Identifier
Mp00016: left-right symmetryBinary trees
Mp00018: left border symmetryBinary trees
Mp00017: to 312-avoiding permutationPermutations
Images
=>
Cc0010;cc-rep-0Cc0010;cc-rep-1Cc0010;cc-rep-2
[.,.]=>[.,.]=>[.,.]=>[1] [.,[.,.]]=>[[.,.],.]=>[[.,.],.]=>[1,2] [[.,.],.]=>[.,[.,.]]=>[.,[.,.]]=>[2,1] [.,[.,[.,.]]]=>[[[.,.],.],.]=>[[[.,.],.],.]=>[1,2,3] [.,[[.,.],.]]=>[[.,[.,.]],.]=>[[.,.],[.,.]]=>[1,3,2] [[.,.],[.,.]]=>[[.,.],[.,.]]=>[[.,[.,.]],.]=>[2,1,3] [[.,[.,.]],.]=>[.,[[.,.],.]]=>[.,[[.,.],.]]=>[2,3,1] [[[.,.],.],.]=>[.,[.,[.,.]]]=>[.,[.,[.,.]]]=>[3,2,1] [.,[.,[.,[.,.]]]]=>[[[[.,.],.],.],.]=>[[[[.,.],.],.],.]=>[1,2,3,4] [.,[.,[[.,.],.]]]=>[[[.,[.,.]],.],.]=>[[[.,.],.],[.,.]]=>[1,2,4,3] [.,[[.,.],[.,.]]]=>[[[.,.],[.,.]],.]=>[[[.,.],[.,.]],.]=>[1,3,2,4] [.,[[.,[.,.]],.]]=>[[.,[[.,.],.]],.]=>[[.,.],[[.,.],.]]=>[1,3,4,2] [.,[[[.,.],.],.]]=>[[.,[.,[.,.]]],.]=>[[.,.],[.,[.,.]]]=>[1,4,3,2] [[.,.],[.,[.,.]]]=>[[[.,.],.],[.,.]]=>[[[.,[.,.]],.],.]=>[2,1,3,4] [[.,.],[[.,.],.]]=>[[.,[.,.]],[.,.]]=>[[.,[.,.]],[.,.]]=>[2,1,4,3] [[.,[.,.]],[.,.]]=>[[.,.],[[.,.],.]]=>[[.,[[.,.],.]],.]=>[2,3,1,4] [[[.,.],.],[.,.]]=>[[.,.],[.,[.,.]]]=>[[.,[.,[.,.]]],.]=>[3,2,1,4] [[.,[.,[.,.]]],.]=>[.,[[[.,.],.],.]]=>[.,[[[.,.],.],.]]=>[2,3,4,1] [[.,[[.,.],.]],.]=>[.,[[.,[.,.]],.]]=>[.,[[.,.],[.,.]]]=>[2,4,3,1] [[[.,.],[.,.]],.]=>[.,[[.,.],[.,.]]]=>[.,[[.,[.,.]],.]]=>[3,2,4,1] [[[.,[.,.]],.],.]=>[.,[.,[[.,.],.]]]=>[.,[.,[[.,.],.]]]=>[3,4,2,1] [[[[.,.],.],.],.]=>[.,[.,[.,[.,.]]]]=>[.,[.,[.,[.,.]]]]=>[4,3,2,1] [.,[.,[.,[.,[.,.]]]]]=>[[[[[.,.],.],.],.],.]=>[[[[[.,.],.],.],.],.]=>[1,2,3,4,5] [.,[.,[.,[[.,.],.]]]]=>[[[[.,[.,.]],.],.],.]=>[[[[.,.],.],.],[.,.]]=>[1,2,3,5,4] [.,[.,[[.,.],[.,.]]]]=>[[[[.,.],[.,.]],.],.]=>[[[[.,.],.],[.,.]],.]=>[1,2,4,3,5] [.,[.,[[.,[.,.]],.]]]=>[[[.,[[.,.],.]],.],.]=>[[[.,.],.],[[.,.],.]]=>[1,2,4,5,3] [.,[.,[[[.,.],.],.]]]=>[[[.,[.,[.,.]]],.],.]=>[[[.,.],.],[.,[.,.]]]=>[1,2,5,4,3] [.,[[.,.],[.,[.,.]]]]=>[[[[.,.],.],[.,.]],.]=>[[[[.,.],[.,.]],.],.]=>[1,3,2,4,5] [.,[[.,.],[[.,.],.]]]=>[[[.,[.,.]],[.,.]],.]=>[[[.,.],[.,.]],[.,.]]=>[1,3,2,5,4] [.,[[.,[.,.]],[.,.]]]=>[[[.,.],[[.,.],.]],.]=>[[[.,.],[[.,.],.]],.]=>[1,3,4,2,5] [.,[[[.,.],.],[.,.]]]=>[[[.,.],[.,[.,.]]],.]=>[[[.,.],[.,[.,.]]],.]=>[1,4,3,2,5] [.,[[.,[.,[.,.]]],.]]=>[[.,[[[.,.],.],.]],.]=>[[.,.],[[[.,.],.],.]]=>[1,3,4,5,2] [.,[[.,[[.,.],.]],.]]=>[[.,[[.,[.,.]],.]],.]=>[[.,.],[[.,.],[.,.]]]=>[1,3,5,4,2] [.,[[[.,.],[.,.]],.]]=>[[.,[[.,.],[.,.]]],.]=>[[.,.],[[.,[.,.]],.]]=>[1,4,3,5,2] [.,[[[.,[.,.]],.],.]]=>[[.,[.,[[.,.],.]]],.]=>[[.,.],[.,[[.,.],.]]]=>[1,4,5,3,2] [.,[[[[.,.],.],.],.]]=>[[.,[.,[.,[.,.]]]],.]=>[[.,.],[.,[.,[.,.]]]]=>[1,5,4,3,2] [[.,.],[.,[.,[.,.]]]]=>[[[[.,.],.],.],[.,.]]=>[[[[.,[.,.]],.],.],.]=>[2,1,3,4,5] [[.,.],[.,[[.,.],.]]]=>[[[.,[.,.]],.],[.,.]]=>[[[.,[.,.]],.],[.,.]]=>[2,1,3,5,4] [[.,.],[[.,.],[.,.]]]=>[[[.,.],[.,.]],[.,.]]=>[[[.,[.,.]],[.,.]],.]=>[2,1,4,3,5] [[.,.],[[.,[.,.]],.]]=>[[.,[[.,.],.]],[.,.]]