Identifier
Mp00018: Binary trees left border symmetryBinary trees
Mp00017: Binary trees to 312-avoiding permutationPermutations
Mp00151: Permutations to cycle type Set partitions
Images
=>
Cc0010;cc-rep-0Cc0010;cc-rep-1Cc0009;cc-rep-3
[.,.]=>[.,.]=>[1]=>{{1}} [.,[.,.]]=>[.,[.,.]]=>[2,1]=>{{1,2}} [[.,.],.]=>[[.,.],.]=>[1,2]=>{{1},{2}} [.,[.,[.,.]]]=>[.,[.,[.,.]]]=>[3,2,1]=>{{1,3},{2}} [.,[[.,.],.]]=>[.,[[.,.],.]]=>[2,3,1]=>{{1,2,3}} [[.,.],[.,.]]=>[[.,[.,.]],.]=>[2,1,3]=>{{1,2},{3}} [[.,[.,.]],.]=>[[.,.],[.,.]]=>[1,3,2]=>{{1},{2,3}} [[[.,.],.],.]=>[[[.,.],.],.]=>[1,2,3]=>{{1},{2},{3}} [.,[.,[.,[.,.]]]]=>[.,[.,[.,[.,.]]]]=>[4,3,2,1]=>{{1,4},{2,3}} [.,[.,[[.,.],.]]]=>[.,[.,[[.,.],.]]]=>[3,4,2,1]=>{{1,2,3,4}} [.,[[.,.],[.,.]]]=>[.,[[.,[.,.]],.]]=>[3,2,4,1]=>{{1,3,4},{2}} [.,[[.,[.,.]],.]]=>[.,[[.,.],[.,.]]]=>[2,4,3,1]=>{{1,2,4},{3}} [.,[[[.,.],.],.]]=>[.,[[[.,.],.],.]]=>[2,3,4,1]=>{{1,2,3,4}} [[.,.],[.,[.,.]]]=>[[.,[.,[.,.]]],.]=>[3,2,1,4]=>{{1,3},{2},{4}} [[.,.],[[.,.],.]]=>[[.,[[.,.],.]],.]=>[2,3,1,4]=>{{1,2,3},{4}} [[.,[.,.]],[.,.]]=>[[.,[.,.]],[.,.]]=>[2,1,4,3]=>{{1,2},{3,4}} [[[.,.],.],[.,.]]=>[[[.,[.,.]],.],.]=>[2,1,3,4]=>{{1,2},{3},{4}} [[.,[.,[.,.]]],.]=>[[.,.],[.,[.,.]]]=>[1,4,3,2]=>{{1},{2,4},{3}} [[.,[[.,.],.]],.]=>[[.,.],[[.,.],.]]=>[1,3,4,2]=>{{1},{2,3,4}} [[[.,.],[.,.]],.]=>[[[.,.],[.,.]],.]=>[1,3,2,4]=>{{1},{2,3},{4}} [[[.,[.,.]],.],.]=>[[[.,.],.],[.,.]]=>[1,2,4,3]=>{{1},{2},{3,4}} [[[[.,.],.],.],.]=>[[[[.,.],.],.],.]=>[1,2,3,4]=>{{1},{2},{3},{4}} [.,[.,[.,[.,[.,.]]]]]=>[.,[.,[.,[.,[.,.]]]]]=>[5,4,3,2,1]=>{{1,5},{2,4},{3}} [.,[.,[.,[[.,.],.]]]]=>[.,[.,[.,[[.,.],.]]]]=>[4,5,3,2,1]=>{{1,2,4,5},{3}} [.,[.,[[.,.],[.,.]]]]=>[.,[.,[[.,[.,.]],.]]]=>[4,3,5,2,1]=>{{1,2,3,4,5}} [.,[.,[[.,[.,.]],.]]]=>[.,[.,[[.,.],[.,.]]]]=>[3,5,4,2,1]=>{{1,2,3,4,5}} [.,[.,[[[.,.],.],.]]]=>[.,[.,[[[.,.],.],.]]]=>[3,4,5,2,1]=>{{1,3,5},{2,4}} [.,[[.,.],[.,[.,.]]]]=>[.,[[.,[.,[.,.]]],.]]=>[4,3,2,5,1]=>{{1,4,5},{2,3}} [.,[[.,.],[[.,.],.]]]=>[.,[[.,[[.,.],.]],.]]=>[3,4,2,5,1]=>{{1,2,3,4,5}} [.,[[.,[.,.]],[.,.]]]=>[.,[[.,[.,.]],[.,.]]]