Identifier
Mp00031:
Dyck paths
—to 312-avoiding permutation⟶
Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Images
=>
Cc0005;cc-rep-0Cc0010;cc-rep-3
[1,0]=>[1]=>[1]=>[.,.]
[1,0,1,0]=>[1,2]=>[1,2]=>[.,[.,.]]
[1,1,0,0]=>[2,1]=>[2,1]=>[[.,.],.]
[1,0,1,0,1,0]=>[1,2,3]=>[1,2,3]=>[.,[.,[.,.]]]
[1,0,1,1,0,0]=>[1,3,2]=>[3,1,2]=>[[.,.],[.,.]]
[1,1,0,0,1,0]=>[2,1,3]=>[2,1,3]=>[[.,.],[.,.]]
[1,1,0,1,0,0]=>[2,3,1]=>[2,3,1]=>[[.,[.,.]],.]
[1,1,1,0,0,0]=>[3,2,1]=>[3,2,1]=>[[[.,.],.],.]
[1,0,1,0,1,0,1,0]=>[1,2,3,4]=>[1,2,3,4]=>[.,[.,[.,[.,.]]]]
[1,0,1,0,1,1,0,0]=>[1,2,4,3]=>[4,1,2,3]=>[[.,.],[.,[.,.]]]
[1,0,1,1,0,0,1,0]=>[1,3,2,4]=>[3,1,2,4]=>[[.,.],[.,[.,.]]]
[1,0,1,1,0,1,0,0]=>[1,3,4,2]=>[3,1,4,2]=>[[.,.],[[.,.],.]]
[1,0,1,1,1,0,0,0]=>[1,4,3,2]=>[4,3,1,2]=>[[[.,.],.],[.,.]]
[1,1,0,0,1,0,1,0]=>[2,1,3,4]=>[2,1,3,4]=>[[.,.],[.,[.,.]]]
[1,1,0,0,1,1,0,0]=>[2,1,4,3]=>[4,2,1,3]=>[[[.,.],.],[.,.]]
[1,1,0,1,0,0,1,0]=>[2,3,1,4]=>[2,3,1,4]=>[[.,[.,.]],[.,.]]
[1,1,0,1,0,1,0,0]=>[2,3,4,1]=>[2,3,4,1]=>[[.,[.,[.,.]]],.]
[1,1,0,1,1,0,0,0]=>[2,4,3,1]=>[4,2,3,1]=>[[[.,.],[.,.]],.]
[1,1,1,0,0,0,1,0]=>[3,2,1,4]=>[3,2,1,4]=>[[[.,.],.],[.,.]]
[1,1,1,0,0,1,0,0]=>[3,2,4,1]=>[3,2,4,1]=>[[[.,.],[.,.]],.]
[1,1,1,0,1,0,0,0]=>[3,4,2,1]=>[3,4,2,1]=>[[[.,[.,.]],.],.]
[1,1,1,1,0,0,0,0]=>[4,3,2,1]=>[4,3,2,1]=>[[[[.,.],.],.],.]
[1,0,1,0,1,0,1,0,1,0]=>[1,2,3,4,5]=>[1,2,3,4,5]=>[.,[.,[.,[.,[.,.]]]]]
[1,0,1,0,1,0,1,1,0,0]=>[1,2,3,5,4]=>[5,1,2,3,4]=>[[.,.],[.,[.,[.,.]]]]
[1,0,1,0,1,1,0,0,1,0]=>[1,2,4,3,5]=>[4,1,2,3,5]=>[[.,.],[.,[.,[.,.]]]]
[1,0,1,0,1,1,0,1,0,0]=>[1,2,4,5,3]=>[4,1,2,5,3]=>[[.,.],[.,[[.,.],.]]]
[1,0,1,0,1,1,1,0,0,0]=>[1,2,5,4,3]=>[5,4,1,2,3]=>[[[.,.],.],[.,[.,.]]]
[1,0,1,1,0,0,1,0,1,0]=>[1,3,2,4,5]=>[3,1,2,4,5]=>[[.,.],[.,[.,[.,.]]]]
[1,0,1,1,0,0,1,1,0,0]=>[1,3,2,5,4]=>[5,3,1,2,4]=>[[[.,.],.],[.,[.,.]]]
[1,0,1,1,0,1,0,0,1,0]=>[1,3,4,2,5]=>[3,1,4,2,5]=>[[.,.],[[.,.],[.,.]]]
[1,0,1,1,0,1,0,1,0,0]=>[1,3,4,5,2]=>[3,1,4,5,2]=>[[.,.],[[.,[.,.]],.]]
[1,0,1,1,0,1,1,0,0,0]=>[1,3,5,4,2]=>[5,3,1,4,2]=>[[[.,.],.],[[.,.],.]]
[1,0,1,1,1,0,0,0,1,0]=>[1,4,3,2,5]=>[4,3,1,2,5]=>[[[.,.],.],[.,[.,.]]]
[1,0,1,1,1,0,0,1,0,0]=>[1,4,3,5,2]=>[4,3,1,5,2]=>[[[.,.],.],[[.,.],.]]
[1,0,1,1,1,0,1,0,0,0]=>[1,4,5,3,2]=>[4,5,1,3,2]=>[[.,[.,.]],[[.,.],.]]
[1,0,1,1,1,1,0,0,0,0]=>[1,5,4,3,2]=>[5,4,3,1,2]=>[[[[.,.],.],.],[.,.]]
[1,1,0,0,1,0,1,0,1,0]=>[2,1,3,4,5]=>[2,1,3,4,5]=>[[.,.],[.,[.,[.,.]]]]
[1,1,0,0,1,0,1,1,0,0]=>[2,1,3,5,4]=>[5,2,1,3,4]=>[[[.,.],.],[.,[.,.]]]
[1,1,0,0,1,1,0,0,1,0]=>[2,1,4,3,5]=>[4,2,1,3,5]=>[[[.,.],.],[.,[.,.]]]
[1,1,0,0,1,1,0,1,0,0]=>[2,1,4,5,3]=>[4,2,1,5,3]=>[[[.,.],.],[[.,.],.]]
[1,1,0,0,1,1,1,0,0,0]=>[2,1,5,4,3]=>[5,4,2,1,3]=>[[[[.,.],.],.],[.,.]]
[1,1,0,1,0,0,1,0,1,0]=>[2,3,1,4,5]=>[2,3,1,4,5]=>[[.,[.,.]],[.,[.,.]]]
[1,1,0,1,0,0,1,1,0,0]=>[2,3,1,5,4]=>[5,2,3,1,4]=>[[[.,.],[.,.]],[.,.]]
[1,1,0,1,0,1,0,0,1,0]=>[2,3,4,1,5]=>[2,3,4,1,5]=>[[.,[.,[.,.]]],[.,.]]
[1,1,0,1,0,1,0,1,0,0]=>[2,3,4,5,1]=>[2,3,4,5,1]=>[[.,[.,[.,[.,.]]]],.]
[1,1,0,1,0,1,1,0,0,0]=>[2,3,5,4,1]=>[5,2,3,4,1]=>[[[.,.],[.,[.,.]]],.]
[1,1,0,1,1,0,0,0,1,0]=>[2,4,3,1,5]=>[4,2,3,1,5]=>[[[.,.],[.,.]],[.,.]]
[1,1,0,1,1,0,0,1,0,0]=>[2,4,3,5,1]=>[4,2,3,5,1]=>[[[.,.],[.,[.,.]]],.]
[1,1,0,1,1,0,1,0,0,0]=>[2,4,5,3,1]=>[4,2,5,3,1]=>[[[.,.],[[.,.],.]],.]
[1,1,0,1,1,1,0,0,0,0]=>[2,5,4,3,1]=>[5,4,2,3,1]=>[[[[.,.],.],[.,.]],.]
[1,1,1,0,0,0,1,0,1,0]=>[3,2,1,4,5]=>[3,2,1,4,5]=>[[[.,.],.],[.,[.,.]]]
[1,1,1,0,0,0,1,1,0,0]=>[3,2,1,5,4]=>[5,3,2,1,4]=>[[[[.,.],.],.],[.,.]]
[1,1,1,0,0,1,0,0,1,0]=>[3,2,4,1,5]=>[3,2,4,1,5]=>[[[.,.],[.,.]],[.,.]]
[1,1,1,0,0,1,0,1,0,0]=>[3,2,4,5,1]=>[3,2,4,5,1]=>[[[.,.],[.,[.,.]]],.]
[1,1,1,0,0,1,1,0,0,0]=>[3,2,5,4,1]=>[5,3,2,4,1]=>[[[[.,.],.],[.,.]],.]
[1,1,1,0,1,0,0,0,1,0]=>[3,4,2,1,5]=>[3,4,2,1,5]=>[[[.,[.,.]],.],[.,.]]
[1,1,1,0,1,0,0,1,0,0]=>[3,4,2,5,1]=>[3,4,2,5,1]=>[[[.,[.,.]],[.,.]],.]
[1,1,1,0,1,0,1,0,0,0]=>[3,4,5,2,1]=>[3,4,5,2,1]=>[[[.,[.,[.,.]]],.],.]
[1,1,1,0,1,1,0,0,0,0]=>[3,5,4,2,1]=>[5,3,4,2,1]=>[[[[.,.],[.,.]],.],.]
[1,1,1,1,0,0,0,0,1,0]=>[4,3,2,1,5]=>[4,3,2,1,5]=>[[[[.,.],.],.],[.,.]]
[1,1,1,1,0,0,0,1,0,0]=>[4,3,2,5,1]=>[4,3,2,5,1]=>[[[[.,.],.],[.,.]],.]
[1,1,1,1,0,0,1,0,0,0]=>[4,3,5,2,1]=>[4,3,5,2,1]=>[[[[.,.],[.,.]],.],.]
[1,1,1,1,0,1,0,0,0,0]=>[4,5,3,2,1]=>[4,5,3,2,1]=>[[[[.,[.,.]],.],.],.]
[1,1,1,1,1,0,0,0,0,0]=>[5,4,3,2,1]=>[5,4,3,2,1]=>[[[[[.,.],.],.],.],.]
[1,0,1,0,1,0,1,0,1,0,1,0]=>[1,2,3,4,5,6]=>[1,2,3,4,5,6]=>[.,[.,[.,[.,[.,[.,.]]]]]]
[1,0,1,0,1,0,1,0,1,1,0,0]=>[1,2,3,4,6,5]=>[6,1,2,3,4,5]=>[[.,.],[.,[.,[.,[.,.]]]]]
[1,0,1,0,1,0,1,1,0,0,1,0]=>[1,2,3,5,4,6]=>[5,1,2,3,4,6]=>[[.,.],[.,[.,[.,[.,.]]]]]
[1,0,1,0,1,0,1,1,0,1,0,0]=>[1,2,3,5,6,4]=>[5,1,2,3,6,4]=>[[.,.],[.,[.,[[.,.],.]]]]
[1,0,1,0,1,0,1,1,1,0,0,0]=>[1,2,3,6,5,4]=>[6,5,1,2,3,4]=>[[[.,.],.],[.,[.,[.,.]]]]
[1,0,1,0,1,1,0,0,1,0,1,0]=>[1,2,4,3,5,6]=>[4,1,2,3,5,6]=>[[.,.],[.,[.,[.,[.,.]]]]]
[1,0,1,0,1,1,0,0,1,1,0,0]=>[1,2,4,3,6,5]=>[6,4,1,2,3,5]=>[[[.,.],.],[.,[.,[.,.]]]]
[1,0,1,0,1,1,0,1,0,0,1,0]=>[1,2,4,5,3,6]=>[4,1,2,5,3,6]=>[[.,.],[.,[[.,.],[.,.]]]]
[1,0,1,0,1,1,0,1,0,1,0,0]=>[1,2,4,5,6,3]=>[4,1,2,5,6,3]=>[[.,.],[.,[[.,[.,.]],.]]]
[1,0,1,0,1,1,0,1,1,0,0,0]=>[1,2,4,6,5,3]=>[6,4,1,2,5,3]=>[[[.,.],.],[.,[[.,.],.]]]
[1,0,1,0,1,1,1,0,0,0,1,0]=>[1,2,5,4,3,6]=>[5,4,1,2,3,6]=>[[[.,.],.],[.,[.,[.,.]]]]
[1,0,1,0,1,1,1,0,0,1,0,0]=>[1,2,5,4,6,3]=>[5,4,1,2,6,3]=>[[[.,.],.],[.,[[.,.],.]]]
[1,0,1,0,1,1,1,0,1,0,0,0]=>[1,2,5,6,4,3]=>[5,6,1,2,4,3]=>[[.,[.,.]],[.,[[.,.],.]]]
[1,0,1,0,1,1,1,1,0,0,0,0]=>[1,2,6,5,4,3]=>[6,5,4,1,2,3]=>[[[[.,.],.],.],[.,[.,.]]]
[1,0,1,1,0,0,1,0,1,0,1,0]=>[1,3,2,4,5,6]=>[3,1,2,4,5,6]=>[[.,.],[.,[.,[.,[.,.]]]]]
[1,0,1,1,0,0,1,0,1,1,0,0]=>[1,3,2,4,6,5]=>[6,3,1,2,4,5]=>[[[.,.],.],[.,[.,[.,.]]]]
[1,0,1,1,0,0,1,1,0,0,1,0]=>[1,3,2,5,4,6]=>[5,3,1,2,4,6]=>[[[.,.],.],[.,[.,[.,.]]]]
[1,0,1,1,0,0,1,1,0,1,0,0]=>[1,3,2,5,6,4]=>[5,3,1,2,6,4]=>[[[.,.],.],[.,[[.,.],.]]]
[1,0,1,1,0,0,1,1,1,0,0,0]=>[1,3,2,6,5,4]=>[6,5,3,1,2,4]=>[[[[.,.],.],.],[.,[.,.]]]
[1,0,1,1,0,1,0,0,1,0,1,0]=>[1,3,4,2,5,6]=>[3,1,4,2,5,6]=>[[.,.],[[.,.],[.,[.,.]]]]
[1,0,1,1,0,1,0,0,1,1,0,0]=>[1,3,4,2,6,5]=>[6,3,1,4,2,5]=>[[[.,.],.],[[.,.],[.,.]]]
[1,0,1,1,0,1,0,1,0,0,1,0]=>[1,3,4,5,2,6]=>[3,1,4,5,2,6]=>[[.,.],[[.,[.,.]],[.,.]]]
[1,0,1,1,0,1,0,1,0,1,0,0]=>[1,3,4,5,6,2]=>[3,1,4,5,6,2]=>[[.,.],[[.,[.,[.,.]]],.]]