=>[[.,[.,.]],[[.,.],.]]=>[2,1,4,5,3] [[.,.],[[[.,.],.],.]]=>[[.,[.,[.,.]]],[.,.]]=>[[.,[.,.]],[.,[.,.]]]=>[2,1,5,4,3] [[.,[.,.]],[.,[.,.]]]=>[[[.,.],.],[[.,.],.]]=>[[[.,[[.,.],.]],.],.]=>[2,3,1,4,5] [[.,[.,.]],[[.,.],.]]=>[[.,[.,.]],[[.,.],.]]=>[[.,[[.,.],.]],[.,.]]=>[2,3,1,5,4] [[[.,.],.],[.,[.,.]]]=>[[[.,.],.],[.,[.,.]]]=>[[[.,[.,[.,.]]],.],.]=>[3,2,1,4,5] [[[.,.],.],[[.,.],.]]=>[[.,[.,.]],[.,[.,.]]]=>[[.,[.,[.,.]]],[.,.]]=>[3,2,1,5,4] [[.,[.,[.,.]]],[.,.]]=>[[.,.],[[[.,.],.],.]]=>[[.,[[[.,.],.],.]],.]=>[2,3,4,1,5] [[.,[[.,.],.]],[.,.]]=>[[.,.],[[.,[.,.]],.]]=>[[.,[[.,.],[.,.]]],.]=>[2,4,3,1,5] [[[.,.],[.,.]],[.,.]]=>[[.,.],[[.,.],[.,.]]]=>[[.,[[.,[.,.]],.]],.]=>[3,2,4,1,5] [[[.,[.,.]],.],[.,.]]=>[[.,.],[.,[[.,.],.]]]=>[[.,[.,[[.,.],.]]],.]=>[3,4,2,1,5] [[[[.,.],.],.],[.,.]]=>[[.,.],[.,[.,[.,.]]]]=>[[.,[.,[.,[.,.]]]],.]=>[4,3,2,1,5] [[.,[.,[.,[.,.]]]],.]=>[.,[[[[.,.],.],.],.]]=>[.,[[[[.,.],.],.],.]]=>[2,3,4,5,1] [[.,[.,[[.,.],.]]],.]=>[.,[[[.,[.,.]],.],.]]=>[.,[[[.,.],.],[.,.]]]=>[2,3,5,4,1] [[.,[[.,.],[.,.]]],.]=>[.,[[[.,.],[.,.]],.]]=>[.,[[[.,.],[.,.]],.]]=>[2,4,3,5,1] [[.,[[.,[.,.]],.]],.]=>[.,[[.,[[.,.],.]],.]]=>[.,[[.,.],[[.,.],.]]]=>[2,4,5,3,1] [[.,[[[.,.],.],.]],.]=>[.,[[.,[.,[.,.]]],.]]=>[.,[[.,.],[.,[.,.]]]]=>[2,5,4,3,1] [[[.,.],[.,[.,.]]],.]=>[.,[[[.,.],.],[.,.]]]=>[.,[[[.,[.,.]],.],.]]=>[3,2,4,5,1] [[[.,.],[[.,.],.]],.]=>[.,[[.,[.,.]],[.,.]]]=>[.,[[.,[.,.]],[.,.]]]=>[3,2,5,4,1] [[[.,[.,.]],[.,.]],.]=>[.,[[.,.],[[.,.],.]]]=>[.,[[.,[[.,.],.]],.]]=>[3,4,2,5,1] [[[[.,.],.],[.,.]],.]=>[.,[[.,.],[.,[.,.]]]]=>[.,[[.,[.,[.,.]]],.]]=>[4,3,2,5,1] [[[.,[.,[.,.]]],.],.]=>[.,[.,[[[.,.],.],.]]]=>[.,[.,[[[.,.],.],.]]]=>[3,4,5,2,1] [[[.,[[.,.],.]],.],.]=>[.,[.,[[.,[.,.]],.]]]=>[.,[.,[[.,.],[.,.]]]]=>[3,5,4,2,1] [[[[.,.],[.,.]],.],.]=>[.,[.,[[.,.],[.,.]]]]=>[.,[.,[[.,[.,.]],.]]]=>[4,3,5,2,1] [[[[.,[.,.]],.],.],.]=>[.,[.,[.,[[.,.],.]]]]=>[.,[.,[.,[[.,.],.]]]]=>[4,5,3,2,1] [[[[[.,.],.],.],.],.]=>[.,[.,[.,[.,[.,.]]]]]=>[.,[.,[.,[.,[.,.]]]]]=>[5,4,3,2,1] [.,[.,[.,[.,[.,[.,.]]]]]]=>[[[[[[.,.],.],.],.],.],.]=>[[[[[[.,.],.],.],.],.],.]=>[1,2,3,4,5,6] [.,[.,[.,[.,[[.,.],.]]]]]=>[[[[[.,[.,.]],.],.],.],.]=>[[[[[.,.],.],.],.],[.,.]]=>[1,2,3,4,6,5] 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[.,[[[.,.],.],[[.,.],.]]]=>[[[.,[.,.]],[.,[.,.]]],.]=>[[[.,.],[.,[.,.]]],[.,.]]=>[1,4,3,2,6,5] [.,[[.,[.,[.,.]]],[.,.]]]=>[[[.,.],[[[.,.],.],.]],.]=>[[[.,.],[[[.,.],.],.]],.]=>[1,3,4,5,2,6] [.,[[.,[[.,.],.]],[.,.]]]=>[[[.,.],[[.,[.,.]],.]],.]=>[[[.,.],[[.,.],[.,.]]],.]=>[1,3,5,4,2,6] [.,[[[.,.],[.,.]],[.,.]]]=>[[[.,.],[[.,.],[.,.]]],.]=>[[[.,.],[[.,[.,.]],.]],.]=>[1,4,3,5,2,6] [.,[[[.,[.,.]],.],[.,.]]]=>[[[.,.],[.,[[.,.],.]]],.]=>[[[.,.],[.,[[.,.],.]]],.]=>[1,4,5,3,2,6] [.,[[[[.,.],.],.],[.,.]]]=>[[[.,.],[.,[.,[.,.]]]],.]=>[[[.,.],[.,[.,[.,.]]]],.]=>[1,5,4,3,2,6] [.,[[.,[.,[.,[.,.]]]],.]]=>[[.,[[[[.,.],.],.],.]],.]=>[[.,.],[[[[.,.],.],.],.]]=>[1,3,4,5,6,2] [.,[[.,[.,[[.,.],.]]],.]]