=>[3,2,5,4,1]=>{{1,3,5},{2},{4}} [.,[[[.,.],.],[.,.]]]=>[.,[[[.,[.,.]],.],.]]=>[3,2,4,5,1]=>{{1,3,4,5},{2}} [.,[[.,[.,[.,.]]],.]]=>[.,[[.,.],[.,[.,.]]]]=>[2,5,4,3,1]=>{{1,2,5},{3,4}} [.,[[.,[[.,.],.]],.]]=>[.,[[.,.],[[.,.],.]]]=>[2,4,5,3,1]=>{{1,2,3,4,5}} [.,[[[.,.],[.,.]],.]]=>[.,[[[.,.],[.,.]],.]]=>[2,4,3,5,1]=>{{1,2,4,5},{3}} [.,[[[.,[.,.]],.],.]]=>[.,[[[.,.],.],[.,.]]]=>[2,3,5,4,1]=>{{1,2,3,5},{4}} [.,[[[[.,.],.],.],.]]=>[.,[[[[.,.],.],.],.]]=>[2,3,4,5,1]=>{{1,2,3,4,5}} [[.,.],[.,[.,[.,.]]]]=>[[.,[.,[.,[.,.]]]],.]=>[4,3,2,1,5]=>{{1,4},{2,3},{5}} [[.,.],[.,[[.,.],.]]]=>[[.,[.,[[.,.],.]]],.]=>[3,4,2,1,5]=>{{1,2,3,4},{5}} [[.,.],[[.,.],[.,.]]]=>[[.,[[.,[.,.]],.]],.]=>[3,2,4,1,5]=>{{1,3,4},{2},{5}} [[.,.],[[.,[.,.]],.]]=>[[.,[[.,.],[.,.]]],.]=>[2,4,3,1,5]=>{{1,2,4},{3},{5}} [[.,.],[[[.,.],.],.]]=>[[.,[[[.,.],.],.]],.]=>[2,3,4,1,5]=>{{1,2,3,4},{5}} [[.,[.,.]],[.,[.,.]]]=>[[.,[.,[.,.]]],[.,.]]=>[3,2,1,5,4]=>{{1,3},{2},{4,5}} [[.,[.,.]],[[.,.],.]]=>[[.,[[.,.],.]],[.,.]]=>[2,3,1,5,4]=>{{1,2,3},{4,5}} [[[.,.],.],[.,[.,.]]]=>[[[.,[.,[.,.]]],.],.]=>[3,2,1,4,5]=>{{1,3},{2},{4},{5}} [[[.,.],.],[[.,.],.]]=>[[[.,[[.,.],.]],.],.]=>[2,3,1,4,5]=>{{1,2,3},{4},{5}} [[.,[.,[.,.]]],[.,.]]=>[[.,[.,.]],[.,[.,.]]]=>[2,1,5,4,3]=>{{1,2},{3,5},{4}} [[.,[[.,.],.]],[.,.]]=>[[.,[.,.]],[[.,.],.]]=>[2,1,4,5,3]=>{{1,2},{3,4,5}} [[[.,.],[.,.]],[.,.]]=>[[[.,[.,.]],[.,.]],.]=>[2,1,4,3,5]=>{{1,2},{3,4},{5}} [[[.,[.,.]],.],[.,.]]=>[[[.,[.,.]],.],[.,.]]=>[2,1,3,5,4]=>{{1,2},{3},{4,5}} [[[[.,.],.],.],[.,.]]=>[[[[.,[.,.]],.],.],.]=>[2,1,3,4,5]=>{{1,2},{3},{4},{5}} [[.,[.,[.,[.,.]]]],.]=>[[.,.],[.,[.,[.,.]]]]=>[1,5,4,3,2]=>{{1},{2,5},{3,4}} [[.,[.,[[.,.],.]]],.]=>[[.,.],[.,[[.,.],.]]]=>[1,4,5,3,2]=>{{1},{2,3,4,5}} [[.,[[.,.],[.,.]]],.]=>[[.,.],[[.,[.,.]],.]]=>[1,4,3,5,2]=>{{1},{2,4,5},{3}} [[.,[[.,[.,.]],.]],.]=>[[.,.],[[.,.],[.,.]]]=>[1,3,5,4,2]=>{{1},{2,3,5},{4}} [[.,[[[.,.],.],.]],.]=>[[.,.],[[[.,.],.],.]]=>[1,3,4,5,2]=>{{1},{2,3,4,5}} [[[.,.],[.,[.,.]]],.]=>[[[.,.],[.,[.,.]]],.]=>[1,4,3,2,5]=>{{1},{2,4},{3},{5}} [[[.,.],[[.,.],.]],.]