[1,0,1,1,0,1,0,1,1,0,0,0]=>[1,3,4,6,5,2]=>[6,3,1,4,5,2]=>[[[.,.],.],[[.,[.,.]],.]]
[1,0,1,1,0,1,1,0,0,0,1,0]=>[1,3,5,4,2,6]=>[5,3,1,4,2,6]=>[[[.,.],.],[[.,.],[.,.]]]
[1,0,1,1,0,1,1,0,0,1,0,0]=>[1,3,5,4,6,2]=>[5,3,1,4,6,2]=>[[[.,.],.],[[.,[.,.]],.]]
[1,0,1,1,0,1,1,0,1,0,0,0]=>[1,3,5,6,4,2]=>[5,3,1,6,4,2]=>[[[.,.],.],[[[.,.],.],.]]
[1,0,1,1,0,1,1,1,0,0,0,0]=>[1,3,6,5,4,2]=>[6,5,3,1,4,2]=>[[[[.,.],.],.],[[.,.],.]]
[1,0,1,1,1,0,0,0,1,0,1,0]=>[1,4,3,2,5,6]=>[4,3,1,2,5,6]=>[[[.,.],.],[.,[.,[.,.]]]]
[1,0,1,1,1,0,0,0,1,1,0,0]=>[1,4,3,2,6,5]=>[6,4,3,1,2,5]=>[[[[.,.],.],.],[.,[.,.]]]
[1,0,1,1,1,0,0,1,0,0,1,0]=>[1,4,3,5,2,6]=>[4,3,1,5,2,6]=>[[[.,.],.],[[.,.],[.,.]]]
[1,0,1,1,1,0,0,1,0,1,0,0]=>[1,4,3,5,6,2]=>[4,3,1,5,6,2]=>[[[.,.],.],[[.,[.,.]],.]]
[1,0,1,1,1,0,0,1,1,0,0,0]=>[1,4,3,6,5,2]=>[6,4,3,1,5,2]=>[[[[.,.],.],.],[[.,.],.]]
[1,0,1,1,1,0,1,0,0,0,1,0]=>[1,4,5,3,2,6]=>[4,5,1,3,2,6]=>[[.,[.,.]],[[.,.],[.,.]]]
[1,0,1,1,1,0,1,0,0,1,0,0]=>[1,4,5,3,6,2]=>[4,5,1,3,6,2]=>[[.,[.,.]],[[.,[.,.]],.]]
[1,0,1,1,1,0,1,0,1,0,0,0]=>[1,4,5,6,3,2]=>[4,5,1,6,3,2]=>[[.,[.,.]],[[[.,.],.],.]]
[1,0,1,1,1,0,1,1,0,0,0,0]=>[1,4,6,5,3,2]=>[6,4,5,1,3,2]=>[[[.,.],[.,.]],[[.,.],.]]
[1,0,1,1,1,1,0,0,0,0,1,0]=>[1,5,4,3,2,6]=>[5,4,3,1,2,6]=>[[[[.,.],.],.],[.,[.,.]]]
[1,0,1,1,1,1,0,0,0,1,0,0]=>[1,5,4,3,6,2]=>[5,4,3,1,6,2]=>[[[[.,.],.],.],[[.,.],.]]
[1,0,1,1,1,1,0,0,1,0,0,0]=>[1,5,4,6,3,2]=>[5,4,6,1,3,2]=>[[[.,.],[.,.]],[[.,.],.]]
[1,0,1,1,1,1,0,1,0,0,0,0]=>[1,5,6,4,3,2]=>[5,6,4,1,3,2]=>[[[.,[.,.]],.],[[.,.],.]]
[1,0,1,1,1,1,1,0,0,0,0,0]=>[1,6,5,4,3,2]=>[6,5,4,3,1,2]=>[[[[[.,.],.],.],.],[.,.]]
[1,1,0,0,1,0,1,0,1,0,1,0]=>[2,1,3,4,5,6]=>[2,1,3,4,5,6]=>[[.,.],[.,[.,[.,[.,.]]]]]
[1,1,0,0,1,0,1,0,1,1,0,0]=>[2,1,3,4,6,5]=>[6,2,1,3,4,5]=>[[[.,.],.],[.,[.,[.,.]]]]
[1,1,0,0,1,0,1,1,0,0,1,0]=>[2,1,3,5,4,6]=>[5,2,1,3,4,6]=>[[[.,.],.],[.,[.,[.,.]]]]
[1,1,0,0,1,0,1,1,0,1,0,0]=>[2,1,3,5,6,4]=>[5,2,1,3,6,4]=>[[[.,.],.],[.,[[.,.],.]]]
[1,1,0,0,1,0,1,1,1,0,0,0]=>[2,1,3,6,5,4]=>[6,5,2,1,3,4]=>[[[[.,.],.],.],[.,[.,.]]]
[1,1,0,0,1,1,0,0,1,0,1,0]=>[2,1,4,3,5,6]=>[4,2,1,3,5,6]=>[[[.,.],.],[.,[.,[.,.]]]]
[1,1,0,0,1,1,0,0,1,1,0,0]=>[2,1,4,3,6,5]=>[6,4,2,1,3,5]=>[[[[.,.],.],.],[.,[.,.]]]
[1,1,0,0,1,1,0,1,0,0,1,0]=>[2,1,4,5,3,6]=>[4,2,1,5,3,6]=>[[[.,.],.],[[.,.],[.,.]]]
[1,1,0,0,1,1,0,1,0,1,0,0]=>[2,1,4,5,6,3]=>[4,2,1,5,6,3]=>[[[.,.],.],[[.,[.,.]],.]]
[1,1,0,0,1,1,0,1,1,0,0,0]=>[2,1,4,6,5,3]=>[6,4,2,1,5,3]=>[[[[.,.],.],.],[[.,.],.]]
[1,1,0,0,1,1,1,0,0,0,1,0]=>[2,1,5,4,3,6]=>[5,4,2,1,3,6]=>[[[[.,.],.],.],[.,[.,.]]]
[1,1,0,0,1,1,1,0,0,1,0,0]=>[2,1,5,4,6,3]=>[5,4,2,1,6,3]=>[[[[.,.],.],.],[[.,.],.]]
[1,1,0,0,1,1,1,0,1,0,0,0]=>[2,1,5,6,4,3]=>[5,6,2,1,4,3]=>[[[.,[.,.]],.],[[.,.],.]]
[1,1,0,0,1,1,1,1,0,0,0,0]=>[2,1,6,5,4,3]=>[6,5,4,2,1,3]=>[[[[[.,.],.],.],.],[.,.]]
[1,1,0,1,0,0,1,0,1,0,1,0]=>[2,3,1,4,5,6]=>[2,3,1,4,5,6]=>[[.,[.,.]],[.,[.,[.,.]]]]
[1,1,0,1,0,0,1,0,1,1,0,0]=>[2,3,1,4,6,5]=>[6,2,3,1,4,5]=>[[[.,.],[.,.]],[.,[.,.]]]
[1,1,0,1,0,0,1,1,0,0,1,0]=>[2,3,1,5,4,6]=>[5,2,3,1,4,6]=>[[[.,.],[.,.]],[.,[.,.]]]
[1,1,0,1,0,0,1,1,0,1,0,0]=>[2,3,1,5,6,4]=>[5,2,3,1,6,4]=>[[[.,.],[.,.]],[[.,.],.]]
[1,1,0,1,0,0,1,1,1,0,0,0]=>[2,3,1,6,5,4]=>[6,5,2,3,1,4]=>[[[[.,.],.],[.,.]],[.,.]]
[1,1,0,1,0,1,0,0,1,0,1,0]=>[2,3,4,1,5,6]=>[2,3,4,1,5,6]=>[[.,[.,[.,.]]],[.,[.,.]]]
[1,1,0,1,0,1,0,0,1,1,0,0]=>[2,3,4,1,6,5]=>[6,2,3,4,1,5]=>[[[.,.],[.,[.,.]]],[.,.]]
[1,1,0,1,0,1,0,1,0,0,1,0]=>[2,3,4,5,1,6]=>[2,3,4,5,1,6]=>[[.,[.,[.,[.,.]]]],[.,.]]
[1,1,0,1,0,1,0,1,0,1,0,0]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>[[.,[.,[.,[.,[.,.]]]]],.]
[1,1,0,1,0,1,0,1,1,0,0,0]=>[2,3,4,6,5,1]=>[6,2,3,4,5,1]=>[[[.,.],[.,[.,[.,.]]]],.]
[1,1,0,1,0,1,1,0,0,0,1,0]=>[2,3,5,4,1,6]=>[5,2,3,4,1,6]=>[[[.,.],[.,[.,.]]],[.,.]]
[1,1,0,1,0,1,1,0,0,1,0,0]=>[2,3,5,4,6,1]=>[5,2,3,4,6,1]=>[[[.,.],[.,[.,[.,.]]]],.]
[1,1,0,1,0,1,1,0,1,0,0,0]=>[2,3,5,6,4,1]=>[5,2,3,6,4,1]=>[[[.,.],[.,[[.,.],.]]],.]
[1,1,0,1,0,1,1,1,0,0,0,0]=>[2,3,6,5,4,1]=>[6,5,2,3,4,1]=>[[[[.,.],.],[.,[.,.]]],.]
[1,1,0,1,1,0,0,0,1,0,1,0]=>[2,4,3,1,5,6]=>[4,2,3,1,5,6]=>[[[.,.],[.,.]],[.,[.,.]]]
[1,1,0,1,1,0,0,0,1,1,0,0]=>[2,4,3,1,6,5]=>[6,4,2,3,1,5]=>[[[[.,.],.],[.,.]],[.,.]]
[1,1,0,1,1,0,0,1,0,0,1,0]=>[2,4,3,5,1,6]=>[4,2,3,5,1,6]=>[[[.,.],[.,[.,.]]],[.,.]]
[1,1,0,1,1,0,0,1,0,1,0,0]=>[2,4,3,5,6,1]=>[4,2,3,5,6,1]=>[[[.,.],[.,[.,[.,.]]]],.]
[1,1,0,1,1,0,0,1,1,0,0,0]=>[2,4,3,6,5,1]=>[6,4,2,3,5,1]=>[[[[.,.],.],[.,[.,.]]],.]
[1,1,0,1,1,0,1,0,0,0,1,0]=>[2,4,5,3,1,6]=>[4,2,5,3,1,6]=>[[[.,.],[[.,.],.]],[.,.]]
[1,1,0,1,1,0,1,0,0,1,0,0]=>[2,4,5,3,6,1]=>[4,2,5,3,6,1]=>[[[.,.],[[.,.],[.,.]]],.]
[1,1,0,1,1,0,1,0,1,0,0,0]=>[2,4,5,6,3,1]=>[4,2,5,6,3,1]=>[[[.,.],[[.,[.,.]],.]],.]
[1,1,0,1,1,0,1,1,0,0,0,0]=>[2,4,6,5,3,1]=>[6,4,2,5,3,1]=>[[[[.,.],.],[[.,.],.]],.]
[1,1,0,1,1,1,0,0,0,0,1,0]=>[2,5,4,3,1,6]=>[5,4,2,3,1,6]=>[[[[.,.],.],[.,.]],[.,.]]
[1,1,0,1,1,1,0,0,0,1,0,0]=>[2,5,4,3,6,1]=>[5,4,2,3,6,1]=>[[[[.,.],.],[.,[.,.]]],.]
[1,1,0,1,1,1,0,0,1,0,0,0]=>[2,5,4,6,3,1]=>[5,4,2,6,3,1]=>[[[[.,.],.],[[.,.],.]],.]
[1,1,0,1,1,1,0,1,0,0,0,0]=>[2,5,6,4,3,1]=>[5,6,2,4,3,1]=>[[[.,[.,.]],[[.,.],.]],.]
[1,1,0,1,1,1,1,0,0,0,0,0]=>[2,6,5,4,3,1]=>[6,5,4,2,3,1]=>[[[[[.,.],.],.],[.,.]],.]
[1,1,1,0,0,0,1,0,1,0,1,0]=>[3,2,1,4,5,6]=>[3,2,1,4,5,6]=>[[[.,.],.],[.,[.,[.,.]]]]
[1,1,1,0,0,0,1,0,1,1,0,0]=>[3,2,1,4,6,5]=>[6,3,2,1,4,5]=>[[[[.,.],.],.],[.,[.,.]]]
[1,1,1,0,0,0,1,1,0,0,1,0]=>[3,2,1,5,4,6]=>[5,3,2,1,4,6]=>[[[[.,.],.],.],[.,[.,.]]]
[1,1,1,0,0,0,1,1,0,1,0,0]=>[3,2,1,5,6,4]=>[5,3,2,1,6,4]=>[[[[.,.],.],.],[[.,.],.]]
[1,1,1,0,0,0,1,1,1,0,0,0]=>[3,2,1,6,5,4]=>[6,5,3,2,1,4]=>[[[[[.,.],.],.],.],[.,.]]
[1,1,1,0,0,1,0,0,1,0,1,0]=>[3,2,4,1,5,6]=>[3,2,4,1,5,6]=>[[[.,.],[.,.]],[.,[.,.]]]
[1,1,1,0,0,1,0,0,1,1,0,0]=>[3,2,4,1,6,5]=>[6,3,2,4,1,5]=>[[[[.,.],.],[.,.]],[.,.]]
[1,1,1,0,0,1,0,1,0,0,1,0]=>[3,2,4,5,1,6]=>[3,2,4,5,1,6]=>[[[.,.],[.,[.,.]]],[.,.]]
[1,1,1,0,0,1,0,1,0,1,0,0]=>[3,2,4,5,6,1]=>[3,2,4,5,6,1]=>[[[.,.],[.,[.,[.,.]]]],.]
[1,1,1,0,0,1,0,1,1,0,0,0]=>[3,2,4,6,5,1]=>[6,3,2,4,5,1]=>[[[[.,.],.],[.,[.,.]]],.]
[1,1,1,0,0,1,1,0,0,0,1,0]=>[3,2,5,4,1,6]=>[5,3,2,4,1,6]=>[[[[.,.],.],[.,.]],[.,.]]
[1,1,1,0,0,1,1,0,0,1,0,0]=>[3,2,5,4,6,1]=>[5,3,2,4,6,1]=>[[[[.,.],.],[.,[.,.]]],.]
[1,1,1,0,0,1,1,0,1,0,0,0]=>[3,2,5,6,4,1]=>[5,3,2,6,4,1]=>[[[[.,.],.],[[.,.],.]],.]
[1,1,1,0,0,1,1,1,0,0,0,0]=>[3,2,6,5,4,1]=>[6,5,3,2,4,1]=>[[[[[.,.],.],.],[.,.]],.]
