Identifier
Mp00018: left border symmetryBinary trees
Mp00017: to 312-avoiding permutationPermutations
Mp00070: Permutations Robinson-Schensted recording tableau
Images
=>
Cc0010;cc-rep-0Cc0010;cc-rep-1Cc0007;cc-rep-3
[.,.]=>[.,.]=>[1]=>[[1]] [.,[.,.]]=>[.,[.,.]]=>[2,1]=>[[1],[2]] [[.,.],.]=>[[.,.],.]=>[1,2]=>[[1,2]] [.,[.,[.,.]]]=>[.,[.,[.,.]]]=>[3,2,1]=>[[1],[2],[3]] [.,[[.,.],.]]=>[.,[[.,.],.]]=>[2,3,1]=>[[1,2],[3]] [[.,.],[.,.]]=>[[.,[.,.]],.]=>[2,1,3]=>[[1,3],[2]] [[.,[.,.]],.]=>[[.,.],[.,.]]=>[1,3,2]=>[[1,2],[3]] [[[.,.],.],.]=>[[[.,.],.],.]=>[1,2,3]=>[[1,2,3]] [.,[.,[.,[.,.]]]]=>[.,[.,[.,[.,.]]]]=>[4,3,2,1]=>[[1],[2],[3],[4]] [.,[.,[[.,.],.]]]=>[.,[.,[[.,.],.]]]=>[3,4,2,1]=>[[1,2],[3],[4]] [.,[[.,.],[.,.]]]=>[.,[[.,[.,.]],.]]=>[3,2,4,1]=>[[1,3],[2],[4]] [.,[[.,[.,.]],.]]=>[.,[[.,.],[.,.]]]=>[2,4,3,1]=>[[1,2],[3],[4]] [.,[[[.,.],.],.]]=>[.,[[[.,.],.],.]]=>[2,3,4,1]=>[[1,2,3],[4]] [[.,.],[.,[.,.]]]=>[[.,[.,[.,.]]],.]=>[3,2,1,4]=>[[1,4],[2],[3]] [[.,.],[[.,.],.]]=>[[.,[[.,.],.]],.]=>[2,3,1,4]=>[[1,2,4],[3]] [[.,[.,.]],[.,.]]=>[[.,[.,.]],[.,.]]=>[2,1,4,3]=>[[1,3],[2,4]] [[[.,.],.],[.,.]]=>[[[.,[.,.]],.],.]=>[2,1,3,4]=>[[1,3,4],[2]] [[.,[.,[.,.]]],.]=>[[.,.],[.,[.,.]]]=>[1,4,3,2]=>[[1,2],[3],[4]] [[.,[[.,.],.]],.]=>[[.,.],[[.,.],.]]=>[1,3,4,2]=>[[1,2,3],[4]] [[[.,.],[.,.]],.]=>[[[.,.],[.,.]],.]=>[1,3,2,4]=>[[1,2,4],[3]] [[[.,[.,.]],.],.]=>[[[.,.],.],[.,.]]=>[1,2,4,3]=>[[1,2,3],[4]] [[[[.,.],.],.],.]=>[[[[.,.],.],.],.]=>[1,2,3,4]=>[[1,2,3,4]] [.,[.,[.,[.,[.,.]]]]]=>[.,[.,[.,[.,[.,.]]]]]=>[5,4,3,2,1]=>[[1],[2],[3],[4],[5]] [.,[.,[.,[[.,.],.]]]]=>[.,[.,[.,[[.,.],.]]]]=>[4,5,3,2,1]=>[[1,2],[3],[4],[5]] [.,[.,[[.,.],[.,.]]]]=>[.,[.,[[.,[.,.]],.]]]=>[4,3,5,2,1]=>[[1,3],[2],[4],[5]] [.,[.,[[.,[.,.]],.]]]=>[.,[.,[[.,.],[.,.]]]]=>[3,5,4,2,1]=>[[1,2],[3],[4],[5]] [.,[.,[[[.,.],.],.]]]=>[.,[.,[[[.,.],.],.]]]=>[3,4,5,2,1]=>[[1,2,3],[4],[5]] [.,[[.,.],[.,[.,.]]]]=>[.,[[.,[.,[.,.]]],.]]=>[4,3,2,5,1]=>[[1,4],[2],[3],[5]] [.,[[.,.],[[.,.],.]]]=>[.,[[.,[[.,.],.]],.]]=>[3,4,2,5,1]=>[[1,2,4],[3],[5]] [.,[[.,[.,.]],[.,.]]]=>[.,[[.,[.,.]],[.,.]]]=>[3,2,5,4,1]=>[[1,3],[2,4],[5]] [.,[[[.,.],.],[.,.]]]=>[.,[[[.,[.,.]],.],.]]=>[3,2,4,5,1]=>[[1,3,4],[2],[5]] [.,[[.,[.,[.,.]]],.]]=>[.,[[.,.],[.,[.,.]]]]=>[2,5,4,3,1]=>[[1,2],[3],[4],[5]] [.,[[.,[[.,.],.]],.]]=>[.,[[.,.],[[.,.],.]]]=>[2,4,5,3,1]=>[[1,2,3],[4],[5]] [.,[[[.,.],[.,.]],.]]=>[.,[[[.,.],[.,.]],.]]=>[2,4,3,5,1]=>[[1,2,4],[3],[5]] [.,[[[.,[.,.]],.],.]]=>[.,[[[.,.],.],[.,.]]]=>[2,3,5,4,1]=>[[1,2,3],[4],[5]] [.,[[[[.,.],.],.],.]]=>[.,[[[[.,.],.],.],.]]=>[2,3,4,5,1]=>[[1,2,3,4],[5]] [[.,.],[.,[.,[.,.]]]]=>[[.,[.,[.,[.,.]]]],.]=>[4,3,2,1,5]=>[[1,5],[2],[3],[4]] [[.,.],[.,[[.,.],.]]]=>[[.,[.,[[.,.],.]]],.]=>[3,4,2,1,5]=>[[1,2,5],[3],[4]] [[.,.],[[.,.],[.,.]]]=>[[.,[[.,[.,.]],.]],.]=>[3,2,4,1,5]=>[[1,3,5],[2],[4]] [[.,.],[[.,[.,.]],.]]=>[[.,[[.,.],[.,.]]],.]=>[2,4,3,1,5]=>[[1,2,5],[3],[4]] [[.,.],[[[.,.],.],.]]=>[[.,[[[.,.],.],.]],.]=>[2,3,4,1,5]=>[[1,2,3,5],[4]] [[.,[.,.]],[.,[.,.]]]