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Identifier
Mp00314: Integer compositions —Foata bijection⟶ Integer compositions
Images
=>
[1]=>[1] [1,1]=>[1,1] [2]=>[2] [1,1,1]=>[1,1,1] [1,2]=>[1,2] [2,1]=>[2,1] [3]=>[3] [1,1,1,1]=>[1,1,1,1] [1,1,2]=>[1,1,2] [1,2,1]=>[2,1,1] [1,3]=>[1,3] [2,1,1]=>[1,2,1] [2,2]=>[2,2] [3,1]=>[3,1] [4]=>[4] [1,1,1,1,1]=>[1,1,1,1,1] [1,1,1,2]=>[1,1,1,2] [1,1,2,1]=>[2,1,1,1] [1,1,3]=>[1,1,3] [1,2,1,1]=>[1,2,1,1] [1,2,2]=>[1,2,2] [1,3,1]=>[3,1,1] [1,4]=>[1,4] [2,1,1,1]=>[1,1,2,1] [2,1,2]=>[2,1,2] [2,2,1]=>[2,2,1] [2,3]=>[2,3] [3,1,1]=>[1,3,1] [3,2]=>[3,2] [4,1]=>[4,1] [5]=>[5] [1,1,1,1,1,1]=>[1,1,1,1,1,1] [1,1,1,1,2]=>[1,1,1,1,2] [1,1,1,2,1]=>[2,1,1,1,1] [1,1,1,3]=>[1,1,1,3] [1,1,2,1,1]=>[1,2,1,1,1] [1,1,2,2]=>[1,1,2,2] [1,1,3,1]=>[3,1,1,1] [1,1,4]=>[1,1,4] [1,2,1,1,1]=>[1,1,2,1,1] [1,2,1,2]=>[2,1,1,2] [1,2,2,1]=>[2,1,2,1] [1,2,3]=>[1,2,3] [1,3,1,1]=>[1,3,1,1] [1,3,2]=>[3,1,2] [1,4,1]=>[4,1,1] [1,5]=>[1,5] [2,1,1,1,1]=>[1,1,1,2,1] [2,1,1,2]=>[1,2,1,2] [2,1,2,1]=>[2,2,1,1] [2,1,3]=>[2,1,3] [2,2,1,1]=>[1,2,2,1] [2,2,2]=>[2,2,2] [2,3,1]=>[2,3,1] [2,4]=>[2,4] [3,1,1,1]=>[1,1,3,1] [3,1,2]=>[1,3,2] [3,2,1]=>[3,2,1] [3,3]=>[3,3] [4,1,1]=>[1,4,1] [4,2]=>[4,2] [5,1]=>[5,1] [6]=>[6] [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] [1,1,1,1,2,1]=>[2,1,1,1,1,1] [1,1,1,1,3]=>[1,1,1,1,3] [1,1,1,2,1,1]=>[1,2,1,1,1,1] [1,1,1,2,2]=>[1,1,1,2,2] [1,1,1,3,1]=>[3,1,1,1,1] [1,1,1,4]=>[1,1,1,4] [1,1,2,1,1,1]=>[1,1,2,1,1,1] [1,1,2,1,2]=>[2,1,1,1,2] [1,1,2,2,1]=>[2,1,1,2,1] [1,1,2,3]=>[1,1,2,3] [1,1,3,1,1]=>[1,3,1,1,1] [1,1,3,2]=>[3,1,1,2] [1,1,4,1]=>[4,1,1,1] [1,1,5]=>[1,1,5] [1,2,1,1,1,1]=>[1,1,1,2,1,1] [1,2,1,1,2]=>[1,2,1,1,2] [1,2,1,2,1]=>[2,2,1,1,1] [1,2,1,3]=>[2,1,1,3] [1,2,2,1,1]=>[1,2,1,2,1] [1,2,2,2]=>[1,2,2,2] [1,2,3,1]=>[2,1,3,1] [1,2,4]=>[1,2,4] [1,3,1,1,1]=>[1,1,3,1,1] [1,3,1,2]=>[1,3,1,2] [1,3,2,1]=>[3,2,1,1] [1,3,3]=>[1,3,3] [1,4,1,1]=>[1,4,1,1] [1,4,2]=>[4,1,2] [1,5,1]=>[5,1,1] [1,6]=>[1,6] [2,1,1,1,1,1]=>[1,1,1,1,2,1] [2,1,1,1,2]=>[1,1,2,1,2] [2,1,1,2,1]=>[2,1,2,1,1] [2,1,1,3]=>[1,2,1,3] [2,1,2,1,1]=>[1,2,2,1,1] [2,1,2,2]=>[2,1,2,2] [2,1,3,1]=>[2,3,1,1] [2,1,4]=>[2,1,4] [2,2,1,1,1]=>[1,1,2,2,1] [2,2,1,2]=>[2,2,1,2] [2,2,2,1]=>[2,2,2,1] [2,2,3]=>[2,2,3] [2,3,1,1]=>[1,2,3,1] [2,3,2]=>[3,2,2] [2,4,1]=>[2,4,1] [2,5]=>[2,5] [3,1,1,1,1]=>[1,1,1,3,1] [3,1,1,2]=>[1,1,3,2] [3,1,2,1]=>[3,1,2,1] [3,1,3]=>[3,1,3] [3,2,1,1]=>[1,3,2,1] [3,2,2]=>[2,3,2] [3,3,1]=>[3,3,1] [3,4]=>[3,4] [4,1,1,1]=>[1,1,4,1] [4,1,2]=>[1,4,2] [4,2,1]=>[4,2,1] [4,3]=>[4,3] [5,1,1]=>[1,5,1] [5,2]=>[5,2] [6,1]=>[6,1] [7]=>[7] [1,1,1,1,1,1,1,1]=>[1,1,1,1,1,1,1,1] [1,1,1,1,1,1,2]=>[1,1,1,1,1,1,2] [1,1,1,1,1,2,1]=>[2,1,1,1,1,1,1] [1,1,1,1,1,3]=>[1,1,1,1,1,3] [1,1,1,1,2,1,1]=>[1,2,1,1,1,1,1] [1,1,1,1,2,2]=>[1,1,1,1,2,2] [1,1,1,1,3,1]=>[3,1,1,1,1,1] [1,1,1,1,4]=>[1,1,1,1,4] [1,1,1,2,1,1,1]=>[1,1,2,1,1,1,1] [1,1,1,2,1,2]=>[2,1,1,1,1,2] [1,1,1,2,2,1]=>[2,1,1,1,2,1] [1,1,1,2,3]=>[1,1,1,2,3] [1,1,1,3,1,1]=>[1,3,1,1,1,1] [1,1,1,3,2]=>[3,1,1,1,2] [1,1,1,4,1]=>[4,1,1,1,1] [1,1,1,5]=>[1,1,1,5] [1,1,2,1,1,1,1]=>[1,1,1,2,1,1,1] [1,1,2,1,1,2]=>[1,2,1,1,1,2] [1,1,2,1,2,1]=>[2,2,1,1,1,1] [1,1,2,1,3]=>[2,1,1,1,3] [1,1,2,2,1,1]=>[1,2,1,1,2,1] [1,1,2,2,2]=>[1,1,2,2,2] [1,1,2,3,1]=>[2,1,1,3,1] [1,1,2,4]=>[1,1,2,4] [1,1,3,1,1,1]=>[1,1,3,1,1,1] [1