=>[[.,[[[.,[.,.]],.],.]],.]=>[[.,.],[[[.,.],.],[.,.]]]=>[1,3,4,6,5,2] [.,[[.,[[.,.],[.,.]]],.]]=>[[.,[[[.,.],[.,.]],.]],.]=>[[.,.],[[[.,.],[.,.]],.]]=>[1,3,5,4,6,2] [.,[[.,[[.,[.,.]],.]],.]]=>[[.,[[.,[[.,.],.]],.]],.]=>[[.,.],[[.,.],[[.,.],.]]]=>[1,3,5,6,4,2] 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[[.,.],[[.,[[.,.],.]],.]]=>[[.,[[.,[.,.]],.]],[.,.]]=>[[.,[.,.]],[[.,.],[.,.]]]=>[2,1,4,6,5,3] [[.,.],[[[.,.],[.,.]],.]]=>[[.,[[.,.],[.,.]]],[.,.]]=>[[.,[.,.]],[[.,[.,.]],.]]=>[2,1,5,4,6,3] [[.,.],[[[.,[.,.]],.],.]]=>[[.,[.,[[.,.],.]]],[.,.]]=>[[.,[.,.]],[.,[[.,.],.]]]=>[2,1,5,6,4,3] [[.,.],[[[[.,.],.],.],.]]=>[[.,[.,[.,[.,.]]]],[.,.]]=>[[.,[.,.]],[.,[.,[.,.]]]]=>[2,1,6,5,4,3] [[.,[.,.]],[.,[.,[.,.]]]]=>[[[[.,.],.],.],[[.,.],.]]=>[[[[.,[[.,.],.]],.],.],.]=>[2,3,1,4,5,6] [[.,[.,.]],[.,[[.,.],.]]]=>[[[.,[.,.]],.],[[.,.],.]]=>[[[.,[[.,.],.]],.],[.,.]]=>[2,3,1,4,6,5] [[.,[.,.]],[[.,.],[.,.]]]=>[[[.,.],[.,.]],[[.,.],.]]=>[[[.,[[.,.],.]],[.,.]],.]=>[2,3,1,5,4,6] [[.,[.,.]],[[.,[.,.]],.]]=>[[.,[[.,.],.]],[[.,.],.]]=>[[.,[[.,.],.]],[[.,.],.]]=>[2,3,1,5,6,4] [[.,[.,.]],[[[.,.],.],.]]=>[[.,[.,[.,.]]],[[.,.],.]]=>[[.,[[.,.],.]],[.,[.,.]]]=>[2,3,1,6,5,4] [[[.,.],.],[.,[.,[.,.]]]]=>[[[[.,.],.],.],[.,[.,.]]]=>[[[[.,[.,[.,.]]],.],.],.]=>[3,2,1,4,5,6] 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[[.,.],[[[.,.],.],[.,[.,.]]]]=>[[[[.,.],.],[.,[.,.]]],[.,.]]=>[[[[.,[.,.]],[.,[.,.]]],.],.]=>[2,1,5,4,3,6,7] [[.,.],[[[.,.],.],[[.,.],.]]]=>[[[.,[.,.]],[.,[.,.]]],[.,.]]=>[[[.,[.,.]],[.,[.,.]]],[.,.]]=>[2,1,5,4,3,7,6] [[.,.],[[.,[.,[.,.]]],[.,.]]]=>[[[.,.],[[[.,.],.],.]],[.,.]]=>[[[.,[.,.]],[[[.,.],.],.]],.]=>[2,1,4,5,6,3,7] [[.,.],[[.,[[.,.],.]],[.,.]]]=>[[[.,.],[[.,[.,.]],.]],[.,.]]=>[[[.,[.,.]],[[.,.],[.,.]]],.]=>[2,1,4,6,5,3,7] [[.,.],[[[.,.],[.,.]],[.,.]]]=>[[[.,.],[[.,.],[.,.]]],[.,.]]=>[[[.,[.,.]],[[.,[.,.]],.]],.]=>[2,1,5,4,6,3,7] [[.,.],[[[.,[.,.]],.],[.,.]]]=>[[[.,.],[.,[[.,.],.]]],[.,.]]=>[[[.,[.,.]],[.,[[.,.],.]]],.]=>[2,1,5,6,4,3,7] [[.,.],[[[[.,.],.],.],[.,.]]]=>[[[.,.],[.,[.,[.,.]]]],[.,.]]=>[[[.,[.,.]],[.,[.,[.,.]]]],.]=>[2,1,6,5,4,3,7] [[.,.],[[.,[.,[.,[.,.]]]],.]]=>[[.,[[[[.,.],.],.],.]],[.,.]]=>[[.,[.,.]],[[[[.,.],.],.],.]]=>[2,1,4,5,6,7,3] [[.,.],[[.,[.,[[.,.],.]]],.]]=>[[.,[[[.,[.,.]],.],.]],[.,.]]=>[[.,[.,.]],[[[.,.],.],[.,.]]]=>[2,1,4,5,7,6,3] [[.,.],[[.,[[.,.],[.,.]]],.]]=>[[.,[[[.,.],[.,.]],.]],[.,.]]=>[[.,[.,.]],[[[.,.],[.,.]],.]]=>[2,1,4,6,5,7,3] [[.,.],[[.,[[.,[.,.]],.]],.]]=>[[.,[[.,[[.,.],.]],.]],[.,.]]=>[[.,[.,.]],[[.,.],[[.,.],.]]]=>[2,1,4,6,7,5,3] [[.,.],[[.,[[[.,.],.],.]],.]]=>[[.,[[.,[.,[.,.]]],.]],[.,.]]=>[[.,[.,.]],[[.,.],[.,[.,.]]]]=>[2,1,4,7,6,5,3] [[.,.],[[[.,.],[.,[.,.]]],.]]=>[[.,[[[.,.],.],[.,.]]],[.,.]]=>[[.,[.,.]],[[[.,[.,.]],.],.]]=>[2,1,5,4,6,7,3] [[.,.],[[[.,.],[[.,.],.]],.]]=>[[.,[[.,[.,.]],[.,.]]],[.,.]]=>[[.,[.,.]],[[.,[.,.]],[.,.]]]=>[2,1,5,4,7,6,3] [[.,.],[[[.,[.,.]],[.,.]],.]]=>[[.,[[.,.],[[.,.],.]]],[.,.]]=>[[.,[.,.]],[[.,[[.,.],.]],.]]