=>[[.,[.,[.,.]]],[.,.]]=>[3,2,1,5,4]=>[[1,4],[2,5],[3]] [[.,[.,.]],[[.,.],.]]=>[[.,[[.,.],.]],[.,.]]=>[2,3,1,5,4]=>[[1,2,4],[3,5]] [[[.,.],.],[.,[.,.]]]=>[[[.,[.,[.,.]]],.],.]=>[3,2,1,4,5]=>[[1,4,5],[2],[3]] [[[.,.],.],[[.,.],.]]=>[[[.,[[.,.],.]],.],.]=>[2,3,1,4,5]=>[[1,2,4,5],[3]] [[.,[.,[.,.]]],[.,.]]=>[[.,[.,.]],[.,[.,.]]]=>[2,1,5,4,3]=>[[1,3],[2,4],[5]] [[.,[[.,.],.]],[.,.]]=>[[.,[.,.]],[[.,.],.]]=>[2,1,4,5,3]=>[[1,3,4],[2,5]] [[[.,.],[.,.]],[.,.]]=>[[[.,[.,.]],[.,.]],.]=>[2,1,4,3,5]=>[[1,3,5],[2,4]] [[[.,[.,.]],.],[.,.]]=>[[[.,[.,.]],.],[.,.]]=>[2,1,3,5,4]=>[[1,3,4],[2,5]] [[[[.,.],.],.],[.,.]]=>[[[[.,[.,.]],.],.],.]=>[2,1,3,4,5]=>[[1,3,4,5],[2]] [[.,[.,[.,[.,.]]]],.]=>[[.,.],[.,[.,[.,.]]]]=>[1,5,4,3,2]=>[[1,2],[3],[4],[5]] [[.,[.,[[.,.],.]]],.]=>[[.,.],[.,[[.,.],.]]]=>[1,4,5,3,2]=>[[1,2,3],[4],[5]] [[.,[[.,.],[.,.]]],.]=>[[.,.],[[.,[.,.]],.]]=>[1,4,3,5,2]=>[[1,2,4],[3],[5]] [[.,[[.,[.,.]],.]],.]=>[[.,.],[[.,.],[.,.]]]=>[1,3,5,4,2]=>[[1,2,3],[4],[5]] [[.,[[[.,.],.],.]],.]=>[[.,.],[[[.,.],.],.]]=>[1,3,4,5,2]=>[[1,2,3,4],[5]] [[[.,.],[.,[.,.]]],.]=>[[[.,.],[.,[.,.]]],.]=>[1,4,3,2,5]=>[[1,2,5],[3],[4]] [[[.,.],[[.,.],.]],.]=>[[[.,.],[[.,.],.]],.]=>[1,3,4,2,5]=>[[1,2,3,5],[4]] [[[.,[.,.]],[.,.]],.]=>[[[.,.],[.,.]],[.,.]]=>[1,3,2,5,4]=>[[1,2,4],[3,5]] [[[[.,.],.],[.,.]],.]=>[[[[.,.],[.,.]],.],.]=>[1,3,2,4,5]=>[[1,2,4,5],[3]] [[[.,[.,[.,.]]],.],.]=>[[[.,.],.],[.,[.,.]]]=>[1,2,5,4,3]=>[[1,2,3],[4],[5]] [[[.,[[.,.],.]],.],.]=>[[[.,.],.],[[.,.],.]]=>[1,2,4,5,3]=>[[1,2,3,4],[5]] [[[[.,.],[.,.]],.],.]=>[[[[.,.],.],[.,.]],.]=>[1,2,4,3,5]=>[[1,2,3,5],[4]] [[[[.,[.,.]],.],.],.]=>[[[[.,.],.],.],[.,.]]=>[1,2,3,5,4]=>[[1,2,3,4],[5]] [[[[[.,.],.],.],.],.]=>[[[[[.,.],.],.],.],.]=>[1,2,3,4,5]=>[[1,2,3,4,5]] [.,[.,[.,[.,[.,[.,.]]]]]]=>[.,[.,[.,[.,[.,[.,.]]]]]]=>[6,5,4,3,2,1]=>[[1],[2],[3],[4],[5],[6]] [.,[.,[.,[.,[[.,.],.]]]]]=>[.,[.,[.,[.,[[.,.],.]]]]]=>[5,6,4,3,2,1]=>[[1,2],[3],[4],[5],[6]] [.,[.,[.,[[.,.],[.,.]]]]]=>[.,[.,[.,[[.,[.,.]],.]]]]=>[5,4,6,3,2,1]=>[[1,3],[2],[4],[5],[6]] [.,[.,[.,[[.,[.,.]],.]]]]=>[.,[.,[.,[[.,.],[.,.]]]]]=>[4,6,5,3,2,1]=>[[1,2],[3],[4],[5],[6]] [.,[.,[.,[[[.,.],.],.]]]]=>[.,[.,[.,[[[.,.],.],.]]]]=>[4,5,6,3,2,1]=>[[1,2,3],[4],[5],[6]] [.,[.,[[.,.],[.,[.,.]]]]]=>[.,[.,[[.,[.,[.,.]]],.]]]=>[5,4,3,6,2,1]=>[[1,4],[2],[3],[5],[6]] [.,[.,[[.,.],[[.,.],.]]]]=>[.,[.,[[.,[[.,.],.]],.]]]=>[4,5,3,6,2,1]=>[[1,2,4],[3],[5],[6]] [.,[.,[[.,[.,.]],[.,.]]]]=>[.,[.,[[.,[.,.]],[.,.]]]]=>[4,3,6,5,2,1]=>[[1,3],[2,4],[5],[6]] [.,[.,[[[.,.],.],[.,.]]]]=>[.,[.,[[[.,[.,.]],.],.]]]=>[4,3,5,6,2,1]=>[[1,3,4],[2],[5],[6]] [.,[.,[[.,[.,[.,.]]],.]]]=>[.,[.,[[.,.],[.,[.,.]]]]]=>[3,6,5,4,2,1]=>[[1,2],[3],[4],[5],[6]] [.,[.,[[.,[[.,.],.]],.]]]=>[.,[.,[[.,.],[[.,.],.]]]]=>[3,5,6,4,2,1]=>[[1,2,3],[4],[5],[6]] [.,[.,[[[.,.],[.,.]],.]]]=>[.,[.,[[[.,.],[.,.]],.]]]=>[3,5,4,6,2,1]=>[[1,2,4],[3],[5],[6]] [.,[.,[[[.,[.,.]],.],.]]]=>[.,[.,[[[.,.],.],[.,.]]]]=>[3,4,6,5,2,1]=>[[1,2,3],[4],[5],[6]] [.,[.,[[[[.,.],.],.],.]]]