,1,3,1,2]=>[1,3,1,1,2] [1,1,3,2,1]=>[3,2,1,1,1] [1,1,3,3]=>[1,1,3,3] [1,1,4,1,1]=>[1,4,1,1,1] [1,1,4,2]=>[4,1,1,2] [1,1,5,1]=>[5,1,1,1] [1,1,6]=>[1,1,6] [1,2,1,1,1,1,1]=>[1,1,1,1,2,1,1] [1,2,1,1,1,2]=>[1,1,2,1,1,2] [1,2,1,1,2,1]=>[2,1,2,1,1,1] [1,2,1,1,3]=>[1,2,1,1,3] [1,2,1,2,1,1]=>[1,2,2,1,1,1] [1,2,1,2,2]=>[2,1,1,2,2] [1,2,1,3,1]=>[2,3,1,1,1] [1,2,1,4]=>[2,1,1,4] [1,2,2,1,1,1]=>[1,1,2,1,2,1] [1,2,2,1,2]=>[2,1,2,1,2] [1,2,2,2,1]=>[2,1,2,2,1] [1,2,2,3]=>[1,2,2,3] [1,2,3,1,1]=>[1,2,1,3,1] [1,2,3,2]=>[3,1,2,2] [1,2,4,1]=>[2,1,4,1] [1,2,5]=>[1,2,5] [1,3,1,1,1,1]=>[1,1,1,3,1,1] [1,3,1,1,2]=>[1,1,3,1,2] [1,3,1,2,1]=>[3,1,2,1,1] [1,3,1,3]=>[3,1,1,3] [1,3,2,1,1]=>[1,3,2,1,1] [1,3,2,2]=>[1,3,2,2] [1,3,3,1]=>[3,1,3,1] [1,3,4]=>[1,3,4] [1,4,1,1,1]=>[1,1,4,1,1] [1,4,1,2]=>[1,4,1,2] [1,4,2,1]=>[4,2,1,1] [1,4,3]=>[4,1,3] [1,5,1,1]=>[1,5,1,1] [1,5,2]=>[5,1,2] [1,6,1]=>[6,1,1] [1,7]=>[1,7] [2,1,1,1,1,1,1]=>[1,1,1,1,1,2,1] [2,1,1,1,1,2]=>[1,1,1,2,1,2] [2,1,1,1,2,1]=>[2,1,1,2,1,1] [2,1,1,1,3]=>[1,1,2,1,3] [2,1,1,2,1,1]=>[1,2,1,2,1,1] [2,1,1,2,2]=>[1,2,1,2,2] [2,1,1,3,1]=>[2,1,3,1,1] [2,1,1,4]=>[1,2,1,4] [2,1,2,1,1,1]=>[1,1,2,2,1,1] [2,1,2,1,2]=>[2,2,1,1,2] [2,1,2,2,1]=>[2,2,1,2,1] [2,1,2,3]=>[2,1,2,3] [2,1,3,1,1]=>[1,2,3,1,1] [2,1,3,2]=>[3,2,1,2] [2,1,4,1]=>[2,4,1,1] [2,1,5]=>[2,1,5] [2,2,1,1,1,1]=>[1,1,1,2,2,1] [2,2,1,1,2]=>[1,2,2,1,2] [2,2,1,2,1]=>[2,2,2,1,1] [2,2,1,3]=>[2,2,1,3] [2,2,2,1,1]=>[1,2,2,2,1] [2,2,2,2]=>[2,2,2,2] [2,2,3,1]=>[2,2,3,1] [2,2,4]=>[2,2,4] [2,3,1,1,1]=>[1,1,2,3,1] [2,3,1,2]=>[2,1,3,2] [2,3,2,1]=>[3,2,2,1] [2,3,3]=>[2,3,3] [2,4,1,1]=>[1,2,4,1] [2,4,2]=>[4,2,2] [2,5,1]=>[2,5,1] [2,6]=>[2,6] [3,1,1,1,1,1]=>[1,1,1,1,3,1] [3,1,1,1,2]=>[1,1,1,3,2] [3,1,1,2,1]=>[3,1,1,2,1] [3,1,1,3]=>[1,3,1,3] [3,1,2,1,1]=>[1,3,1,2,1] [3,1,2,2]=>[1,2,3,2] [3,1,3,1]=>[3,3,1,1] [3,1,4]=>[3,1,4] [3,2,1,1,1]=>[1,1,3,2,1] [3,2,1,2]=>[2,3,1,2] [3,2,2,1]=>[2,3,2,1] [3,2,3]=>[3,2,3] [3,3,1,1]=>[1,3,3,1] [3,3,2]=>[3,3,2] [3,4,1]=>[3,4,1] [3,5]=>[3,5] [4,1,1,1,1]=>[1,1,1,4,1] [4,1,1,2]=>[1,1,4,2] [4,1,2,1]=>[4,1,2,1] [4,1,3]=>[1,4,3] [4,2,1,1]=>[1,4,2,1] [4,2,2]=>[2,4,2] [4,3,1]=>[4,3,1] [4,4]=>[4,4] [5,1,1,1]=>[1,1,5,1] [5,1,2]=>[1,5,2] [5,2,1]=>[5,2,1] [5,3]=>[5,3] [6,1,1]=>[1,6,1] [6,2]=>[6,2] [7,1]=>[7,1] [8]=>[8] [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]=>[1,1,1,1,1,1,1,2] [1,1,1,1,1,1,2,1]=>[2,1,1,1,1,1,1,1] [1,1,1,1,1,1,3]=>[1,1,1,1,1,1,3] [1,1,1,1,1,2,1,1]=>[1,2,1,1,1,1,1,1] [1,1,1,1,1,2,2]=>[1,1,1,1,1,2,2] [1,1,1,1,1,3,1]=>[3,1,1,1,1,1,1] [1,1,1,1,1,4]=>[1,1,1,1,1,4] [1,1,1,1,2,1,1,1]=>[1,1,2,1,1,1,1,1] [1,1,1,1,2,1,2]=>[2,1,1,1,1,1,2] [1,1,1,1,2,2,1]=>[2,1,1,1,1,2,1] [1,1,1,1,2,3]=>[1,1,1,1,2,3] [1,1,1,1,3,1,1]=>[1,3,1,1,1,1,1] [1,1,1,1,3,2]=>[3,1,1,1,1,2] [1,1,1,1,4,1]=>[4,1,1,1,1,1] [1,1,1,1,5]=>[1,1,1,1,5] [1,1,1,2,1,1,1,1]=>[1,1,1,2,1,1,1,1] [1,1,1,2,1,1,2]=>[1,2,1,1,1,1,2] [1,1,1,2,1,2,1]=>[2,2,1,1,1,1,1] [1,1,1,2,1,3]=>[2,1,1,1,1,3] [1,1,1,2,2,1,1]=>[1,2,1,1,1,2,1] [1,1,1,2,2,2]=>[1,1,1,2,2,2] [1,1,1,2,3,1]=>[2,1,1,1,3,1] [1,1,1,2,4]=>[1,1,1,2,4] [1,1,1,3,1,1,1]=>[1,1,3,1,1,1,1] [1,1,1,3,1,2]=>[1,3,1,1,1,2] [1,1,1,3,2,1]=>[3,2,1,1,1,1] [1,1,1,3,3]=>[1,1,1,3,3] [1,1,1,4,1,1]=>[1,4,1,1,1,1] [1,1,1,4,2]=>[4,1,1,1,2] [1,1,1,5,1]=>[5,1,1,1,1] [1,1,1,6]=>[1,1,1,6] [1,1,2,1,1,1,1,1]=>[1,1,1,1,2,1,1,1] [1,1,2,1,1,1,2]=