Identifier
Mp00017:
Binary trees
—to 312-avoiding permutation⟶
Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Images
=>
Cc0010;cc-rep-0Cc0010;cc-rep-3
[.,.]=>[1]=>[1]=>[.,.]
[.,[.,.]]=>[2,1]=>[2,1]=>[[.,.],.]
[[.,.],.]=>[1,2]=>[1,2]=>[.,[.,.]]
[.,[.,[.,.]]]=>[3,2,1]=>[3,2,1]=>[[[.,.],.],.]
[.,[[.,.],.]]=>[2,3,1]=>[3,1,2]=>[[.,.],[.,.]]
[[.,.],[.,.]]=>[1,3,2]=>[2,3,1]=>[[.,[.,.]],.]
[[.,[.,.]],.]=>[2,1,3]=>[2,1,3]=>[[.,.],[.,.]]
[[[.,.],.],.]=>[1,2,3]=>[1,2,3]=>[.,[.,[.,.]]]
[.,[.,[.,[.,.]]]]=>[4,3,2,1]=>[4,3,2,1]=>[[[[.,.],.],.],.]
[.,[.,[[.,.],.]]]=>[3,4,2,1]=>[4,3,1,2]=>[[[.,.],.],[.,.]]
[.,[[.,.],[.,.]]]=>[2,4,3,1]=>[4,2,3,1]=>[[[.,.],[.,.]],.]
[.,[[.,[.,.]],.]]=>[3,2,4,1]=>[4,2,1,3]=>[[[.,.],.],[.,.]]
[.,[[[.,.],.],.]]=>[2,3,4,1]=>[4,1,2,3]=>[[.,.],[.,[.,.]]]
[[.,.],[.,[.,.]]]=>[1,4,3,2]=>[3,4,2,1]=>[[[.,[.,.]],.],.]
[[.,.],[[.,.],.]]=>[1,3,4,2]=>[2,4,1,3]=>[[.,[.,.]],[.,.]]
[[.,[.,.]],[.,.]]=>[2,1,4,3]=>[3,2,4,1]=>[[[.,.],[.,.]],.]
[[[.,.],.],[.,.]]=>[1,2,4,3]=>[2,3,4,1]=>[[.,[.,[.,.]]],.]
[[.,[.,[.,.]]],.]=>[3,2,1,4]=>[3,2,1,4]=>[[[.,.],.],[.,.]]
[[.,[[.,.],.]],.]=>[2,3,1,4]=>[3,1,2,4]=>[[.,.],[.,[.,.]]]
[[[.,.],[.,.]],.]=>[1,3,2,4]=>[2,3,1,4]=>[[.,[.,.]],[.,.]]
[[[.,[.,.]],.],.]=>[2,1,3,4]=>[2,1,3,4]=>[[.,.],[.,[.,.]]]
[[[[.,.],.],.],.]=>[1,2,3,4]=>[1,2,3,4]=>[.,[.,[.,[.,.]]]]
[.,[.,[.,[.,[.,.]]]]]=>[5,4,3,2,1]=>[5,4,3,2,1]=>[[[[[.,.],.],.],.],.]
[.,[.,[.,[[.,.],.]]]]=>[4,5,3,2,1]=>[5,4,3,1,2]=>[[[[.,.],.],.],[.,.]]
[.,[.,[[.,.],[.,.]]]]=>[3,5,4,2,1]=>[5,4,2,3,1]=>[[[[.,.],.],[.,.]],.]
[.,[.,[[.,[.,.]],.]]]=>[4,3,5,2,1]=>[5,4,2,1,3]=>[[[[.,.],.],.],[.,.]]
[.,[.,[[[.,.],.],.]]]=>[3,4,5,2,1]=>[5,4,1,2,3]=>[[[.,.],.],[.,[.,.]]]
[.,[[.,.],[.,[.,.]]]]=>[2,5,4,3,1]=>[5,3,4,2,1]=>[[[[.,.],[.,.]],.],.]
[.,[[.,.],[[.,.],.]]]=>[2,4,5,3,1]=>[5,2,4,1,3]=>[[[.,.],[.,.]],[.,.]]
[.,[[.,[.,.]],[.,.]]]=>[3,2,5,4,1]=>[5,3,2,4,1]=>[[[[.,.],.],[.,.]],.]
[.,[[[.,.],.],[.,.]]]=>[2,3,5,4,1]=>[5,2,3,4,1]=>[[[.,.],[.,[.,.]]],.]
[.,[[.,[.,[.,.]]],.]]=>[4,3,2,5,1]=>[5,3,2,1,4]=>[[[[.,.],.],.],[.,.]]
[.,[[.,[[.,.],.]],.]]=>[3,4,2,5,1]=>[5,3,1,2,4]=>[[[.,.],.],[.,[.,.]]]