=>[[[.,.],[[.,.],.]],.]=>[1,3,4,2,5]=>{{1},{2,3,4},{5}} [[[.,[.,.]],[.,.]],.]=>[[[.,.],[.,.]],[.,.]]=>[1,3,2,5,4]=>{{1},{2,3},{4,5}} [[[[.,.],.],[.,.]],.]=>[[[[.,.],[.,.]],.],.]=>[1,3,2,4,5]=>{{1},{2,3},{4},{5}} [[[.,[.,[.,.]]],.],.]=>[[[.,.],.],[.,[.,.]]]=>[1,2,5,4,3]=>{{1},{2},{3,5},{4}} [[[.,[[.,.],.]],.],.]=>[[[.,.],.],[[.,.],.]]=>[1,2,4,5,3]=>{{1},{2},{3,4,5}} [[[[.,.],[.,.]],.],.]=>[[[[.,.],.],[.,.]],.]=>[1,2,4,3,5]=>{{1},{2},{3,4},{5}} [[[[.,[.,.]],.],.],.]=>[[[[.,.],.],.],[.,.]]=>[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]=>{{1,6},{2,5},{3,4}} [.,[.,[.,[.,[[.,.],.]]]]]=>[.,[.,[.,[.,[[.,.],.]]]]]=>[5,6,4,3,2,1]=>{{1,2,5,6},{3,4}} [.,[.,[.,[[.,.],[.,.]]]]]=>[.,[.,[.,[[.,[.,.]],.]]]]=>[5,4,6,3,2,1]=>{{1,2,3,4,5,6}} [.,[.,[.,[[.,[.,.]],.]]]]=>[.,[.,[.,[[.,.],[.,.]]]]]=>[4,6,5,3,2,1]=>{{1,2,3,4,5,6}} [.,[.,[.,[[[.,.],.],.]]]]=>[.,[.,[.,[[[.,.],.],.]]]]=>[4,5,6,3,2,1]=>{{1,3,4,6},{2,5}} [.,[.,[[.,.],[.,[.,.]]]]]=>[.,[.,[[.,[.,[.,.]]],.]]]=>[5,4,3,6,2,1]=>{{1,2,4,5,6},{3}} [.,[.,[[.,.],[[.,.],.]]]]=>[.,[.,[[.,[[.,.],.]],.]]]=>[4,5,3,6,2,1]=>{{1,4,6},{2,5},{3}} [.,[.,[[.,[.,.]],[.,.]]]]=>[.,[.,[[.,[.,.]],[.,.]]]]=>[4,3,6,5,2,1]=>{{1,2,3,4,5,6}} [.,[.,[[[.,.],.],[.,.]]]]=>[.,[.,[[[.,[.,.]],.],.]]]=>[4,3,5,6,2,1]=>{{1,4,6},{2,3,5}} [.,[.,[[.,[.,[.,.]]],.]]]=>[.,[.,[[.,.],[.,[.,.]]]]]=>[3,6,5,4,2,1]=>{{1,2,3,5,6},{4}} [.,[.,[[.,[[.,.],.]],.]]]=>[.,[.,[[.,.],[[.,.],.]]]]=>[3,5,6,4,2,1]=>{{1,3,6},{2,5},{4}} [.,[.,[[[.,.],[.,.]],.]]]=>[.,[.,[[[.,.],[.,.]],.]]]=>[3,5,4,6,2,1]=>{{1,3,4,6},{2,5}} [.,[.,[[[.,[.,.]],.],.]]]=>[.,[.,[[[.,.],.],[.,.]]]]=>[3,4,6,5,2,1]=>{{1,3,6},{2,4,5}} [.,[.,[[[[.,.],.],.],.]]]=>[.,[.,[[[[.,.],.],.],.]]]=>[3,4,5,6,2,1]=>{{1,2,3,4,5,6}} [.,[[.,.],[.,[.,[.,.]]]]]=>[.,[[.,[.,[.,[.,.]]]],.]]=>[5,4,3,2,6,1]=>{{1,5,6},{2,4},{3}} [.,[[.,.],[.,[[.,.],.]]]]=>[.,[[.,[.,[[.,.],.]]],.]]=>[4,5,3,2,6,1]=>{{1,2,4,5,6},{3}} [.,[[.,.],[[.,.],[.,.]]]]=>[.,[[.,[[.,[.,.]],.]],.]]=>[4,3,5,2,6,1]=>{{1,2,3,4,5,6}} [.,[[.,.],[[.,[.,.]],.]]]=>[.,[[.,[[.,.],[.,.]]],.]]