3,1]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3] [1,1,2,3,1,1]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3] [1,1,3,1,1,2]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3] [1,1,3,1,2,1]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3] [1,1,3,2,1,1]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3] [1,2,1,1,1,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3] [1,2,1,1,3,1]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3] [1,2,1,3,1,1]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3] [1,2,3,1,1,1]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3] [1,3,1,1,1,2]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3] [1,3,1,1,2,1]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3] [1,3,1,2,1,1]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3] [1,3,2,1,1,1]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3] [2,1,1,1,1,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3] [2,1,1,1,3,1]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3] [2,1,1,3,1,1]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3] [2,1,3,1,1,1]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3] [2,3,1,1,1,1]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3] [3,1,1,1,1,2]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3] [3,1,1,1,2,1]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3] [3,1,1,2,1,1]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3] [3,1,2,1,1,1]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3] [3,2,1,1,1,1]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3]=>[1,1,1,1,2,3] [1,1,1,1,2,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4] [1,1,1,1,4,2]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4] [1,1,1,2,1,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4] [1,1,1,2,4,1]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4] [1,1,1,4,1,2]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4] [1,1,1,4,2,1]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4] [1,1,2,1,1,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4] [1,1,2,1,4,1]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4] [1,1,2,4,1,1]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4] [1,1,4,1,1,2]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4] [1,1,4,1,2,1]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4] [1,1,4,2,1,1]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4] [1,2,1,1,1,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4] [1,2,1,1,4,1]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4] [1,2,1,4,1,1]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4] [1,2,4,1,1,1]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4] [1,4,1,1,1,2]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4] [1,4,1,1,2,1]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4] [1,4,1,2,1,1]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4] [1,4,2,1,1,1]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4] [2,1,1,1,1,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4] [2,1,1,1,4,1]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4] [2,1,1,4,1,1]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4] [2,1,4,1,1,1]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4] [2,4,1,1,1,1]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4] [4,1,1,1,1,2]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4] [4,1,1,1,2,1]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4] [4,1,1,2,1,1]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4] [4,1,2,1,1,1]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4] [4,2,1,1,1,1]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4]=>[1,1,1,1,2,4] [1,1,1,1,3,3]=>[1,1,1,1,3,3]=>[1,1,1,1,3,3]=>[1,1,1,1,3,3] [1,1,1,3,1,3]=>[1,1,1,1,3,3]=>[1,1,1,1,3,3]=>[1,1,1,1,3,3] [1,1,1,3,3,1]=>[1,1,1,1,3,3]=>[1,1,1,1,3,3]=>[1,1,1,1,3,3] [1,1,3,1,1,3]=>[1,1,1,1,3,3]=>[1,1,1,1,3,3]=>[1,1,1,1,3,3] [1,1,3,1,3,1]=>[1,1,1,1,3,3]=>[1,1,1,1,3,3]=>[1,1,1,1,3,3] [1,1,3,3,1,1]=>[1,1,1,1,3,3]=>[1,1,1,1,3,3]=>[1,1,1,1,3,3] [1,3,1,1,1,3]=>[1,1,1,1,3,3]=>[1,1,1,1,3,3]=>[1,1,1,1,3,3] [1,3,1,1,3,1]=>[1,1,1,1,3,3]=>[1,1,1,1,3,3]=>[1,1,1,1,3,3] [1,3,1,3,1,1]=>[1,1,1,1,3,3]=>[1,1,1,1,3,3]=>[1,1,1,1,3,3] [1,3,3,1,1,1]=>[1,1,1,1,3,3]=>[1,1,1,1,3,3]=>[1,1,1,1,3,3] [3,1,1,1,1,3]=>[1,1,1,1,3,3]=>[1,1,1,1,3,3]=>[1,1,1,1,3,3] [3,1,1,1,3,1]=>[1,1,1,1,3,3]=>[1,1,1,1,3,3]=>[1,1,1,1,3,3] [3,1,1,3,1,1]=>[1,1,1,1,3,3]=>[1,1,1,1,3,3]=>[1,1,1,1,3,3] [3,1,3,1,1,1]=>[1,1,1,1,3,3]=>[1,1,1,1,3,3]=>[1,1,1,1,3,3] [3,3,1,1,1,1]=>[1,1,1,1,3,3]=>[1,1,1,1,3,3]=>[1,1,1,1,3,3] [1,1,1,2,2,2]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2] [1,1,2,1,2,2]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2] [1,1,2,2,1,2]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2] [1,1,2,2,2,1]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2] [1,2,1,1,2,2]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2] [1,2,1,2,1,2]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2] [1,2,1,2,2,1]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2] [1,2,2,1,1,2]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2] [1,2,2,1,2,1]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2] [1,2,2,2,1,1]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2] [2,1,1,1,2,2]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2] [2,1,1,2,1,2]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2] [2,1,1,2,2,1]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2] [2,1,2,1,1,2]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2] [2,1,2,1,2,1]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2] [2,1,2,2,1,1]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2] [2,2,1,1,1,2]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2] [2,2,1,1,2,1]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2] [2,2,1,2,1,1]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2] [2,2,2,1,1,1]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2]=>[1