[.,[[[.,.],[.,.]],.]]=>[2,4,3,5,1]=>[5,2,3,1,4]=>[[[.,.],[.,.]],[.,.]]
[.,[[[.,[.,.]],.],.]]=>[3,2,4,5,1]=>[5,2,1,3,4]=>[[[.,.],.],[.,[.,.]]]
[.,[[[[.,.],.],.],.]]=>[2,3,4,5,1]=>[5,1,2,3,4]=>[[.,.],[.,[.,[.,.]]]]
[[.,.],[.,[.,[.,.]]]]=>[1,5,4,3,2]=>[4,5,3,2,1]=>[[[[.,[.,.]],.],.],.]
[[.,.],[.,[[.,.],.]]]=>[1,4,5,3,2]=>[3,5,4,1,2]=>[[.,[[.,.],.]],[.,.]]
[[.,.],[[.,.],[.,.]]]=>[1,3,5,4,2]=>[3,5,2,4,1]=>[[[.,[.,.]],[.,.]],.]
[[.,.],[[.,[.,.]],.]]=>[1,4,3,5,2]=>[3,5,2,1,4]=>[[[.,[.,.]],.],[.,.]]
[[.,.],[[[.,.],.],.]]=>[1,3,4,5,2]=>[2,5,1,3,4]=>[[.,[.,.]],[.,[.,.]]]
[[.,[.,.]],[.,[.,.]]]=>[2,1,5,4,3]=>[4,3,5,2,1]=>[[[[.,.],[.,.]],.],.]
[[.,[.,.]],[[.,.],.]]=>[2,1,4,5,3]=>[3,2,5,1,4]=>[[[.,.],[.,.]],[.,.]]
[[[.,.],.],[.,[.,.]]]=>[1,2,5,4,3]=>[3,4,5,2,1]=>[[[.,[.,[.,.]]],.],.]
[[[.,.],.],[[.,.],.]]=>[1,2,4,5,3]=>[2,3,5,1,4]=>[[.,[.,[.,.]]],[.,.]]
[[.,[.,[.,.]]],[.,.]]=>[3,2,1,5,4]=>[4,3,2,5,1]=>[[[[.,.],.],[.,.]],.]
[[.,[[.,.],.]],[.,.]]=>[2,3,1,5,4]=>[4,2,3,5,1]=>[[[.,.],[.,[.,.]]],.]
[[[.,.],[.,.]],[.,.]]=>[1,3,2,5,4]=>[3,4,2,5,1]=>[[[.,[.,.]],[.,.]],.]
[[[.,[.,.]],.],[.,.]]=>[2,1,3,5,4]=>[3,2,4,5,1]=>[[[.,.],[.,[.,.]]],.]
[[[[.,.],.],.],[.,.]]=>[1,2,3,5,4]=>[2,3,4,5,1]=>[[.,[.,[.,[.,.]]]],.]
[[.,[.,[.,[.,.]]]],.]=>[4,3,2,1,5]=>[4,3,2,1,5]=>[[[[.,.],.],.],[.,.]]
[[.,[.,[[.,.],.]]],.]=>[3,4,2,1,5]=>[4,3,1,2,5]=>[[[.,.],.],[.,[.,.]]]
[[.,[[.,.],[.,.]]],.]=>[2,4,3,1,5]=>[4,2,3,1,5]=>[[[.,.],[.,.]],[.,.]]
[[.,[[.,[.,.]],.]],.]=>[3,2,4,1,5]=>[4,2,1,3,5]=>[[[.,.],.],[.,[.,.]]]
[[.,[[[.,.],.],.]],.]=>[2,3,4,1,5]=>[4,1,2,3,5]=>[[.,.],[.,[.,[.,.]]]]
[[[.,.],[.,[.,.]]],.]=>[1,4,3,2,5]=>[3,4,2,1,5]=>[[[.,[.,.]],.],[.,.]]
[[[.,.],[[.,.],.]],.]=>[1,3,4,2,5]=>[2,4,1,3,5]=>[[.,[.,.]],[.,[.,.]]]
[[[.,[.,.]],[.,.]],.]=>[2,1,4,3,5]=>[3,2,4,1,5]=>[[[.,.],[.,.]],[.,.]]
[[[[.,.],.],[.,.]],.]=>[1,2,4,3,5]=>[2,3,4,1,5]=>[[.,[.,[.,.]]],[.,.]]
[[[.,[.,[.,.]]],.],.]=>[3,2,1,4,5]=>[3,2,1,4,5]=>[[[.,.],.],[.,[.,.]]]
[[[.,[[.,.],.]],.],.]=>[2,3,1,4,5]=>[3,1,2,4,5]=>[[.,.],[.,[.,[.,.]]]]