=>[2,1,5,6,4,7,3] [[.,.],[[[[.,.],.],[.,.]],.]]=>[[.,[[.,.],[.,[.,.]]]],[.,.]]=>[[.,[.,.]],[[.,[.,[.,.]]],.]]=>[2,1,6,5,4,7,3] [[.,.],[[[.,[.,[.,.]]],.],.]]=>[[.,[.,[[[.,.],.],.]]],[.,.]]=>[[.,[.,.]],[.,[[[.,.],.],.]]]=>[2,1,5,6,7,4,3] [[.,.],[[[.,[[.,.],.]],.],.]]=>[[.,[.,[[.,[.,.]],.]]],[.,.]]=>[[.,[.,.]],[.,[[.,.],[.,.]]]]=>[2,1,5,7,6,4,3] 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[[.,[.,.]],[[.,.],[[.,.],.]]]=>[[[.,[.,.]],[.,.]],[[.,.],.]]=>[[[.,[[.,.],.]],[.,.]],[.,.]]=>[2,3,1,5,4,7,6] [[.,[.,.]],[[.,[.,.]],[.,.]]]=>[[[.,.],[[.,.],.]],[[.,.],.]]=>[[[.,[[.,.],.]],[[.,.],.]],.]=>[2,3,1,5,6,4,7] [[.,[.,.]],[[[.,.],.],[.,.]]]=>[[[.,.],[.,[.,.]]],[[.,.],.]]=>[[[.,[[.,.],.]],[.,[.,.]]],.]=>[2,3,1,6,5,4,7] [[.,[.,.]],[[.,[.,[.,.]]],.]]=>[[.,[[[.,.],.],.]],[[.,.],.]]=>[[.,[[.,.],.]],[[[.,.],.],.]]=>[2,3,1,5,6,7,4] [[.,[.,.]],[[.,[[.,.],.]],.]]=>[[.,[[.,[.,.]],.]],[[.,.],.]]=>[[.,[[.,.],.]],[[.,.],[.,.]]]=>[2,3,1,5,7,6,4] [[.,[.,.]],[[[.,.],[.,.]],.]]=>[[.,[[.,.],[.,.]]],[[.,.],.]]=>[[.,[[.,.],.]],[[.,[.,.]],.]]=>[2,3,1,6,5,7,4] [[.,[.,.]],[[[.,[.,.]],.],.]]=>[[.,[.,[[.,.],.]]],[[.,.],.]]=>[[.,[[.,.],.]],[.,[[.,.],.]]]=>[2,3,1,6,7,5,4] [[.,[.,.]],[[[[.,.],.],.],.]]=>[[.,[.,[.,[.,.]]]],[[.,.],.]]=>[[.,[[.,.],.]],[.,[.,[.,.]]]]=>[2,3,1,7,6,5,4] [[[.,.],.],[.,[.,[.,[.,.]]]]]=>[[[[[.,.],.],.],.],[.,[.,.]]]=>[[[[[.,[.,[.,.]]],.],.],.],.]=>[3,2,1,4,5,6,7] [[[.,.],.],[.,[.,[[.,.],.]]]]=>[[[[.,[.,.]],.],.],[.,[.,.]]]=>[[[[.,[.,[.,.]]],.],.],[.,.]]=>[3,2,1,4,5,7,6] [[[.,.],.],[.,[[.,.],[.,.]]]]=>[[[[.,.],[.,.]],.],[.,[.,.]]]=>[[[[.,[.,[.,.]]],.],[.,.]],.]=>[3,2,1,4,6,5,7] [[[.,.],.],[.,[[.,[.,.]],.]]]=>[[[.,[[.,.],.]],.],[.,[.,.]]]=>[[[.,[.,[.,.]]],.],[[.,.],.]]=>[3,2,1,4,6,7,5] [[[.,.],.],[.,[[[.,.],.],.]]]=>[[[.,[.,[.,.]]],.],[.,[.,.]]]=>[[[.,[.,[.,.]]],.],[.,[.,.]]]=>[3,2,1,4,7,6,5] [[[.,.],.],[[.,.],[.,[.,.]]]]=>[[[[.,.],.],[.,.]],[.,[.,.]]]=>[[[[.,[.,[.,.]]],[.,.]],.],.]=>[3,2,1,5,4,6,7] [[[.,.],.],[[.,.],[[.,.],.]]]=>[[[.,[.,.]],[.,.]],[.,[.,.]]]=>[[[.,[.,[.,.]]],[.,.]],[.,.]]=>[3,2,1,5,4,7,6] [[[.,.],.],[[.,[.,.]],[.,.]]]=>[[[.,.],[[.,.],.]],[.,[.,.]]]=>[[[.,[.,[.,.]]],[[.,.],.]],.]=>[3,2,1,5,6,4,7] [[[.,.],.],[[[.,.],.],[.,.]]]=>[[[.,.],[.,[.,.]]],[.,[.,.]]]=>[[[.,[.,[.,.]]],[.,[.,.]]],.]=>[3,2,1,6,5,4,7] [[[.,.],.],[[.,[.,[.,.]]],.]]=>[[.,[[[.,.],.],.]],[.,[.,.]]]=>[[.,[.,[.,.]]],[[[.,.],.],.]]=>[3,2,1,5,6,7,4] [[[.,.],.],[[.,[[.,.],.]],.]]=>[[.,[[.,[.,.]],.]],[.,[.,.]]]=>[[.,[.,[.,.]]],[[.,.],[.,.]]]=>[3,2,1,5,7,6,4] [[[.,.],.],[[[.,.],[.,.]],.]]=>[[.,[[.,.],[.,.]]],[.,[.,.]]]=>[[.,[.,[.,.]]],[[.,[.,.]],.]]=>[3,2,1,6,5,7,4] [[[.,.],.],[[[.,[.,.]],.],.]]=>[[.,[.,[[.,.],.]]],[.,[.,.]]]=>[[.,[.,[.,.]]],[.,[[.,.],.]]]=>[3,2,1,6,7,5,4] [[[.,.],.],[[[[.,.],.],.],.]]=>[[.,[.,[.,[.,.]]]],[.,[.,.]]]=>[[.,[.,[.,.]]],[.,[.,[.,.]]]]=>[3,2,1,7,6,5,4] [[.,[.,[.,.]]],[.,[.,[.,.]]]]=>[[[[.,.],.],.],[[[.,.],.],.]]=>[[[[.,[[[.,.],.],.]],.],.],.]=>[2,3,4,1,5,6,7] [[.,[.,[.,.]]],[.,[[.,.],.]]]