Identifier
Mp00014:
Binary trees
—to 132-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]=>[[.,.],[.,.]]
[[.,.],[.,.]]=>[3,1,2]=>[1,3,2]=>[.,[[.,.],.]]
[[.,[.,.]],.]=>[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]=>[[[.,.],.],[.,.]]
[.,[[.,.],[.,.]]]=>[4,2,3,1]=>[4,1,3,2]=>[[.,.],[[.,.],.]]
[.,[[.,[.,.]],.]]=>[3,2,4,1]=>[4,2,1,3]=>[[[.,.],.],[.,.]]
[.,[[[.,.],.],.]]=>[2,3,4,1]=>[4,1,2,3]=>[[.,.],[.,[.,.]]]
[[.,.],[.,[.,.]]]=>[4,3,1,2]=>[1,4,3,2]=>[.,[[[.,.],.],.]]
[[.,.],[[.,.],.]]=>[3,4,1,2]=>[1,4,2,3]=>[.,[[.,.],[.,.]]]
[[.,[.,.]],[.,.]]=>[4,2,1,3]=>[3,1,4,2]=>[[.,.],[[.,.],.]]
[[[.,.],.],[.,.]]=>[4,1,2,3]=>[1,2,4,3]=>[.,[.,[[.,.],.]]]
[[.,[.,[.,.]]],.]=>[3,2,1,4]=>[3,2,1,4]=>[[[.,.],.],[.,.]]
[[.,[[.,.],.]],.]=>[2,3,1,4]=>[3,1,2,4]=>[[.,.],[.,[.,.]]]
[[[.,.],[.,.]],.]=>[3,1,2,4]=>[1,3,2,4]=>[.,[[.,.],[.,.]]]
[[[.,[.,.]],.],.]=>[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]=>[[[[.,.],.],.],[.,.]]
[.,[.,[[.,.],[.,.]]]]=>[5,3,4,2,1]=>[5,4,1,3,2]=>[[[.,.],.],[[.,.],.]]
[.,[.,[[.,[.,.]],.]]]=>[4,3,5,2,1]=>[5,4,2,1,3]=>[[[[.,.],.],.],[.,.]]
[.,[.,[[[.,.],.],.]]]=>[3,4,5,2,1]=>[5,4,1,2,3]=>[[[.,.],.],[.,[.,.]]]
[.,[[.,.],[.,[.,.]]]]=>[5,4,2,3,1]=>[5,1,4,3,2]=>[[.,.],[[[.,.],.],.]]
[.,[[.,.],[[.,.],.]]]=>[4,5,2,3,1]=>[5,1,4,2,3]=>[[.,.],[[.,.],[.,.]]]
[.,[[.,[.,.]],[.,.]]]=>[5,3,2,4,1]=>[5,3,1,4,2]=>[[[.,.],.],[[.,.],.]]
[.,[[[.,.],.],[.,.]]]=>[5,2,3,4,1]=>[5,1,2,4,3]=>[[.,.],[.,[[.,.],.]]]
[.,[[.,[.,[.,.]]],.]]=>[4,3,2,5,1]=>[5,3,2,1,4]=>[[[[.,.],.],.],[.,.]]
[.,[[.,[[.,.],.]],.]]=>[3,4,2,5,1]=>[5,3,1,2,4]=>[[[.,.],.],[.,[.,.]]]
[.,[[[.,.],[.,.]],.]]=>[4,2,3,5,1]=>[5,1,3,2,4]=>[[.,.],[[.,.],[.,.]]]
[.,[[[.,[.,.]],.],.]]=>[3,2,4,5,1]=>[5,2,1,3,4]=>[[[.,.],.],[.,[.,.]]]
[.,[[[[.,.],.],.],.]]=>[2,3,4,5,1]=>[5,1,2,3,4]=>[[.,.],[.,[.,[.,.]]]]
[[.,.],[.,[.,[.,.]]]]=>[5,4,3,1,2]=>[1,5,4,3,2]=>[.,[[[[.,.],.],.],.]]
[[.,.],[.,[[.,.],.]]]=>[4,5,3,1,2]=>[1,5,4,2,3]=>[.,[[[.,.],.],[.,.]]]
[[.,.],[[.,.],[.,.]]]=>[5,3,4,1,2]=>[1,5,2,4,3]=>[.,[[.,.],[[.,.],.]]]
[[.,.],[[.,[.,.]],.]]=>[4,3,5,1,2]=>[1,5,3,2,4]=>[.,[[[.,.],.],[.,.]]]
[[.,.],[[[.,.],.],.]]=>[3,4,5,1,2]=>[1,5,2,3,4]=>[.,[[.,.],[.,[.,.]]]]
[[.,[.,.]],[.,[.,.]]]=>[5,4,2,1,3]=>[4,1,5,3,2]=>[[.,.],[[[.,.],.],.]]
[[.,[.,.]],[[.,.],.]]=>[4,5,2,1,3]=>[4,1,5,2,3]=>[[.,.],[[.,.],[.,.]]]
[[[.,.],.],[.,[.,.]]]=>[5,4,1,2,3]=>[1,2,5,4,3]=>[.,[.,[[[.,.],.],.]]]