[[[[.,.],[.,.]],.],.]=>[1,3,2,4,5]=>[2,3,1,4,5]=>[[.,[.,.]],[.,[.,.]]]
[[[[.,[.,.]],.],.],.]=>[2,1,3,4,5]=>[2,1,3,4,5]=>[[.,.],[.,[.,[.,.]]]]
[[[[[.,.],.],.],.],.]=>[1,2,3,4,5]=>[1,2,3,4,5]=>[.,[.,[.,[.,[.,.]]]]]
[.,[.,[.,[.,[.,[.,.]]]]]]=>[6,5,4,3,2,1]=>[6,5,4,3,2,1]=>[[[[[[.,.],.],.],.],.],.]
[.,[.,[.,[.,[[.,.],.]]]]]=>[5,6,4,3,2,1]=>[6,5,4,3,1,2]=>[[[[[.,.],.],.],.],[.,.]]
[.,[.,[.,[[.,.],[.,.]]]]]=>[4,6,5,3,2,1]=>[6,5,4,2,3,1]=>[[[[[.,.],.],.],[.,.]],.]
[.,[.,[.,[[.,[.,.]],.]]]]=>[5,4,6,3,2,1]=>[6,5,4,2,1,3]=>[[[[[.,.],.],.],.],[.,.]]
[.,[.,[.,[[[.,.],.],.]]]]=>[4,5,6,3,2,1]=>[6,5,4,1,2,3]=>[[[[.,.],.],.],[.,[.,.]]]
[.,[.,[[.,.],[.,[.,.]]]]]=>[3,6,5,4,2,1]=>[6,5,3,4,2,1]=>[[[[[.,.],.],[.,.]],.],.]
[.,[.,[[.,.],[[.,.],.]]]]=>[3,5,6,4,2,1]=>[6,5,2,4,1,3]=>[[[[.,.],.],[.,.]],[.,.]]
[.,[.,[[.,[.,.]],[.,.]]]]=>[4,3,6,5,2,1]=>[6,5,3,2,4,1]=>[[[[[.,.],.],.],[.,.]],.]
[.,[.,[[[.,.],.],[.,.]]]]=>[3,4,6,5,2,1]=>[6,5,2,3,4,1]=>[[[[.,.],.],[.,[.,.]]],.]
[.,[.,[[.,[.,[.,.]]],.]]]=>[5,4,3,6,2,1]=>[6,5,3,2,1,4]=>[[[[[.,.],.],.],.],[.,.]]
[.,[.,[[.,[[.,.],.]],.]]]=>[4,5,3,6,2,1]=>[6,5,3,1,2,4]=>[[[[.,.],.],.],[.,[.,.]]]
[.,[.,[[[.,.],[.,.]],.]]]=>[3,5,4,6,2,1]=>[6,5,2,3,1,4]=>[[[[.,.],.],[.,.]],[.,.]]
[.,[.,[[[.,[.,.]],.],.]]]=>[4,3,5,6,2,1]=>[6,5,2,1,3,4]=>[[[[.,.],.],.],[.,[.,.]]]
[.,[.,[[[[.,.],.],.],.]]]=>[3,4,5,6,2,1]=>[6,5,1,2,3,4]=>[[[.,.],.],[.,[.,[.,.]]]]
[.,[[.,.],[.,[.,[.,.]]]]]=>[2,6,5,4,3,1]=>[6,4,5,3,2,1]=>[[[[[.,.],[.,.]],.],.],.]
[.,[[.,.],[.,[[.,.],.]]]]=>[2,5,6,4,3,1]=>[6,3,5,4,1,2]=>[[[.,.],[[.,.],.]],[.,.]]
[.,[[.,.],[[.,.],[.,.]]]]=>[2,4,6,5,3,1]=>[6,3,5,2,4,1]=>[[[[.,.],[.,.]],[.,.]],.]
[.,[[.,.],[[.,[.,.]],.]]]=>[2,5,4,6,3,1]=>[6,3,5,2,1,4]=>[[[[.,.],[.,.]],.],[.,.]]
[.,[[.,.],[[[.,.],.],.]]]=>[2,4,5,6,3,1]=>[6,2,5,1,3,4]=>[[[.,.],[.,.]],[.,[.,.]]]
[.,[[.,[.,.]],[.,[.,.]]]]=>[3,2,6,5,4,1]=>[6,4,3,5,2,1]=>[[[[[.,.],.],[.,.]],.],.]
[.,[[.,[.,.]],[[.,.],.]]]=>[3,2,5,6,4,1]=>[6,3,2,5,1,4]=>[[[[.,.],.],[.,.]],[.,.]]
[.,[[[.,.],.],[.,[.,.]]]]=>[2,3,6,5,4,1]=>[6,3,4,5,2,1]=>[[[[.,.],[.,[.,.]]],.],.]