=>[[[.,[.,.]],.],[[[.,.],.],.]]=>[[[.,[[[.,.],.],.]],.],[.,.]]=>[2,3,4,1,5,7,6] [[.,[.,[.,.]]],[[.,.],[.,.]]]=>[[[.,.],[.,.]],[[[.,.],.],.]]=>[[[.,[[[.,.],.],.]],[.,.]],.]=>[2,3,4,1,6,5,7] [[.,[.,[.,.]]],[[.,[.,.]],.]]=>[[.,[[.,.],.]],[[[.,.],.],.]]=>[[.,[[[.,.],.],.]],[[.,.],.]]=>[2,3,4,1,6,7,5] [[.,[.,[.,.]]],[[[.,.],.],.]]=>[[.,[.,[.,.]]],[[[.,.],.],.]]=>[[.,[[[.,.],.],.]],[.,[.,.]]]=>[2,3,4,1,7,6,5] [[.,[[.,.],.]],[.,[.,[.,.]]]]=>[[[[.,.],.],.],[[.,[.,.]],.]]=>[[[[.,[[.,.],[.,.]]],.],.],.]=>[2,4,3,1,5,6,7] [[.,[[.,.],.]],[.,[[.,.],.]]]=>[[[.,[.,.]],.],[[.,[.,.]],.]]=>[[[.,[[.,.],[.,.]]],.],[.,.]]=>[2,4,3,1,5,7,6] [[.,[[.,.],.]],[[.,.],[.,.]]]=>[[[.,.],[.,.]],[[.,[.,.]],.]]=>[[[.,[[.,.],[.,.]]],[.,.]],.]=>[2,4,3,1,6,5,7] [[.,[[.,.],.]],[[.,[.,.]],.]]=>[[.,[[.,.],.]],[[.,[.,.]],.]]=>[[.,[[.,.],[.,.]]],[[.,.],.]]=>[2,4,3,1,6,7,5] [[.,[[.,.],.]],[[[.,.],.],.]]=>[[.,[.,[.,.]]],[[.,[.,.]],.]]=>[[.,[[.,.],[.,.]]],[.,[.,.]]]=>[2,4,3,1,7,6,5] [[[.,.],[.,.]],[.,[.,[.,.]]]]=>[[[[.,.],.],.],[[.,.],[.,.]]]=>[[[[.,[[.,[.,.]],.]],.],.],.]=>[3,2,4,1,5,6,7] [[[.,.],[.,.]],[.,[[.,.],.]]]=>[[[.,[.,.]],.],[[.,.],[.,.]]]=>[[[.,[[.,[.,.]],.]],.],[.,.]]=>[3,2,4,1,5,7,6] [[[.,.],[.,.]],[[.,.],[.,.]]]=>[[[.,.],[.,.]],[[.,.],[.,.]]]=>[[[.,[[.,[.,.]],.]],[.,.]],.]=>[3,2,4,1,6,5,7] [[[.,.],[.,.]],[[.,[.,.]],.]]=>[[.,[[.,.],.]],[[.,.],[.,.]]]=>[[.,[[.,[.,.]],.]],[[.,.],.]]=>[3,2,4,1,6,7,5] [[[.,.],[.,.]],[[[.,.],.],.]]=>[[.,[.,[.,.]]],[[.,.],[.,.]]]=>[[.,[[.,[.,.]],.]],[.,[.,.]]]=>[3,2,4,1,7,6,5] [[[.,[.,.]],.],[.,[.,[.,.]]]]=>[[[[.,.],.],.],[.,[[.,.],.]]]=>[[[[.,[.,[[.,.],.]]],.],.],.]=>[3,4,2,1,5,6,7] [[[.,[.,.]],.],[.,[[.,.],.]]]=>[[[.,[.,.]],.],[.,[[.,.],.]]]=>[[[.,[.,[[.,.],.]]],.],[.,.]]=>[3,4,2,1,5,7,6] [[[.,[.,.]],.],[[.,.],[.,.]]]=>[[[.,.],[.,.]],[.,[[.,.],.]]]=>[[[.,[.,[[.,.],.]]],[.,.]],.]=>[3,4,2,1,6,5,7] [[[.,[.,.]],.],[[.,[.,.]],.]]=>[[.,[[.,.],.]],[.,[[.,.],.]]]=>[[.,[.,[[.,.],.]]],[[.,.],.]]=>[3,4,2,1,6,7,5] [[[.,[.,.]],.],[[[.,.],.],.]]=>[[.,[.,[.,.]]],[.,[[.,.],.]]]=>[[.,[.,[[.,.],.]]],[.,[.,.]]]=>[3,4,2,1,7,6,5] [[[[.,.],.],.],[.,[.,[.,.]]]]=>[[[[.,.],.],.],[.,[.,[.,.]]]]=>[[[[.,[.,[.,[.,.]]]],.],.],.]=>[4,3,2,1,5,6,7] [[[[.,.],.],.],[.,[[.,.],.]]]=>[[[.,[.,.]],.],[.,[.,[.,.]]]]=>[[[.,[.,[.,[.,.]]]],.],[.,.]]=>[4,3,2,1,5,7,6] [[[[.,.],.],.],[[.,.],[.,.]]]=>[[[.,.],[.,.]],[.,[.,[.,.]]]]=>[[[.,[.,[.,[.,.]]]],[.,.]],.]=>[4,3,2,1,6,5,7] [[[[.,.],.],.],[[.,[.,.]],.]]=>[[.,[[.,.],.]],[.,[.,[.,.]]]]=>[[.,[.,[.,[.,.]]]],[[.,.],.]]=>[4,3,2,1,6,7,5] [[[[.,.],.],.],[[[.,.],.],.]]=>[[.,[.,[.,.]]],[.,[.,[.,.]]]]=>[[.,[.,[.,[.,.]]]],[.,[.,.]]]=>[4,3,2,1,7,6,5] [[.,[.,[.,[.,.]]]],[.,[.,.]]]=>[[[.,.],.],[[[[.,.],.],.],.]]=>[[[.,[[[[.,.],.],.],.]],.],.]=>[2,3,4,5,1,6,7] [[.,[.,[.,[.,.]]]],[[.,.],.]]=>[[.,[.,.]],[[[[.,.],.],.],.]]=>[[.,[[[[.,.],.],.],.]],[.,.]]=>[2,3,4,5,1,7,6] [[.,[.,[[.,.],.]]],[.,[.,.]]]=>[[[.,.],.],[[[.,[.,.]],.],.]]=>[[[.,[[[.,.],.],[.,.]]],.],.]