=>[.,[.,[[[[.,.],.],.],.]]]=>[3,4,5,6,2,1]=>[[1,2,3,4],[5],[6]] [.,[[.,.],[.,[.,[.,.]]]]]=>[.,[[.,[.,[.,[.,.]]]],.]]=>[5,4,3,2,6,1]=>[[1,5],[2],[3],[4],[6]] 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[[.,[.,.]],[[[[[.,.],[.,.]],[.,.]],[.,.]],[.,.]]]=>[[.,[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],.]],[.,.]]=>[3,2,5,4,7,6,9,8,10,1,12,11]=>[[1,3,5,7,9,11],[2,4,6,8,12],[10]] [[.,[[.,.],[.,.]]],[[[.,.],[.,.]],[[.,.],[.,.]]]]=>[[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],[[.,[.,.]],.]]=>[4,3,5,2,7,6,8,1,11,10,12,9]=>[[1,3,5,7,9,11],[2,6,10],[4,12],[8]] [[[.,[.,.]],[.,.]],[[[.,.],[.,.]],[[.,.],[.,.]]]]=>[[[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],[.,.]],[.,.]]=>[4,3,5,2,7,6,8,1,10,9,12,11]=>[[1,3,5,7,9,11],[2,6,10,12],[4],[8]] [[[.,[.,.]],[[[[.,.],[.,.]],[.,.]],[.,.]]],[.,.]]=>[[[.,[.,.]],[[[[.,[.,.]],[.,.]],[.,.]],.]],[.,.]]=>[2,1,5,4,7,6,9,8,10,3,12,11]=>[[1,3,5,7,9,11],[2,4,6,8,12],[10]] [[[.,[.,.]],[[.,.],[.,.]]],[[[.,.],[.,.]],[.,.]]]=>[[[.,[[[.,[.,.]],[.,.]],.]],[[.,[.,.]],.]],[.,.]]=>[3,2,5,4,6,1,9,8,10,7,12,11]=>[[1,3,5,7,9,11],[2,4,8,12],[6,10]] 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[[.,[[[[.,.],[.,.]],[.,.]],[.,.]]],[[.,.],[.,.]]]=>[[.,[[.,[.,.]],.]],[[[[.,[.,.]],[.,.]],[.,.]],.]]=>[3,2,4,1,7,6,9,8,11,10,12,5]=>[[1,3,5,7,9,11],[2,6,8,10],[4,12]] [[[.,[[.,.],[[.,.],[[.,.],[.,.]]]]],[.,.]],[.,.]]=>[[[.,[.,.]],[.,.]],[[.,[[.,[[.,[.,.]],.]],.]],.]]=>[2,1,4,3,9,8,10,7,11,6,12,5]=>[[1,3,5,7,9,11],[2,4,6],[8],[10],[12]] [[[.,[[.,.],[[[.,.],[.,.]],[.,.]]]],[.,.]],[.,.]]=>[[[.,[.,.]],[.,.]],[[.,[[[.,[.,.]],[.,.]],.]],.]]=>[2,1,4,3,8,7,10,9,11,6,12,5]=>[[1,3,5,7,9,11],[2,4,6,8],[10],[12]] [[[.,[[[.,.],[[.,.],[.,.]]],[.,.]]],[.,.]],[.,.]]=>[[[.,[.,.]],[.,.]],[[[.,[.,.]],[[.,[.,.]],.]],.]]=>[2,1,4,3,7,6,10,9,11,8,12,5]=>[[1,3,5,7,9,11],[2,4,6,8],[10],[12]] [[[.,[[[.,.],[.,.]],[[.,.],[.,.]]]],[.,.]],[.,.]]=>[[[.,[.,.]],[.,.]],[[[.,[[.,[.,.]],.]],[.,.]],.]]=>[2,1,4,3,8,7,9,6,11,10,12,5]=>[[1,3,5,7,9,11],[2,4,6,10],[8],[12]] [[[.,[[[[.,.],[.,.]],[.,.]],[.,.]]],[.,.]],[.,.]]=>[[[.,[.,.]],[.,.]],[[[[.,[.,.]],[.,.]],[.,.]],.]]=>[2,1,4,3,7,6,9,8,11,10,12,5]=>[[1,3,5,7,9,11],[2,4,6,8,10],[12]] [[.,[[.,.],[[.,.],[.,.]]]],[[.,.],[[.,.],[.,.]]]]=>[[.,[[.,[[.,[.,.]],.]],.]],[[.,[[.,[.,.]],.]],.]]=>[4,3,5,2,6,1,10,9,11,8,12,7]=>[[1,3,5,7,9,11],[2,8],[4,10],[6,12]] [[.,[[[.,.],[.,.]],[.,.]]],[[.,.],[[.,.],[.,.]]]]=>[[.,[[.,[[.,[.,.]],.]],.]],[[[.,[.,.]],[.,.]],.]]=>[4,3,5,2,6,1,9,8,11,10,12,7]=>[[1,3,5,7,9,11],[2,8,10],[4,12],[6]] [[.,[[.,.],[[.,.],[.,.]]]],[[[.,.],[.,.]],[.,.]]]=>[[.,[[[.,[.,.]],[.,.]],.]],[[.,[[.,[.,.]],.]],.]]=>[3,2,5,4,6,1,10,9,11,8,12,7]=>[[1,3,5,7,9,11],[2,4,8],[6,10],[12]] [[.,[[[.,.],[.,.]],[.,.]]],[[[.,.],[.,.]],[.,.]]]=>[[.,[[[.,[.,.]],[.,.]],.]],[[[.,[.,.]],[.,.]],.]]=>[3,2,5,4,6,1,9,8,11,10,12,7]=>[[1,3,5,7,9,11],[2,4,8,10],[6,12]] [[[.,[[.,.],[[.,.],[.,.]]]],[[.,.],[.,.]]],[.,.]]=>[[[.,[.,.]],[[.,[.,.]],.]],[[.,[[.,[.,.]],.]],.]]