[[[.,.],.],[[.,.],.]]=>[4,5,1,2,3]=>[1,2,5,3,4]=>[.,[.,[[.,.],[.,.]]]]
[[.,[.,[.,.]]],[.,.]]=>[5,3,2,1,4]=>[4,3,1,5,2]=>[[[.,.],.],[[.,.],.]]
[[.,[[.,.],.]],[.,.]]=>[5,2,3,1,4]=>[4,1,2,5,3]=>[[.,.],[.,[[.,.],.]]]
[[[.,.],[.,.]],[.,.]]=>[5,3,1,2,4]=>[1,4,2,5,3]=>[.,[[.,.],[[.,.],.]]]
[[[.,[.,.]],.],[.,.]]=>[5,2,1,3,4]=>[3,1,2,5,4]=>[[.,.],[.,[[.,.],.]]]
[[[[.,.],.],.],[.,.]]=>[5,1,2,3,4]=>[1,2,3,5,4]=>[.,[.,[.,[[.,.],.]]]]
[[.,[.,[.,[.,.]]]],.]=>[4,3,2,1,5]=>[4,3,2,1,5]=>[[[[.,.],.],.],[.,.]]
[[.,[.,[[.,.],.]]],.]=>[3,4,2,1,5]=>[4,3,1,2,5]=>[[[.,.],.],[.,[.,.]]]
[[.,[[.,.],[.,.]]],.]=>[4,2,3,1,5]=>[4,1,3,2,5]=>[[.,.],[[.,.],[.,.]]]
[[.,[[.,[.,.]],.]],.]=>[3,2,4,1,5]=>[4,2,1,3,5]=>[[[.,.],.],[.,[.,.]]]
[[.,[[[.,.],.],.]],.]=>[2,3,4,1,5]=>[4,1,2,3,5]=>[[.,.],[.,[.,[.,.]]]]
[[[.,.],[.,[.,.]]],.]=>[4,3,1,2,5]=>[1,4,3,2,5]=>[.,[[[.,.],.],[.,.]]]
[[[.,.],[[.,.],.]],.]=>[3,4,1,2,5]=>[1,4,2,3,5]=>[.,[[.,.],[.,[.,.]]]]
[[[.,[.,.]],[.,.]],.]=>[4,2,1,3,5]=>[3,1,4,2,5]=>[[.,.],[[.,.],[.,.]]]
[[[[.,.],.],[.,.]],.]=>[4,1,2,3,5]=>[1,2,4,3,5]=>[.,[.,[[.,.],[.,.]]]]
[[[.,[.,[.,.]]],.],.]=>[3,2,1,4,5]=>[3,2,1,4,5]=>[[[.,.],.],[.,[.,.]]]
[[[.,[[.,.],.]],.],.]=>[2,3,1,4,5]=>[3,1,2,4,5]=>[[.,.],[.,[.,[.,.]]]]
[[[[.,.],[.,.]],.],.]=>[3,1,2,4,5]=>[1,3,2,4,5]=>[.,[[.,.],[.,[.,.]]]]
[[[[.,[.,.]],.],.],.]=>[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]=>[[[[[.,.],.],.],.],[.,.]]
[.,[.,[.,[[.,.],[.,.]]]]]=>[6,4,5,3,2,1]=>[6,5,4,1,3,2]=>[[[[.,.],.],.],[[.,.],.]]
[.,[.,[.,[[.,[.,.]],.]]]]=>[5,4,6,3,2,1]=>[6,5,4,2,1,3]=>[[[[[.,.],.],.],.],[.,.]]
[.,[.,[.,[[[.,.],.],.]]]]=>[4,5,6,3,2,1]=>[6,5,4,1,2,3]=>[[[[.,.],.],.],[.,[.,.]]]
[.,[.,[[.,.],[.,[.,.]]]]]=>[6,5,3,4,2,1]=>[6,5,1,4,3,2]=>[[[.,.],.],[[[.,.],.],.]]
[.,[.,[[.,.],[[.,.],.]]]]=>[5,6,3,4,2,1]=>[6,5,1,4,2,3]=>[[[.,.],.],[[.,.],[.,.]]]
[.,[.,[[.,[.,.]],[.,.]]]]=>[6,4,3,5,2,1]=>[6,5,3,1,4,2]=>[[[[.,.],.],.],[[.,.],.]]
[.,[.,[[[.,.],.],[.,.]]]]=>[6,3,4,5,2,1]=>[6,5,1,2,4,3]=>[[[.,.],.],[.,[[.,.],.]]]
[.,[.,[[.,[.,[.,.]]],.]]]=>[5,4,3,6,2,1]=>[6,5,3,2,1,4]=>[[[[[.,.],.],.],.],[.,.]]
[.,[.,[[.,[[.,.],.]],.]]]=>[4,5,3,6,2,1]=>[6,5,3,1,2,4]=>[[[[.,.],.],.],[.,[.,.]]]
[.,[.,[[[.,.],[.,.]],.]]]=>[5,3,4,6,2,1]=>[6,5,1,3,2,4]=>[[[.,.],.],[[.,.],[.,.]]]
[.,[.,[[[.,[.,.]],.],.]]]=>[4,3,5,6,2,1]=>[6,5,2,1,3,4]=>[[[[.,.],.],.],[.,[.,.]]]