>[1,1,2,1,1,1,2] [1,1,2,1,1,2,1]=>[2,1,2,1,1,1,1] [1,1,2,1,1,3]=>[1,2,1,1,1,3] [1,1,2,1,2,1,1]=>[1,2,2,1,1,1,1] [1,1,2,1,2,2]=>[2,1,1,1,2,2] [1,1,2,1,3,1]=>[2,3,1,1,1,1] [1,1,2,1,4]=>[2,1,1,1,4] [1,1,2,2,1,1,1]=>[1,1,2,1,1,2,1] [1,1,2,2,1,2]=>[2,1,1,2,1,2] [1,1,2,2,2,1]=>[2,1,1,2,2,1] [1,1,2,2,3]=>[1,1,2,2,3] [1,1,2,3,1,1]=>[1,2,1,1,3,1] [1,1,2,3,2]=>[3,1,1,2,2] [1,1,2,4,1]=>[2,1,1,4,1] [1,1,2,5]=>[1,1,2,5] [1,1,3,1,1,1,1]=>[1,1,1,3,1,1,1] [1,1,3,1,1,2]=>[1,1,3,1,1,2] [1,1,3,1,2,1]=>[3,1,2,1,1,1] [1,1,3,1,3]=>[3,1,1,1,3] [1,1,3,2,1,1]=>[1,3,2,1,1,1] [1,1,3,2,2]=>[1,3,1,2,2] [1,1,3,3,1]=>[3,1,1,3,1] [1,1,3,4]=>[1,1,3,4] [1,1,4,1,1,1]=>[1,1,4,1,1,1] [1,1,4,1,2]=>[1,4,1,1,2] [1,1,4,2,1]=>[4,2,1,1,1] [1,1,4,3]=>[4,1,1,3] [1,1,5,1,1]=>[1,5,1,1,1] [1,1,5,2]=>[5,1,1,2] [1,1,6,1]=>[6,1,1,1] [1,1,7]=>[1,1,7] [1,2,1,1,1,1,1,1]=>[1,1,1,1,1,2,1,1] [1,2,1,1,1,1,2]=>[1,1,1,2,1,1,2] [1,2,1,1,1,2,1]=>[2,1,1,2,1,1,1] [1,2,1,1,1,3]=>[1,1,2,1,1,3] [1,2,1,1,2,1,1]=>[1,2,1,2,1,1,1] [1,2,1,1,2,2]=>[1,2,1,1,2,2] [1,2,1,1,3,1]=>[2,1,3,1,1,1] [1,2,1,1,4]=>[1,2,1,1,4] [1,2,1,2,1,1,1]=>[1,1,2,2,1,1,1] [1,2,1,2,1,2]=>[2,2,1,1,1,2] [1,2,1,2,2,1]=>[2,2,1,1,2,1] [1,2,1,2,3]=>[2,1,1,2,3] [1,2,1,3,1,1]=>[1,2,3,1,1,1] [1,2,1,3,2]=>[3,2,1,1,2] [1,2,1,4,1]=>[2,4,1,1,1] [1,2,1,5]=>[2,1,1,5] [1,2,2,1,1,1,1]=>[1,1,1,2,1,2,1] [1,2,2,1,1,2]=>[1,2,1,2,1,2] [1,2,2,1,2,1]=>[2,2,1,2,1,1] [1,2,2,1,3]=>[2,1,2,1,3] [1,2,2,2,1,1]=>[1,2,1,2,2,1] [1,2,2,2,2]=>[1,2,2,2,2] [1,2,2,3,1]=>[2,1,2,3,1] [1,2,2,4]=>[1,2,2,4] [1,2,3,1,1,1]=>[1,1,2,1,3,1] [1,2,3,1,2]=>[2,1,1,3,2] [1,2,3,2,1]=>[3,2,1,2,1] [1,2,3,3]=>[1,2,3,3] [1,2,4,1,1]=>[1,2,1,4,1] [1,2,4,2]=>[4,1,2,2] [1,2,5,1]=>[2,1,5,1] [1,2,6]=>[1,2,6] [1,3,1,1,1,1,1]=>[1,1,1,1,3,1,1] [1,3,1,1,1,2]=>[1,1,1,3,1,2] [1,3,1,1,2,1]=>[3,1,1,2,1,1] [1,3,1,1,3]=>[1,3,1,1,3] [1,3,1,2,1,1]=>[1,3,1,2,1,1] [1,3,1,2,2]=>[1,1,3,2,2] [1,3,1,3,1]=>[3,3,1,1,1] [1,3,1,4]=>[3,1,1,4] [1,3,2,1,1,1]=>[1,1,3,2,1,1] [1,3,2,1,2]=>[2,3,1,1,2] [1,3,2,2,1]=>[3,1,2,2,1] [1,3,2,3]=>[3,1,2,3] [1,3,3,1,1]=>[1,3,1,3,1] [1,3,3,2]=>[3,1,3,2] [1,3,4,1]=>[3,1,4,1] [1,3,5]=>[1,3,5] [1,4,1,1,1,1]=>[1,1,1,4,1,1] [1,4,1,1,2]=>[1,1,4,1,2] [1,4,1,2,1]=>[4,1,2,1,1] [1,4,1,3]=>[1,4,1,3] [1,4,2,1,1]=>[1,4,2,1,1] [1,4,2,2]=>[1,4,2,2] [1,4,3,1]=>[4,3,1,1] [1,4,4]=>[1,4,4] [1,5,1,1,1]=>[1,1,5,1,1] [1,5,1,2]=>[1,5,1,2] [1,5,2,1]=>[5,2,1,1] [1,5,3]=>[5,1,3] [1,6,1,1]=>[1,6,1,1] [1,6,2]=>[6,1,2] [1,7,1]=>[7,1,1] [1,8]=>[1,8] [2,1,1,1,1,1,1,1]=>[1,1,1,1,1,1,2,1] [2,1,1,1,1,1,2]=>[1,1,1,1,2,1,2] [2,1,1,1,1,2,1]=>[2,1,1,1,2,1,1] [2,1,1,1,1,3]=>[1,1,1,2,1,3] [2,1,1,1,2,1,1]=>[1,2,1,1,2,1,1] [2,1,1,1,2,2]=>[1,1,2,1,2,2] [2,1,1,1,3,1]=>[2,1,1,3,1,1] [2,1,1,1,4]=>[1,1,2,1,4] [2,1,1,2,1,1,1]=>[1,1,2,1,2,1,1] [2,1,1,2,1,2]=>[2,1,2,1,1,2] [2,1,1,2,2,1]=>[2,1,2,1,2,1] [2,1,1,2,3]=>[1,2,1,2,3] [2,1,1,3,1,1]=>[1,2,1,3,1,1] [2,1,1,3,2]=>[3,1,2,1,2] [2,1,1,4,1]=>[2,1,4,1,1] [2,1,1,5]=>[1,2,1,5] [2,1,2,1,1,1,1]=>[1,1,1,2,2,1,1] [2,1,2,1,1,2]=>[1,2,2,1,1,2] [2,1,2,1,2,1]=>[2,2,2,1,1,1] [2,1,2,1,3]=>[2,2,1,1,3] [2,1,2,2,1,1]=>[1,2,2,1,2,1] [2,1,2,2,2]=>[2,1,2,2,2] [2,1,2,3,1]=>[2,2,1,3,1] [2,1,2,4]=>[2,1,2,4] [2,1,3,1,1,1]=>[1,1,2,3,1,1] [2,1,3,1,2]=>[2,1,3,1,2] [2,1,3,2,1]=>[3,2,2,1,1] [2,1,3,3]=>[2,1,3,3] [2,1,4,1,1]=>[1,2,4,1,1] [2,1,4,2]=>[4,2,1,2] [2,1,5,1]=>[2,5,1,1] [2,1,6]=>[2,1,6] [