=>[2,3,5,4,1,6,7] [[.,[.,[[.,.],.]]],[[.,.],.]]=>[[.,[.,.]],[[[.,[.,.]],.],.]]=>[[.,[[[.,.],.],[.,.]]],[.,.]]=>[2,3,5,4,1,7,6] [[.,[[.,.],[.,.]]],[.,[.,.]]]=>[[[.,.],.],[[[.,.],[.,.]],.]]=>[[[.,[[[.,.],[.,.]],.]],.],.]=>[2,4,3,5,1,6,7] [[.,[[.,.],[.,.]]],[[.,.],.]]=>[[.,[.,.]],[[[.,.],[.,.]],.]]=>[[.,[[[.,.],[.,.]],.]],[.,.]]=>[2,4,3,5,1,7,6] [[.,[[.,[.,.]],.]],[.,[.,.]]]=>[[[.,.],.],[[.,[[.,.],.]],.]]=>[[[.,[[.,.],[[.,.],.]]],.],.]=>[2,4,5,3,1,6,7] 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[[[[[[[[.,.],.],.],.],.],.],[[.,.],.]],.]=>[.,[[.,[.,.]],[.,[.,[.,[.,[.,[.,.]]]]]]]]=>[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]]=>[8,7,6,5,4,3,2,10,9,1] [[.,[[[[[[.,[.,.]],.],.],.],.],.]],.]=>[.,[[.,[.,[.,[.,[.,[[.,.],.]]]]]],.]]=>[.,[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]]=>[2,8,9,7,6,5,4,3,1] [[[[[[[.,.],.],.],.],[.,[.,.]]],.],.]=>[.,[.,[[[.,.],.],[.,[.,[.,[.,.]]]]]]]=>[.,[.,[[[.,[.,[.,[.,[.,.]]]]],.],.]]]=>[7,6,5,4,3,8,9,2,1] [.,[.,[.,[.,[.,[[[.,.],.],[.,.]]]]]]]=>[[[[[[[.,.],[.,[.,.]]],.],.],.],.],.]=>[[[[[[[.,.],.],.],.],.],[.,[.,.]]],.]=>[1,2,3,4,5,8,7,6,9] [.,[[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]]=>[[[[[[[.,.],.],.],.],.],[.,[.,.]]],.]=>[[[[[[[.,.],[.,[.,.]]],.],.],.],.],.]=>[1,4,3,2,5,6,7,8,9] [[.,.],[[.,[.,[.,[.,[.,[.,.]]]]]],.]]=>[[.,[[[[[[.,.],.],.],.],.],.]],[.,.]]=>[[.,[.,.]],[[[[[[.,.],.],.],.],.],.]]=>[2,1,4,5,6,7,8,9,3] [[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]],.]=>[.,[[[[[[[.,.],.],.],.],.],[.,.]],.]]=>[.,[[[[[[[.,.],[.,.]],.],.],.],.],.]]=>[2,4,3,5,6,7,8,9,1] [[.,[[[[[[.,.],.],.],.],.],.]],[.,.]]=>[[.,.],[[.,[.,[.,[.,[.,[.,.]]]]]],.]]=>[[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]],.]=>[2,8,7,6,5,4,3,1,9] [[.,[.,.]],[[.,.],[.,[.,[.,[.,.]]]]]]=>[[[[[[.,.],.],.],.],[.,.]],[[.,.],.]]=>[[[[[[.,[[.,.],.]],[.,.]],.],.],.],.]=>[2,3,1,5,4,6,7,8,9] [[.,[[.,.],[[[[[.,.],.],.],.],.]]],.]=>[.,[[[.,[.,[.,[.,[.,.]]]]],[.,.]],.]]=>[.,[[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]]=>[2,4,3,9,8,7,6,5,1] [[[[.,.],.],[.,[.,[.,[.,[.,.]]]]]],.]=>[.,[[[[[[.,.],.],.],.],.],[.,[.,.]]]]=>[.,[[[[[[.,[.,[.,.]]],.],.],.],.],.]]=>[4,3,2,5,6,7,8,9,1] [[[[[.,.],.],.],[.,[.,[.,[.,.]]]]],.]=>[.,[[[[[.,.],.],.],.],[.,[.,[.,.]]]]]=>[.,[[[[[.,[.,[.,[.,.]]]],.],.],.],.]]=>[5,4,3,2,6,7,8,9,1] [[[[[[.,.],.],.],.],[.,[.,[.,.]]]],.]=>[.,[[[[.,.],.],.],[.,[.,[.,[.,.]]]]]]=>[.,[[[[.,[.,[.,[.,[.,.]]]]],.],.],.]]=>[6,5,4,3,2,7,8,9,1] [.,[[[[[[.,.],.],.],.],.],[.,[.,.]]]]=>[[[[.,.],.],[.,[.,[.,[.,[.,.]]]]]],.]=>[[[[.,.],[.,[.,[.,[.,[.,.]]]]]],.],.]=>[1,7,6,5,4,3,2,8,9] [.,[[[[[.,.],.],.],.],[.,[.,[.,.]]]]]=>[[[[[.,.],.],.],[.,[.,[.,[.,.]]]]],.]=>[[[[[.,.],[.,[.,[.,[.,.]]]]],.],.],.]=>[1,6,5,4,3,2,7,8,9] [[[.,[.,.]],.],[[[[[.,.],.],.],.],.]]=>[[.,[.,[.,[.,[.,.]]]]],[.,[[.,.],.]]]=>[[.,[.,[[.,.],.]]],[.,[.,[.,[.,.]]]]]=>[3,4,2,1,9,8,7,6,5] [.,[.,[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]]]=>[[[[[[[[.,[.,.]],[.,.]],.],.],.],.],.],.]