[.,[.,[[[[.,.],.],.],.]]]=>[3,4,5,6,2,1]=>[6,5,1,2,3,4]=>[[[.,.],.],[.,[.,[.,.]]]]
[.,[[.,.],[.,[.,[.,.]]]]]=>[6,5,4,2,3,1]=>[6,1,5,4,3,2]=>[[.,.],[[[[.,.],.],.],.]]
[.,[[.,.],[.,[[.,.],.]]]]=>[5,6,4,2,3,1]=>[6,1,5,4,2,3]=>[[.,.],[[[.,.],.],[.,.]]]
[.,[[.,.],[[.,.],[.,.]]]]=>[6,4,5,2,3,1]=>[6,1,5,2,4,3]=>[[.,.],[[.,.],[[.,.],.]]]
[.,[[.,.],[[.,[.,.]],.]]]=>[5,4,6,2,3,1]=>[6,1,5,3,2,4]=>[[.,.],[[[.,.],.],[.,.]]]
[.,[[.,.],[[[.,.],.],.]]]=>[4,5,6,2,3,1]=>[6,1,5,2,3,4]=>[[.,.],[[.,.],[.,[.,.]]]]
[.,[[.,[.,.]],[.,[.,.]]]]=>[6,5,3,2,4,1]=>[6,4,1,5,3,2]=>[[[.,.],.],[[[.,.],.],.]]
[.,[[.,[.,.]],[[.,.],.]]]=>[5,6,3,2,4,1]=>[6,4,1,5,2,3]=>[[[.,.],.],[[.,.],[.,.]]]
[.,[[[.,.],.],[.,[.,.]]]]=>[6,5,2,3,4,1]=>[6,1,2,5,4,3]=>[[.,.],[.,[[[.,.],.],.]]]
[.,[[[.,.],.],[[.,.],.]]]=>[5,6,2,3,4,1]=>[6,1,2,5,3,4]=>[[.,.],[.,[[.,.],[.,.]]]]
[.,[[.,[.,[.,.]]],[.,.]]]=>[6,4,3,2,5,1]=>[6,4,3,1,5,2]=>[[[[.,.],.],.],[[.,.],.]]
[.,[[.,[[.,.],.]],[.,.]]]=>[6,3,4,2,5,1]=>[6,4,1,2,5,3]=>[[[.,.],.],[.,[[.,.],.]]]
[.,[[[.,.],[.,.]],[.,.]]]=>[6,4,2,3,5,1]=>[6,1,4,2,5,3]=>[[.,.],[[.,.],[[.,.],.]]]
[.,[[[.,[.,.]],.],[.,.]]]=>[6,3,2,4,5,1]=>[6,3,1,2,5,4]=>[[[.,.],.],[.,[[.,.],.]]]
[.,[[[[.,.],.],.],[.,.]]]=>[6,2,3,4,5,1]=>[6,1,2,3,5,4]=>[[.,.],[.,[.,[[.,.],.]]]]
[.,[[.,[.,[.,[.,.]]]],.]]=>[5,4,3,2,6,1]=>[6,4,3,2,1,5]=>[[[[[.,.],.],.],.],[.,.]]
[.,[[.,[.,[[.,.],.]]],.]]=>[4,5,3,2,6,1]=>[6,4,3,1,2,5]=>[[[[.,.],.],.],[.,[.,.]]]
[.,[[.,[[.,.],[.,.]]],.]]=>[5,3,4,2,6,1]=>[6,4,1,3,2,5]=>[[[.,.],.],[[.,.],[.,.]]]
[.,[[.,[[.,[.,.]],.]],.]]=>[4,3,5,2,6,1]=>[6,4,2,1,3,5]=>[[[[.,.],.],.],[.,[.,.]]]
[.,[[.,[[[.,.],.],.]],.]]=>[3,4,5,2,6,1]=>[6,4,1,2,3,5]=>[[[.,.],.],[.,[.,[.,.]]]]
[.,[[[.,.],[.,[.,.]]],.]]=>[5,4,2,3,6,1]=>[6,1,4,3,2,5]=>[[.,.],[[[.,.],.],[.,.]]]
[.,[[[.,.],[[.,.],.]],.]]=>[4,5,2,3,6,1]=>[6,1,4,2,3,5]=>[[.,.],[[.,.],[.,[.,.]]]]
[.,[[[.,[.,.]],[.,.]],.]]=>[5,3,2,4,6,1]=>[6,3,1,4,2,5]=>[[[.,.],.],[[.,.],[.,.]]]
[.,[[[[.,.],.],[.,.]],.]]=>[5,2,3,4,6,1]=>[6,1,2,4,3,5]=>[[.,.],[.,[[.,.],[.,.]]]]
[.,[[[.,[.,[.,.]]],.],.]]=>[4,3,2,5,6,1]=>[6,3,2,1,4,5]=>[[[[.,.],.],.],[.,[.,.]]]
[.,[[[.,[[.,.],.]],.],.]]=>[3,4,2,5,6,1]=>[6,3,1,2,4,5]=>[[[.,.],.],[.,[.,[.,.]]]]
[.,[[[[.,.],[.,.]],.],.]]=>[4,2,3,5,6,1]=>[6,1,3,2,4,5]=>[[.,.],[[.,.],[.,[.,.]]]]