2,2,1,1,1,1,1]=>[1,1,1,1,2,2,1] [2,2,1,1,1,2]=>[1,1,2,2,1,2] [2,2,1,1,2,1]=>[2,1,2,2,1,1] [2,2,1,1,3]=>[1,2,2,1,3] [2,2,1,2,1,1]=>[1,2,2,2,1,1] [2,2,1,2,2]=>[2,2,1,2,2] [2,2,1,3,1]=>[2,2,3,1,1] [2,2,1,4]=>[2,2,1,4] [2,2,2,1,1,1]=>[1,1,2,2,2,1] [2,2,2,1,2]=>[2,2,2,1,2] [2,2,2,2,1]=>[2,2,2,2,1] [2,2,2,3]=>[2,2,2,3] [2,2,3,1,1]=>[1,2,2,3,1] [2,2,3,2]=>[3,2,2,2] [2,2,4,1]=>[2,2,4,1] [2,2,5]=>[2,2,5] [2,3,1,1,1,1]=>[1,1,1,2,3,1] [2,3,1,1,2]=>[1,2,1,3,2] [2,3,1,2,1]=>[2,3,1,2,1] [2,3,1,3]=>[2,3,1,3] [2,3,2,1,1]=>[1,3,2,2,1] [2,3,2,2]=>[2,3,2,2] [2,3,3,1]=>[2,3,3,1] [2,3,4]=>[2,3,4] [2,4,1,1,1]=>[1,1,2,4,1] [2,4,1,2]=>[2,1,4,2] [2,4,2,1]=>[4,2,2,1] [2,4,3]=>[4,2,3] [2,5,1,1]=>[1,2,5,1] [2,5,2]=>[5,2,2] [2,6,1]=>[2,6,1] [2,7]=>[2,7] [3,1,1,1,1,1,1]=>[1,1,1,1,1,3,1] [3,1,1,1,1,2]=>[1,1,1,1,3,2] [3,1,1,1,2,1]=>[3,1,1,1,2,1] [3,1,1,1,3]=>[1,1,3,1,3] [3,1,1,2,1,1]=>[1,3,1,1,2,1] [3,1,1,2,2]=>[1,1,2,3,2] [3,1,1,3,1]=>[3,1,3,1,1] [3,1,1,4]=>[1,3,1,4] [3,1,2,1,1,1]=>[1,1,3,1,2,1] [3,1,2,1,2]=>[1,3,2,1,2] [3,1,2,2,1]=>[2,1,3,2,1] [3,1,2,3]=>[1,3,2,3] [3,1,3,1,1]=>[1,3,3,1,1] [3,1,3,2]=>[3,3,1,2] [3,1,4,1]=>[3,4,1,1] [3,1,5]=>[3,1,5] [3,2,1,1,1,1]=>[1,1,1,3,2,1] [3,2,1,1,2]=>[1,2,3,1,2] [3,2,1,2,1]=>[2,3,2,1,1] [3,2,1,3]=>[3,2,1,3] [3,2,2,1,1]=>[1,2,3,2,1] [3,2,2,2]=>[2,2,3,2] [3,2,3,1]=>[3,2,3,1] [3,2,4]=>[3,2,4] [3,3,1,1,1]=>[1,1,3,3,1] [3,3,1,2]=>[1,3,3,2] [3,3,2,1]=>[3,3,2,1] [3,3,3]=>[3,3,3] [3,4,1,1]=>[1,3,4,1] [3,4,2]=>[3,4,2] [3,5,1]=>[3,5,1] [3,6]=>[3,6] [4,1,1,1,1,1]=>[1,1,1,1,4,1] [4,1,1,1,2]=>[1,1,1,4,2] [4,1,1,2,1]=>[4,1,1,2,1] [4,1,1,3]=>[1,1,4,3] [4,1,2,1,1]=>[1,4,1,2,1] [4,1,2,2]=>[1,2,4,2] [4,1,3,1]=>[4,1,3,1] [4,1,4]=>[4,1,4] [4,2,1,1,1]=>[1,1,4,2,1] [4,2,1,2]=>[2,4,1,2] [4,2,2,1]=>[2,4,2,1] [4,2,3]=>[2,4,3] [4,3,1,1]=>[1,4,3,1] [4,3,2]=>[4,3,2] [4,4,1]=>[4,4,1] [4,5]=>[4,5] [5,1,1,1,1]=>[1,1,1,5,1] [5,1,1,2]=>[1,1,5,2] [5,1,2,1]=>[5,1,2,1] [5,1,3]=>[1,5,3] [5,2,1,1]=>[1,5,2,1] [5,2,2]=>[2,5,2] [5,3,1]=>[5,3,1] [5,4]=>[5,4] [6,1,1,1]=>[1,1,6,1] [6,1,2]=>[1,6,2] [6,2,1]=>[6,2,1] [6,3]=>[6,3] [7,1,1]=>[1,7,1] [7,2]=>[7,2] [8,1]=>[8,1] [9]=>[9] [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]=>[1,1,1,1,1,1,1,1,2] [1,1,1,1,1,1,1,2,1]=>[2,1,1,1,1,1,1,1,1] [1,1,1,1,1,1,1,3]=>[1,1,1,1,1,1,1,3] [1,1,1,1,1,1,2,1,1]=>[1,2,1,1,1,1,1,1,1] [1,1,1,1,1,1,2,2]=>[1,1,1,1,1,1,2,2] [1,1,1,1,1,1,3,1]=>[3,1,1,1,1,1,1,1] [1,1,1,1,1,1,4]=>[1,1,1,1,1,1,4] [1,1,1,1,1,2,1,1,1]=>[1,1,2,1,1,1,1,1,1] [1,1,1,1,1,2,1,2]=>[2,1,1,1,1,1,1,2] [1,1,1,1,1,2,2,1]=>[2,1,1,1,1,1,2,1] [1,1,1,1,1,2,3]=>[1,1,1,1,1,2,3] [1,1,1,1,1,3,1,1]=>[1,3,1,1,1,1,1,1] [1,1,1,1,1,3,2]=>[3,1,1,1,1,1,2] [1,1,1,1,1,4,1]=>[4,1,1,1,1,1,1] [1,1,1,1,1,5]=>[1,1,1,1,1,5] [1,1,1,1,2,1,1,1,1]=>[1,1,1,2,1,1,1,1,1] [1,1,1,1,2,1,1,2]=>[1,2,1,1,1,1,1,2] [1,1,1,1,2,1,2,1]=>[2,2,1,1,1,1,1,1] [1,1,1,1,2,1,3]=>[2,1,1,1,1,1,3] [1,1,1,1,2,2,1,1]=>[1,2,1,1,1,1,2,1] [1,1,1,1,2,2,2]=>[1,1,1,1,2,2,2] [1,1,1,1,2,3,1]=>[2,1,1,1,1,3,1] [1,1,1,1,2,4]=>[1,1,1,1,2,4] [1,1,1,1,3,1,1,1]=>[1,1,3,1,1,1,1,1] [1,1,1,1,3,1,2]=>[1,3,1,1,1,1,2] [1,1,1,1,3,2,1]=>[3,2,1,1,1,1,1] [1,1,1,1,3,3]=>[1,1,1,1,3,3] [1,1,1,1,4,1,1]=>[1,4,1,1,1,1,1] [1,1,1,1,4,2]=>[4,1,1,1,1,2] [1,1,1,1,5,1]=>[5,1,1,1,1,1] [1,1,1,1,6]=>[1,1,1,1,6] [1,1,1,2,1,1,1,1,1]=>[1,1,1,1,2,1,1,1,1] [1,1,1,2,1,1,1,2]=>[1,1,2,1,1,1,1,2] [1,1,1,