[1,1,1,0,1,0,0,0,1,0,1,0]=>[3,4,2,1,5,6]=>[3,4,2,1,5,6]=>[[[.,[.,.]],.],[.,[.,.]]]
[1,1,1,0,1,0,0,0,1,1,0,0]=>[3,4,2,1,6,5]=>[6,3,4,2,1,5]=>[[[[.,.],[.,.]],.],[.,.]]
[1,1,1,0,1,0,0,1,0,0,1,0]=>[3,4,2,5,1,6]=>[3,4,2,5,1,6]=>[[[.,[.,.]],[.,.]],[.,.]]
[1,1,1,0,1,0,0,1,0,1,0,0]=>[3,4,2,5,6,1]=>[3,4,2,5,6,1]=>[[[.,[.,.]],[.,[.,.]]],.]
[1,1,1,0,1,0,0,1,1,0,0,0]=>[3,4,2,6,5,1]=>[6,3,4,2,5,1]=>[[[[.,.],[.,.]],[.,.]],.]
[1,1,1,0,1,0,1,0,0,0,1,0]=>[3,4,5,2,1,6]=>[3,4,5,2,1,6]=>[[[.,[.,[.,.]]],.],[.,.]]
[1,1,1,0,1,0,1,0,0,1,0,0]=>[3,4,5,2,6,1]=>[3,4,5,2,6,1]=>[[[.,[.,[.,.]]],[.,.]],.]
[1,1,1,0,1,0,1,0,1,0,0,0]=>[3,4,5,6,2,1]=>[3,4,5,6,2,1]=>[[[.,[.,[.,[.,.]]]],.],.]
[1,1,1,0,1,0,1,1,0,0,0,0]=>[3,4,6,5,2,1]=>[6,3,4,5,2,1]=>[[[[.,.],[.,[.,.]]],.],.]
[1,1,1,0,1,1,0,0,0,0,1,0]=>[3,5,4,2,1,6]=>[5,3,4,2,1,6]=>[[[[.,.],[.,.]],.],[.,.]]
[1,1,1,0,1,1,0,0,0,1,0,0]=>[3,5,4,2,6,1]=>[5,3,4,2,6,1]=>[[[[.,.],[.,.]],[.,.]],.]
[1,1,1,0,1,1,0,0,1,0,0,0]=>[3,5,4,6,2,1]=>[5,3,4,6,2,1]=>[[[[.,.],[.,[.,.]]],.],.]
[1,1,1,0,1,1,0,1,0,0,0,0]=>[3,5,6,4,2,1]=>[5,3,6,4,2,1]=>[[[[.,.],[[.,.],.]],.],.]
[1,1,1,0,1,1,1,0,0,0,0,0]=>[3,6,5,4,2,1]=>[6,5,3,4,2,1]=>[[[[[.,.],.],[.,.]],.],.]
[1,1,1,1,0,0,0,0,1,0,1,0]=>[4,3,2,1,5,6]=>[4,3,2,1,5,6]=>[[[[.,.],.],.],[.,[.,.]]]
[1,1,1,1,0,0,0,0,1,1,0,0]=>[4,3,2,1,6,5]=>[6,4,3,2,1,5]=>[[[[[.,.],.],.],.],[.,.]]
[1,1,1,1,0,0,0,1,0,0,1,0]=>[4,3,2,5,1,6]=>[4,3,2,5,1,6]=>[[[[.,.],.],[.,.]],[.,.]]
[1,1,1,1,0,0,0,1,0,1,0,0]=>[4,3,2,5,6,1]=>[4,3,2,5,6,1]=>[[[[.,.],.],[.,[.,.]]],.]
[1,1,1,1,0,0,0,1,1,0,0,0]=>[4,3,2,6,5,1]=>[6,4,3,2,5,1]=>[[[[[.,.],.],.],[.,.]],.]
[1,1,1,1,0,0,1,0,0,0,1,0]=>[4,3,5,2,1,6]=>[4,3,5,2,1,6]=>[[[[.,.],[.,.]],.],[.,.]]
[1,1,1,1,0,0,1,0,0,1,0,0]=>[4,3,5,2,6,1]=>[4,3,5,2,6,1]=>[[[[.,.],[.,.]],[.,.]],.]
[1,1,1,1,0,0,1,0,1,0,0,0]=>[4,3,5,6,2,1]=>[4,3,5,6,2,1]=>[[[[.,.],[.,[.,.]]],.],.]
[1,1,1,1,0,0,1,1,0,0,0,0]=>[4,3,6,5,2,1]=>[6,4,3,5,2,1]=>[[[[[.,.],.],[.,.]],.],.]
[1,1,1,1,0,1,0,0,0,0,1,0]=>[4,5,3,2,1,6]=>[4,5,3,2,1,6]=>[[[[.,[.,.]],.],.],[.,.]]
[1,1,1,1,0,1,0,0,0,1,0,0]=>[4,5,3,2,6,1]=>[4,5,3,2,6,1]=>[[[[.,[.,.]],.],[.,.]],.]
[1,1,1,1,0,1,0,0,1,0,0,0]=>[4,5,3,6,2,1]=>[4,5,3,6,2,1]=>[[[[.,[.,.]],[.,.]],.],.]
[1,1,1,1,0,1,0,1,0,0,0,0]=>[4,5,6,3,2,1]=>[4,5,6,3,2,1]=>[[[[.,[.,[.,.]]],.],.],.]
[1,1,1,1,0,1,1,0,0,0,0,0]=>[4,6,5,3,2,1]=>[6,4,5,3,2,1]=>[[[[[.,.],[.,.]],.],.],.]
[1,1,1,1,1,0,0,0,0,0,1,0]=>[5,4,3,2,1,6]=>[5,4,3,2,1,6]=>[[[[[.,.],.],.],.],[.,.]]
[1,1,1,1,1,0,0,0,0,1,0,0]=>[5,4,3,2,6,1]=>[5,4,3,2,6,1]=>[[[[[.,.],.],.],[.,.]],.]
[1,1,1,1,1,0,0,0,1,0,0,0]=>[5,4,3,6,2,1]=>[5,4,3,6,2,1]=>[[[[[.,.],.],[.,.]],.],.]
[1,1,1,1,1,0,0,1,0,0,0,0]=>[5,4,6,3,2,1]=>[5,4,6,3,2,1]=>[[[[[.,.],[.,.]],.],.],.]
[1,1,1,1,1,0,1,0,0,0,0,0]=>[5,6,4,3,2,1]=>[5,6,4,3,2,1]=>[[[[[.,[.,.]],.],.],.],.]
[1,1,1,1,1,1,0,0,0,0,0,0]=>[6,5,4,3,2,1]=>[6,5,4,3,2,1]=>[[[[[[.,.],.],.],.],.],.]
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[1,2,3,4,5,6,7]=>[1,2,3,4,5,6,7]=>[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[1,2,3,4,5,7,6]=>[7,1,2,3,4,5,6]=>[[.,.],[.,[.,[.,[.,[.,.]]]]]]
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[1,2,3,4,6,5,7]=>[6,1,2,3,4,5,7]=>[[.,.],[.,[.,[.,[.,[.,.]]]]]]
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>[1,2,3,4,6,7,5]=>[6,1,2,3,4,7,5]=>[[.,.],[.,[.,[.,[[.,.],.]]]]]
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[1,2,3,4,7,6,5]=>[7,6,1,2,3,4,5]=>[[[.,.],.],[.,[.,[.,[.,.]]]]]
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[1,2,3,5,4,6,7]=>[5,1,2,3,4,6,7]=>[[.,.],[.,[.,[.,[.,[.,.]]]]]]
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[1,2,3,5,4,7,6]=>[7,5,1,2,3,4,6]=>[[[.,.],.],[.,[.,[.,[.,.]]]]]
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]=>[1,2,3,5,6,4,7]=>[5,1,2,3,6,4,7]=>[[.,.],[.,[.,[[.,.],[.,.]]]]]
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]=>[1,2,3,5,6,7,4]=>[5,1,2,3,6,7,4]=>[[.,.],[.,[.,[[.,[.,.]],.]]]]
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]=>[1,2,3,5,7,6,4]=>[7,5,1,2,3,6,4]=>[[[.,.],.],[.,[.,[[.,.],.]]]]
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[1,2,3,6,5,4,7]=>[6,5,1,2,3,4,7]=>[[[.,.],.],[.,[.,[.,[.,.]]]]]
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]=>[1,2,3,6,5,7,4]=>[6,5,1,2,3,7,4]=>[[[.,.],.],[.,[.,[[.,.],.]]]]
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]=>[1,2,3,6,7,5,4]=>[6,7,1,2,3,5,4]=>[[.,[.,.]],[.,[.,[[.,.],.]]]]
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[1,2,3,7,6,5,4]=>[7,6,5,1,2,3,4]=>[[[[.,.],.],.],[.,[.,[.,.]]]]
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[1,2,4,3,5,6,7]=>[4,1,2,3,5,6,7]=>[[.,.],[.,[.,[.,[.,[.,.]]]]]]
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[1,2,4,3,5,7,6]=>[7,4,1,2,3,5,6]=>[[[.,.],.],[.,[.,[.,[.,.]]]]]
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[1,2,4,3,6,5,7]=>[6,4,1,2,3,5,7]=>[[[.,.],.],[.,[.,[.,[.,.]]]]]
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]=>[1,2,4,3,6,7,5]=>[6,4,1,2,3,7,5]=>[[[.,.],.],[.,[.,[[.,.],.]]]]
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[1,2,4,3,7,6,5]=>[7,6,4,1,2,3,5]=>[[[[.,.],.],.],[.,[.,[.,.]]]]
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]=>[1,2,4,5,3,6,7]=>[4,1,2,5,3,6,7]=>[[.,.],[.,[[.,.],[.,[.,.]]]]]
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]=>[1,2,4,5,3,7,6]=>[7,4,1,2,5,3,6]=>[[[.,.],.],[.,[[.,.],[.,.]]]]
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]=>[1,2,4,5,6,3,7]=>[4,1,2,5,6,3,7]=>[[.,.],[.,[[.,[.,.]],[.,.]]]]
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]=>[1,2,4,5,6,7,3]=>[4,1,2,5,6,7,3]=>[[.,.],[.,[[.,[.,[.,.]]],.]]]
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]=>[1,2,4,5,7,6,3]=>[7,4,1,2,5,6,3]=>[[[.,.],.],[.,[[.,[.,.]],.]]]
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]=>[1,2,4,6,5,3,7]=>[6,4,1,2,5,3,7]=>[[[.,.],.],[.,[[.,.],[.,.]]]]
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]=>[1,2,4,6,5,7,3]=>[6,4,1,2,5,7,3]=>[[[.,.],.],[.,[[.,[.,.]],.]]]
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]=>[1,2,4,6,7,5,3]=>[6,4,1,2,7,5,3]=>[[[.,.],.],[.,[[[.,.],.],.]]]
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]=>[1,2,4,7,6,5,3]=>[7,6,4,1,2,5,3]=>[[[[.,.],.],.],[.,[[.,.],.]]]
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>[1,2,5,4,3,6,7]=>[5,4,1,2,3,6,7]=>[[[.,.],.],[.,[.,[.,[.,.]]]]]
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>[1,2,5,4,3,7,6]=>[7,5,4,1,2,3,6]=>[[[[.,.],.],.],[.,[.,[.,.]]]]
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]=>[1,2,5,4,6,3,7]=>[5,4,1,2,6,3,7]=>[[[.,.],.],[.,[[.,.],[.,.]]]]
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]=>[1,2,5,4,6,7,3]=>[5,4,1,2,6,7,3]=>[[[.,.],.],[.,[[.,[.,.]],.]]]
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]=>[1,2,5,4,7,6,3]=>[7,5,4,1,2,6,3]=>[[[[.,.],.],.],[.,[[.,.],.]]]
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]=>[1,2,5,6,4,3,7]=>[5,6,1,2,4,3,7]=>[[.,[.,.]],[.,[[.,.],[.,.]]]]
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]=>[1,2,5,6,4,7,3]=>[5,6,1,2,4,7,3]=>[[.,[.,.]],[.,[[.,[.,.]],.]]]
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]=>[1,2,5,6,7,4,3]=>[5,6,1,2,7,4,3]=>[[.,[.,.]],[.,[[[.,.],.],.]]]
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]=>[1,2,5,7,6,4,3]=>[7,5,6,1,2,4,3]=>[[[.,.],[.,.]],[.,[[.,.],.]]]
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[1,2,6,5,4,3,7]=>[6,5,4,1,2,3,7]=>[[[[.,.],.],.],[.,[.,[.,.]]]]
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]=>[1,2,6,5,4,7,3]=>[6,5,4,1,2,7,3]=>[[[[.,.],.],.],[.,[[.,.],.]]]
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]=>[1,2,6,5,7,4,3]=>[6,5,7,1,2,4,3]=>[[[.,.],[.,.]],[.,[[.,.],.]]]
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]=>[1,2,6,7,5,4,3]=>[6,7,5,1,2,4,3]=>[[[.,[.,.]],.],[.,[[.,.],.]]]
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[1,2,7,6,5,4,3]=>[7,6,5,4,1,2,3]=>[[[[[.,.],.],.],.],[.,[.,.]]]
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[1,3,2,4,5,6,7]=>[3,1,2,4,5,6,7]=>[[.,.],[.,[.,[.,[.,[.,.]]]]]]
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[1,3,2,4,5,7,6]=>[7,3,1,2,4,5,6]=>[[[.,.],.],[.,[.,[.,[.,.]]]]]
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[1,3,2,4,6,5,7]=>[6,3,1,2,4,5,7]=>[[[.,.],.],[.,[.,[.,[.,.]]]]]
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]=>[1,3,2,4,6,7,5]=>[6,3,1,2,4,7,5]=>[[[.,.],.],[.,[.,[[.,.],.]]]]
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]=>[1,3,2,4,7,6,5]=>[7,6,3,1,2,4,5]=>[[[[.,.],.],.],[.,[.,[.,.]]]]
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[1,3,2,5,4,6,7]=>[5,3,1,2,4,6,7]=>[[[.,.],.],[.,[.,[.,[.,.]]]]]
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[1,3,2,5,4,7,6]=>[7,5,3,1,2,4,6]=>[[[[.,.],.],.],[.,[.,[.,.]]]]
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]=>[1,3,2,5,6,4,7]=>[5,3,1,2,6,4,7]=>[[[.,.],.],[.,[[.,.],[.,.]]]]
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]=>[1,3,2,5,6,7,4]=>[5,3,1,2,6,7,4]=>[[[.,.],.],[.,[[.,[.,.]],.]]]