=>[2,1,5,4,6,3,10,9,11,8,12,7]=>[[1,3,5,7,9,11],[2,4,8],[6,10],[12]] [[[.,[[[.,.],[.,.]],[.,.]]],[[.,.],[.,.]]],[.,.]]=>[[[.,[.,.]],[[.,[.,.]],.]],[[[.,[.,.]],[.,.]],.]]=>[2,1,5,4,6,3,9,8,11,10,12,7]=>[[1,3,5,7,9,11],[2,4,8,10],[6,12]] [[[.,[[.,.],[[.,.],[.,.]]]],[.,.]],[[.,.],[.,.]]]=>[[[.,[[.,[.,.]],.]],[.,.]],[[.,[[.,[.,.]],.]],.]]=>[3,2,4,1,6,5,10,9,11,8,12,7]=>[[1,3,5,7,9,11],[2,6,8],[4,10],[12]] [[[.,[[[.,.],[.,.]],[.,.]]],[.,.]],[[.,.],[.,.]]]=>[[[.,[[.,[.,.]],.]],[.,.]],[[[.,[.,.]],[.,.]],.]]=>[3,2,4,1,6,5,9,8,11,10,12,7]=>[[1,3,5,7,9,11],[2,6,8,10],[4,12]] [[[[.,[[.,.],[[.,.],[.,.]]]],[.,.]],[.,.]],[.,.]]=>[[[[.,[.,.]],[.,.]],[.,.]],[[.,[[.,[.,.]],.]],.]]=>[2,1,4,3,6,5,10,9,11,8,12,7]=>[[1,3,5,7,9,11],[2,4,6,8],[10],[12]] [[[[.,[[[.,.],[.,.]],[.,.]]],[.,.]],[.,.]],[.,.]]=>[[[[.,[.,.]],[.,.]],[.,.]],[[[.,[.,.]],[.,.]],.]]=>[2,1,4,3,6,5,9,8,11,10,12,7]=>[[1,3,5,7,9,11],[2,4,6,8,10],[12]] [[.,[[.,.],[.,.]]],[[.,.],[[.,.],[[.,.],[.,.]]]]]=>[[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],[[.,[.,.]],.]]=>[5,4,6,3,7,2,8,1,11,10,12,9]=>[[1,3,5,7,9,11],[2,10],[4,12],[6],[8]] [[.,[[.,.],[.,.]]],[[.,.],[[[.,.],[.,.]],[.,.]]]]=>[[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],[[.,[.,.]],.]]=>[4,3,6,5,7,2,8,1,11,10,12,9]=>[[1,3,5,7,9,11],[2,4,10],[6,12],[8]] [[.,[[.,.],[.,.]]],[[[.,.],[[.,.],[.,.]]],[.,.]]]=>[[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],[[.,[.,.]],.]]=>[3,2,6,5,7,4,8,1,11,10,12,9]=>[[1,3,5,7,9,11],[2,4,10],[6,12],[8]] [[.,[[.,.],[.,.]]],[[[[.,.],[.,.]],[.,.]],[.,.]]]=>[[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],[[.,[.,.]],.]]=>[3,2,5,4,7,6,8,1,11,10,12,9]=>[[1,3,5,7,9,11],[2,4,6,10],[8,12]] [[[.,[[.,.],[.,.]]],[[.,.],[[.,.],[.,.]]]],[.,.]]=>[[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],[[.,[.,.]],.]]=>[2,1,6,5,7,4,8,3,11,10,12,9]=>[[1,3,5,7,9,11],[2,4,10],[6,12],[8]] [[[.,[[.,.],[.,.]]],[[[.,.],[.,.]],[.,.]]],[.,.]]=>[[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],[[.,[.,.]],.]]=>[2,1,5,4,7,6,8,3,11,10,12,9]=>[[1,3,5,7,9,11],[2,4,6,10],[8,12]] [[[.,[[.,.],[.,.]]],[[.,.],[.,.]]],[[.,.],[.,.]]]=>[[[.,[[.,[.,.]],.]],[[.,[.,.]],.]],[[.,[.,.]],.]]=>[3,2,4,1,7,6,8,5,11,10,12,9]=>[[1,3,5,7,9,11],[2,6,10],[4,8,12]] [[[[.,[[.,.],[.,.]]],[[.,.],[.,.]]],[.,.]],[.,.]]=>[[[[.,[.,.]],[.,.]],[[.,[.,.]],.]],[[.,[.,.]],.]]=>[2,1,4,3,7,6,8,5,11,10,12,9]=>[[1,3,5,7,9,11],[2,4,6,10],[8,12]] [[[.,[[.,.],[.,.]]],[.,.]],[[.,.],[[.,.],[.,.]]]]=>[[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],[[.,[.,.]],.]]=>[4,3,5,2,6,1,8,7,11,10,12,9]=>[[1,3,5,7,9,11],[2,8,10],[4,12],[6]] [[[.,[[.,.],[.,.]]],[.,.]],[[[.,.],[.,.]],[.,.]]]=>[[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],[[.,[.,.]],.]]=>[3,2,5,4,6,1,8,7,11,10,12,9]=>[[1,3,5,7,9,11],[2,4,8,10],[6,12]] [[[[.,[[.,.],[.,.]]],[.,.]],[[.,.],[.,.]]],[.,.]]=>[[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],[[.,[.,.]],.]]=>[2,1,5,4,6,3,8,7,11,10,12,9]=>[[1,3,5,7,9,11],[2,4,8,10],[6,12]] [[[[.,[[.,.],[.,.]]],[.,.]],[.,.]],[[.,.],[.,.]]]