=>[[[[[[[[.,.],.],.],.],.],.],[.,.]],[.,.]]=>[1,2,3,4,5,6,8,7,10,9] [.,[.,[[.,.],[.,[[.,[[.,.],.]],.]]]]]=>[[[[[.,[[.,[.,.]],.]],.],[.,.]],.],.]=>[[[[[.,.],.],[.,.]],.],[[.,.],[.,.]]]=>[1,2,4,3,5,7,9,8,6] [[.,[.,[[[[[[.,.],.],.],.],.],.]]],.]=>[.,[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]]=>[.,[[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]]=>[2,3,9,8,7,6,5,4,1] [.,[[[.,.],.],[[.,.],[[.,[[.,.],.]],.]]]]=>[[[[.,[[.,[.,.]],.]],[.,.]],[.,[.,.]]],.]=>[[[[.,.],[.,[.,.]]],[.,.]],[[.,.],[.,.]]]=>[1,4,3,2,6,5,8,10,9,7] [.,[[[.,[[.,.],[[.,.],.]]],.],[[.,.],.]]]=>[[[.,[.,.]],[.,[[[.,[.,.]],[.,.]],.]]],.]=>[[[.,.],[.,[[[.,.],[.,.]],[.,.]]]],[.,.]]=>[1,4,6,5,8,7,3,2,10,9] [.,[[[.,[[.,.],[[.,[[.,.],.]],.]]],.],.]]=>[[.,[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]],.]=>[[.,.],[.,[[[.,.],[.,.]],[[.,.],[.,.]]]]]=>[1,4,6,5,8,10,9,7,3,2] [[[[[[.,.],.],.],.],.],[[.,.],[.,.]]]=>[[[.,.],[.,.]],[.,[.,[.,[.,[.,.]]]]]]=>[[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]],.]=>[6,5,4,3,2,1,8,7,9] [[.,.],[.,[.,[.,[[.,[.,[.,.]]],.]]]]]=>[[[[[.,[[[.,.],.],.]],.],.],.],[.,.]]=>[[[[[.,[.,.]],.],.],.],[[[.,.],.],.]]=>[2,1,3,4,5,7,8,9,6] [[.,[.,[.,[.,[.,.]]]]],[.,[[.,.],.]]]=>[[[.,[.,.]],.],[[[[[.,.],.],.],.],.]]=>[[[.,[[[[[.,.],.],.],.],.]],.],[.,.]]=>[2,3,4,5,6,1,7,9,8] [[.,.],[.,[.,[[.,[.,[.,[.,.]]]],.]]]]=>[[[[.,[[[[.,.],.],.],.]],.],.],[.,.]]=>[[[[.,[.,.]],.],.],[[[[.,.],.],.],.]]=>[2,1,3,4,6,7,8,9,5] [[[[[.,[.,[.,[.,.]]]],.],[.,.]],.],.]=>[.,[.,[[.,.],[.,[[[[.,.],.],.],.]]]]]=>[.,[.,[[.,[.,[[[[.,.],.],.],.]]],.]]]=>[5,6,7,8,4,3,9,2,1] [.,[.,[[[[[[.,.],.],.],.],.],[.,.]]]]=>[[[[.,.],[.,[.,[.,[.,[.,.]]]]]],.],.]=>[[[[.,.],.],[.,[.,[.,[.,[.,.]]]]]],.]=>[1,2,8,7,6,5,4,3,9] [.,[.,[.,[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]]]]=>[[[[[[[[[[.,.],[.,.]],.],.],.],.],.],.],.],.]=>[[[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]],.]=>[1,2,3,4,5,6,7,8,10,9,11] [[[[[[.,.],.],.],.],.],[.,[[.,.],.]]]=>[[[.,[.,.]],.],[.,[.,[.,[.,[.,.]]]]]]=>[[[.,[.,[.,[.,[.,[.,.]]]]]],.],[.,.]]=>[6,5,4,3,2,1,7,9,8] [[[[[.,.],.],.],.],[.,[[[.,.],.],.]]]=>[[[.,[.,[.,.]]],.],[.,[.,[.,[.,.]]]]]=>[[[.,[.,[.,[.,[.,.]]]]],.],[.,[.,.]]]=>[5,4,3,2,1,6,9,8,7] [[[[.,.],.],.],[.,[[[[.,.],.],.],.]]]=>[[[.,[.,[.,[.,.]]]],.],[.,[.,[.,.]]]]=>[[[.,[.,[.,[.,.]]]],.],[.,[.,[.,.]]]]=>[4,3,2,1,5,9,8,7,6] [[[.,.],.],[.,[[[[[.,.],.],.],.],.]]]=>[[[.,[.,[.,[.,[.,.]]]]],.],[.,[.,.]]]=>[[[.,[.,[.,.]]],.],[.,[.,[.,[.,.]]]]]=>[3,2,1,4,9,8,7,6,5] [.,[[.,.],[[.,.],[[[.,.],[[.,.],.]],.]]]]=>[[[[.,[[.,[.,.]],[.,.]]],[.,.]],[.,.]],.]=>[[[[.,.],[.,.]],[.,.]],[[.,[.,.]],[.,.]]]=>[1,3,2,5,4,8,7,10,9,6] [[[[[[.,.],.],.],.],[[.,.],.]],[.,.]]=>[[.,.],[[.,[.,.]],[.,[.,[.,[.,.]]]]]]=>[[.,[[.,[.,[.,[.,[.,.]]]]],[.,.]]],.]=>[6,5,4,3,2,8,7,1,9] [[[[[[.,.],[.,[.,.]]],.],.],.],[.,.]]=>[[.,.],[.,[.,[.,[[[.,.],.],[.,.]]]]]]=>[[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]],.]