[.,[[[.,.],.],[[.,.],.]]]=>[2,3,5,6,4,1]=>[6,2,3,5,1,4]=>[[[.,.],[.,[.,.]]],[.,.]]
[.,[[.,[.,[.,.]]],[.,.]]]=>[4,3,2,6,5,1]=>[6,4,3,2,5,1]=>[[[[[.,.],.],.],[.,.]],.]
[.,[[.,[[.,.],.]],[.,.]]]=>[3,4,2,6,5,1]=>[6,4,2,3,5,1]=>[[[[.,.],.],[.,[.,.]]],.]
[.,[[[.,.],[.,.]],[.,.]]]=>[2,4,3,6,5,1]=>[6,3,4,2,5,1]=>[[[[.,.],[.,.]],[.,.]],.]
[.,[[[.,[.,.]],.],[.,.]]]=>[3,2,4,6,5,1]=>[6,3,2,4,5,1]=>[[[[.,.],.],[.,[.,.]]],.]
[.,[[[[.,.],.],.],[.,.]]]=>[2,3,4,6,5,1]=>[6,2,3,4,5,1]=>[[[.,.],[.,[.,[.,.]]]],.]
[.,[[.,[.,[.,[.,.]]]],.]]=>[5,4,3,2,6,1]=>[6,4,3,2,1,5]=>[[[[[.,.],.],.],.],[.,.]]
[.,[[.,[.,[[.,.],.]]],.]]=>[4,5,3,2,6,1]=>[6,4,3,1,2,5]=>[[[[.,.],.],.],[.,[.,.]]]
[.,[[.,[[.,.],[.,.]]],.]]=>[3,5,4,2,6,1]=>[6,4,2,3,1,5]=>[[[[.,.],.],[.,.]],[.,.]]
[.,[[.,[[.,[.,.]],.]],.]]=>[4,3,5,2,6,1]=>[6,4,2,1,3,5]=>[[[[.,.],.],.],[.,[.,.]]]
[.,[[.,[[[.,.],.],.]],.]]=>[3,4,5,2,6,1]=>[6,4,1,2,3,5]=>[[[.,.],.],[.,[.,[.,.]]]]
[.,[[[.,.],[.,[.,.]]],.]]=>[2,5,4,3,6,1]=>[6,3,4,2,1,5]=>[[[[.,.],[.,.]],.],[.,.]]
[.,[[[.,.],[[.,.],.]],.]]=>[2,4,5,3,6,1]=>[6,2,4,1,3,5]=>[[[.,.],[.,.]],[.,[.,.]]]
[.,[[[.,[.,.]],[.,.]],.]]=>[3,2,5,4,6,1]=>[6,3,2,4,1,5]=>[[[[.,.],.],[.,.]],[.,.]]
[.,[[[[.,.],.],[.,.]],.]]=>[2,3,5,4,6,1]=>[6,2,3,4,1,5]=>[[[.,.],[.,[.,.]]],[.,.]]
[.,[[[.,[.,[.,.]]],.],.]]=>[4,3,2,5,6,1]=>[6,3,2,1,4,5]=>[[[[.,.],.],.],[.,[.,.]]]
[.,[[[.,[[.,.],.]],.],.]]=>[3,4,2,5,6,1]=>[6,3,1,2,4,5]=>[[[.,.],.],[.,[.,[.,.]]]]
[.,[[[[.,.],[.,.]],.],.]]=>[2,4,3,5,6,1]=>[6,2,3,1,4,5]=>[[[.,.],[.,.]],[.,[.,.]]]
[.,[[[[.,[.,.]],.],.],.]]=>[3,2,4,5,6,1]=>[6,2,1,3,4,5]=>[[[.,.],.],[.,[.,[.,.]]]]
[.,[[[[[.,.],.],.],.],.]]=>[2,3,4,5,6,1]=>[6,1,2,3,4,5]=>[[.,.],[.,[.,[.,[.,.]]]]]
[[.,.],[.,[.,[.,[.,.]]]]]=>[1,6,5,4,3,2]=>[5,6,4,3,2,1]=>[[[[[.,[.,.]],.],.],.],.]
[[.,.],[.,[.,[[.,.],.]]]]=>[1,5,6,4,3,2]=>[4,6,5,3,1,2]=>[[[.,[[.,.],.]],.],[.,.]]
[[.,.],[.,[[.,.],[.,.]]]]=>[1,4,6,5,3,2]=>[4,6,5,2,3,1]=>[[[.,[[.,.],.]],[.,.]],.]
[[.,.],[.,[[.,[.,.]],.]]]=>[1,5,4,6,3,2]=>[4,6,5,2,1,3]=>[[[.,[[.,.],.]],.],[.,.]]