[.,[[[[.,[.,.]],.],.],.]]=>[3,2,4,5,6,1]=>[6,2,1,3,4,5]=>[[[.,.],.],[.,[.,[.,.]]]]
[.,[[[[[.,.],.],.],.],.]]=>[2,3,4,5,6,1]=>[6,1,2,3,4,5]=>[[.,.],[.,[.,[.,[.,.]]]]]
[[.,.],[.,[.,[.,[.,.]]]]]=>[6,5,4,3,1,2]=>[1,6,5,4,3,2]=>[.,[[[[[.,.],.],.],.],.]]
[[.,.],[.,[.,[[.,.],.]]]]=>[5,6,4,3,1,2]=>[1,6,5,4,2,3]=>[.,[[[[.,.],.],.],[.,.]]]
[[.,.],[.,[[.,.],[.,.]]]]=>[6,4,5,3,1,2]=>[1,6,5,2,4,3]=>[.,[[[.,.],.],[[.,.],.]]]
[[.,.],[.,[[.,[.,.]],.]]]=>[5,4,6,3,1,2]=>[1,6,5,3,2,4]=>[.,[[[[.,.],.],.],[.,.]]]
[[.,.],[.,[[[.,.],.],.]]]=>[4,5,6,3,1,2]=>[1,6,5,2,3,4]=>[.,[[[.,.],.],[.,[.,.]]]]
[[.,.],[[.,.],[.,[.,.]]]]=>[6,5,3,4,1,2]=>[1,6,2,5,4,3]=>[.,[[.,.],[[[.,.],.],.]]]
[[.,.],[[.,.],[[.,.],.]]]=>[5,6,3,4,1,2]=>[1,6,2,5,3,4]=>[.,[[.,.],[[.,.],[.,.]]]]
[[.,.],[[.,[.,.]],[.,.]]]=>[6,4,3,5,1,2]=>[1,6,4,2,5,3]=>[.,[[[.,.],.],[[.,.],.]]]
[[.,.],[[[.,.],.],[.,.]]]=>[6,3,4,5,1,2]=>[1,6,2,3,5,4]=>[.,[[.,.],[.,[[.,.],.]]]]
[[.,.],[[.,[.,[.,.]]],.]]=>[5,4,3,6,1,2]=>[1,6,4,3,2,5]=>[.,[[[[.,.],.],.],[.,.]]]
[[.,.],[[.,[[.,.],.]],.]]=>[4,5,3,6,1,2]=>[1,6,4,2,3,5]=>[.,[[[.,.],.],[.,[.,.]]]]
[[.,.],[[[.,.],[.,.]],.]]=>[5,3,4,6,1,2]=>[1,6,2,4,3,5]=>[.,[[.,.],[[.,.],[.,.]]]]
[[.,.],[[[.,[.,.]],.],.]]=>[4,3,5,6,1,2]=>[1,6,3,2,4,5]=>[.,[[[.,.],.],[.,[.,.]]]]
[[.,.],[[[[.,.],.],.],.]]=>[3,4,5,6,1,2]=>[1,6,2,3,4,5]=>[.,[[.,.],[.,[.,[.,.]]]]]
[[.,[.,.]],[.,[.,[.,.]]]]=>[6,5,4,2,1,3]=>[5,1,6,4,3,2]=>[[.,.],[[[[.,.],.],.],.]]
[[.,[.,.]],[.,[[.,.],.]]]=>[5,6,4,2,1,3]=>[5,1,6,4,2,3]=>[[.,.],[[[.,.],.],[.,.]]]
[[.,[.,.]],[[.,.],[.,.]]]=>[6,4,5,2,1,3]=>[5,1,6,2,4,3]=>[[.,.],[[.,.],[[.,.],.]]]
[[.,[.,.]],[[.,[.,.]],.]]=>[5,4,6,2,1,3]=>[5,1,6,3,2,4]=>[[.,.],[[[.,.],.],[.,.]]]
[[.,[.,.]],[[[.,.],.],.]]=>[4,5,6,2,1,3]=>[5,1,6,2,3,4]=>[[.,.],[[.,.],[.,[.,.]]]]
[[[.,.],.],[.,[.,[.,.]]]]=>[6,5,4,1,2,3]=>[1,2,6,5,4,3]=>[.,[.,[[[[.,.],.],.],.]]]
[[[.,.],.],[.,[[.,.],.]]]=>[5,6,4,1,2,3]=>[1,2,6,5,3,4]=>[.,[.,[[[.,.],.],[.,.]]]]
[[[.,.],.],[[.,.],[.,.]]]=>[6,4,5,1,2,3]=>[1,2,6,3,5,4]=>[.,[.,[[.,.],[[.,.],.]]]]
[[[.,.],.],[[.,[.,.]],.]]=>[5,4,6,1,2,3]=>[1,2,6,4,3,5]=>[.,[.,[[[.,.],.],[.,.]]]]
[[[.,.],.],[[[.,.],.],.]]=>[4,5,6,1,2,3]=>[1,2,6,3,4,5]=>[.,[.,[[.,.],[.,[.,.]]]]]
[[.,[.,[.,.]]],[.,[.,.]]]=>[6,5,3,2,1,4]=>[5,4,1,6,3,2]=>[[[.,.],.],[[[.,.],.],.]]