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]=>[1,3,2,5,7,6,4]=>[7,5,3,1,2,6,4]=>[[[[.,.],.],.],[.,[[.,.],.]]]
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]=>[1,3,2,6,5,4,7]=>[6,5,3,1,2,4,7]=>[[[[.,.],.],.],[.,[.,[.,.]]]]
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]=>[1,3,2,6,5,7,4]=>[6,5,3,1,2,7,4]=>[[[[.,.],.],.],[.,[[.,.],.]]]
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]=>[1,3,2,6,7,5,4]=>[6,7,3,1,2,5,4]=>[[[.,[.,.]],.],[.,[[.,.],.]]]
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[1,3,2,7,6,5,4]=>[7,6,5,3,1,2,4]=>[[[[[.,.],.],.],.],[.,[.,.]]]
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]=>[1,3,4,2,5,6,7]=>[3,1,4,2,5,6,7]=>[[.,.],[[.,.],[.,[.,[.,.]]]]]
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]=>[1,3,4,2,5,7,6]=>[7,3,1,4,2,5,6]=>[[[.,.],.],[[.,.],[.,[.,.]]]]
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]=>[1,3,4,2,6,5,7]=>[6,3,1,4,2,5,7]=>[[[.,.],.],[[.,.],[.,[.,.]]]]
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]=>[1,3,4,2,6,7,5]=>[6,3,1,4,2,7,5]=>[[[.,.],.],[[.,.],[[.,.],.]]]
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]=>[1,3,4,2,7,6,5]=>[7,6,3,1,4,2,5]=>[[[[.,.],.],.],[[.,.],[.,.]]]
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]=>[1,3,4,5,2,6,7]=>[3,1,4,5,2,6,7]=>[[.,.],[[.,[.,.]],[.,[.,.]]]]
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]=>[1,3,4,5,2,7,6]=>[7,3,1,4,5,2,6]=>[[[.,.],.],[[.,[.,.]],[.,.]]]
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]=>[1,3,4,5,6,2,7]=>[3,1,4,5,6,2,7]=>[[.,.],[[.,[.,[.,.]]],[.,.]]]
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]=>[1,3,4,5,6,7,2]=>[3,1,4,5,6,7,2]=>[[.,.],[[.,[.,[.,[.,.]]]],.]]
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]=>[1,3,4,5,7,6,2]=>[7,3,1,4,5,6,2]=>[[[.,.],.],[[.,[.,[.,.]]],.]]
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]=>[1,3,4,6,5,2,7]=>[6,3,1,4,5,2,7]=>[[[.,.],.],[[.,[.,.]],[.,.]]]
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]=>[1,3,4,6,5,7,2]=>[6,3,1,4,5,7,2]=>[[[.,.],.],[[.,[.,[.,.]]],.]]
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]=>[1,3,4,6,7,5,2]=>[6,3,1,4,7,5,2]=>[[[.,.],.],[[.,[[.,.],.]],.]]
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]=>[1,3,4,7,6,5,2]=>[7,6,3,1,4,5,2]=>[[[[.,.],.],.],[[.,[.,.]],.]]
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]=>[1,3,5,4,2,6,7]=>[5,3,1,4,2,6,7]=>[[[.,.],.],[[.,.],[.,[.,.]]]]
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]=>[1,3,5,4,2,7,6]=>[7,5,3,1,4,2,6]=>[[[[.,.],.],.],[[.,.],[.,.]]]
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]=>[1,3,5,4,6,2,7]=>[5,3,1,4,6,2,7]=>[[[.,.],.],[[.,[.,.]],[.,.]]]
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]=>[1,3,5,4,6,7,2]=>[5,3,1,4,6,7,2]=>[[[.,.],.],[[.,[.,[.,.]]],.]]
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]=>[1,3,5,4,7,6,2]=>[7,5,3,1,4,6,2]=>[[[[.,.],.],.],[[.,[.,.]],.]]
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]=>[1,3,5,6,4,2,7]=>[5,3,1,6,4,2,7]=>[[[.,.],.],[[[.,.],.],[.,.]]]
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]=>[1,3,5,6,4,7,2]=>[5,3,1,6,4,7,2]=>[[[.,.],.],[[[.,.],[.,.]],.]]
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]=>[1,3,5,6,7,4,2]=>[5,3,1,6,7,4,2]=>[[[.,.],.],[[[.,[.,.]],.],.]]
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]=>[1,3,5,7,6,4,2]=>[7,5,3,1,6,4,2]=>[[[[.,.],.],.],[[[.,.],.],.]]
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]=>[1,3,6,5,4,2,7]=>[6,5,3,1,4,2,7]=>[[[[.,.],.],.],[[.,.],[.,.]]]
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]=>[1,3,6,5,4,7,2]=>[6,5,3,1,4,7,2]=>[[[[.,.],.],.],[[.,[.,.]],.]]
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]=>[1,3,6,5,7,4,2]=>[6,5,3,1,7,4,2]=>[[[[.,.],.],.],[[[.,.],.],.]]
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]=>[1,3,6,7,5,4,2]=>[6,7,3,1,5,4,2]=>[[[.,[.,.]],.],[[[.,.],.],.]]
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]=>[1,3,7,6,5,4,2]=>[7,6,5,3,1,4,2]=>[[[[[.,.],.],.],.],[[.,.],.]]
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[1,4,3,2,5,6,7]=>[4,3,1,2,5,6,7]=>[[[.,.],.],[.,[.,[.,[.,.]]]]]
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]=>[1,4,3,2,5,7,6]=>[7,4,3,1,2,5,6]=>[[[[.,.],.],.],[.,[.,[.,.]]]]
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]=>[1,4,3,2,6,5,7]=>[6,4,3,1,2,5,7]=>[[[[.,.],.],.],[.,[.,[.,.]]]]
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]=>[1,4,3,2,6,7,5]=>[6,4,3,1,2,7,5]=>[[[[.,.],.],.],[.,[[.,.],.]]]
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[1,4,3,2,7,6,5]=>[7,6,4,3,1,2,5]=>[[[[[.,.],.],.],.],[.,[.,.]]]
[1,0,1,1,1,0,0,1,0,0,1,0,1,0]=>[1,4,3,5,2,6,7]=>[4,3,1,5,2,6,7]=>[[[.,.],.],[[.,.],[.,[.,.]]]]
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]=>[1,4,3,5,2,7,6]=>[7,4,3,1,5,2,6]=>[[[[.,.],.],.],[[.,.],[.,.]]]
[1,0,1,1,1,0,0,1,0,1,0,0,1,0]=>[1,4,3,5,6,2,7]=>[4,3,1,5,6,2,7]=>[[[.,.],.],[[.,[.,.]],[.,.]]]
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]=>[1,4,3,5,6,7,2]=>[4,3,1,5,6,7,2]=>[[[.,.],.],[[.,[.,[.,.]]],.]]
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]=>[1,4,3,5,7,6,2]=>[7,4,3,1,5,6,2]=>[[[[.,.],.],.],[[.,[.,.]],.]]
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]=>[1,4,3,6,5,2,7]=>[6,4,3,1,5,2,7]=>[[[[.,.],.],.],[[.,.],[.,.]]]
[1,0,1,1,1,0,0,1,1,0,0,1,0,0]=>[1,4,3,6,5,7,2]=>[6,4,3,1,5,7,2]=>[[[[.,.],.],.],[[.,[.,.]],.]]
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]=>[1,4,3,6,7,5,2]=>[6,4,3,1,7,5,2]=>[[[[.,.],.],.],[[[.,.],.],.]]
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]=>[1,4,3,7,6,5,2]=>[7,6,4,3,1,5,2]=>[[[[[.,.],.],.],.],[[.,.],.]]
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]=>[1,4,5,3,2,6,7]=>[4,5,1,3,2,6,7]=>[[.,[.,.]],[[.,.],[.,[.,.]]]]
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]=>[1,4,5,3,2,7,6]=>[7,4,5,1,3,2,6]=>[[[.,.],[.,.]],[[.,.],[.,.]]]
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]=>[1,4,5,3,6,2,7]=>[4,5,1,3,6,2,7]=>[[.,[.,.]],[[.,[.,.]],[.,.]]]
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]=>[1,4,5,3,6,7,2]=>[4,5,1,3,6,7,2]=>[[.,[.,.]],[[.,[.,[.,.]]],.]]
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]=>[1,4,5,3,7,6,2]=>[7,4,5,1,3,6,2]=>[[[.,.],[.,.]],[[.,[.,.]],.]]
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]=>[1,4,5,6,3,2,7]=>[4,5,1,6,3,2,7]=>[[.,[.,.]],[[[.,.],.],[.,.]]]
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]=>[1,4,5,6,3,7,2]=>[4,5,1,6,3,7,2]=>[[.,[.,.]],[[[.,.],[.,.]],.]]
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]=>[1,4,5,6,7,3,2]=>[4,5,1,6,7,3,2]=>[[.,[.,.]],[[[.,[.,.]],.],.]]
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]=>[1,4,5,7,6,3,2]=>[7,4,5,1,6,3,2]=>[[[.,.],[.,.]],[[[.,.],.],.]]
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]=>[1,4,6,5,3,2,7]=>[6,4,5,1,3,2,7]=>[[[.,.],[.,.]],[[.,.],[.,.]]]
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]=>[1,4,6,5,3,7,2]=>[6,4,5,1,3,7,2]=>[[[.,.],[.,.]],[[.,[.,.]],.]]
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]=>[1,4,6,5,7,3,2]=>[6,4,5,1,7,3,2]=>[[[.,.],[.,.]],[[[.,.],.],.]]
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]=>[1,4,6,7,5,3,2]=>[6,4,7,1,5,3,2]=>[[[.,.],[.,.]],[[[.,.],.],.]]
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]=>[1,4,7,6,5,3,2]=>[7,6,4,5,1,3,2]=>[[[[.,.],.],[.,.]],[[.,.],.]]
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[1,5,4,3,2,6,7]=>[5,4,3,1,2,6,7]=>[[[[.,.],.],.],[.,[.,[.,.]]]]
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[1,5,4,3,2,7,6]=>[7,5,4,3,1,2,6]=>[[[[[.,.],.],.],.],[.,[.,.]]]
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]=>[1,5,4,3,6,2,7]=>[5,4,3,1,6,2,7]=>[[[[.,.],.],.],[[.,.],[.,.]]]
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]=>[1,5,4,3,6,7,2]=>[5,4,3,1,6,7,2]=>[[[[.,.],.],.],[[.,[.,.]],.]]
[1,0,1,1,1,1,0,0,0,1,1,0,0,0]=>[1,5,4,3,7,6,2]=>[7,5,4,3,1,6,2]=>[[[[[.,.],.],.],.],[[.,.],.]]
[1,0,1,1,1,1,0,0,1,0,0,0,1,0]=>[1,5,4,6,3,2,7]=>[5,4,6,1,3,2,7]=>[[[.,.],[.,.]],[[.,.],[.,.]]]
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]=>[1,5,4,6,3,7,2]=>[5,4,6,1,3,7,2]=>[[[.,.],[.,.]],[[.,[.,.]],.]]
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]=>[1,5,4,6,7,3,2]=>[5,4,6,1,7,3,2]=>[[[.,.],[.,.]],[[[.,.],.],.]]
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]=>[1,5,4,7,6,3,2]=>[7,5,4,6,1,3,2]=>[[[[.,.],.],[.,.]],[[.,.],.]]
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]=>[1,5,6,4,3,2,7]=>[5,6,4,1,3,2,7]=>[[[.,[.,.]],.],[[.,.],[.,.]]]
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]=>[1,5,6,4,3,7,2]=>[5,6,4,1,3,7,2]=>[[[.,[.,.]],.],[[.,[.,.]],.]]
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]=>[1,5,6,4,7,3,2]=>[5,6,4,1,7,3,2]=>[[[.,[.,.]],.],[[[.,.],.],.]]
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]=>[1,5,6,7,4,3,2]=>[5,6,7,1,4,3,2]=>[[.,[.,[.,.]]],[[[.,.],.],.]]
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]=>[1,5,7,6,4,3,2]=>[7,5,6,4,1,3,2]=>[[[[.,.],[.,.]],.],[[.,.],.]]
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[1,6,5,4,3,2,7]=>[6,5,4,3,1,2,7]=>[[[[[.,.],.],.],.],[.,[.,.]]]
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]=>[1,6,5,4,3,7,2]=>[6,5,4,3,1,7,2]=>[[[[[.,.],.],.],.],[[.,.],.]]
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]=>[1,6,5,4,7,3,2]=>[6,5,4,7,1,3,2]=>[[[[.,.],.],[.,.]],[[.,.],.]]
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]=>[1,6,5,7,4,3,2]=>[6,5,7,4,1,3,2]=>[[[[.,.],[.,.]],.],[[.,.],.]]
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]=>[1,6,7,5,4,3,2]=>[6,7,5,4,1,3,2]=>[[[[.,[.,.]],.],.],[[.,.],.]]
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[1,7,6,5,4,3,2]=>[7,6,5,4,3,1,2]=>[[[[[[.,.],.],.],.],.],[.,.]]
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[2,1,3,4,5,6,7]=>[2,1,3,4,5,6,7]=>[[.,.],[.,[.,[.,[.,[.,.]]]]]]
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[2,1,3,4,5,7,6]=>[7,2,1,3,4,5,6]=>[[[.,.],.],[.,[.,[.,[.,.]]]]]
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>[2,1,3,4,6,5,7]=>[6,2,1,3,4,5,7]=>[[[.,.],.],[.,[.,[.,[.,.]]]]]
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]=>[2,1,3,4,6,7,5]=>[6,2,1,3,4,7,5]=>[[[.,.],.],[.,[.,[[.,.],.]]]]
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]=>[2,1,3,4,7,6,5]=>[7,6,2,1,3,4,5]=>[[[[.,.],.],.],[.,[.,[.,.]]]]
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>[2,1,3,5,4,6,7]=>[5,2,1,3,4,6,7]=>[[[.,.],.],[.,[.,[.,[.,.]]]]]
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]=>[2,1,3,5,4,7,6]=>[7,5,2,1,3,4,6]=>[[[[.,.],.],.],[.,[.,[.,.]]]]
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]=>[2,1,3,5,6,4,7]=>[5,2,1,3,6,4,7]=>[[[.,.],.],[.,[[.,.],[.,.]]]]
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]=>[2,1,3,5,6,7,4]=>[5,2,1,3,6,7,4]=>[[[.,.],.],[.,[[.,[.,.]],.]]]