[[.,.],[.,[[[.,.],.],.]]]=>[1,4,5,6,3,2]=>[3,6,5,1,2,4]=>[[.,[[.,.],.]],[.,[.,.]]]
[[.,.],[[.,.],[.,[.,.]]]]=>[1,3,6,5,4,2]=>[4,6,3,5,2,1]=>[[[[.,[.,.]],[.,.]],.],.]
[[.,.],[[.,.],[[.,.],.]]]=>[1,3,5,6,4,2]=>[3,6,2,5,1,4]=>[[[.,[.,.]],[.,.]],[.,.]]
[[.,.],[[.,[.,.]],[.,.]]]=>[1,4,3,6,5,2]=>[4,6,3,2,5,1]=>[[[[.,[.,.]],.],[.,.]],.]
[[.,.],[[[.,.],.],[.,.]]]=>[1,3,4,6,5,2]=>[3,6,2,4,5,1]=>[[[.,[.,.]],[.,[.,.]]],.]
[[.,.],[[.,[.,[.,.]]],.]]=>[1,5,4,3,6,2]=>[4,6,3,2,1,5]=>[[[[.,[.,.]],.],.],[.,.]]
[[.,.],[[.,[[.,.],.]],.]]=>[1,4,5,3,6,2]=>[3,6,4,1,2,5]=>[[.,[[.,.],.]],[.,[.,.]]]
[[.,.],[[[.,.],[.,.]],.]]=>[1,3,5,4,6,2]=>[3,6,2,4,1,5]=>[[[.,[.,.]],[.,.]],[.,.]]
[[.,.],[[[.,[.,.]],.],.]]=>[1,4,3,5,6,2]=>[3,6,2,1,4,5]=>[[[.,[.,.]],.],[.,[.,.]]]
[[.,.],[[[[.,.],.],.],.]]=>[1,3,4,5,6,2]=>[2,6,1,3,4,5]=>[[.,[.,.]],[.,[.,[.,.]]]]
[[.,[.,.]],[.,[.,[.,.]]]]=>[2,1,6,5,4,3]=>[5,4,6,3,2,1]=>[[[[[.,.],[.,.]],.],.],.]
[[.,[.,.]],[.,[[.,.],.]]]=>[2,1,5,6,4,3]=>[4,3,6,5,1,2]=>[[[.,.],[[.,.],.]],[.,.]]
[[.,[.,.]],[[.,.],[.,.]]]=>[2,1,4,6,5,3]=>[4,3,6,2,5,1]=>[[[[.,.],[.,.]],[.,.]],.]
[[.,[.,.]],[[.,[.,.]],.]]=>[2,1,5,4,6,3]=>[4,3,6,2,1,5]=>[[[[.,.],[.,.]],.],[.,.]]
[[.,[.,.]],[[[.,.],.],.]]=>[2,1,4,5,6,3]=>[3,2,6,1,4,5]=>[[[.,.],[.,.]],[.,[.,.]]]
[[[.,.],.],[.,[.,[.,.]]]]=>[1,2,6,5,4,3]=>[4,5,6,3,2,1]=>[[[[.,[.,[.,.]]],.],.],.]
[[[.,.],.],[.,[[.,.],.]]]=>[1,2,5,6,4,3]=>[3,4,6,5,1,2]=>[[.,[.,[[.,.],.]]],[.,.]]
[[[.,.],.],[[.,.],[.,.]]]=>[1,2,4,6,5,3]=>[3,4,6,2,5,1]=>[[[.,[.,[.,.]]],[.,.]],.]
[[[.,.],.],[[.,[.,.]],.]]=>[1,2,5,4,6,3]=>[3,4,6,2,1,5]=>[[[.,[.,[.,.]]],.],[.,.]]
[[[.,.],.],[[[.,.],.],.]]=>[1,2,4,5,6,3]=>[2,3,6,1,4,5]=>[[.,[.,[.,.]]],[.,[.,.]]]
[[.,[.,[.,.]]],[.,[.,.]]]=>[3,2,1,6,5,4]=>[5,4,3,6,2,1]=>[[[[[.,.],.],[.,.]],.],.]
[[.,[.,[.,.]]],[[.,.],.]]=>[3,2,1,5,6,4]=>[4,3,2,6,1,5]=>[[[[.,.],.],[.,.]],[.,.]]
[[.,[[.,.],.]],[.,[.,.]]]=>[2,3,1,6,5,4]=>[5,3,4,6,2,1]=>[[[[.,.],[.,[.,.]]],.],.]
[[.,[[.,.],.]],[[.,.],.]]=>[2,3,1,5,6,4]=>[4,2,3,6,1,5]=>[[[.,.],[.,[.,.]]],[.,.]]