2,1,1,2,1]=>[2,1,2,1,1,1,1,1] [1,1,1,2,1,1,3]=>[1,2,1,1,1,1,3] [1,1,1,2,1,2,1,1]=>[1,2,2,1,1,1,1,1] [1,1,1,2,1,2,2]=>[2,1,1,1,1,2,2] [1,1,1,2,1,3,1]=>[2,3,1,1,1,1,1] [1,1,1,2,1,4]=>[2,1,1,1,1,4] [1,1,1,2,2,1,1,1]=>[1,1,2,1,1,1,2,1] [1,1,1,2,2,1,2]=>[2,1,1,1,2,1,2] [1,1,1,2,2,2,1]=>[2,1,1,1,2,2,1] [1,1,1,2,2,3]=>[1,1,1,2,2,3] [1,1,1,2,3,1,1]=>[1,2,1,1,1,3,1] [1,1,1,2,3,2]=>[3,1,1,1,2,2] [1,1,1,2,4,1]=>[2,1,1,1,4,1] [1,1,1,2,5]=>[1,1,1,2,5] [1,1,1,3,1,1,1,1]=>[1,1,1,3,1,1,1,1] [1,1,1,3,1,1,2]=>[1,1,3,1,1,1,2] [1,1,1,3,1,2,1]=>[3,1,2,1,1,1,1] [1,1,1,3,1,3]=>[3,1,1,1,1,3] [1,1,1,3,2,1,1]=>[1,3,2,1,1,1,1] [1,1,1,3,2,2]=>[1,3,1,1,2,2] [1,1,1,3,3,1]=>[3,1,1,1,3,1] [1,1,1,3,4]=>[1,1,1,3,4] [1,1,1,4,1,1,1]=>[1,1,4,1,1,1,1] [1,1,1,4,1,2]=>[1,4,1,1,1,2] [1,1,1,4,2,1]=>[4,2,1,1,1,1] [1,1,1,4,3]=>[4,1,1,1,3] [1,1,1,5,1,1]=>[1,5,1,1,1,1] [1,1,1,5,2]=>[5,1,1,1,2] [1,1,1,6,1]=>[6,1,1,1,1] [1,1,1,7]=>[1,1,1,7] [1,1,2,1,1,1,1,1,1]=>[1,1,1,1,1,2,1,1,1] [1,1,2,1,1,1,1,2]=>[1,1,1,2,1,1,1,2] [1,1,2,1,1,1,2,1]=>[2,1,1,2,1,1,1,1] [1,1,2,1,1,1,3]=>[1,1,2,1,1,1,3] [1,1,2,1,1,2,1,1]=>[1,2,1,2,1,1,1,1] [1,1,2,1,1,2,2]=>[1,2,1,1,1,2,2] [1,1,2,1,1,3,1]=>[2,1,3,1,1,1,1] [1,1,2,1,1,4]=>[1,2,1,1,1,4] [1,1,2,1,2,1,1,1]=>[1,1,2,2,1,1,1,1] [1,1,2,1,2,1,2]=>[2,2,1,1,1,1,2] [1,1,2,1,2,2,1]=>[2,2,1,1,1,2,1] [1,1,2,1,2,3]=>[2,1,1,1,2,3] [1,1,2,1,3,1,1]=>[1,2,3,1,1,1,1] [1,1,2,1,3,2]=>[3,2,1,1,1,2] [1,1,2,1,4,1]=>[2,4,1,1,1,1] [1,1,2,1,5]=>[2,1,1,1,5] [1,1,2,2,1,1,1,1]=>[1,1,1,2,1,1,2,1] [1,1,2,2,1,1,2]=>[1,2,1,1,2,1,2] [1,1,2,2,1,2,1]=>[2,2,1,1,2,1,1] [1,1,2,2,1,3]=>[2,1,1,2,1,3] [1,1,2,2,2,1,1]=>[1,2,1,1,2,2,1] [1,1,2,2,2,2]=>[1,1,2,2,2,2] [1,1,2,2,3,1]=>[2,1,1,2,3,1] [1,1,2,2,4]=>[1,1,2,2,4] [1,1,2,3,1,1,1]=>[1,1,2,1,1,3,1] [1,1,2,3,1,2]=>[2,1,1,1,3,2] [1,1,2,3,2,1]=>[3,2,1,1,2,1] [1,1,2,3,3]=>[1,1,2,3,3] [1,1,2,4,1,1]=>[1,2,1,1,4,1] [1,1,2,4,2]=>[4,1,1,2,2] [1,1,2,5,1]=>[2,1,1,5,1] [1,1,2,6]=>[1,1,2,6] [1,1,3,1,1,1,1,1]=>[1,1,1,1,3,1,1,1] [1,1,3,1,1,1,2]=>[1,1,1,3,1,1,2] [1,1,3,1,1,2,1]=>[3,1,1,2,1,1,1] [1,1,3,1,1,3]=>[1,3,1,1,1,3] [1,1,3,1,2,1,1]=>[1,3,1,2,1,1,1] [1,1,3,1,2,2]=>[1,1,3,1,2,2] [1,1,3,1,3,1]=>[3,3,1,1,1,1] [1,1,3,1,4]=>[3,1,1,1,4] [1,1,3,2,1,1,1]=>[1,1,3,2,1,1,1] [1,1,3,2,1,2]=>[2,3,1,1,1,2] [1,1,3,2,2,1]=>[3,1,2,1,2,1] [1,1,3,2,3]=>[3,1,1,2,3] [1,1,3,3,1,1]=>[1,3,1,1,3,1] [1,1,3,3,2]=>[3,1,1,3,2] [1,1,3,4,1]=>[3,1,1,4,1] [1,1,3,5]=>[1,1,3,5] [1,1,4,1,1,1,1]=>[1,1,1,4,1,1,1] [1,1,4,1,1,2]=>[1,1,4,1,1,2] [1,1,4,1,2,1]=>[4,1,2,1,1,1] [1,1,4,1,3]=>[1,4,1,1,3] [1,1,4,2,1,1]=>[1,4,2,1,1,1] [1,1,4,2,2]=>[1,4,1,2,2] [1,1,4,3,1]=>[4,3,1,1,1] [1,1,4,4]=>[1,1,4,4] [1,1,5,1,1,1]=>[1,1,5,1,1,1] [1,1,5,1,2]=>[1,5,1,1,2] [1,1,5,2,1]=>[5,2,1,1,1] [1,1,5,3]=>[5,1,1,3] [1,1,6,1,1]=>[1,6,1,1,1] [1,1,6,2]=>[6,1,1,2] [1,1,7,1]=>[7,1,1,1] [1,1,8]=>[1,1,8] [1,2,1,1,1,1,1,1,1]=>[1,1,1,1,1,1,2,1,1] [1,2,1,1,1,1,1,2]=>[1,1,1,1,2,1,1,2] [1,2,1,1,1,1,2,1]=>[2,1,1,1,2,1,1,1] [1,2,1,1,1,1,3]=>[1,1,1,2,1,1,3] [1,2,1,1,1,2,1,1]=>[1,2,1,1,2,1,1,1] [1,2,1,1,1,2,2]=>[1,1,2,1,1,2,2] [1,2,1,1,1,3,1]=>[2,1,1,3,1,1,1] [1,2,1,1,1,4]=>[1,1,2,1,1,4] [1,2,1,1,2,1,1,1]=>[1,1,2,1,2,1,1,1] [1,2,1,1,2,1,2]=>[2,1,2,1,1,1,2] [1,2,1,1,2,2,1]=>[2,1,2,1,1,2,1] [1,2,1,1,2,3]=>[1,2,1,1,2,3] [1,2,1,1,3,1,1]=>[1,2,1,3,1,1,1] [1,2,1,1,3,2]=>[3,1,2,1,1,2] [1,2,1,1,4,1]=>[2,1,4,1,1,1] [1,2,1,1,