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]=>[2,1,3,5,7,6,4]=>[7,5,2,1,3,6,4]=>[[[[.,.],.],.],[.,[[.,.],.]]]
[1,1,0,0,1,0,1,1,1,0,0,0,1,0]=>[2,1,3,6,5,4,7]=>[6,5,2,1,3,4,7]=>[[[[.,.],.],.],[.,[.,[.,.]]]]
[1,1,0,0,1,0,1,1,1,0,0,1,0,0]=>[2,1,3,6,5,7,4]=>[6,5,2,1,3,7,4]=>[[[[.,.],.],.],[.,[[.,.],.]]]
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]=>[2,1,3,6,7,5,4]=>[6,7,2,1,3,5,4]=>[[[.,[.,.]],.],[.,[[.,.],.]]]
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>[2,1,3,7,6,5,4]=>[7,6,5,2,1,3,4]=>[[[[[.,.],.],.],.],[.,[.,.]]]
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[2,1,4,3,5,6,7]=>[4,2,1,3,5,6,7]=>[[[.,.],.],[.,[.,[.,[.,.]]]]]
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]=>[2,1,4,3,5,7,6]=>[7,4,2,1,3,5,6]=>[[[[.,.],.],.],[.,[.,[.,.]]]]
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[2,1,4,3,6,5,7]=>[6,4,2,1,3,5,7]=>[[[[.,.],.],.],[.,[.,[.,.]]]]
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]=>[2,1,4,3,6,7,5]=>[6,4,2,1,3,7,5]=>[[[[.,.],.],.],[.,[[.,.],.]]]
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>[2,1,4,3,7,6,5]=>[7,6,4,2,1,3,5]=>[[[[[.,.],.],.],.],[.,[.,.]]]
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]=>[2,1,4,5,3,6,7]=>[4,2,1,5,3,6,7]=>[[[.,.],.],[[.,.],[.,[.,.]]]]
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]=>[2,1,4,5,3,7,6]=>[7,4,2,1,5,3,6]=>[[[[.,.],.],.],[[.,.],[.,.]]]
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]=>[2,1,4,5,6,3,7]=>[4,2,1,5,6,3,7]=>[[[.,.],.],[[.,[.,.]],[.,.]]]
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]=>[2,1,4,5,6,7,3]=>[4,2,1,5,6,7,3]=>[[[.,.],.],[[.,[.,[.,.]]],.]]
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]=>[2,1,4,5,7,6,3]=>[7,4,2,1,5,6,3]=>[[[[.,.],.],.],[[.,[.,.]],.]]
[1,1,0,0,1,1,0,1,1,0,0,0,1,0]=>[2,1,4,6,5,3,7]=>[6,4,2,1,5,3,7]=>[[[[.,.],.],.],[[.,.],[.,.]]]
[1,1,0,0,1,1,0,1,1,0,0,1,0,0]=>[2,1,4,6,5,7,3]=>[6,4,2,1,5,7,3]=>[[[[.,.],.],.],[[.,[.,.]],.]]
[1,1,0,0,1,1,0,1,1,0,1,0,0,0]=>[2,1,4,6,7,5,3]=>[6,4,2,1,7,5,3]=>[[[[.,.],.],.],[[[.,.],.],.]]
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]=>[2,1,4,7,6,5,3]=>[7,6,4,2,1,5,3]=>[[[[[.,.],.],.],.],[[.,.],.]]
[1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>[2,1,5,4,3,6,7]=>[5,4,2,1,3,6,7]=>[[[[.,.],.],.],[.,[.,[.,.]]]]
[1,1,0,0,1,1,1,0,0,0,1,1,0,0]=>[2,1,5,4,3,7,6]=>[7,5,4,2,1,3,6]=>[[[[[.,.],.],.],.],[.,[.,.]]]
[1,1,0,0,1,1,1,0,0,1,0,0,1,0]=>[2,1,5,4,6,3,7]=>[5,4,2,1,6,3,7]=>[[[[.,.],.],.],[[.,.],[.,.]]]
[1,1,0,0,1,1,1,0,0,1,0,1,0,0]=>[2,1,5,4,6,7,3]=>[5,4,2,1,6,7,3]=>[[[[.,.],.],.],[[.,[.,.]],.]]
[1,1,0,0,1,1,1,0,0,1,1,0,0,0]=>[2,1,5,4,7,6,3]=>[7,5,4,2,1,6,3]=>[[[[[.,.],.],.],.],[[.,.],.]]
[1,1,0,0,1,1,1,0,1,0,0,0,1,0]=>[2,1,5,6,4,3,7]=>[5,6,2,1,4,3,7]=>[[[.,[.,.]],.],[[.,.],[.,.]]]
[1,1,0,0,1,1,1,0,1,0,0,1,0,0]=>[2,1,5,6,4,7,3]=>[5,6,2,1,4,7,3]=>[[[.,[.,.]],.],[[.,[.,.]],.]]
[1,1,0,0,1,1,1,0,1,0,1,0,0,0]=>[2,1,5,6,7,4,3]=>[5,6,2,1,7,4,3]=>[[[.,[.,.]],.],[[[.,.],.],.]]
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]=>[2,1,5,7,6,4,3]=>[7,5,6,2,1,4,3]=>[[[[.,.],[.,.]],.],[[.,.],.]]
[1,1,0,0,1,1,1,1,0,0,0,0,1,0]=>[2,1,6,5,4,3,7]=>[6,5,4,2,1,3,7]=>[[[[[.,.],.],.],.],[.,[.,.]]]
[1,1,0,0,1,1,1,1,0,0,0,1,0,0]=>[2,1,6,5,4,7,3]=>[6,5,4,2,1,7,3]=>[[[[[.,.],.],.],.],[[.,.],.]]
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]=>[2,1,6,5,7,4,3]=>[6,5,7,2,1,4,3]=>[[[[.,.],[.,.]],.],[[.,.],.]]
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]=>[2,1,6,7,5,4,3]=>[6,7,5,2,1,4,3]=>[[[[.,[.,.]],.],.],[[.,.],.]]
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[2,1,7,6,5,4,3]=>[7,6,5,4,2,1,3]=>[[[[[[.,.],.],.],.],.],[.,.]]
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]=>[2,3,1,4,5,6,7]=>[2,3,1,4,5,6,7]=>[[.,[.,.]],[.,[.,[.,[.,.]]]]]
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]=>[2,3,1,4,5,7,6]=>[7,2,3,1,4,5,6]=>[[[.,.],[.,.]],[.,[.,[.,.]]]]
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]=>[2,3,1,4,6,5,7]=>[6,2,3,1,4,5,7]=>[[[.,.],[.,.]],[.,[.,[.,.]]]]
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]=>[2,3,1,4,6,7,5]=>[6,2,3,1,4,7,5]=>[[[.,.],[.,.]],[.,[[.,.],.]]]
[1,1,0,1,0,0,1,0,1,1,1,0,0,0]=>[2,3,1,4,7,6,5]=>[7,6,2,3,1,4,5]=>[[[[.,.],.],[.,.]],[.,[.,.]]]
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]=>[2,3,1,5,4,6,7]=>[5,2,3,1,4,6,7]=>[[[.,.],[.,.]],[.,[.,[.,.]]]]
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]=>[2,3,1,5,4,7,6]=>[7,5,2,3,1,4,6]=>[[[[.,.],.],[.,.]],[.,[.,.]]]
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]=>[2,3,1,5,6,4,7]=>[5,2,3,1,6,4,7]=>[[[.,.],[.,.]],[[.,.],[.,.]]]
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]=>[2,3,1,5,6,7,4]=>[5,2,3,1,6,7,4]=>[[[.,.],[.,.]],[[.,[.,.]],.]]
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]=>[2,3,1,5,7,6,4]=>[7,5,2,3,1,6,4]=>[[[[.,.],.],[.,.]],[[.,.],.]]
[1,1,0,1,0,0,1,1,1,0,0,0,1,0]=>[2,3,1,6,5,4,7]=>[6,5,2,3,1,4,7]=>[[[[.,.],.],[.,.]],[.,[.,.]]]
[1,1,0,1,0,0,1,1,1,0,0,1,0,0]=>[2,3,1,6,5,7,4]=>[6,5,2,3,1,7,4]=>[[[[.,.],.],[.,.]],[[.,.],.]]
[1,1,0,1,0,0,1,1,1,0,1,0,0,0]=>[2,3,1,6,7,5,4]=>[6,7,2,3,1,5,4]=>[[[.,[.,.]],[.,.]],[[.,.],.]]
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]=>[2,3,1,7,6,5,4]=>[7,6,5,2,3,1,4]=>[[[[[.,.],.],.],[.,.]],[.,.]]
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]=>[2,3,4,1,5,6,7]=>[2,3,4,1,5,6,7]=>[[.,[.,[.,.]]],[.,[.,[.,.]]]]
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]=>[2,3,4,1,5,7,6]=>[7,2,3,4,1,5,6]=>[[[.,.],[.,[.,.]]],[.,[.,.]]]
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]=>[2,3,4,1,6,5,7]=>[6,2,3,4,1,5,7]=>[[[.,.],[.,[.,.]]],[.,[.,.]]]
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]=>[2,3,4,1,6,7,5]=>[6,2,3,4,1,7,5]=>[[[.,.],[.,[.,.]]],[[.,.],.]]
[1,1,0,1,0,1,0,0,1,1,1,0,0,0]=>[2,3,4,1,7,6,5]=>[7,6,2,3,4,1,5]=>[[[[.,.],.],[.,[.,.]]],[.,.]]
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]=>[2,3,4,5,1,6,7]=>[2,3,4,5,1,6,7]=>[[.,[.,[.,[.,.]]]],[.,[.,.]]]
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]=>[2,3,4,5,1,7,6]=>[7,2,3,4,5,1,6]=>[[[.,.],[.,[.,[.,.]]]],[.,.]]
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]=>[2,3,4,5,6,1,7]=>[2,3,4,5,6,1,7]=>[[.,[.,[.,[.,[.,.]]]]],[.,.]]
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[2,3,4,5,6,7,1]=>[2,3,4,5,6,7,1]=>[[.,[.,[.,[.,[.,[.,.]]]]]],.]
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]=>[2,3,4,5,7,6,1]=>[7,2,3,4,5,6,1]=>[[[.,.],[.,[.,[.,[.,.]]]]],.]
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]=>[2,3,4,6,5,1,7]=>[6,2,3,4,5,1,7]=>[[[.,.],[.,[.,[.,.]]]],[.,.]]
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]=>[2,3,4,6,5,7,1]=>[6,2,3,4,5,7,1]=>[[[.,.],[.,[.,[.,[.,.]]]]],.]
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]=>[2,3,4,6,7,5,1]=>[6,2,3,4,7,5,1]=>[[[.,.],[.,[.,[[.,.],.]]]],.]
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]=>[2,3,4,7,6,5,1]=>[7,6,2,3,4,5,1]=>[[[[.,.],.],[.,[.,[.,.]]]],.]
[1,1,0,1,0,1,1,0,0,0,1,0,1,0]=>[2,3,5,4,1,6,7]=>[5,2,3,4,1,6,7]=>[[[.,.],[.,[.,.]]],[.,[.,.]]]
[1,1,0,1,0,1,1,0,0,0,1,1,0,0]=>[2,3,5,4,1,7,6]=>[7,5,2,3,4,1,6]=>[[[[.,.],.],[.,[.,.]]],[.,.]]
[1,1,0,1,0,1,1,0,0,1,0,0,1,0]=>[2,3,5,4,6,1,7]=>[5,2,3,4,6,1,7]=>[[[.,.],[.,[.,[.,.]]]],[.,.]]
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]=>[2,3,5,4,6,7,1]=>[5,2,3,4,6,7,1]=>[[[.,.],[.,[.,[.,[.,.]]]]],.]
[1,1,0,1,0,1,1,0,0,1,1,0,0,0]=>[2,3,5,4,7,6,1]=>[7,5,2,3,4,6,1]=>[[[[.,.],.],[.,[.,[.,.]]]],.]
[1,1,0,1,0,1,1,0,1,0,0,0,1,0]=>[2,3,5,6,4,1,7]=>[5,2,3,6,4,1,7]=>[[[.,.],[.,[[.,.],.]]],[.,.]]
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]=>[2,3,5,6,4,7,1]=>[5,2,3,6,4,7,1]=>[[[.,.],[.,[[.,.],[.,.]]]],.]
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]=>[2,3,5,6,7,4,1]=>[5,2,3,6,7,4,1]=>[[[.,.],[.,[[.,[.,.]],.]]],.]
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]=>[2,3,5,7,6,4,1]=>[7,5,2,3,6,4,1]=>[[[[.,.],.],[.,[[.,.],.]]],.]
[1,1,0,1,0,1,1,1,0,0,0,0,1,0]=>[2,3,6,5,4,1,7]=>[6,5,2,3,4,1,7]=>[[[[.,.],.],[.,[.,.]]],[.,.]]
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]=>[2,3,6,5,4,7,1]=>[6,5,2,3,4,7,1]=>[[[[.,.],.],[.,[.,[.,.]]]],.]
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]=>[2,3,6,5,7,4,1]=>[6,5,2,3,7,4,1]=>[[[[.,.],.],[.,[[.,.],.]]],.]
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]=>[2,3,6,7,5,4,1]=>[6,7,2,3,5,4,1]=>[[[.,[.,.]],[.,[[.,.],.]]],.]
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]=>[2,3,7,6,5,4,1]=>[7,6,5,2,3,4,1]=>[[[[[.,.],.],.],[.,[.,.]]],.]
[1,1,0,1,1,0,0,0,1,0,1,0,1,0]=>[2,4,3,1,5,6,7]=>[4,2,3,1,5,6,7]=>[[[.,.],[.,.]],[.,[.,[.,.]]]]
[1,1,0,1,1,0,0,0,1,0,1,1,0,0]=>[2,4,3,1,5,7,6]=>[7,4,2,3,1,5,6]=>[[[[.,.],.],[.,.]],[.,[.,.]]]
[1,1,0,1,1,0,0,0,1,1,0,0,1,0]=>[2,4,3,1,6,5,7]=>[6,4,2,3,1,5,7]=>[[[[.,.],.],[.,.]],[.,[.,.]]]
[1,1,0,1,1,0,0,0,1,1,0,1,0,0]=>[2,4,3,1,6,7,5]=>[6,4,2,3,1,7,5]=>[[[[.,.],.],[.,.]],[[.,.],.]]
[1,1,0,1,1,0,0,0,1,1,1,0,0,0]=>[2,4,3,1,7,6,5]=>[7,6,4,2,3,1,5]=>[[[[[.,.],.],.],[.,.]],[.,.]]