[[[.,.],[.,.]],[.,[.,.]]]=>[1,3,2,6,5,4]=>[4,5,3,6,2,1]=>[[[[.,[.,.]],[.,.]],.],.]
[[[.,.],[.,.]],[[.,.],.]]=>[1,3,2,5,6,4]=>[3,4,2,6,1,5]=>[[[.,[.,.]],[.,.]],[.,.]]
[[[.,[.,.]],.],[.,[.,.]]]=>[2,1,3,6,5,4]=>[4,3,5,6,2,1]=>[[[[.,.],[.,[.,.]]],.],.]
[[[.,[.,.]],.],[[.,.],.]]=>[2,1,3,5,6,4]=>[3,2,4,6,1,5]=>[[[.,.],[.,[.,.]]],[.,.]]
[[[[.,.],.],.],[.,[.,.]]]=>[1,2,3,6,5,4]=>[3,4,5,6,2,1]=>[[[.,[.,[.,[.,.]]]],.],.]
[[[[.,.],.],.],[[.,.],.]]=>[1,2,3,5,6,4]=>[2,3,4,6,1,5]=>[[.,[.,[.,[.,.]]]],[.,.]]
[[.,[.,[.,[.,.]]]],[.,.]]=>[4,3,2,1,6,5]=>[5,4,3,2,6,1]=>[[[[[.,.],.],.],[.,.]],.]
[[.,[.,[[.,.],.]]],[.,.]]=>[3,4,2,1,6,5]=>[5,4,2,3,6,1]=>[[[[.,.],.],[.,[.,.]]],.]
[[.,[[.,.],[.,.]]],[.,.]]=>[2,4,3,1,6,5]=>[5,3,4,2,6,1]=>[[[[.,.],[.,.]],[.,.]],.]
[[.,[[.,[.,.]],.]],[.,.]]=>[3,2,4,1,6,5]=>[5,3,2,4,6,1]=>[[[[.,.],.],[.,[.,.]]],.]
[[.,[[[.,.],.],.]],[.,.]]=>[2,3,4,1,6,5]=>[5,2,3,4,6,1]=>[[[.,.],[.,[.,[.,.]]]],.]
[[[.,.],[.,[.,.]]],[.,.]]=>[1,4,3,2,6,5]=>[4,5,3,2,6,1]=>[[[[.,[.,.]],.],[.,.]],.]
[[[.,.],[[.,.],.]],[.,.]]=>[1,3,4,2,6,5]=>[3,5,2,4,6,1]=>[[[.,[.,.]],[.,[.,.]]],.]
[[[.,[.,.]],[.,.]],[.,.]]=>[2,1,4,3,6,5]=>[4,3,5,2,6,1]=>[[[[.,.],[.,.]],[.,.]],.]
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[.,[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]]=>[7,6,5,4,3,2,8,9,1]=>[9,6,5,4,3,2,1,7,8]=>[[[[[[[.,.],.],.],.],.],.],[.,[.,.]]]
[.,[[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],.]]=>[8,7,6,5,4,3,2,9,10,1]=>[10,7,6,5,4,3,2,1,8,9]=>[[[[[[[[.,.],.],.],.],.],.],.],[.,[.,.]]]
[[.,[.,[.,[.,[.,[.,.]]]]]],[[.,.],.]]=>[6,5,4,3,2,1,8,9,7]=>[7,6,5,4,3,2,9,1,8]=>[[[[[[[.,.],.],.],.],.],[.,.]],[.,.]]
[[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.],.]=>[1,2,3,4,5,6,7,8,9,10,11]=>[1,2,3,4,5,6,7,8,9,10,11]=>[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]]
[[[.,.],.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]=>[1,2,10,9,8,7,6,5,4,3]=>[8,9,10,7,6,5,4,3,2,1]=>[[[[[[[[.,[.,[.,.]]],.],.],.],.],.],.],.]
[[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]],.]=>[2,1,8,7,6,5,4,3,9]=>[7,6,8,5,4,3,2,1,9]=>[[[[[[[.,.],[.,.]],.],.],.],.],[.,.]]
[[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]=>[1,9,8,7,6,5,4,3,2,10]=>[8,9,7,6,5,4,3,2,1,10]=>[[[[[[[[.,[.,.]],.],.],.],.],.],.],[.,.]]
[[[[[[[.,.],.],[[.,.],.]],.],.],.],.]=>[1,2,4,5,3,6,7,8,9]=>[2,3,5,1,4,6,7,8,9]=>[[.,[.,[.,.]]],[.,[.,[.,[.,[.,.]]]]]]
[[[[[[.,[.,.]],[[.,.],.]],.],.],.],.]=>[2,1,4,5,3,6,7,8,9]=>[3,2,5,1,4,6,7,8,9]=>[[[.,.],[.,.]],[.,[.,[.,[.,[.,.]]]]]]