[[.,[.,[.,.]]],[[.,.],.]]=>[5,6,3,2,1,4]=>[5,4,1,6,2,3]=>[[[.,.],.],[[.,.],[.,.]]]
[[.,[[.,.],.]],[.,[.,.]]]=>[6,5,2,3,1,4]=>[5,1,2,6,4,3]=>[[.,.],[.,[[[.,.],.],.]]]
[[.,[[.,.],.]],[[.,.],.]]=>[5,6,2,3,1,4]=>[5,1,2,6,3,4]=>[[.,.],[.,[[.,.],[.,.]]]]
[[[.,.],[.,.]],[.,[.,.]]]=>[6,5,3,1,2,4]=>[1,5,2,6,4,3]=>[.,[[.,.],[[[.,.],.],.]]]
[[[.,.],[.,.]],[[.,.],.]]=>[5,6,3,1,2,4]=>[1,5,2,6,3,4]=>[.,[[.,.],[[.,.],[.,.]]]]
[[[.,[.,.]],.],[.,[.,.]]]=>[6,5,2,1,3,4]=>[4,1,2,6,5,3]=>[[.,.],[.,[[[.,.],.],.]]]
[[[.,[.,.]],.],[[.,.],.]]=>[5,6,2,1,3,4]=>[4,1,2,6,3,5]=>[[.,.],[.,[[.,.],[.,.]]]]
[[[[.,.],.],.],[.,[.,.]]]=>[6,5,1,2,3,4]=>[1,2,3,6,5,4]=>[.,[.,[.,[[[.,.],.],.]]]]
[[[[.,.],.],.],[[.,.],.]]=>[5,6,1,2,3,4]=>[1,2,3,6,4,5]=>[.,[.,[.,[[.,.],[.,.]]]]]
[[.,[.,[.,[.,.]]]],[.,.]]=>[6,4,3,2,1,5]=>[5,4,3,1,6,2]=>[[[[.,.],.],.],[[.,.],.]]
[[.,[.,[[.,.],.]]],[.,.]]=>[6,3,4,2,1,5]=>[5,4,1,2,6,3]=>[[[.,.],.],[.,[[.,.],.]]]
[[.,[[.,.],[.,.]]],[.,.]]=>[6,4,2,3,1,5]=>[5,1,4,2,6,3]=>[[.,.],[[.,.],[[.,.],.]]]
[[.,[[.,[.,.]],.]],[.,.]]=>[6,3,2,4,1,5]=>[5,3,1,2,6,4]=>[[[.,.],.],[.,[[.,.],.]]]
[[.,[[[.,.],.],.]],[.,.]]=>[6,2,3,4,1,5]=>[5,1,2,3,6,4]=>[[.,.],[.,[.,[[.,.],.]]]]
[[[.,.],[.,[.,.]]],[.,.]]=>[6,4,3,1,2,5]=>[1,5,4,2,6,3]=>[.,[[[.,.],.],[[.,.],.]]]
[[[.,.],[[.,.],.]],[.,.]]=>[6,3,4,1,2,5]=>[1,5,2,3,6,4]=>[.,[[.,.],[.,[[.,.],.]]]]
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[.,[[.,[.,[.,[.,.]]]],[.,.]]]=>[7,5,4,3,2,6,1]=>[7,5,4,3,1,6,2]=>[[[[[.,.],.],.],.],[[.,.],.]]
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[.,[[.,[[.,.],[.,.]]],[.,.]]]=>[7,5,3,4,2,6,1]=>[7,5,1,4,2,6,3]=>[[[.,.],.],[[.,.],[[.,.],.]]]
[.,[[.,[[.,[.,.]],.]],[.,.]]]=>[7,4,3,5,2,6,1]=>[7,5,3,1,2,6,4]=>[[[[.,.],.],.],[.,[[.,.],.]]]
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[.,[[[.,.],[[.,.],.]],[.,.]]]=>[7,4,5,2,3,6,1]=>[7,1,5,2,3,6,4]=>[[.,.],[[.,.],[.,[[.,.],.]]]]
[.,[[[.,[.,.]],[.,.]],[.,.]]]=>[7,5,3,2,4,6,1]=>[7,4,1,5,2,6,3]=>[[[.,.],.],[[.,.],[[.,.],.]]]
[.,[[[[.,.],.],[.,.]],[.,.]]]=>[7,5,2,3,4,6,1]=>[7,1,2,5,3,6,4]=>[[.,.],[.,[[.,.],[[.,.],.]]]]
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[.,[[[.,[[.,.],.]],.],[.,.]]]=>[7,3,4,2,5,6,1]=>[7,4,1,2,3,6,5]=>[[[.,.],.],[.,[.,[[.,.],.]]]]
[.,[[[[.,.],[.,.]],.],[.,.]]]=>[7,4,2,3,5,6,1]=>[7,1,4,2,3,6,5]=>[[.,.],[[.,.],[.,[[.,.],.]]]]
[.,[[[[.,[.,.]],.],.],[.,.]]]=>[7,3,2,4,5,6,1]=>[7,3,1,2,4,6,5]=>[[[.,.],.],[.,[.,[[.,.],.]]]]