5]=>[1,2,1,1,5] [1,2,1,2,1,1,1,1]=>[1,1,1,2,2,1,1,1] [1,2,1,2,1,1,2]=>[1,2,2,1,1,1,2] [1,2,1,2,1,2,1]=>[2,2,2,1,1,1,1] [1,2,1,2,1,3]=>[2,2,1,1,1,3] [1,2,1,2,2,1,1]=>[1,2,2,1,1,2,1] [1,2,1,2,2,2]=>[2,1,1,2,2,2] [1,2,1,2,3,1]=>[2,2,1,1,3,1] [1,2,1,2,4]=>[2,1,1,2,4] [1,2,1,3,1,1,1]=>[1,1,2,3,1,1,1] [1,2,1,3,1,2]=>[2,1,3,1,1,2] [1,2,1,3,2,1]=>[3,2,2,1,1,1] [1,2,1,3,3]=>[2,1,1,3,3] [1,2,1,4,1,1]=>[1,2,4,1,1,1] [1,2,1,4,2]=>[4,2,1,1,2] [1,2,1,5,1]=>[2,5,1,1,1] [1,2,1,6]=>[2,1,1,6] [1,2,2,1,1,1,1,1]=>[1,1,1,1,2,1,2,1] [1,2,2,1,1,1,2]=>[1,1,2,1,2,1,2] [1,2,2,1,1,2,1]=>[2,1,2,1,2,1,1] [1,2,2,1,1,3]=>[1,2,1,2,1,3] [1,2,2,1,2,1,1]=>[1,2,2,1,2,1,1] [1,2,2,1,2,2]=>[2,1,2,1,2,2] [1,2,2,1,3,1]=>[2,2,1,3,1,1] [1,2,2,1,4]=>[2,1,2,1,4] [1,2,2,2,1,1,1]=>[1,1,2,1,2,2,1] [1,2,2,2,1,2]=>[2,1,2,2,1,2] [1,2,2,2,2,1]=>[2,1,2,2,2,1] [1,2,2,2,3]=>[1,2,2,2,3] [1,2,2,3,1,1]=>[1,2,1,2,3,1] [1,2,2,3,2]=>[3,1,2,2,2] [1,2,2,4,1]=>[2,1,2,4,1] [1,2,2,5]=>[1,2,2,5] [1,2,3,1,1,1,1]=>[1,1,1,2,1,3,1] [1,2,3,1,1,2]=>[1,2,1,1,3,2] [1,2,3,1,2,1]=>[2,3,1,1,2,1] [1,2,3,1,3]=>[2,1,3,1,3] [1,2,3,2,1,1]=>[1,3,2,1,2,1] [1,2,3,2,2]=>[1,3,2,2,2] [1,2,3,3,1]=>[2,1,3,3,1] [1,2,3,4]=>[1,2,3,4] [1,2,4,1,1,1]=>[1,1,2,1,4,1] [1,2,4,1,2]=>[2,1,1,4,2] [1,2,4,2,1]=>[4,2,1,2,1] [1,2,4,3]=>[4,1,2,3] [1,2,5,1,1]=>[1,2,1,5,1] [1,2,5,2]=>[5,1,2,2] [1,2,6,1]=>[2,1,6,1] [1,2,7]=>[1,2,7] [1,3,1,1,1,1,1,1]=>[1,1,1,1,1,3,1,1] [1,3,1,1,1,1,2]=>[1,1,1,1,3,1,2] [1,3,1,1,1,2,1]=>[3,1,1,1,2,1,1] [1,3,1,1,1,3]=>[1,1,3,1,1,3] [1,3,1,1,2,1,1]=>[1,3,1,1,2,1,1] [1,3,1,1,2,2]=>[1,1,1,3,2,2] [1,3,1,1,3,1]=>[3,1,3,1,1,1] [1,3,1,1,4]=>[1,3,1,1,4] [1,3,1,2,1,1,1]=>[1,1,3,1,2,1,1] [1,3,1,2,1,2]=>[1,3,2,1,1,2] [1,3,1,2,2,1]=>[3,1,1,2,2,1] [1,3,1,2,3]=>[1,3,1,2,3] [1,3,1,3,1,1]=>[1,3,3,1,1,1] [1,3,1,3,2]=>[3,3,1,1,2] [1,3,1,4,1]=>[3,4,1,1,1] [1,3,1,5]=>[3,1,1,5] [1,3,2,1,1,1,1]=>[1,1,1,3,2,1,1] [1,3,2,1,1,2]=>[1,2,3,1,1,2] [1,3,2,1,2,1]=>[2,3,2,1,1,1] [1,3,2,1,3]=>[3,2,1,1,3] [1,3,2,2,1,1]=>[1,3,1,2,2,1] [1,3,2,2,2]=>[1,2,3,2,2] [1,3,2,3,1]=>[3,2,1,3,1] [1,3,2,4]=>[3,1,2,4] [1,3,3,1,1,1]=>[1,1,3,1,3,1] [1,3,3,1,2]=>[1,3,1,3,2] [1,3,3,2,1]=>[3,3,1,2,1] [1,3,3,3]=>[1,3,3,3] [1,3,4,1,1]=>[1,3,1,4,1] [1,3,4,2]=>[3,1,4,2] [1,3,5,1]=>[3,1,5,1] [1,3,6]=>[1,3,6] [1,4,1,1,1,1,1]=>[1,1,1,1,4,1,1] [1,4,1,1,1,2]=>[1,1,1,4,1,2] [1,4,1,1,2,1]=>[4,1,1,2,1,1] [1,4,1,1,3]=>[1,1,4,1,3] [1,4,1,2,1,1]=>[1,4,1,2,1,1] [1,4,1,2,2]=>[1,1,4,2,2] [1,4,1,3,1]=>[4,1,3,1,1] [1,4,1,4]=>[4,1,1,4] [1,4,2,1,1,1]=>[1,1,4,2,1,1] [1,4,2,1,2]=>[2,4,1,1,2] [1,4,2,2,1]=>[4,1,2,2,1] [1,4,2,3]=>[1,4,2,3] [1,4,3,1,1]=>[1,4,3,1,1] [1,4,3,2]=>[4,3,1,2] [1,4,4,1]=>[4,1,4,1] [1,4,5]=>[1,4,5] [1,5,1,1,1,1]=>[1,1,1,5,1,1] [1,5,1,1,2]=>[1,1,5,1,2] [1,5,1,2,1]=>[5,1,2,1,1] [1,5,1,3]=>[1,5,1,3] [1,5,2,1,1]=>[1,5,2,1,1] [1,5,2,2]=>[1,5,2,2] [1,5,3,1]=>[5,3,1,1] [1,5,4]=>[5,1,4] [1,6,1,1,1]=>[1,1,6,1,1] [1,6,1,2]=>[1,6,1,2] [1,6,2,1]=>[6,2,1,1] [1,6,3]=>[6,1,3] [1,7,1,1]=>[1,7,1,1] [1,7,2]=>[7,1,2] [1,8,1]=>[8,1,1] [1,9]=>[1,9] [2,1,1,1,1,1,1,1,1]=>[1,1,1,1,1,1,1,2,1] [2,1,1,1,1,1,1,2]=>[1,1,1,1,1,2,1,2] [2,1,1,1,1,1,2,1]=>[2,1,1,1,1,2,1,1] [2,1,1,1,1,1,3]=>[1,1,1,1,2,1,3] [2,1,1,1,1,2,1,1]=>[1,2,1,1,1,2,1,1] [2,1,1,1,1,2,2]=>[1,1,1,2,1,2,2] [2,1,1,1,1,3,1]=>[2,1,1,1,3,1,1] [2,1,1,1,1,4]=>[1,1,1,2,1,4] [2,1,1,1,2,1,1,1]=>[1,1,2,1,1,2,1,1] [2,1,1,1,2,1,2]