[[[[[[[.,.],[.,.]],[.,.]],.],.],.],.]=>[1,3,2,5,4,6,7,8,9]=>[3,4,2,5,1,6,7,8,9]=>[[[.,[.,.]],[.,.]],[.,[.,[.,[.,.]]]]]
[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]]=>[8,7,6,5,4,3,2,10,9,1]=>[10,8,7,6,5,4,3,2,9,1]=>[[[[[[[[[.,.],.],.],.],.],.],.],[.,.]],.]
[[.,[.,.]],[[[[[[.,.],.],.],.],.],.]]=>[2,1,4,5,6,7,8,9,3]=>[3,2,9,1,4,5,6,7,8]=>[[[.,.],[.,.]],[.,[.,[.,[.,[.,.]]]]]]
[[[.,.],[[[[[[.,.],.],.],.],.],.]],.]=>[1,3,4,5,6,7,8,2,9]=>[2,8,1,3,4,5,6,7,9]=>[[.,[.,.]],[.,[.,[.,[.,[.,[.,.]]]]]]]
[[[[[[[[.,.],.],.],.],.],.],.],[.,[.,.]]]=>[1,2,3,4,5,6,7,10,9,8]=>[3,4,5,6,7,8,9,10,2,1]=>[[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.],.]
[[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]],.]=>[6,7,8,5,4,3,2,1,9]=>[8,7,6,5,4,1,2,3,9]=>[[[[[[.,.],.],.],.],.],[.,[.,[.,.]]]]
[[.,[.,[.,[.,[.,.]]]]],[[[.,.],.],.]]=>[5,4,3,2,1,7,8,9,6]=>[6,5,4,3,2,9,1,7,8]=>[[[[[[.,.],.],.],.],[.,.]],[.,[.,.]]]
[[[[[[.,.],.],.],.],.],[.,[.,[.,.]]]]=>[1,2,3,4,5,9,8,7,6]=>[4,5,6,7,8,9,3,2,1]=>[[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.]
[[[[[.,.],.],.],.],[.,[.,[.,[.,.]]]]]=>[1,2,3,4,9,8,7,6,5]=>[5,6,7,8,9,4,3,2,1]=>[[[[[.,[.,[.,[.,[.,.]]]]],.],.],.],.]
[[[[.,.],.],.],[.,[.,[.,[.,[.,.]]]]]]=>[1,2,3,9,8,7,6,5,4]=>[6,7,8,9,5,4,3,2,1]=>[[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.]
[[[[[[[.,.],.],.],.],.],[.,.]],[.,.]]=>[1,2,3,4,5,7,6,9,8]=>[3,4,5,6,7,8,2,9,1]=>[[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]],.]
[[[[[[[.,.],.],.],.],.],.],[.,[.,[.,.]]]]=>[1,2,3,4,5,6,10,9,8,7]=>[4,5,6,7,8,9,10,3,2,1]=>[[[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],.],.]
[[[[[[.,.],.],.],.],.],[.,[.,[.,[.,.]]]]]=>[1,2,3,4,5,10,9,8,7,6]=>[5,6,7,8,9,10,4,3,2,1]=>[[[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.],.]
[[[[[.,.],.],.],.],[.,[.,[.,[.,[.,.]]]]]]=>[1,2,3,4,10,9,8,7,6,5]=>[6,7,8,9,10,5,4,3,2,1]=>[[[[[[.,[.,[.,[.,[.,.]]]]],.],.],.],.],.]
[[[[.,.],.],.],[.,[.,[.,[.,[.,[.,.]]]]]]]=>[1,2,3,10,9,8,7,6,5,4]=>[7,8,9,10,6,5,4,3,2,1]=>[[[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.],.]
[.,[.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]]=>[10,11,9,8,7,6,5,4,3,2,1]=>[11,10,9,8,7,6,5,4,3,1,2]=>[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]
[.,[[.,[.,[.,[.,[.,[[.,.],.]]]]]],.]]=>[7,8,6,5,4,3,2,9,1]=>[9,7,6,5,4,3,1,2,8]=>[[[[[[[.,.],.],.],.],.],.],[.,[.,.]]]
[[[[[[.,[.,[[.,.],.]]],.],.],.],.],.]=>[3,4,2,1,5,6,7,8,9]=>[4,3,1,2,5,6,7,8,9]=>[[[.,.],.],[.,[.,[.,[.,[.,[.,.]]]]]]]
[[[[[.,[[.,[[.,.],.]],.]],.],.],.],.]=>[3,4,2,5,1,6,7,8,9]=>[5,3,1,2,4,6,7,8,9]=>[[[.,.],.],[.,[.,[.,[.,[.,[.,.]]]]]]]