[.,[[[[[.,.],.],.],.],[.,.]]]=>[7,2,3,4,5,6,1]=>[7,1,2,3,4,6,5]=>[[.,.],[.,[.,[.,[[.,.],.]]]]]
[.,[[.,[.,[.,[.,[.,.]]]]],.]]=>[6,5,4,3,2,7,1]=>[7,5,4,3,2,1,6]=>[[[[[[.,.],.],.],.],.],[.,.]]
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[.,[[.,[.,[[.,[.,.]],.]]],.]]=>[5,4,6,3,2,7,1]=>[7,5,4,2,1,3,6]=>[[[[[.,.],.],.],.],[.,[.,.]]]
[.,[[.,[.,[[[.,.],.],.]]],.]]=>[4,5,6,3,2,7,1]=>[7,5,4,1,2,3,6]=>[[[[.,.],.],.],[.,[.,[.,.]]]]
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[.,[[.,[[.,[[.,.],.]],.]],.]]=>[4,5,3,6,2,7,1]=>[7,5,3,1,2,4,6]=>[[[[.,.],.],.],[.,[.,[.,.]]]]
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[.,[[[.,[.,.]],[.,[.,.]]],.]]=>[6,5,3,2,4,7,1]=>[7,4,1,5,3,2,6]=>[[[.,.],.],[[[.,.],.],[.,.]]]
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[.,[[[.,[.,[.,.]]],[.,.]],.]]=>[6,4,3,2,5,7,1]=>[7,4,3,1,5,2,6]=>[[[[.,.],.],.],[[.,.],[.,.]]]
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[[.,.],[.,[.,[.,[.,[.,.]]]]]]=>[7,6,5,4,3,1,2]=>[1,7,6,5,4,3,2]=>[.,[[[[[[.,.],.],.],.],.],.]]
[[.,.],[.,[.,[.,[[.,.],.]]]]]=>[6,7,5,4,3,1,2]=>[1,7,6,5,4,2,3]=>[.,[[[[[.,.],.],.],.],[.,.]]]
[[.,.],[.,[.,[[.,.],[.,.]]]]]=>[7,5,6,4,3,1,2]=>[1,7,6,5,2,4,3]=>[.,[[[[.,.],.],.],[[.,.],.]]]
[[.,.],[.,[.,[[.,[.,.]],.]]]]=>[6,5,7,4,3,1,2]=>[1,7,6,5,3,2,4]=>[.,[[[[[.,.],.],.],.],[.,.]]]
[[.,.],[.,[.,[[[.,.],.],.]]]]=>[5,6,7,4,3,1,2]=>[1,7,6,5,2,3,4]=>[.,[[[[.,.],.],.],[.,[.,.]]]]
[[.,.],[.,[[.,.],[.,[.,.]]]]]=>[7,6,4,5,3,1,2]=>[1,7,6,2,5,4,3]=>[.,[[[.,.],.],[[[.,.],.],.]]]
[[.,.],[.,[[.,.],[[.,.],.]]]]=>[6,7,4,5,3,1,2]=>[1,7,6,2,5,3,4]=>[.,[[[.,.],.],[[.,.],[.,.]]]]
[[.,.],[.,[[.,[.,.]],[.,.]]]]=>[7,5,4,6,3,1,2]=>[1,7,6,4,2,5,3]=>[.,[[[[.,.],.],.],[[.,.],.]]]
[[.,.],[.,[[[.,.],.],[.,.]]]]=>[7,4,5,6,3,1,2]=>[1,7,6,2,3,5,4]=>[.,[[[.,.],.],[.,[[.,.],.]]]]
[[.,.],[.,[[.,[.,[.,.]]],.]]]=>[6,5,4,7,3,1,2]=>[1,7,6,4,3,2,5]=>[.,[[[[[.,.],.],.],.],[.,.]]]
[[.,.],[.,[[.,[[.,.],.]],.]]]=>[5,6,4,7,3,1,2]=>[1,7,6,4,2,3,5]=>[.,[[[[.,.],.],.],[.,[.,.]]]]
[[.,.],[.,[[[.,.],[.,.]],.]]]=>[6,4,5,7,3,1,2]=>[1,7,6,2,4,3,5]=>[.,[[[.,.],.],[[.,.],[.,.]]]]
[[.,.],[.,[[[.,[.,.]],.],.]]]=>[5,4,6,7,3,1,2]=>[1,7,6,3,2,4,5]=>[.,[[[[.,.],.],.],[.,[.,.]]]]
[[.,.],[.,[[[[.,.],.],.],.]]]=>[4,5,6,7,3,1,2]=>[1,7,6,2,3,4,5]=>[.,[[[.,.],.],[.,[.,[.,.]]]]]
[[.,.],[[.,.],[.,[.,[.,.]]]]]=>[7,6,5,3,4,1,2]=>[1,7,2,6,5,4,3]=>[.,[[.,.],[[[[.,.],.],.],.]]]
[[.,.],[[.,.],[.,[[.,.],.]]]]=>[6,7,5,3,4,1,2]=>[1,7,2,6,5,3,4]=>[.,[[.,.],[[[.,.],.],[.,.]]]]