=>[[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],[[.,[.,.]],.]]=>[3,2,4,1,6,5,8,7,11,10,12,9]=>[[1,3,5,7,9,11],[2,6,8,10],[4,12]] [[[[[.,[[.,.],[.,.]]],[.,.]],[.,.]],[.,.]],[.,.]]=>[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],[[.,[.,.]],.]]=>[2,1,4,3,6,5,8,7,11,10,12,9]=>[[1,3,5,7,9,11],[2,4,6,8,10],[12]] [[.,[.,.]],[[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]]]=>[[.,[[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],.]],[.,.]]=>[6,5,7,4,8,3,9,2,10,1,12,11]=>[[1,3,5,7,9,11],[2,12],[4],[6],[8],[10]] [[.,[.,.]],[[.,.],[[.,.],[[[.,.],[.,.]],[.,.]]]]]=>[[.,[[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],.]],[.,.]]=>[5,4,7,6,8,3,9,2,10,1,12,11]=>[[1,3,5,7,9,11],[2,4,12],[6],[8],[10]] [[.,[.,.]],[[.,.],[[[.,.],[[.,.],[.,.]]],[.,.]]]]=>[[.,[[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],.]],[.,.]]=>[4,3,7,6,8,5,9,2,10,1,12,11]=>[[1,3,5,7,9,11],[2,4,12],[6],[8],[10]] [[.,[.,.]],[[.,.],[[[.,.],[.,.]],[[.,.],[.,.]]]]]=>[[.,[[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],.]],[.,.]]=>[5,4,6,3,8,7,9,2,10,1,12,11]=>[[1,3,5,7,9,11],[2,6,12],[4],[8],[10]] [[.,[.,.]],[[.,.],[[[[.,.],[.,.]],[.,.]],[.,.]]]]=>[[.,[[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],.]],[.,.]]=>[4,3,6,5,8,7,9,2,10,1,12,11]=>[[1,3,5,7,9,11],[2,4,6,12],[8],[10]] [[.,[.,.]],[[[.,.],[[.,.],[[.,.],[.,.]]]],[.,.]]]=>[[.,[[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],.]],[.,.]]=>[3,2,7,6,8,5,9,4,10,1,12,11]=>[[1,3,5,7,9,11],[2,4,12],[6],[8],[10]] [[.,[.,.]],[[[.,.],[[[.,.],[.,.]],[.,.]]],[.,.]]]=>[[.,[[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],.]],[.,.]]=>[3,2,6,5,8,7,9,4,10,1,12,11]=>[[1,3,5,7,9,11],[2,4,6,12],[8],[10]] [[.,[.,.]],[[[.,.],[[.,.],[.,.]]],[[.,.],[.,.]]]]=>[[.,[[[.,[[.,[.,.]],.]],[[.,[.,.]],.]],.]],[.,.]]=>[4,3,5,2,8,7,9,6,10,1,12,11]=>[[1,3,5,7,9,11],[2,6,12],[4,8],[10]] [[.,[.,.]],[[[[.,.],[[.,.],[.,.]]],[.,.]],[.,.]]]=>[[.,[[[[.,[.,.]],[.,.]],[[.,[.,.]],.]],.]],[.,.]]=>[3,2,5,4,8,7,9,6,10,1,12,11]=>[[1,3,5,7,9,11],[2,4,6,12],[8],[10]] [[.,[.,.]],[[[.,.],[.,.]],[[.,.],[[.,.],[.,.]]]]]=>[[.,[[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],.]],[.,.]]=>[5,4,6,3,7,2,9,8,10,1,12,11]=>[[1,3,5,7,9,11],[2,8,12],[4],[6],[10]] [[[.,[.,.]],[[.,.],[[.,.],[[.,.],[.,.]]]]],[.,.]]=>[[[.,[.,.]],[[.,[[.,[[.,[.,.]],.]],.]],.]],[.,.]]=>[2,1,7,6,8,5,9,4,10,3,12,11]=>[[1,3,5,7,9,11],[2,4,12],[6],[8],[10]] [[[.,[.,.]],[[.,.],[[[.,.],[.,.]],[.,.]]]],[.,.]]=>[[[.,[.,.]],[[.,[[[.,[.,.]],[.,.]],.]],.]],[.,.]]=>[2,1,6,5,8,7,9,4,10,3,12,11]=>[[1,3,5,7,9,11],[2,4,6,12],[8],[10]] [[[.,[.,.]],[[[.,.],[[.,.],[.,.]]],[.,.]]],[.,.]]=>[[[.,[.,.]],[[[.,[.,.]],[[.,[.,.]],.]],.]],[.,.]]=>[2,1,5,4,8,7,9,6,10,3,12,11]=>[[1,3,5,7,9,11],[2,4,6,12],[8],[10]] [[[.,[.,.]],[[[.,.],[.,.]],[[.,.],[.,.]]]],[.,.]]=>[[[.,[.,.]],[[[.,[[.,[.,.]],.]],[.,.]],.]],[.,.]]=>[2,1,6,5,7,4,9,8,10,3,12,11]=>[[1,3,5,7,9,11],[2,4,8,12],[6],[10]] [[[.,[.,.]],[[.,.],[[.,.],[.,.]]]],[[.,.],[.,.]]]