[[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]],.]=>[7,6,5,4,3,2,8,1,9]=>[8,6,5,4,3,2,1,7,9]=>[[[[[[[.,.],.],.],.],.],.],[.,[.,.]]]
[[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]],.]=>[7,6,5,8,4,3,2,1,9]=>[8,7,6,5,3,2,1,4,9]=>[[[[[[[.,.],.],.],.],.],.],[.,[.,.]]]
[[[[[.,[.,[.,[[.,.],.]]]],.],.],.],.]=>[4,5,3,2,1,6,7,8,9]=>[5,4,3,1,2,6,7,8,9]=>[[[[.,.],.],.],[.,[.,[.,[.,[.,.]]]]]]
[[.,[[[[[[.,[.,.]],.],.],.],.],.]],.]=>[3,2,4,5,6,7,8,1,9]=>[8,2,1,3,4,5,6,7,9]=>[[[.,.],.],[.,[.,[.,[.,[.,[.,.]]]]]]]
[[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]],.]=>[7,6,5,4,3,2,1,9,8,10]=>[8,7,6,5,4,3,2,9,1,10]=>[[[[[[[[.,.],.],.],.],.],.],[.,.]],[.,.]]
[.,[[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]]=>[2,3,9,8,7,6,5,4,1]=>[9,6,7,8,5,4,3,2,1]=>[[[[[[[.,.],[.,[.,.]]],.],.],.],.],.]
[[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]],.]=>[2,8,7,6,5,4,3,1,9]=>[8,6,7,5,4,3,2,1,9]=>[[[[[[[.,.],[.,.]],.],.],.],.],[.,.]]
[[.,[[[[[[.,.],.],.],.],.],.]],[.,.]]=>[2,3,4,5,6,7,1,9,8]=>[8,2,3,4,5,6,7,9,1]=>[[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]],.]
[[[[[[[.,.],[[.,.],.]],.],.],.],.],.]=>[1,3,4,2,5,6,7,8,9]=>[2,4,1,3,5,6,7,8,9]=>[[.,[.,.]],[.,[.,[.,[.,[.,[.,.]]]]]]]
[[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]],.]=>[8,7,6,5,4,3,2,9,1,10]=>[9,7,6,5,4,3,2,1,8,10]=>[[[[[[[[.,.],.],.],.],.],.],.],[.,[.,.]]]
[[[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]],.]=>[1,2,3,4,5,6,7,8,10,9,11]=>[2,3,4,5,6,7,8,9,10,1,11]=>[[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],[.,.]]
[[[[.,[.,[.,[.,[[.,.],.]]]]],.],.],.]=>[5,6,4,3,2,1,7,8,9]=>[6,5,4,3,1,2,7,8,9]=>[[[[[.,.],.],.],.],[.,[.,[.,[.,.]]]]]
[[[[[.,[.,[.,[.,[[.,.],.]]]]],.],.],.],.]=>[5,6,4,3,2,1,7,8,9,10]=>[6,5,4,3,1,2,7,8,9,10]=>[[[[[.,.],.],.],.],[.,[.,[.,[.,[.,.]]]]]]
[[[[[.,[.,[.,[.,[.,[[.,.],.]]]]]],.],.],.],.]=>[6,7,5,4,3,2,1,8,9,10,11]=>[7,6,5,4,3,1,2,8,9,10,11]=>[[[[[[.,.],.],.],.],.],[.,[.,[.,[.,[.,.]]]]]]
[[[[.,[.,[.,[.,[.,[[.,.],.]]]]]],.],.],.]=>[6,7,5,4,3,2,1,8,9,10]=>[7,6,5,4,3,1,2,8,9,10]=>[[[[[[.,.],.],.],.],.],[.,[.,[.,[.,.]]]]]
[[[.,[[.,[.,[.,[[.,.],.]]]],.]],.],.]=>[5,6,4,3,2,7,1,8,9]=>[7,5,4,3,1,2,6,8,9]=>[[[[[.,.],.],.],.],[.,[.,[.,[.,.]]]]]
[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[[.,.],.]]=>[7,6,5,4,3,2,1,9,10,8]=>[8,7,6,5,4,3,2,10,1,9]=>[[[[[[[[.,.],.],.],.],.],.],[.,.]],[.,.]]
Map
to 312-avoiding permutation
Description
Return a 312-avoiding permutation corresponding to a binary tree.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the minimal element of this Sylvester class.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the minimal element of this Sylvester class.
Map
major-index to inversion-number bijection
Description
Return the permutation whose Lehmer code equals the major code of the preimage.
This map sends the major index to the number of inversions.
This map sends the major index to the number of inversions.
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.
searching the database
Sorry, this map was not found in the database.