[[.,.],[[.,.],[[.,.],[.,.]]]]=>[7,5,6,3,4,1,2]=>[1,7,2,6,3,5,4]=>[.,[[.,.],[[.,.],[[.,.],.]]]]
[[.,.],[[.,.],[[.,[.,.]],.]]]=>[6,5,7,3,4,1,2]=>[1,7,2,6,4,3,5]=>[.,[[.,.],[[[.,.],.],[.,.]]]]
[[.,.],[[.,.],[[[.,.],.],.]]]=>[5,6,7,3,4,1,2]=>[1,7,2,6,3,4,5]=>[.,[[.,.],[[.,.],[.,[.,.]]]]]
[[.,.],[[.,[.,.]],[.,[.,.]]]]=>[7,6,4,3,5,1,2]=>[1,7,5,2,6,4,3]=>[.,[[[.,.],.],[[[.,.],.],.]]]
[[.,.],[[.,[.,.]],[[.,.],.]]]=>[6,7,4,3,5,1,2]=>[1,7,5,2,6,3,4]=>[.,[[[.,.],.],[[.,.],[.,.]]]]
[[.,.],[[[.,.],.],[.,[.,.]]]]=>[7,6,3,4,5,1,2]=>[1,7,2,3,6,5,4]=>[.,[[.,.],[.,[[[.,.],.],.]]]]
[[.,.],[[[.,.],.],[[.,.],.]]]=>[6,7,3,4,5,1,2]=>[1,7,2,3,6,4,5]=>[.,[[.,.],[.,[[.,.],[.,.]]]]]
[[.,.],[[.,[.,[.,.]]],[.,.]]]=>[7,5,4,3,6,1,2]=>[1,7,5,4,2,6,3]=>[.,[[[[.,.],.],.],[[.,.],.]]]
[[.,.],[[.,[[.,.],.]],[.,.]]]=>[7,4,5,3,6,1,2]=>[1,7,5,2,3,6,4]=>[.,[[[.,.],.],[.,[[.,.],.]]]]
[[.,.],[[[.,.],[.,.]],[.,.]]]=>[7,5,3,4,6,1,2]=>[1,7,2,5,3,6,4]=>[.,[[.,.],[[.,.],[[.,.],.]]]]
[[.,.],[[[.,[.,.]],.],[.,.]]]=>[7,4,3,5,6,1,2]=>[1,7,4,2,3,6,5]=>[.,[[[.,.],.],[.,[[.,.],.]]]]
[[.,.],[[[[.,.],.],.],[.,.]]]=>[7,3,4,5,6,1,2]=>[1,7,2,3,4,6,5]=>[.,[[.,.],[.,[.,[[.,.],.]]]]]
[[.,.],[[.,[.,[.,[.,.]]]],.]]=>[6,5,4,3,7,1,2]=>[1,7,5,4,3,2,6]=>[.,[[[[[.,.],.],.],.],[.,.]]]
[[.,.],[[.,[.,[[.,.],.]]],.]]=>[5,6,4,3,7,1,2]=>[1,7,5,4,2,3,6]=>[.,[[[[.,.],.],.],[.,[.,.]]]]
[[.,.],[[.,[[.,.],[.,.]]],.]]=>[6,4,5,3,7,1,2]=>[1,7,5,2,4,3,6]=>[.,[[[.,.],.],[[.,.],[.,.]]]]
[[.,.],[[.,[[.,[.,.]],.]],.]]=>[5,4,6,3,7,1,2]=>[1,7,5,3,2,4,6]=>[.,[[[[.,.],.],.],[.,[.,.]]]]
[[.,.],[[.,[[[.,.],.],.]],.]]=>[4,5,6,3,7,1,2]=>[1,7,5,2,3,4,6]=>[.,[[[.,.],.],[.,[.,[.,.]]]]]
[[.,.],[[[.,.],[.,[.,.]]],.]]=>[6,5,3,4,7,1,2]=>[1,7,2,5,4,3,6]=>[.,[[.,.],[[[.,.],.],[.,.]]]]
[[.,.],[[[.,.],[[.,.],.]],.]]=>[5,6,3,4,7,1,2]=>[1,7,2,5,3,4,6]=>[.,[[.,.],[[.,.],[.,[.,.]]]]]
[[.,.],[[[.,[.,.]],[.,.]],.]]=>[6,4,3,5,7,1,2]=>[1,7,4,2,5,3,6]=>[.,[[[.,.],.],[[.,.],[.,.]]]]
[[.,.],[[[[.,.],.],[.,.]],.]]=>[6,3,4,5,7,1,2]=>[1,7,2,3,5,4,6]=>[.,[[.,.],[.,[[.,.],[.,.]]]]]
[[.,.],[[[.,[.,[.,.]]],.],.]]=>[5,4,3,6,7,1,2]=>[1,7,4,3,2,5,6]=>[.,[[[[.,.],.],.],[.,[.,.]]]]
[[.,.],[[[.,[[.,.],.]],.],.]]=>[4,5,3,6,7,1,2]=>[1,7,4,2,3,5,6]=>[.,[[[.,.],.],[.,[.,[.,.]]]]]
[[.,.],[[[[.,.],[.,.]],.],.]]=>[5,3,4,6,7,1,2]=>[1,7,2,4,3,5,6]=>[.,[[.,.],[[.,.],[.,[.,.]]]]]