=>[[[.,[[.,[.,.]],.]],[[.,[[.,[.,.]],.]],.]],[.,.]]=>[3,2,4,1,8,7,9,6,10,5,12,11]=>[[1,3,5,7,9,11],[2,6,12],[4,8],[10]] [[[[.,[.,.]],[[.,.],[[.,.],[.,.]]]],[.,.]],[.,.]]=>[[[[.,[.,.]],[.,.]],[[.,[[.,[.,.]],.]],.]],[.,.]]=>[2,1,4,3,8,7,9,6,10,5,12,11]=>[[1,3,5,7,9,11],[2,4,6,12],[8],[10]] [[[.,[.,.]],[[.,.],[.,.]]],[[.,.],[[.,.],[.,.]]]]=>[[[.,[[.,[[.,[.,.]],.]],.]],[[.,[.,.]],.]],[.,.]]=>[4,3,5,2,6,1,9,8,10,7,12,11]=>[[1,3,5,7,9,11],[2,8,12],[4,10],[6]] [[[.,[.,.]],[.,.]],[[.,.],[[.,.],[[.,.],[.,.]]]]]=>[[[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],[.,.]],[.,.]]=>[5,4,6,3,7,2,8,1,10,9,12,11]=>[[1,3,5,7,9,11],[2,10,12],[4],[6],[8]] [[[.,[.,.]],[.,.]],[[.,.],[[[.,.],[.,.]],[.,.]]]]=>[[[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],[.,.]],[.,.]]=>[4,3,6,5,7,2,8,1,10,9,12,11]=>[[1,3,5,7,9,11],[2,4,10,12],[6],[8]] [[[.,[.,.]],[.,.]],[[[.,.],[[.,.],[.,.]]],[.,.]]]=>[[[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],[.,.]],[.,.]]=>[3,2,6,5,7,4,8,1,10,9,12,11]=>[[1,3,5,7,9,11],[2,4,10,12],[6],[8]] [[[.,[.,.]],[.,.]],[[[[.,.],[.,.]],[.,.]],[.,.]]]=>[[[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],[.,.]],[.,.]]=>[3,2,5,4,7,6,8,1,10,9,12,11]=>[[1,3,5,7,9,11],[2,4,6,10,12],[8]] [[[[.,[.,.]],[.,.]],[[.,.],[[.,.],[.,.]]]],[.,.]]=>[[[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],[.,.]],[.,.]]=>[2,1,6,5,7,4,8,3,10,9,12,11]=>[[1,3,5,7,9,11],[2,4,10,12],[6],[8]] [[[[.,[.,.]],[.,.]],[[[.,.],[.,.]],[.,.]]],[.,.]]=>[[[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],[.,.]],[.,.]]=>[2,1,5,4,7,6,8,3,10,9,12,11]=>[[1,3,5,7,9,11],[2,4,6,10,12],[8]] [[[[.,[.,.]],[.,.]],[[.,.],[.,.]]],[[.,.],[.,.]]]=>[[[[.,[[.,[.,.]],.]],[[.,[.,.]],.]],[.,.]],[.,.]]=>[3,2,4,1,7,6,8,5,10,9,12,11]=>[[1,3,5,7,9,11],[2,6,10,12],[4,8]] [[[[[.,[.,.]],[.,.]],[[.,.],[.,.]]],[.,.]],[.,.]]=>[[[[[.,[.,.]],[.,.]],[[.,[.,.]],.]],[.,.]],[.,.]]=>[2,1,4,3,7,6,8,5,10,9,12,11]=>[[1,3,5,7,9,11],[2,4,6,10,12],[8]] [[[[.,[.,.]],[.,.]],[.,.]],[[.,.],[[.,.],[.,.]]]]=>[[[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],[.,.]],[.,.]]=>[4,3,5,2,6,1,8,7,10,9,12,11]=>[[1,3,5,7,9,11],[2,8,10,12],[4],[6]] [[[[.,[.,.]],[.,.]],[.,.]],[[[.,.],[.,.]],[.,.]]]=>[[[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],[.,.]],[.,.]]=>[3,2,5,4,6,1,8,7,10,9,12,11]=>[[1,3,5,7,9,11],[2,4,8,10,12],[6]] [[[[[.,[.,.]],[.,.]],[.,.]],[[.,.],[.,.]]],[.,.]]=>[[[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],[.,.]],[.,.]]=>[2,1,5,4,6,3,8,7,10,9,12,11]=>[[1,3,5,7,9,11],[2,4,8,10,12],[6]] [[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],[[.,.],[.,.]]]=>[[[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],[.,.]],[.,.]]=>[3,2,4,1,6,5,8,7,10,9,12,11]=>[[1,3,5,7,9,11],[2,6,8,10,12],[4]] [.,[[[[[[[.,.],.],.],.],.],.],[.,.]]]=>[.,[[[[[[[.,[.,.]],.],.],.],.],.],.]]=>[3,2,4,5,6,7,8,9,1]=>[[1,3,4,5,6,7,8],[2],[9]] [[[[[[[[[[.,[.,.]],.],.],.],.],.],.],.],.],.]