=>[2,1,1,2,1,1,2] [2,1,1,1,2,2,1]=>[2,1,1,2,1,2,1] [2,1,1,1,2,3]=>[1,1,2,1,2,3] [2,1,1,1,3,1,1]=>[1,2,1,1,3,1,1] [2,1,1,1,3,2]=>[3,1,1,2,1,2] [2,1,1,1,4,1]=>[2,1,1,4,1,1] [2,1,1,1,5]=>[1,1,2,1,5] [2,1,1,2,1,1,1,1]=>[1,1,1,2,1,2,1,1] [2,1,1,2,1,1,2]=>[1,2,1,2,1,1,2] [2,1,1,2,1,2,1]=>[2,2,1,2,1,1,1] [2,1,1,2,1,3]=>[2,1,2,1,1,3] [2,1,1,2,2,1,1]=>[1,2,1,2,1,2,1] [2,1,1,2,2,2]=>[1,2,1,2,2,2] [2,1,1,2,3,1]=>[2,1,2,1,3,1] [2,1,1,2,4]=>[1,2,1,2,4] [2,1,1,3,1,1,1]=>[1,1,2,1,3,1,1] [2,1,1,3,1,2]=>[2,1,1,3,1,2] [2,1,1,3,2,1]=>[3,2,1,2,1,1] [2,1,1,3,3]=>[1,2,1,3,3] [2,1,1,4,1,1]=>[1,2,1,4,1,1] [2,1,1,4,2]=>[4,1,2,1,2] [2,1,1,5,1]=>[2,1,5,1,1] [2,1,1,6]=>[1,2,1,6] [2,1,2,1,1,1,1,1]=>[1,1,1,1,2,2,1,1] [2,1,2,1,1,1,2]=>[1,1,2,2,1,1,2] [2,1,2,1,1,2,1]=>[2,1,2,2,1,1,1] [2,1,2,1,1,3]=>[1,2,2,1,1,3] [2,1,2,1,2,1,1]=>[1,2,2,2,1,1,1] [2,1,2,1,2,2]=>[2,2,1,1,2,2] [2,1,2,1,3,1]=>[2,2,3,1,1,1] [2,1,2,1,4]=>[2,2,1,1,4] [2,1,2,2,1,1,1]=>[1,1,2,2,1,2,1] [2,1,2,2,1,2]=>[2,2,1,2,1,2] [2,1,2,2,2,1]=>[2,2,1,2,2,1] [2,1,2,2,3]=>[2,1,2,2,3] [2,1,2,3,1,1]=>[1,2,2,1,3,1] [2,1,2,3,2]=>[3,2,1,2,2] [2,1,2,4,1]=>[2,2,1,4,1] [2,1,2,5]=>[2,1,2,5] [2,1,3,1,1,1,1]=>[1,1,1,2,3,1,1] [2,1,3,1,1,2]=>[1,2,1,3,1,2] [2,1,3,1,2,1]=>[2,3,1,2,1,1] [2,1,3,1,3]=>[2,3,1,1,3] [2,1,3,2,1,1]=>[1,3,2,2,1,1] [2,1,3,2,2]=>[2,3,1,2,2] [2,1,3,3,1]=>[2,3,1,3,1] [2,1,3,4]=>[2,1,3,4] [2,1,4,1,1,1]=>[1,1,2,4,1,1] [2,1,4,1,2]=>[2,1,4,1,2] [2,1,4,2,1]=>[4,2,2,1,1] [2,1,4,3]=>[4,2,1,3] [2,1,5,1,1]=>[1,2,5,1,1] [2,1,5,2]=>[5,2,1,2] [2,1,6,1]=>[2,6,1,1] [2,1,7]=>[2,1,7] [2,2,1,1,1,1,1,1]=>[1,1,1,1,1,2,2,1] [2,2,1,1,1,1,2]=>[1,1,1,2,2,1,2] [2,2,1,1,1,2,1]=>[2,1,1,2,2,1,1] [2,2,1,1,1,3]=>[1,1,2,2,1,3] [2,2,1,1,2,1,1]=>[1,2,1,2,2,1,1] [2,2,1,1,2,2]=>[1,2,2,1,2,2] [2,2,1,1,3,1]=>[2,1,2,3,1,1] [2,2,1,1,4]=>[1,2,2,1,4] [2,2,1,2,1,1,1]=>[1,1,2,2,2,1,1] [2,2,1,2,1,2]=>[2,2,2,1,1,2] [2,2,1,2,2,1]=>[2,2,2,1,2,1] [2,2,1,2,3]=>[2,2,1,2,3] [2,2,1,3,1,1]=>[1,2,2,3,1,1] [2,2,1,3,2]=>[3,2,2,1,2] [2,2,1,4,1]=>[2,2,4,1,1] [2,2,1,5]=>[2,2,1,5] [2,2,2,1,1,1,1]=>[1,1,1,2,2,2,1] [2,2,2,1,1,2]=>[1,2,2,2,1,2] [2,2,2,1,2,1]=>[2,2,2,2,1,1] [2,2,2,1,3]=>[2,2,2,1,3] [2,2,2,2,1,1]=>[1,2,2,2,2,1] [2,2,2,2,2]=>[2,2,2,2,2] [2,2,2,3,1]=>[2,2,2,3,1] [2,2,2,4]=>[2,2,2,4] [2,2,3,1,1,1]=>[1,1,2,2,3,1] [2,2,3,1,2]=>[2,2,1,3,2] [2,2,3,2,1]=>[3,2,2,2,1] [2,2,3,3]=>[2,2,3,3] [2,2,4,1,1]=>[1,2,2,4,1] [2,2,4,2]=>[4,2,2,2] [2,2,5,1]=>[2,2,5,1] [2,2,6]=>[2,2,6] [2,3,1,1,1,1,1]=>[1,1,1,1,2,3,1] [2,3,1,1,1,2]=>[1,1,2,1,3,2] [2,3,1,1,2,1]=>[2,1,3,1,2,1] [2,3,1,1,3]=>[1,2,3,1,3] [2,3,1,2,1,1]=>[1,2,3,1,2,1] [2,3,1,2,2]=>[2,1,2,3,2] [2,3,1,3,1]=>[2,3,3,1,1] [2,3,1,4]=>[2,3,1,4] [2,3,2,1,1,1]=>[1,1,3,2,2,1] [2,3,2,1,2]=>[2,3,2,1,2] [2,3,2,2,1]=>[2,3,2,2,1] [2,3,2,3]=>[3,2,2,3] [2,3,3,1,1]=>[1,2,3,3,1] [2,3,3,2]=>[3,2,3,2] [2,3,4,1]=>[2,3,4,1] [2,3,5]=>[2,3,5] [2,4,1,1,1,1]=>[1,1,1,2,4,1] [2,4,1,1,2]=>[1,2,1,4,2] [2,4,1,2,1]=>[2,4,1,2,1] [2,4,1,3]=>[2,1,4,3] [2,4,2,1,1]=>[1,4,2,2,1] [2,4,2,2]=>[2,4,2,2] [2,4,3,1]=>[4,2,3,1] [2,4,4]=>[2,4,4] [2,5,1,1,1]=>[1,1,2,5,1] [2,5,1,2]=>[2,1,5,2] [2,5,2,1]=>[5,2,2,1] [2,5,3]=>[5,2,3] [2,6,1,1]=>[1,2,6,1] [2,6,2]=>[6,2,2] [2,7,1]=>[2,7,1] [2,8]=>[2,8] [3,1,1,1,1,1,1,1]=>[1,1,1,1,1,1,3,1] [3,1,1,1,1,1,2]=>[1,1,1,1,1,3,2] [3,1,1,1,1,2,1]=>[3,1,1,1,1,2,1] [3,1,1,1,1,3]=>[1,1,1,3,1,3] [3,1,1,1,2,1,1]=>[1