,1,1,2,2,2] [1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [1,1,1,2,3,2]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [1,1,1,3,2,2]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [1,1,2,1,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [1,1,2,1,3,2]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [1,1,2,2,1,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [1,1,2,2,3,1]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [1,1,2,3,1,2]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [1,1,2,3,2,1]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [1,1,3,1,2,2]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [1,1,3,2,1,2]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [1,1,3,2,2,1]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [1,2,1,1,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [1,2,1,1,3,2]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [1,2,1,2,1,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [1,2,1,2,3,1]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [1,2,1,3,1,2]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [1,2,1,3,2,1]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [1,2,2,1,1,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [1,2,2,1,3,1]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [1,2,2,3,1,1]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [1,2,3,1,1,2]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [1,2,3,1,2,1]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [1,2,3,2,1,1]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [1,3,1,1,2,2]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [1,3,1,2,1,2]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [1,3,1,2,2,1]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [1,3,2,1,1,2]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [1,3,2,1,2,1]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [1,3,2,2,1,1]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [2,1,1,1,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [2,1,1,1,3,2]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [2,1,1,2,1,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [2,1,1,2,3,1]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [2,1,1,3,1,2]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [2,1,1,3,2,1]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [2,1,2,1,1,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [2,1,2,1,3,1]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [2,1,2,3,1,1]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [2,1,3,1,1,2]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [2,1,3,1,2,1]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [2,1,3,2,1,1]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [2,2,1,1,1,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [2,2,1,1,3,1]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [2,2,1,3,1,1]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [2,2,3,1,1,1]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [2,3,1,1,1,2]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [2,3,1,1,2,1]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [2,3,1,2,1,1]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [2,3,2,1,1,1]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [3,1,1,1,2,2]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [3,1,1,2,1,2]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [3,1,1,2,2,1]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [3,1,2,1,1,2]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [3,1,2,1,2,1]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [3,1,2,2,1,1]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [3,2,1,1,1,2]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [3,2,1,1,2,1]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [3,2,1,2,1,1]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [3,2,2,1,1,1]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3]=>[1,1,1,2,2,3] [1,1,2,2,2,2]=>[1,1,2,2,2,2]=>[1,1,2,2,2,2]=>[1,1,2,2,2,2] [1,2,1,2,2,2]=>[1,1,2,2,2,2]=>[1,1,2,2,2,2]=>[1,1,2,2,2,2] [1,2,2,1,2,2]=>[1,1,2,2,2,2]=>[1,1,2,2,2,2]=>[1,1,2,2,2,2] [1,2,2,2,1,2]=>[1,1,2,2,2,2]=>[1,1,2,2,2,2]=>[1,1,2,2,2,2] [1,2,2,2,2,1]=>[1,1,2,2,2,2]=>[1,1,2,2,2,2]=>[1,1,2,2,2,2] [2,1,1,2,2,2]=>[1,1,2,2,2,2]=>[1,1,2,2,2,2]=>[1,1,2,2,2,2] [2,1,2,1,2,2]=>[1,1,2,2,2,2]=>[1,1,2,2,2,2]=>[1,1,2,2,2,2] [2,1,2,2,1,2]=>[1,1,2,2,2,2]=>[1,1,2,2,2,2]=>[1,1,2,2,2,2] [2,1,2,2,2,1]=>[1,1,2,2,2,2]=>[1,1,2,2,2,2]=>[1,1,2,2,2,2] [2,2,1,1,2,2]=>[1,1,2,2,2,2]=>[1,1,2,2,2,2]=>[1,1,2,2,2,2] [2,2,1,2,1,2]=>[1,1,2,2,2,2]=>[1,1,2,2,2,2]=>[1,1,2,2,2,2] [2,2,1,2,2,1]=>[1,1,2,2,2,2]=>[1,1,2,2,2,2]=>[1,1,2,2,2,2] [2,2,2,1,1,2]=>[1,1,2,2,2,2]=>[1,1,2,2,2,2]=>[1,1,2,2,2,2] [2,2,2,1,2,1]=>[1,1,2,2,2,2]=>[1,1,2,2,2,2]=>[1,1,2,2,2,2] [2,2,2,2,1,1]=>[1,1,2,2,2,2]=>[1,1,2,2,2,2]=>[1,1,2,2,2,2]
Map
to non-decreasing parking function
Description
Return the non-decreasing parking function which underlies the parking function.
Sorts the parking function into an increasing sequence.
Map
to composition
Description
Return the parking function interpreted as an integer composition.
Map
Foata bijection
Description
The Foata bijection for compositions.
The Foata bijection $\phi$ is a bijection on the set of words whose letters are positive integers. 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 $w_{i+1} < v_k$.
In either case, place a vertical line at the start of the word as well. Now, within each block between vertical lines, cyclically shift the entries one place to the right.
To compute $\phi([1,4,2,5,3])$, the sequence of words is
  • $1$
  • $|1|4 \to 14$
  • $|14|2 \to 412$
  • $|4|1|2|5 \to 4125$
  • $|4|125|3 \to 45123.$
In total, this gives $\phi([1,4,2,5,3]) = [4,5,1,2,3]$.
This bijection sends the major index St000769The major index of a composition regarded as a word. to the number of inversions St000766The number of inversions of an integer composition..