=>[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]=>[1,2,3,4,5,6,7,8,9,11,10]=>[[1,2,3,4,5,6,7,8,9,10],[11]] [[[[[[[.,[.,.]],.],.],.],.],[.,.]],.]=>[[[[[[[.,.],[.,.]],.],.],.],.],[.,.]]=>[1,3,2,4,5,6,7,9,8]=>[[1,2,4,5,6,7,8],[3,9]] [[[[[[.,[.,.]],.],.],.],.],[[.,.],.]]=>[[[[[[.,[[.,.],.]],.],.],.],.],[.,.]]=>[2,3,1,4,5,6,7,9,8]=>[[1,2,4,5,6,7,8],[3,9]] [.,[[[[[[[.,.],.],.],.],.],[.,.]],.]]=>[.,[[[[[[[.,.],[.,.]],.],.],.],.],.]]=>[2,4,3,5,6,7,8,9,1]=>[[1,2,4,5,6,7,8],[3],[9]] [.,[[[[[[.,.],.],.],.],.],[[.,.],.]]]=>[.,[[[[[[.,[[.,.],.]],.],.],.],.],.]]=>[3,4,2,5,6,7,8,9,1]=>[[1,2,4,5,6,7,8],[3],[9]] [.,[[[[[[[[.,.],.],.],.],.],.],.],[.,.]]]=>[.,[[[[[[[[.,[.,.]],.],.],.],.],.],.],.]]=>[3,2,4,5,6,7,8,9,10,1]=>[[1,3,4,5,6,7,8,9],[2],[10]] [[[[[[[.,[.,.]],[.,.]],.],.],.],.],.]=>[[[[[[[.,.],.],.],.],.],[.,.]],[.,.]]=>[1,2,3,4,5,7,6,9,8]=>[[1,2,3,4,5,6,8],[7,9]] [[[[[[.,[.,[.,.]]],[.,.]],.],.],.],.]=>[[[[[[.,.],.],.],.],[.,.]],[.,[.,.]]]=>[1,2,3,4,6,5,9,8,7]=>[[1,2,3,4,5,7],[6,8],[9]] [[[[[.,[.,[.,[.,.]]]],[.,.]],.],.],.]=>[[[[[.,.],.],.],[.,.]],[.,[.,[.,.]]]]=>[1,2,3,5,4,9,8,7,6]=>[[1,2,3,4,6],[5,7],[8],[9]] 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[[.,[[[[[[.,[.,.]],.],.],.],.],.]],.]=>[[.,.],[[[[[[.,.],.],.],.],.],[.,.]]]=>[1,3,4,5,6,7,9,8,2]=>[[1,2,3,4,5,6,7],[8],[9]] [.,[[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]]=>[.,[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]]=>[7,6,5,4,3,2,8,9,1]=>[[1,7,8],[2],[3],[4],[5],[6],[9]] [[.,.],[[.,[.,[.,[.,[.,[.,.]]]]]],.]]=>[[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]],.]=>[2,8,7,6,5,4,3,1,9]=>[[1,2,9],[3],[4],[5],[6],[7],[8]] [[.,[[[[[[.,.],.],.],.],.],.]],[.,.]]=>[[.,[.,.]],[[[[[[.,.],.],.],.],.],.]]=>[2,1,4,5,6,7,8,9,3]=>[[1,3,4,5,6,7,8],[2,9]] [[.,[[.,.],[[[[[.,.],.],.],.],.]]],.]=>[[.,.],[[.,[[[[[.,.],.],.],.],.]],.]]=>[1,4,5,6,7,8,3,9,2]=>[[1,2,3,4,5,6,8],[7],[9]] [[[[[[[[[.,.],.],.],.],.],.],.],.],[[.,.],.]]=>[[[[[[[[[.,[[.,.],.]],.],.],.],.],.],.],.],.]=>[2,3,1,4,5,6,7,8,9,10,11]=>[[1,2,4,5,6,7,8,9,10,11],[3]] [.,[[[[[[.,.],.],.],.],.],[.,[.,.]]]]=>[.,[[[[[[.,[.,[.,.]]],.],.],.],.],.]]=>[4,3,2,5,6,7,8,9,1]=>[[1,4,5,6,7,8],[2],[3],[9]] [.,[[[[[.,.],.],.],.],[.,[.,[.,.]]]]]=>[.,[[[[[.,[.,[.,[.,.]]]],.],.],.],.]]=>[5,4,3,2,6,7,8,9,1]=>[[1,5,6,7,8],[2],[3],[4],[9]] [[[[[.,.],.],.],.],[.,[.,[.,[[.,.],.]]]]]=>[[[[[.,[.,[.,[.,[[.,.],.]]]]],.],.],.],.]=>[5,6,4,3,2,1,7,8,9,10]=>[[1,2,7,8,9,10],[3],[4],[5],[6]] [[[.,[[[[[.,.],.],.],.],.]],.],[.,.]]=>[[[.,[.,.]],.],[[[[[.,.],.],.],.],.]]=>[2,1,3,5,6,7,8,9,4]=>[[1,3,4,5,6,7,8],[2,9]] [[.,.],[.,[.,[.,[[[.,.],.],[.,.]]]]]]=>[[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]],.]=>[6,5,7,8,4,3,2,1,9]=>[[1,3,4,9],[2],[5],[6],[7],[8]] [[.,.],[.,[.,[[[.,.],.],[.,[.,.]]]]]]=>[[.,[.,[.,[[[.,[.,[.,.]]],.],.]]]],.]=>[6,5,4,7,8,3,2,1,9]=>[[1,4,5,9],[2],[3],[6],[7],[8]] [[.,.],[[[.,.],.],[.,[.,[.,[.,.]]]]]]=>[[.,[[[.,[.,[.,[.,[.,.]]]]],.],.]],.]=>[6,5,4,3,2,7,8,1,9]=>[[1,6,7,9],[2],[3],[4],[5],[8]] [[.,.],[.,[.,[.,[.,[[[.,.],.],[.,.]]]]]]]