[1,1,0,1,1,0,0,1,0,0,1,0,1,0]=>[2,4,3,5,1,6,7]=>[4,2,3,5,1,6,7]=>[[[.,.],[.,[.,.]]],[.,[.,.]]]
[1,1,0,1,1,0,0,1,0,0,1,1,0,0]=>[2,4,3,5,1,7,6]=>[7,4,2,3,5,1,6]=>[[[[.,.],.],[.,[.,.]]],[.,.]]
[1,1,0,1,1,0,0,1,0,1,0,0,1,0]=>[2,4,3,5,6,1,7]=>[4,2,3,5,6,1,7]=>[[[.,.],[.,[.,[.,.]]]],[.,.]]
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]=>[2,4,3,5,6,7,1]=>[4,2,3,5,6,7,1]=>[[[.,.],[.,[.,[.,[.,.]]]]],.]
[1,1,0,1,1,0,0,1,0,1,1,0,0,0]=>[2,4,3,5,7,6,1]=>[7,4,2,3,5,6,1]=>[[[[.,.],.],[.,[.,[.,.]]]],.]
[1,1,0,1,1,0,0,1,1,0,0,0,1,0]=>[2,4,3,6,5,1,7]=>[6,4,2,3,5,1,7]=>[[[[.,.],.],[.,[.,.]]],[.,.]]
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]=>[2,4,3,6,5,7,1]=>[6,4,2,3,5,7,1]=>[[[[.,.],.],[.,[.,[.,.]]]],.]
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]=>[2,4,3,6,7,5,1]=>[6,4,2,3,7,5,1]=>[[[[.,.],.],[.,[[.,.],.]]],.]
[1,1,0,1,1,0,0,1,1,1,0,0,0,0]=>[2,4,3,7,6,5,1]=>[7,6,4,2,3,5,1]=>[[[[[.,.],.],.],[.,[.,.]]],.]
[1,1,0,1,1,0,1,0,0,0,1,0,1,0]=>[2,4,5,3,1,6,7]=>[4,2,5,3,1,6,7]=>[[[.,.],[[.,.],.]],[.,[.,.]]]
[1,1,0,1,1,0,1,0,0,0,1,1,0,0]=>[2,4,5,3,1,7,6]=>[7,4,2,5,3,1,6]=>[[[[.,.],.],[[.,.],.]],[.,.]]
[1,1,0,1,1,0,1,0,0,1,0,0,1,0]=>[2,4,5,3,6,1,7]=>[4,2,5,3,6,1,7]=>[[[.,.],[[.,.],[.,.]]],[.,.]]
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]=>[2,4,5,3,6,7,1]=>[4,2,5,3,6,7,1]=>[[[.,.],[[.,.],[.,[.,.]]]],.]
[1,1,0,1,1,0,1,0,0,1,1,0,0,0]=>[2,4,5,3,7,6,1]=>[7,4,2,5,3,6,1]=>[[[[.,.],.],[[.,.],[.,.]]],.]
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]=>[2,4,5,6,3,1,7]=>[4,2,5,6,3,1,7]=>[[[.,.],[[.,[.,.]],.]],[.,.]]
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]=>[2,4,5,6,3,7,1]=>[4,2,5,6,3,7,1]=>[[[.,.],[[.,[.,.]],[.,.]]],.]
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]=>[2,4,5,6,7,3,1]=>[4,2,5,6,7,3,1]=>[[[.,.],[[.,[.,[.,.]]],.]],.]
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]=>[2,4,5,7,6,3,1]=>[7,4,2,5,6,3,1]=>[[[[.,.],.],[[.,[.,.]],.]],.]
[1,1,0,1,1,0,1,1,0,0,0,0,1,0]=>[2,4,6,5,3,1,7]=>[6,4,2,5,3,1,7]=>[[[[.,.],.],[[.,.],.]],[.,.]]
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]=>[2,4,6,5,3,7,1]=>[6,4,2,5,3,7,1]=>[[[[.,.],.],[[.,.],[.,.]]],.]
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]=>[2,4,6,5,7,3,1]=>[6,4,2,5,7,3,1]=>[[[[.,.],.],[[.,[.,.]],.]],.]
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]=>[2,4,6,7,5,3,1]=>[6,4,2,7,5,3,1]=>[[[[.,.],.],[[[.,.],.],.]],.]
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]=>[2,4,7,6,5,3,1]=>[7,6,4,2,5,3,1]=>[[[[[.,.],.],.],[[.,.],.]],.]
[1,1,0,1,1,1,0,0,0,0,1,0,1,0]=>[2,5,4,3,1,6,7]=>[5,4,2,3,1,6,7]=>[[[[.,.],.],[.,.]],[.,[.,.]]]
[1,1,0,1,1,1,0,0,0,0,1,1,0,0]=>[2,5,4,3,1,7,6]=>[7,5,4,2,3,1,6]=>[[[[[.,.],.],.],[.,.]],[.,.]]
[1,1,0,1,1,1,0,0,0,1,0,0,1,0]=>[2,5,4,3,6,1,7]=>[5,4,2,3,6,1,7]=>[[[[.,.],.],[.,[.,.]]],[.,.]]
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]=>[2,5,4,3,6,7,1]=>[5,4,2,3,6,7,1]=>[[[[.,.],.],[.,[.,[.,.]]]],.]
[1,1,0,1,1,1,0,0,0,1,1,0,0,0]=>[2,5,4,3,7,6,1]=>[7,5,4,2,3,6,1]=>[[[[[.,.],.],.],[.,[.,.]]],.]
[1,1,0,1,1,1,0,0,1,0,0,0,1,0]=>[2,5,4,6,3,1,7]=>[5,4,2,6,3,1,7]=>[[[[.,.],.],[[.,.],.]],[.,.]]
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]=>[2,5,4,6,3,7,1]=>[5,4,2,6,3,7,1]=>[[[[.,.],.],[[.,.],[.,.]]],.]
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]=>[2,5,4,6,7,3,1]=>[5,4,2,6,7,3,1]=>[[[[.,.],.],[[.,[.,.]],.]],.]
[1,1,0,1,1,1,0,0,1,1,0,0,0,0]=>[2,5,4,7,6,3,1]=>[7,5,4,2,6,3,1]=>[[[[[.,.],.],.],[[.,.],.]],.]
[1,1,0,1,1,1,0,1,0,0,0,0,1,0]=>[2,5,6,4,3,1,7]=>[5,6,2,4,3,1,7]=>[[[.,[.,.]],[[.,.],.]],[.,.]]
[1,1,0,1,1,1,0,1,0,0,0,1,0,0]=>[2,5,6,4,3,7,1]=>[5,6,2,4,3,7,1]=>[[[.,[.,.]],[[.,.],[.,.]]],.]
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]=>[2,5,6,4,7,3,1]=>[5,6,2,4,7,3,1]=>[[[.,[.,.]],[[.,[.,.]],.]],.]
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]=>[2,5,6,7,4,3,1]=>[5,6,2,7,4,3,1]=>[[[.,[.,.]],[[[.,.],.],.]],.]
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]=>[2,5,7,6,4,3,1]=>[7,5,6,2,4,3,1]=>[[[[.,.],[.,.]],[[.,.],.]],.]
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]=>[2,6,5,4,3,1,7]=>[6,5,4,2,3,1,7]=>[[[[[.,.],.],.],[.,.]],[.,.]]
[1,1,0,1,1,1,1,0,0,0,0,1,0,0]=>[2,6,5,4,3,7,1]=>[6,5,4,2,3,7,1]=>[[[[[.,.],.],.],[.,[.,.]]],.]
[1,1,0,1,1,1,1,0,0,0,1,0,0,0]=>[2,6,5,4,7,3,1]=>[6,5,4,2,7,3,1]=>[[[[[.,.],.],.],[[.,.],.]],.]
[1,1,0,1,1,1,1,0,0,1,0,0,0,0]=>[2,6,5,7,4,3,1]=>[6,5,7,2,4,3,1]=>[[[[.,.],[.,.]],[[.,.],.]],.]
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]=>[2,6,7,5,4,3,1]=>[6,7,5,2,4,3,1]=>[[[[.,[.,.]],.],[[.,.],.]],.]
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]=>[2,7,6,5,4,3,1]=>[7,6,5,4,2,3,1]=>[[[[[[.,.],.],.],.],[.,.]],.]
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>[3,2,1,4,5,6,7]=>[3,2,1,4,5,6,7]=>[[[.,.],.],[.,[.,[.,[.,.]]]]]
[1,1,1,0,0,0,1,0,1,0,1,1,0,0]=>[3,2,1,4,5,7,6]=>[7,3,2,1,4,5,6]=>[[[[.,.],.],.],[.,[.,[.,.]]]]
[1,1,1,0,0,0,1,0,1,1,0,0,1,0]=>[3,2,1,4,6,5,7]=>[6,3,2,1,4,5,7]=>[[[[.,.],.],.],[.,[.,[.,.]]]]
[1,1,1,0,0,0,1,0,1,1,0,1,0,0]=>[3,2,1,4,6,7,5]=>[6,3,2,1,4,7,5]=>[[[[.,.],.],.],[.,[[.,.],.]]]
[1,1,1,0,0,0,1,0,1,1,1,0,0,0]=>[3,2,1,4,7,6,5]=>[7,6,3,2,1,4,5]=>[[[[[.,.],.],.],.],[.,[.,.]]]
[1,1,1,0,0,0,1,1,0,0,1,0,1,0]=>[3,2,1,5,4,6,7]=>[5,3,2,1,4,6,7]=>[[[[.,.],.],.],[.,[.,[.,.]]]]
[1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>[3,2,1,5,4,7,6]=>[7,5,3,2,1,4,6]=>[[[[[.,.],.],.],.],[.,[.,.]]]
[1,1,1,0,0,0,1,1,0,1,0,0,1,0]=>[3,2,1,5,6,4,7]=>[5,3,2,1,6,4,7]=>[[[[.,.],.],.],[[.,.],[.,.]]]
[1,1,1,0,0,0,1,1,0,1,0,1,0,0]=>[3,2,1,5,6,7,4]=>[5,3,2,1,6,7,4]=>[[[[.,.],.],.],[[.,[.,.]],.]]
[1,1,1,0,0,0,1,1,0,1,1,0,0,0]=>[3,2,1,5,7,6,4]=>[7,5,3,2,1,6,4]=>[[[[[.,.],.],.],.],[[.,.],.]]
[1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>[3,2,1,6,5,4,7]=>[6,5,3,2,1,4,7]=>[[[[[.,.],.],.],.],[.,[.,.]]]
[1,1,1,0,0,0,1,1,1,0,0,1,0,0]=>[3,2,1,6,5,7,4]=>[6,5,3,2,1,7,4]=>[[[[[.,.],.],.],.],[[.,.],.]]
[1,1,1,0,0,0,1,1,1,0,1,0,0,0]=>[3,2,1,6,7,5,4]=>[6,7,3,2,1,5,4]=>[[[[.,[.,.]],.],.],[[.,.],.]]
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>[3,2,1,7,6,5,4]=>[7,6,5,3,2,1,4]=>[[[[[[.,.],.],.],.],.],[.,.]]
[1,1,1,0,0,1,0,0,1,0,1,0,1,0]=>[3,2,4,1,5,6,7]=>[3,2,4,1,5,6,7]=>[[[.,.],[.,.]],[.,[.,[.,.]]]]
[1,1,1,0,0,1,0,0,1,0,1,1,0,0]=>[3,2,4,1,5,7,6]=>[7,3,2,4,1,5,6]=>[[[[.,.],.],[.,.]],[.,[.,.]]]
[1,1,1,0,0,1,0,0,1,1,0,0,1,0]=>[3,2,4,1,6,5,7]=>[6,3,2,4,1,5,7]=>[[[[.,.],.],[.,.]],[.,[.,.]]]
[1,1,1,0,0,1,0,0,1,1,0,1,0,0]=>[3,2,4,1,6,7,5]=>[6,3,2,4,1,7,5]=>[[[[.,.],.],[.,.]],[[.,.],.]]
[1,1,1,0,0,1,0,0,1,1,1,0,0,0]=>[3,2,4,1,7,6,5]=>[7,6,3,2,4,1,5]=>[[[[[.,.],.],.],[.,.]],[.,.]]
[1,1,1,0,0,1,0,1,0,0,1,0,1,0]=>[3,2,4,5,1,6,7]=>[3,2,4,5,1,6,7]=>[[[.,.],[.,[.,.]]],[.,[.,.]]]
[1,1,1,0,0,1,0,1,0,0,1,1,0,0]=>[3,2,4,5,1,7,6]=>[7,3,2,4,5,1,6]=>[[[[.,.],.],[.,[.,.]]],[.,.]]
[1,1,1,0,0,1,0,1,0,1,0,0,1,0]=>[3,2,4,5,6,1,7]=>[3,2,4,5,6,1,7]=>[[[.,.],[.,[.,[.,.]]]],[.,.]]
[1,1,1,0,0,1,0,1,0,1,0,1,0,0]=>[3,2,4,5,6,7,1]=>[3,2,4,5,6,7,1]=>[[[.,.],[.,[.,[.,[.,.]]]]],.]
[1,1,1,0,0,1,0,1,0,1,1,0,0,0]=>[3,2,4,5,7,6,1]=>[7,3,2,4,5,6,1]=>[[[[.,.],.],[.,[.,[.,.]]]],.]
[1,1,1,0,0,1,0,1,1,0,0,0,1,0]=>[3,2,4,6,5,1,7]=>[6,3,2,4,5,1,7]=>[[[[.,.],.],[.,[.,.]]],[.,.]]
[1,1,1,0,0,1,0,1,1,0,0,1,0,0]=>[3,2,4,6,5,7,1]=>[6,3,2,4,5,7,1]=>[[[[.,.],.],[.,[.,[.,.]]]],.]
[1,1,1,0,0,1,0,1,1,0,1,0,0,0]=>[3,2,4,6,7,5,1]=>[6,3,2,4,7,5,1]=>[[[[.,.],.],[.,[[.,.],.]]],.]
[1,1,1,0,0,1,0,1,1,1,0,0,0,0]=>[3,2,4,7,6,5,1]=>[7,6,3,2,4,5,1]=>[[[[[.,.],.],.],[.,[.,.]]],.]
[1,1,1,0,0,1,1,0,0,0,1,0,1,0]=>[3,2,5,4,1,6,7]=>[5,3,2,4,1,6,7]=>[[[[.,.],.],[.,.]],[.,[.,.]]]