,3,1,1,1,2,1] [3,1,1,1,2,2]=>[1,1,1,2,3,2] [3,1,1,1,3,1]=>[3,1,1,3,1,1] [3,1,1,1,4]=>[1,1,3,1,4] [3,1,1,2,1,1,1]=>[1,1,3,1,1,2,1] [3,1,1,2,1,2]=>[1,3,1,2,1,2] [3,1,1,2,2,1]=>[2,1,1,3,2,1] [3,1,1,2,3]=>[1,1,3,2,3] [3,1,1,3,1,1]=>[1,3,1,3,1,1] [3,1,1,3,2]=>[3,1,3,1,2] [3,1,1,4,1]=>[3,1,4,1,1] [3,1,1,5]=>[1,3,1,5] [3,1,2,1,1,1,1]=>[1,1,1,3,1,2,1] [3,1,2,1,1,2]=>[1,1,3,2,1,2] [3,1,2,1,2,1]=>[3,1,2,2,1,1] [3,1,2,1,3]=>[3,1,2,1,3] [3,1,2,2,1,1]=>[1,2,1,3,2,1] [3,1,2,2,2]=>[1,2,2,3,2] [3,1,2,3,1]=>[3,1,2,3,1] [3,1,2,4]=>[1,3,2,4] [3,1,3,1,1,1]=>[1,1,3,3,1,1] [3,1,3,1,2]=>[1,3,3,1,2] [3,1,3,2,1]=>[3,3,2,1,1] [3,1,3,3]=>[3,1,3,3] [3,1,4,1,1]=>[1,3,4,1,1] [3,1,4,2]=>[3,4,1,2] [3,1,5,1]=>[3,5,1,1] [3,1,6]=>[3,1,6] [3,2,1,1,1,1,1]=>[1,1,1,1,3,2,1] [3,2,1,1,1,2]=>[1,1,2,3,1,2] [3,2,1,1,2,1]=>[2,1,3,2,1,1] [3,2,1,1,3]=>[1,3,2,1,3] [3,2,1,2,1,1]=>[1,2,3,2,1,1] [3,2,1,2,2]=>[2,1,3,2,2] [3,2,1,3,1]=>[3,2,3,1,1] [3,2,1,4]=>[3,2,1,4] [3,2,2,1,1,1]=>[1,1,2,3,2,1] [3,2,2,1,2]=>[2,2,3,1,2] [3,2,2,2,1]=>[2,2,3,2,1] [3,2,2,3]=>[2,3,2,3] [3,2,3,1,1]=>[1,3,2,3,1] [3,2,3,2]=>[3,3,2,2] [3,2,4,1]=>[3,2,4,1] [3,2,5]=>[3,2,5] [3,3,1,1,1,1]=>[1,1,1,3,3,1] [3,3,1,1,2]=>[1,1,3,3,2] [3,3,1,2,1]=>[3,1,3,2,1] [3,3,1,3]=>[3,3,1,3] [3,3,2,1,1]=>[1,3,3,2,1] [3,3,2,2]=>[2,3,3,2] [3,3,3,1]=>[3,3,3,1] [3,3,4]=>[3,3,4] [3,4,1,1,1]=>[1,1,3,4,1] [3,4,1,2]=>[1,3,4,2] [3,4,2,1]=>[3,4,2,1] [3,4,3]=>[4,3,3] [3,5,1,1]=>[1,3,5,1] [3,5,2]=>[3,5,2] [3,6,1]=>[3,6,1] [3,7]=>[3,7] [4,1,1,1,1,1,1]=>[1,1,1,1,1,4,1] [4,1,1,1,1,2]=>[1,1,1,1,4,2] [4,1,1,1,2,1]=>[4,1,1,1,2,1] [4,1,1,1,3]=>[1,1,1,4,3] [4,1,1,2,1,1]=>[1,4,1,1,2,1] [4,1,1,2,2]=>[1,1,2,4,2] [4,1,1,3,1]=>[4,1,1,3,1] [4,1,1,4]=>[1,4,1,4] [4,1,2,1,1,1]=>[1,1,4,1,2,1] [4,1,2,1,2]=>[1,4,2,1,2] [4,1,2,2,1]=>[2,1,4,2,1] [4,1,2,3]=>[1,2,4,3] [4,1,3,1,1]=>[1,4,1,3,1] [4,1,3,2]=>[4,1,3,2] [4,1,4,1]=>[4,4,1,1] [4,1,5]=>[4,1,5] [4,2,1,1,1,1]=>[1,1,1,4,2,1] [4,2,1,1,2]=>[1,2,4,1,2] [4,2,1,2,1]=>[2,4,2,1,1] [4,2,1,3]=>[2,4,1,3] [4,2,2,1,1]=>[1,2,4,2,1] [4,2,2,2]=>[2,2,4,2] [4,2,3,1]=>[2,4,3,1] [4,2,4]=>[4,2,4] [4,3,1,1,1]=>[1,1,4,3,1] [4,3,1,2]=>[1,4,3,2] [4,3,2,1]=>[4,3,2,1] [4,3,3]=>[3,4,3] [4,4,1,1]=>[1,4,4,1] [4,4,2]=>[4,4,2] [4,5,1]=>[4,5,1] [4,6]=>[4,6] [5,1,1,1,1,1]=>[1,1,1,1,5,1] [5,1,1,1,2]=>[1,1,1,5,2] [5,1,1,2,1]=>[5,1,1,2,1] [5,1,1,3]=>[1,1,5,3] [5,1,2,1,1]=>[1,5,1,2,1] [5,1,2,2]=>[1,2,5,2] [5,1,3,1]=>[5,1,3,1] [5,1,4]=>[1,5,4] [5,2,1,1,1]=>[1,1,5,2,1] [5,2,1,2]=>[2,5,1,2] [5,2,2,1]=>[2,5,2,1] [5,2,3]=>[2,5,3] [5,3,1,1]=>[1,5,3,1] [5,3,2]=>[5,3,2] [5,4,1]=>[5,4,1] [5,5]=>[5,5] [6,1,1,1,1]=>[1,1,1,6,1] [6,1,1,2]=>[1,1,6,2] [6,1,2,1]=>[6,1,2,1] [6,1,3]=>[1,6,3] [6,2,1,1]=>[1,6,2,1] [6,2,2]=>[2,6,2] [6,3,1]=>[6,3,1] [6,4]=>[6,4] [7,1,1,1]=>[1,1,7,1] [7,1,2]=>[1,7,2] [7,2,1]=>[7,2,1] [7,3]=>[7,3] [8,1,1]=>[1,8,1] [8,2]=>[8,2] [9,1]=>[9,1] [10]=>[10]
click to show experimental identities (only identities of compositions of up to three maps are shown)
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..
References
[1] Foata, D., Schützenberger, M.-P. Major index and inversion number of permutations MathSciNet:0506852
Properties
bijective, graded
Sage code
def mapping(c):
    return Composition(Word(c).foata_bijection())
Weight
26
Created
Sep 09, 2023 at 20:39 by Martin Rubey
Updated
Sep 09, 2023 at 20:39 by Martin Rubey