=>[6,5,7,8,4,3,2,1,9] [[[[[.,.],.],.],.],[[[.,.],.],[.,.]]]=>[[[.,.],[.,[.,.]]],[.,[.,[.,[.,.]]]]]=>[[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]],.]=>[5,4,3,2,1,8,7,6,9] [[[.,.],.],[[.,.],[.,[.,[.,[.,.]]]]]]=>[[[[[[.,.],.],.],.],[.,.]],[.,[.,.]]]=>[[[[[[.,[.,[.,.]]],[.,.]],.],.],.],.]=>[3,2,1,5,4,6,7,8,9] [[[[[[[.,.],.],.],.],.],.],[[.,.],[.,.]]]=>[[[.,.],[.,.]],[.,[.,[.,[.,[.,[.,.]]]]]]]=>[[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]],.]=>[7,6,5,4,3,2,1,9,8,10] [[[[[[.,.],.],.],.],[.,.]],[[.,.],.]]=>[[.,[.,.]],[[.,.],[.,[.,[.,[.,.]]]]]]=>[[.,[[.,[.,[.,[.,[.,.]]]]],.]],[.,.]]=>[6,5,4,3,2,7,1,9,8] [[[[[[[.,.],.],.],.],[.,.]],[.,.]],.]=>[.,[[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]]=>[.,[[.,[[.,[.,[.,[.,[.,.]]]]],.]],.]]=>[7,6,5,4,3,8,2,9,1] [[[.,[.,.]],.],[.,[.,[.,[.,[.,[.,.]]]]]]]=>[[[[[[[.,.],.],.],.],.],.],[.,[[.,.],.]]]=>[[[[[[[.,[.,[[.,.],.]]],.],.],.],.],.],.]=>[3,4,2,1,5,6,7,8,9,10] [[.,[[.,[[[[[.,.],.],.],.],.]],.]],.]=>[.,[[.,[[.,[.,[.,[.,[.,.]]]]],.]],.]]=>[.,[[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]]=>[2,4,9,8,7,6,5,3,1] [[[[[[.,.],.],.],.],[[.,.],[.,.]]],.]=>[.,[[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]]=>[.,[[[.,[.,[.,[.,[.,.]]]]],[.,.]],.]]=>[6,5,4,3,2,8,7,9,1] [[[[[[[[[.,[.,[.,.]]],.],.],.],.],.],.],.],.]=>[.,[.,[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]]]=>[.,[.,[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]]]=>[9,10,11,8,7,6,5,4,3,2,1] [[[[.,.],[[[[[.,.],.],.],.],.]],.],.]=>[.,[.,[[.,[.,[.,[.,[.,.]]]]],[.,.]]]]=>[.,[.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]]=>[4,3,9,8,7,6,5,2,1] [[[[[[[.,.],[[[.,.],.],.]],.],.],.],.],.]=>[.,[.,[.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]]]=>[.,[.,[.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]]]=>[7,6,10,9,8,5,4,3,2,1] [[[[[[[[.,.],.],[[.,.],.]],.],.],.],.],.]=>[.,[.,[.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]]]=>[.,[.,[.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]]]=>[8,7,6,10,9,5,4,3,2,1] [[[[[[.,[.,[.,.]]],.],.],.],.],[.,.]]=>[[.,.],[.,[.,[.,[.,[[[.,.],.],.]]]]]]=>[[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]],.]=>[6,7,8,5,4,3,2,1,9] [[[[[[[.,[.,[.,.]]],.],.],.],.],.],[.,.]]=>[[.,.],[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]=>[[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]],.]=>[7,8,9,6,5,4,3,2,1,10] [[[[[[.,[.,.]],[.,.]],.],.],.],[.,.]]=>[[.,.],[.,[.,[.,[[.,.],[[.,.],.]]]]]]=>[[.,[.,[.,[.,[[.,[[.,.],.]],.]]]]],.]=>[6,7,5,8,4,3,2,1,9]
Map
left-right symmetry
Description
Return the left-right symmetrized tree of a binary tree.
Map
left border symmetry
Description
Return the tree where a symmetry has been applied recursively on all left borders. If a tree is made of three trees $T_1, T_2, T_3$ on its left border, it becomes $T_3, T_2, T_1$ where same symmetry has been applied to $T_1, T_2, T_3$.
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.