[1,1,1,0,0,1,1,0,0,0,1,1,0,0]=>[3,2,5,4,1,7,6]=>[7,5,3,2,4,1,6]=>[[[[[.,.],.],.],[.,.]],[.,.]]
[1,1,1,0,0,1,1,0,0,1,0,0,1,0]=>[3,2,5,4,6,1,7]=>[5,3,2,4,6,1,7]=>[[[[.,.],.],[.,[.,.]]],[.,.]]
[1,1,1,0,0,1,1,0,0,1,0,1,0,0]=>[3,2,5,4,6,7,1]=>[5,3,2,4,6,7,1]=>[[[[.,.],.],[.,[.,[.,.]]]],.]
[1,1,1,0,0,1,1,0,0,1,1,0,0,0]=>[3,2,5,4,7,6,1]=>[7,5,3,2,4,6,1]=>[[[[[.,.],.],.],[.,[.,.]]],.]
[1,1,1,0,0,1,1,0,1,0,0,0,1,0]=>[3,2,5,6,4,1,7]=>[5,3,2,6,4,1,7]=>[[[[.,.],.],[[.,.],.]],[.,.]]
[1,1,1,0,0,1,1,0,1,0,0,1,0,0]=>[3,2,5,6,4,7,1]=>[5,3,2,6,4,7,1]=>[[[[.,.],.],[[.,.],[.,.]]],.]
[1,1,1,0,0,1,1,0,1,0,1,0,0,0]=>[3,2,5,6,7,4,1]=>[5,3,2,6,7,4,1]=>[[[[.,.],.],[[.,[.,.]],.]],.]
[1,1,1,0,0,1,1,0,1,1,0,0,0,0]=>[3,2,5,7,6,4,1]=>[7,5,3,2,6,4,1]=>[[[[[.,.],.],.],[[.,.],.]],.]
[1,1,1,0,0,1,1,1,0,0,0,0,1,0]=>[3,2,6,5,4,1,7]=>[6,5,3,2,4,1,7]=>[[[[[.,.],.],.],[.,.]],[.,.]]
[1,1,1,0,0,1,1,1,0,0,0,1,0,0]=>[3,2,6,5,4,7,1]=>[6,5,3,2,4,7,1]=>[[[[[.,.],.],.],[.,[.,.]]],.]
[1,1,1,0,0,1,1,1,0,0,1,0,0,0]=>[3,2,6,5,7,4,1]=>[6,5,3,2,7,4,1]=>[[[[[.,.],.],.],[[.,.],.]],.]
[1,1,1,0,0,1,1,1,0,1,0,0,0,0]=>[3,2,6,7,5,4,1]=>[6,7,3,2,5,4,1]=>[[[[.,[.,.]],.],[[.,.],.]],.]
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]=>[3,2,7,6,5,4,1]=>[7,6,5,3,2,4,1]=>[[[[[[.,.],.],.],.],[.,.]],.]
[1,1,1,0,1,0,0,0,1,0,1,0,1,0]=>[3,4,2,1,5,6,7]=>[3,4,2,1,5,6,7]=>[[[.,[.,.]],.],[.,[.,[.,.]]]]
[1,1,1,0,1,0,0,0,1,0,1,1,0,0]=>[3,4,2,1,5,7,6]=>[7,3,4,2,1,5,6]=>[[[[.,.],[.,.]],.],[.,[.,.]]]
[1,1,1,0,1,0,0,0,1,1,0,0,1,0]=>[3,4,2,1,6,5,7]=>[6,3,4,2,1,5,7]=>[[[[.,.],[.,.]],.],[.,[.,.]]]
[1,1,1,0,1,0,0,0,1,1,0,1,0,0]=>[3,4,2,1,6,7,5]=>[6,3,4,2,1,7,5]=>[[[[.,.],[.,.]],.],[[.,.],.]]
[1,1,1,0,1,0,0,0,1,1,1,0,0,0]=>[3,4,2,1,7,6,5]=>[7,6,3,4,2,1,5]=>[[[[[.,.],.],[.,.]],.],[.,.]]
[1,1,1,0,1,0,0,1,0,0,1,0,1,0]=>[3,4,2,5,1,6,7]=>[3,4,2,5,1,6,7]=>[[[.,[.,.]],[.,.]],[.,[.,.]]]
[1,1,1,0,1,0,0,1,0,0,1,1,0,0]=>[3,4,2,5,1,7,6]=>[7,3,4,2,5,1,6]=>[[[[.,.],[.,.]],[.,.]],[.,.]]
[1,1,1,0,1,0,0,1,0,1,0,0,1,0]=>[3,4,2,5,6,1,7]=>[3,4,2,5,6,1,7]=>[[[.,[.,.]],[.,[.,.]]],[.,.]]
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]=>[3,4,2,5,6,7,1]=>[3,4,2,5,6,7,1]=>[[[.,[.,.]],[.,[.,[.,.]]]],.]
[1,1,1,0,1,0,0,1,0,1,1,0,0,0]=>[3,4,2,5,7,6,1]=>[7,3,4,2,5,6,1]=>[[[[.,.],[.,.]],[.,[.,.]]],.]
[1,1,1,0,1,0,0,1,1,0,0,0,1,0]=>[3,4,2,6,5,1,7]=>[6,3,4,2,5,1,7]=>[[[[.,.],[.,.]],[.,.]],[.,.]]
[1,1,1,0,1,0,0,1,1,0,0,1,0,0]=>[3,4,2,6,5,7,1]=>[6,3,4,2,5,7,1]=>[[[[.,.],[.,.]],[.,[.,.]]],.]
[1,1,1,0,1,0,0,1,1,0,1,0,0,0]=>[3,4,2,6,7,5,1]=>[6,3,4,2,7,5,1]=>[[[[.,.],[.,.]],[[.,.],.]],.]
[1,1,1,0,1,0,0,1,1,1,0,0,0,0]=>[3,4,2,7,6,5,1]=>[7,6,3,4,2,5,1]=>[[[[[.,.],.],[.,.]],[.,.]],.]
[1,1,1,0,1,0,1,0,0,0,1,0,1,0]=>[3,4,5,2,1,6,7]=>[3,4,5,2,1,6,7]=>[[[.,[.,[.,.]]],.],[.,[.,.]]]
[1,1,1,0,1,0,1,0,0,0,1,1,0,0]=>[3,4,5,2,1,7,6]=>[7,3,4,5,2,1,6]=>[[[[.,.],[.,[.,.]]],.],[.,.]]
[1,1,1,0,1,0,1,0,0,1,0,0,1,0]=>[3,4,5,2,6,1,7]=>[3,4,5,2,6,1,7]=>[[[.,[.,[.,.]]],[.,.]],[.,.]]
[1,1,1,0,1,0,1,0,0,1,0,1,0,0]=>[3,4,5,2,6,7,1]=>[3,4,5,2,6,7,1]=>[[[.,[.,[.,.]]],[.,[.,.]]],.]
[1,1,1,0,1,0,1,0,0,1,1,0,0,0]=>[3,4,5,2,7,6,1]=>[7,3,4,5,2,6,1]=>[[[[.,.],[.,[.,.]]],[.,.]],.]
[1,1,1,0,1,0,1,0,1,0,0,0,1,0]=>[3,4,5,6,2,1,7]=>[3,4,5,6,2,1,7]=>[[[.,[.,[.,[.,.]]]],.],[.,.]]
[1,1,1,0,1,0,1,0,1,0,0,1,0,0]=>[3,4,5,6,2,7,1]=>[3,4,5,6,2,7,1]=>[[[.,[.,[.,[.,.]]]],[.,.]],.]
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]=>[3,4,5,6,7,2,1]=>[3,4,5,6,7,2,1]=>[[[.,[.,[.,[.,[.,.]]]]],.],.]
[1,1,1,0,1,0,1,0,1,1,0,0,0,0]=>[3,4,5,7,6,2,1]=>[7,3,4,5,6,2,1]=>[[[[.,.],[.,[.,[.,.]]]],.],.]
[1,1,1,0,1,0,1,1,0,0,0,0,1,0]=>[3,4,6,5,2,1,7]=>[6,3,4,5,2,1,7]=>[[[[.,.],[.,[.,.]]],.],[.,.]]
[1,1,1,0,1,0,1,1,0,0,0,1,0,0]=>[3,4,6,5,2,7,1]=>[6,3,4,5,2,7,1]=>[[[[.,.],[.,[.,.]]],[.,.]],.]
[1,1,1,0,1,0,1,1,0,0,1,0,0,0]=>[3,4,6,5,7,2,1]=>[6,3,4,5,7,2,1]=>[[[[.,.],[.,[.,[.,.]]]],.],.]
[1,1,1,0,1,0,1,1,0,1,0,0,0,0]=>[3,4,6,7,5,2,1]=>[6,3,4,7,5,2,1]=>[[[[.,.],[.,[[.,.],.]]],.],.]
[1,1,1,0,1,0,1,1,1,0,0,0,0,0]=>[3,4,7,6,5,2,1]=>[7,6,3,4,5,2,1]=>[[[[[.,.],.],[.,[.,.]]],.],.]
[1,1,1,0,1,1,0,0,0,0,1,0,1,0]=>[3,5,4,2,1,6,7]=>[5,3,4,2,1,6,7]=>[[[[.,.],[.,.]],.],[.,[.,.]]]
[1,1,1,0,1,1,0,0,0,0,1,1,0,0]=>[3,5,4,2,1,7,6]=>[7,5,3,4,2,1,6]=>[[[[[.,.],.],[.,.]],.],[.,.]]
[1,1,1,0,1,1,0,0,0,1,0,0,1,0]=>[3,5,4,2,6,1,7]=>[5,3,4,2,6,1,7]=>[[[[.,.],[.,.]],[.,.]],[.,.]]
[1,1,1,0,1,1,0,0,0,1,0,1,0,0]=>[3,5,4,2,6,7,1]=>[5,3,4,2,6,7,1]=>[[[[.,.],[.,.]],[.,[.,.]]],.]
[1,1,1,0,1,1,0,0,0,1,1,0,0,0]=>[3,5,4,2,7,6,1]=>[7,5,3,4,2,6,1]=>[[[[[.,.],.],[.,.]],[.,.]],.]
[1,1,1,0,1,1,0,0,1,0,0,0,1,0]=>[3,5,4,6,2,1,7]=>[5,3,4,6,2,1,7]=>[[[[.,.],[.,[.,.]]],.],[.,.]]
[1,1,1,0,1,1,0,0,1,0,0,1,0,0]=>[3,5,4,6,2,7,1]=>[5,3,4,6,2,7,1]=>[[[[.,.],[.,[.,.]]],[.,.]],.]
[1,1,1,0,1,1,0,0,1,0,1,0,0,0]=>[3,5,4,6,7,2,1]=>[5,3,4,6,7,2,1]=>[[[[.,.],[.,[.,[.,.]]]],.],.]
[1,1,1,0,1,1,0,0,1,1,0,0,0,0]=>[3,5,4,7,6,2,1]=>[7,5,3,4,6,2,1]=>[[[[[.,.],.],[.,[.,.]]],.],.]
[1,1,1,0,1,1,0,1,0,0,0,0,1,0]=>[3,5,6,4,2,1,7]=>[5,3,6,4,2,1,7]=>[[[[.,.],[[.,.],.]],.],[.,.]]
[1,1,1,0,1,1,0,1,0,0,0,1,0,0]=>[3,5,6,4,2,7,1]=>[5,3,6,4,2,7,1]=>[[[[.,.],[[.,.],.]],[.,.]],.]
[1,1,1,0,1,1,0,1,0,0,1,0,0,0]=>[3,5,6,4,7,2,1]=>[5,3,6,4,7,2,1]=>[[[[.,.],[[.,.],[.,.]]],.],.]
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]=>[3,5,6,7,4,2,1]=>[5,3,6,7,4,2,1]=>[[[[.,.],[[.,[.,.]],.]],.],.]
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]=>[3,5,7,6,4,2,1]=>[7,5,3,6,4,2,1]=>[[[[[.,.],.],[[.,.],.]],.],.]
[1,1,1,0,1,1,1,0,0,0,0,0,1,0]=>[3,6,5,4,2,1,7]=>[6,5,3,4,2,1,7]=>[[[[[.,.],.],[.,.]],.],[.,.]]
[1,1,1,0,1,1,1,0,0,0,0,1,0,0]=>[3,6,5,4,2,7,1]=>[6,5,3,4,2,7,1]=>[[[[[.,.],.],[.,.]],[.,.]],.]
[1,1,1,0,1,1,1,0,0,0,1,0,0,0]=>[3,6,5,4,7,2,1]=>[6,5,3,4,7,2,1]=>[[[[[.,.],.],[.,[.,.]]],.],.]
[1,1,1,0,1,1,1,0,0,1,0,0,0,0]=>[3,6,5,7,4,2,1]=>[6,5,3,7,4,2,1]=>[[[[[.,.],.],[[.,.],.]],.],.]
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]=>[3,6,7,5,4,2,1]=>[6,7,3,5,4,2,1]=>[[[[.,[.,.]],[[.,.],.]],.],.]
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]=>[3,7,6,5,4,2,1]=>[7,6,5,3,4,2,1]=>[[[[[[.,.],.],.],[.,.]],.],.]
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>[4,3,2,1,5,6,7]=>[4,3,2,1,5,6,7]=>[[[[.,.],.],.],[.,[.,[.,.]]]]
[1,1,1,1,0,0,0,0,1,0,1,1,0,0]=>[4,3,2,1,5,7,6]=>[7,4,3,2,1,5,6]=>[[[[[.,.],.],.],.],[.,[.,.]]]
[1,1,1,1,0,0,0,0,1,1,0,0,1,0]=>[4,3,2,1,6,5,7]=>[6,4,3,2,1,5,7]=>[[[[[.,.],.],.],.],[.,[.,.]]]
[1,1,1,1,0,0,0,0,1,1,0,1,0,0]=>[4,3,2,1,6,7,5]=>[6,4,3,2,1,7,5]=>[[[[[.,.],.],.],.],[[.,.],.]]
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>[4,3,2,1,7,6,5]=>[7,6,4,3,2,1,5]=>[[[[[[.,.],.],.],.],.],[.,.]]
[1,1,1,1,0,0,0,1,0,0,1,0,1,0]=>[4,3,2,5,1,6,7]=>[4,3,2,5,1,6,7]=>[[[[.,.],.],[.,.]],[.,[.,.]]]
[1,1,1,1,0,0,0,1,0,0,1,1,0,0]=>[4,3,2,5,1,7,6]=>[7,4,3,2,5,1,6]=>[[[[[.,.],.],.],[.,.]],[.,.]]