=>[[.,[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]],.]=>[7,6,8,9,5,4,3,2,1,10]=>[[1,3,4,10],[2],[5],[6],[7],[8],[9]] [[.,.],[.,[.,[.,[[[.,.],.],[.,[.,.]]]]]]]=>[[.,[.,[.,[.,[[[.,[.,[.,.]]],.],.]]]]],.]=>[7,6,5,8,9,4,3,2,1,10]=>[[1,4,5,10],[2],[3],[6],[7],[8],[9]] [[.,.],[.,[[[.,.],.],[.,[.,[.,[.,.]]]]]]]=>[[.,[.,[[[.,[.,[.,[.,[.,.]]]]],.],.]]],.]=>[7,6,5,4,3,8,9,2,1,10]=>[[1,6,7,10],[2],[3],[4],[5],[8],[9]] [[.,.],[[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]]=>[[.,[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]],.]=>[7,6,5,4,3,2,8,9,1,10]=>[[1,7,8,10],[2],[3],[4],[5],[6],[9]] [[.,[.,[.,[.,[.,.]]]]],[.,[[.,.],.]]]=>[[.,[.,[[.,.],.]]],[.,[.,[.,[.,.]]]]]=>[3,4,2,1,9,8,7,6,5]=>[[1,2,5],[3,6],[4,7],[8],[9]] [[.,[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]],.]=>[[.,.],[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]]=>[1,9,8,10,7,6,5,4,3,2]=>[[1,2,4],[3],[5],[6],[7],[8],[9],[10]] [.,[[.,[.,.]],[.,[.,[.,[.,[.,[.,.]]]]]]]]=>[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]]=>[8,7,6,5,4,3,2,10,9,1]=>[[1,8],[2,9],[3],[4],[5],[6],[7],[10]] [.,[.,[.,[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]]]]=>[.,[.,[.,[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]]]]=>[10,9,11,8,7,6,5,4,3,2,1]=>[[1,3],[2],[4],[5],[6],[7],[8],[9],[10],[11]] [[[[[[.,.],.],.],.],.],[.,[[.,.],.]]]=>[[[[[[.,[.,[[.,.],.]]],.],.],.],.],.]=>[3,4,2,1,5,6,7,8,9]=>[[1,2,5,6,7,8,9],[3],[4]] [[[[[[[.,.],.],.],.],.],.],[.,[[.,.],.]]]=>[[[[[[[.,[.,[[.,.],.]]],.],.],.],.],.],.]=>[3,4,2,1,5,6,7,8,9,10]=>[[1,2,5,6,7,8,9,10],[3],[4]] [[[[[[.,.],.],.],.],[[.,.],.]],[.,.]]=>[[[[[[.,[.,.]],[[.,.],.]],.],.],.],.]=>[2,1,4,5,3,6,7,8,9]=>[[1,3,4,6,7,8,9],[2,5]] [[[[[.,.],.],.],.],[[.,.],[[.,.],.]]]=>[[[[[.,[[.,[[.,.],.]],.]],.],.],.],.]=>[3,4,2,5,1,6,7,8,9]=>[[1,2,4,6,7,8,9],[3],[5]] [[[[[[.,.],.],.],.],[.,.]],[[.,.],.]]=>[[[[[[.,[[.,.],.]],[.,.]],.],.],.],.]=>[2,3,1,5,4,6,7,8,9]=>[[1,2,4,6,7,8,9],[3,5]] [[[[[[[.,.],.],.],.],[.,.]],[.,.]],.]=>[[[[[[[.,.],[.,.]],[.,.]],.],.],.],.]=>[1,3,2,5,4,6,7,8,9]=>[[1,2,4,6,7,8,9],[3,5]] [[[[[[[.,.],.],.],.],[[.,.],.]],.],.]=>[[[[[[[.,.],.],[[.,.],.]],.],.],.],.]=>[1,2,4,5,3,6,7,8,9]=>[[1,2,3,4,6,7,8,9],[5]] [.,[[[[.,.],.],.],[.,[.,[.,[.,.]]]]]]=>[.,[[[[.,[.,[.,[.,[.,.]]]]],.],.],.]]=>[6,5,4,3,2,7,8,9,1]=>[[1,6,7,8],[2],[3],[4],[5],[9]] [[[.,[[[.,.],.],[[[.,.],.],.]]],.],.]=>[[[.,.],.],[[[.,[[[.,.],.],.]],.],.]]=>[1,2,5,6,7,4,8,9,3]=>[[1,2,3,4,5,7,8],[6],[9]]
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.
Map
Robinson-Schensted recording tableau
Description
Sends a permutation to its Robinson-Schensted recording tableau.
The Robinson-Schensted corrspondence is a bijection between permutations of length $n$ and pairs of standard Young tableaux of the same shape and of size $n$, see [1]. These two tableaux are the insertion tableau and the recording tableau.
This map sends a permutation to its corresponding recording tableau.