[[.,.],[[[[.,[.,.]],.],.],.]]=>[4,3,5,6,7,1,2]=>[1,7,3,2,4,5,6]=>[.,[[[.,.],.],[.,[.,[.,.]]]]]
[[.,.],[[[[[.,.],.],.],.],.]]=>[3,4,5,6,7,1,2]=>[1,7,2,3,4,5,6]=>[.,[[.,.],[.,[.,[.,[.,.]]]]]]
[[.,[.,.]],[.,[.,[.,[.,.]]]]]=>[7,6,5,4,2,1,3]=>[6,1,7,5,4,3,2]=>[[.,.],[[[[[.,.],.],.],.],.]]
[[.,[.,.]],[.,[.,[[.,.],.]]]]=>[6,7,5,4,2,1,3]=>[6,1,7,5,4,2,3]=>[[.,.],[[[[.,.],.],.],[.,.]]]
[[.,[.,.]],[.,[[.,.],[.,.]]]]=>[7,5,6,4,2,1,3]=>[6,1,7,5,2,4,3]=>[[.,.],[[[.,.],.],[[.,.],.]]]
[[.,[.,.]],[.,[[.,[.,.]],.]]]=>[6,5,7,4,2,1,3]=>[6,1,7,5,3,2,4]=>[[.,.],[[[[.,.],.],.],[.,.]]]
[[.,[.,.]],[.,[[[.,.],.],.]]]=>[5,6,7,4,2,1,3]=>[6,1,7,5,2,3,4]=>[[.,.],[[[.,.],.],[.,[.,.]]]]
[[.,[.,.]],[[.,.],[.,[.,.]]]]=>[7,6,4,5,2,1,3]=>[6,1,7,2,5,4,3]=>[[.,.],[[.,.],[[[.,.],.],.]]]
[[.,[.,.]],[[.,.],[[.,.],.]]]=>[6,7,4,5,2,1,3]=>[6,1,7,2,5,3,4]=>[[.,.],[[.,.],[[.,.],[.,.]]]]
[[.,[.,.]],[[.,[.,.]],[.,.]]]=>[7,5,4,6,2,1,3]=>[6,1,7,4,2,5,3]=>[[.,.],[[[.,.],.],[[.,.],.]]]
[[.,[.,.]],[[[.,.],.],[.,.]]]=>[7,4,5,6,2,1,3]=>[6,1,7,2,3,5,4]=>[[.,.],[[.,.],[.,[[.,.],.]]]]
[[.,[.,.]],[[.,[.,[.,.]]],.]]=>[6,5,4,7,2,1,3]=>[6,1,7,4,3,2,5]=>[[.,.],[[[[.,.],.],.],[.,.]]]
[[.,[.,.]],[[.,[[.,.],.]],.]]=>[5,6,4,7,2,1,3]=>[6,1,7,4,2,3,5]=>[[.,.],[[[.,.],.],[.,[.,.]]]]
[[.,[.,.]],[[[.,.],[.,.]],.]]=>[6,4,5,7,2,1,3]=>[6,1,7,2,4,3,5]=>[[.,.],[[.,.],[[.,.],[.,.]]]]
[[.,[.,.]],[[[.,[.,.]],.],.]]=>[5,4,6,7,2,1,3]=>[6,1,7,3,2,4,5]=>[[.,.],[[[.,.],.],[.,[.,.]]]]
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[[[[[.,[.,.]],[.,.]],.],.],.]=>[4,2,1,3,5,6,7]=>[3,1,4,2,5,6,7]=>[[.,.],[[.,.],[.,[.,[.,.]]]]]
[[[[[[.,.],.],[.,.]],.],.],.]=>[4,1,2,3,5,6,7]=>[1,2,4,3,5,6,7]=>[.,[.,[[.,.],[.,[.,[.,.]]]]]]
[[[[[.,[.,[.,.]]],.],.],.],.]=>[3,2,1,4,5,6,7]=>[3,2,1,4,5,6,7]=>[[[.,.],.],[.,[.,[.,[.,.]]]]]
[[[[[.,[[.,.],.]],.],.],.],.]=>[2,3,1,4,5,6,7]=>[3,1,2,4,5,6,7]=>[[.,.],[.,[.,[.,[.,[.,.]]]]]]
[[[[[[.,.],[.,.]],.],.],.],.]=>[3,1,2,4,5,6,7]=>[1,3,2,4,5,6,7]=>[.,[[.,.],[.,[.,[.,[.,.]]]]]]
[[[[[[.,[.,.]],.],.],.],.],.]=>[2,1,3,4,5,6,7]=>[2,1,3,4,5,6,7]=>[[.,.],[.,[.,[.,[.,[.,.]]]]]]
[[[[[[[.,.],.],.],.],.],.],.]=>[1,2,3,4,5,6,7]=>[1,2,3,4,5,6,7]=>[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>[[[[[[[[.,.],.],.],.],.],.],.],.]
[[[[[[[[.,.],.],.],.],.],.],.],.]=>[1,2,3,4,5,6,7,8]=>[1,2,3,4,5,6,7,8]=>[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
Map
to 132-avoiding permutation
Description
Return a 132-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 maximal element of the 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 maximal element of the 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.