[1,1,1,1,0,0,0,1,0,1,0,0,1,0]=>[4,3,2,5,6,1,7]=>[4,3,2,5,6,1,7]=>[[[[.,.],.],[.,[.,.]]],[.,.]]
[1,1,1,1,0,0,0,1,0,1,0,1,0,0]=>[4,3,2,5,6,7,1]=>[4,3,2,5,6,7,1]=>[[[[.,.],.],[.,[.,[.,.]]]],.]
[1,1,1,1,0,0,0,1,0,1,1,0,0,0]=>[4,3,2,5,7,6,1]=>[7,4,3,2,5,6,1]=>[[[[[.,.],.],.],[.,[.,.]]],.]
[1,1,1,1,0,0,0,1,1,0,0,0,1,0]=>[4,3,2,6,5,1,7]=>[6,4,3,2,5,1,7]=>[[[[[.,.],.],.],[.,.]],[.,.]]
[1,1,1,1,0,0,0,1,1,0,0,1,0,0]=>[4,3,2,6,5,7,1]=>[6,4,3,2,5,7,1]=>[[[[[.,.],.],.],[.,[.,.]]],.]
[1,1,1,1,0,0,0,1,1,0,1,0,0,0]=>[4,3,2,6,7,5,1]=>[6,4,3,2,7,5,1]=>[[[[[.,.],.],.],[[.,.],.]],.]
[1,1,1,1,0,0,0,1,1,1,0,0,0,0]=>[4,3,2,7,6,5,1]=>[7,6,4,3,2,5,1]=>[[[[[[.,.],.],.],.],[.,.]],.]
[1,1,1,1,0,0,1,0,0,0,1,0,1,0]=>[4,3,5,2,1,6,7]=>[4,3,5,2,1,6,7]=>[[[[.,.],[.,.]],.],[.,[.,.]]]
[1,1,1,1,0,0,1,0,0,0,1,1,0,0]=>[4,3,5,2,1,7,6]=>[7,4,3,5,2,1,6]=>[[[[[.,.],.],[.,.]],.],[.,.]]
[1,1,1,1,0,0,1,0,0,1,0,0,1,0]=>[4,3,5,2,6,1,7]=>[4,3,5,2,6,1,7]=>[[[[.,.],[.,.]],[.,.]],[.,.]]
[1,1,1,1,0,0,1,0,0,1,0,1,0,0]=>[4,3,5,2,6,7,1]=>[4,3,5,2,6,7,1]=>[[[[.,.],[.,.]],[.,[.,.]]],.]
[1,1,1,1,0,0,1,0,0,1,1,0,0,0]=>[4,3,5,2,7,6,1]=>[7,4,3,5,2,6,1]=>[[[[[.,.],.],[.,.]],[.,.]],.]
[1,1,1,1,0,0,1,0,1,0,0,0,1,0]=>[4,3,5,6,2,1,7]=>[4,3,5,6,2,1,7]=>[[[[.,.],[.,[.,.]]],.],[.,.]]
[1,1,1,1,0,0,1,0,1,0,0,1,0,0]=>[4,3,5,6,2,7,1]=>[4,3,5,6,2,7,1]=>[[[[.,.],[.,[.,.]]],[.,.]],.]
[1,1,1,1,0,0,1,0,1,0,1,0,0,0]=>[4,3,5,6,7,2,1]=>[4,3,5,6,7,2,1]=>[[[[.,.],[.,[.,[.,.]]]],.],.]
[1,1,1,1,0,0,1,0,1,1,0,0,0,0]=>[4,3,5,7,6,2,1]=>[7,4,3,5,6,2,1]=>[[[[[.,.],.],[.,[.,.]]],.],.]
[1,1,1,1,0,0,1,1,0,0,0,0,1,0]=>[4,3,6,5,2,1,7]=>[6,4,3,5,2,1,7]=>[[[[[.,.],.],[.,.]],.],[.,.]]
[1,1,1,1,0,0,1,1,0,0,0,1,0,0]=>[4,3,6,5,2,7,1]=>[6,4,3,5,2,7,1]=>[[[[[.,.],.],[.,.]],[.,.]],.]
[1,1,1,1,0,0,1,1,0,0,1,0,0,0]=>[4,3,6,5,7,2,1]=>[6,4,3,5,7,2,1]=>[[[[[.,.],.],[.,[.,.]]],.],.]
[1,1,1,1,0,0,1,1,0,1,0,0,0,0]=>[4,3,6,7,5,2,1]=>[6,4,3,7,5,2,1]=>[[[[[.,.],.],[[.,.],.]],.],.]
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]=>[4,3,7,6,5,2,1]=>[7,6,4,3,5,2,1]=>[[[[[[.,.],.],.],[.,.]],.],.]
[1,1,1,1,0,1,0,0,0,0,1,0,1,0]=>[4,5,3,2,1,6,7]=>[4,5,3,2,1,6,7]=>[[[[.,[.,.]],.],.],[.,[.,.]]]
[1,1,1,1,0,1,0,0,0,0,1,1,0,0]=>[4,5,3,2,1,7,6]=>[7,4,5,3,2,1,6]=>[[[[[.,.],[.,.]],.],.],[.,.]]
[1,1,1,1,0,1,0,0,0,1,0,0,1,0]=>[4,5,3,2,6,1,7]=>[4,5,3,2,6,1,7]=>[[[[.,[.,.]],.],[.,.]],[.,.]]
[1,1,1,1,0,1,0,0,0,1,0,1,0,0]=>[4,5,3,2,6,7,1]=>[4,5,3,2,6,7,1]=>[[[[.,[.,.]],.],[.,[.,.]]],.]
[1,1,1,1,0,1,0,0,0,1,1,0,0,0]=>[4,5,3,2,7,6,1]=>[7,4,5,3,2,6,1]=>[[[[[.,.],[.,.]],.],[.,.]],.]
[1,1,1,1,0,1,0,0,1,0,0,0,1,0]=>[4,5,3,6,2,1,7]=>[4,5,3,6,2,1,7]=>[[[[.,[.,.]],[.,.]],.],[.,.]]
[1,1,1,1,0,1,0,0,1,0,0,1,0,0]=>[4,5,3,6,2,7,1]=>[4,5,3,6,2,7,1]=>[[[[.,[.,.]],[.,.]],[.,.]],.]
[1,1,1,1,0,1,0,0,1,0,1,0,0,0]=>[4,5,3,6,7,2,1]=>[4,5,3,6,7,2,1]=>[[[[.,[.,.]],[.,[.,.]]],.],.]
[1,1,1,1,0,1,0,0,1,1,0,0,0,0]=>[4,5,3,7,6,2,1]=>[7,4,5,3,6,2,1]=>[[[[[.,.],[.,.]],[.,.]],.],.]
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]=>[4,5,6,3,2,1,7]=>[4,5,6,3,2,1,7]=>[[[[.,[.,[.,.]]],.],.],[.,.]]
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]=>[4,5,6,3,2,7,1]=>[4,5,6,3,2,7,1]=>[[[[.,[.,[.,.]]],.],[.,.]],.]
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]=>[4,5,6,3,7,2,1]=>[4,5,6,3,7,2,1]=>[[[[.,[.,[.,.]]],[.,.]],.],.]
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]=>[4,5,6,7,3,2,1]=>[4,5,6,7,3,2,1]=>[[[[.,[.,[.,[.,.]]]],.],.],.]
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]=>[4,5,7,6,3,2,1]=>[7,4,5,6,3,2,1]=>[[[[[.,.],[.,[.,.]]],.],.],.]
[1,1,1,1,0,1,1,0,0,0,0,0,1,0]=>[4,6,5,3,2,1,7]=>[6,4,5,3,2,1,7]=>[[[[[.,.],[.,.]],.],.],[.,.]]
[1,1,1,1,0,1,1,0,0,0,0,1,0,0]=>[4,6,5,3,2,7,1]=>[6,4,5,3,2,7,1]=>[[[[[.,.],[.,.]],.],[.,.]],.]
[1,1,1,1,0,1,1,0,0,0,1,0,0,0]=>[4,6,5,3,7,2,1]=>[6,4,5,3,7,2,1]=>[[[[[.,.],[.,.]],[.,.]],.],.]
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]=>[4,6,5,7,3,2,1]=>[6,4,5,7,3,2,1]=>[[[[[.,.],[.,[.,.]]],.],.],.]
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]=>[4,6,7,5,3,2,1]=>[6,4,7,5,3,2,1]=>[[[[[.,.],[[.,.],.]],.],.],.]
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]=>[4,7,6,5,3,2,1]=>[7,6,4,5,3,2,1]=>[[[[[[.,.],.],[.,.]],.],.],.]
[1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>[5,4,3,2,1,6,7]=>[5,4,3,2,1,6,7]=>[[[[[.,.],.],.],.],[.,[.,.]]]
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>[5,4,3,2,1,7,6]=>[7,5,4,3,2,1,6]=>[[[[[[.,.],.],.],.],.],[.,.]]
[1,1,1,1,1,0,0,0,0,1,0,0,1,0]=>[5,4,3,2,6,1,7]=>[5,4,3,2,6,1,7]=>[[[[[.,.],.],.],[.,.]],[.,.]]
[1,1,1,1,1,0,0,0,0,1,0,1,0,0]=>[5,4,3,2,6,7,1]=>[5,4,3,2,6,7,1]=>[[[[[.,.],.],.],[.,[.,.]]],.]
[1,1,1,1,1,0,0,0,0,1,1,0,0,0]=>[5,4,3,2,7,6,1]=>[7,5,4,3,2,6,1]=>[[[[[[.,.],.],.],.],[.,.]],.]
[1,1,1,1,1,0,0,0,1,0,0,0,1,0]=>[5,4,3,6,2,1,7]=>[5,4,3,6,2,1,7]=>[[[[[.,.],.],[.,.]],.],[.,.]]
[1,1,1,1,1,0,0,0,1,0,0,1,0,0]=>[5,4,3,6,2,7,1]=>[5,4,3,6,2,7,1]=>[[[[[.,.],.],[.,.]],[.,.]],.]
[1,1,1,1,1,0,0,0,1,0,1,0,0,0]=>[5,4,3,6,7,2,1]=>[5,4,3,6,7,2,1]=>[[[[[.,.],.],[.,[.,.]]],.],.]
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]=>[5,4,3,7,6,2,1]=>[7,5,4,3,6,2,1]=>[[[[[[.,.],.],.],[.,.]],.],.]
[1,1,1,1,1,0,0,1,0,0,0,0,1,0]=>[5,4,6,3,2,1,7]=>[5,4,6,3,2,1,7]=>[[[[[.,.],[.,.]],.],.],[.,.]]
[1,1,1,1,1,0,0,1,0,0,0,1,0,0]=>[5,4,6,3,2,7,1]=>[5,4,6,3,2,7,1]=>[[[[[.,.],[.,.]],.],[.,.]],.]
[1,1,1,1,1,0,0,1,0,0,1,0,0,0]=>[5,4,6,3,7,2,1]=>[5,4,6,3,7,2,1]=>[[[[[.,.],[.,.]],[.,.]],.],.]
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]=>[5,4,6,7,3,2,1]=>[5,4,6,7,3,2,1]=>[[[[[.,.],[.,[.,.]]],.],.],.]
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]=>[5,4,7,6,3,2,1]=>[7,5,4,6,3,2,1]=>[[[[[[.,.],.],[.,.]],.],.],.]
[1,1,1,1,1,0,1,0,0,0,0,0,1,0]=>[5,6,4,3,2,1,7]=>[5,6,4,3,2,1,7]=>[[[[[.,[.,.]],.],.],.],[.,.]]
[1,1,1,1,1,0,1,0,0,0,0,1,0,0]=>[5,6,4,3,2,7,1]=>[5,6,4,3,2,7,1]=>[[[[[.,[.,.]],.],.],[.,.]],.]
[1,1,1,1,1,0,1,0,0,0,1,0,0,0]=>[5,6,4,3,7,2,1]=>[5,6,4,3,7,2,1]=>[[[[[.,[.,.]],.],[.,.]],.],.]
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]=>[5,6,4,7,3,2,1]=>[5,6,4,7,3,2,1]=>[[[[[.,[.,.]],[.,.]],.],.],.]
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]=>[5,6,7,4,3,2,1]=>[5,6,7,4,3,2,1]=>[[[[[.,[.,[.,.]]],.],.],.],.]
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]=>[5,7,6,4,3,2,1]=>[7,5,6,4,3,2,1]=>[[[[[[.,.],[.,.]],.],.],.],.]
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[6,5,4,3,2,1,7]=>[6,5,4,3,2,1,7]=>[[[[[[.,.],.],.],.],.],[.,.]]
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]=>[6,5,4,3,2,7,1]=>[6,5,4,3,2,7,1]=>[[[[[[.,.],.],.],.],[.,.]],.]
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]=>[6,5,4,3,7,2,1]=>[6,5,4,3,7,2,1]=>[[[[[[.,.],.],.],[.,.]],.],.]
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]=>[6,5,4,7,3,2,1]=>[6,5,4,7,3,2,1]=>[[[[[[.,.],.],[.,.]],.],.],.]
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]=>[6,5,7,4,3,2,1]=>[6,5,7,4,3,2,1]=>[[[[[[.,.],[.,.]],.],.],.],.]
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]=>[6,7,5,4,3,2,1]=>[6,7,5,4,3,2,1]=>[[[[[[.,[.,.]],.],.],.],.],.]
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[7,6,5,4,3,2,1]=>[7,6,5,4,3,2,1]=>[[[[[[[.,.],.],.],.],.],.],.]
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[1,2,3,4,5,6,7,8]=>[1,2,3,4,5,6,7,8]=>[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>[[[[[[[[.,.],.],.],.],.],.],.],.]
Map
to 312-avoiding permutation
Description
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.).
Map
to increasing tree
Description
Sends a permutation to its associated increasing tree.
This tree is recursively obtained by sending the unique permutation of length $0$ to the empty tree, and sending a permutation $\sigma$ of length $n \geq 1$ to a root node with two subtrees $L$ and $R$ by splitting $\sigma$ at the index $\sigma^{-1}(1)$, normalizing both sides again to permutations and sending the permutations on the left and on the right of $\sigma^{-1}(1)$ to the trees $L$ and $R$, respectively.
This tree is recursively obtained by sending the unique permutation of length $0$ to the empty tree, and sending a permutation $\sigma$ of length $n \geq 1$ to a root node with two subtrees $L$ and $R$ by splitting $\sigma$ at the index $\sigma^{-1}(1)$, normalizing both sides again to permutations and sending the permutations on the left and on the right of $\sigma^{-1}(1)$ to the trees $L$ and $R$, respectively.
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