=>[3,5,4,2,6,1]=>{{1,2,3,4,5,6}} [.,[[.,.],[[[.,.],.],.]]]=>[.,[[.,[[[.,.],.],.]],.]]=>[3,4,5,2,6,1]=>{{1,3,5,6},{2,4}} [.,[[.,[.,.]],[.,[.,.]]]]=>[.,[[.,[.,[.,.]]],[.,.]]]=>[4,3,2,6,5,1]=>{{1,4,6},{2,3},{5}} [.,[[.,[.,.]],[[.,.],.]]]=>[.,[[.,[[.,.],.]],[.,.]]]=>[3,4,2,6,5,1]=>{{1,2,3,4,6},{5}} [.,[[[.,.],.],[.,[.,.]]]]=>[.,[[[.,[.,[.,.]]],.],.]]=>[4,3,2,5,6,1]=>{{1,4,5,6},{2,3}} [.,[[[.,.],.],[[.,.],.]]]=>[.,[[[.,[[.,.],.]],.],.]]=>[3,4,2,5,6,1]=>{{1,2,3,4,5,6}} [.,[[.,[.,[.,.]]],[.,.]]]=>[.,[[.,[.,.]],[.,[.,.]]]]=>[3,2,6,5,4,1]=>{{1,3,6},{2},{4,5}} [.,[[.,[[.,.],.]],[.,.]]]=>[.,[[.,[.,.]],[[.,.],.]]]=>[3,2,5,6,4,1]=>{{1,3,4,5,6},{2}} [.,[[[.,.],[.,.]],[.,.]]]=>[.,[[[.,[.,.]],[.,.]],.]]=>[3,2,5,4,6,1]=>{{1,3,5,6},{2},{4}} [.,[[[.,[.,.]],.],[.,.]]]=>[.,[[[.,[.,.]],.],[.,.]]]=>[3,2,4,6,5,1]=>{{1,3,4,6},{2},{5}} 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[[[.,.],[[.,[.,[.,.]]],.]],.]=>[[[.,.],[[.,.],[.,[.,.]]]],.]=>[1,3,6,5,4,2,7]=>{{1},{2,3,6},{4,5},{7}} [[[.,.],[[.,[[.,.],.]],.]],.]=>[[[.,.],[[.,.],[[.,.],.]]],.]=>[1,3,5,6,4,2,7]=>{{1},{2,3,4,5,6},{7}} [[[.,.],[[[.,.],[.,.]],.]],.]=>[[[.,.],[[[.,.],[.,.]],.]],.]=>[1,3,5,4,6,2,7]=>{{1},{2,3,5,6},{4},{7}} [[[.,.],[[[.,[.,.]],.],.]],.]=>[[[.,.],[[[.,.],.],[.,.]]],.]=>[1,3,4,6,5,2,7]=>{{1},{2,3,4,6},{5},{7}} [[[.,.],[[[[.,.],.],.],.]],.]=>[[[.,.],[[[[.,.],.],.],.]],.]=>[1,3,4,5,6,2,7]=>{{1},{2,3,4,5,6},{7}} [[[.,[.,.]],[.,[.,[.,.]]]],.]=>[[[.,.],[.,[.,[.,.]]]],[.,.]]=>[1,5,4,3,2,7,6]=>{{1},{2,5},{3,4},{6,7}} [[[.,[.,.]],[.,[[.,.],.]]],.]=>[[[.,.],[.,[[.,.],.]]],[.,.]]=>[1,4,5,3,2,7,6]=>{{1},{2,3,4,5},{6,7}} [[[.,[.,.]],[[.,.],[.,.]]],.]=>[[[.,.],[[.,[.,.]],.]],[.,.]]=>[1,4,3,5,2,7,6]=>{{1},{2,4,5},{3},{6,7}} [[[.,[.,.]],[[.,[.,.]],.]],.]=>[[[.,.],[[.,.],[.,.]]],[.,.]]=>[1,3,5,4,2,7,6]=>{{1},{2,3,5},{4},{6,7}} [[[.,[.,.]],[[[.,.],.],.]],.]=>[[[.,.],[[[.,.],.],.]],[.,.]]=>[1,3,4,5,2,7,6]=>{{1},{2,3,4,5},{6,7}} [[[[.,.],.],[.,[.,[.,.]]]],.]=>[[[[.,.],[.,[.,[.,.]]]],.],.]=>[1,5,4,3,2,6,7]=>{{1},{2,5},{3,4},{6},{7}} [[[[.,.],.],[.,[[.,.],.]]],.]=>[[[[.,.],[.,[[.,.],.]]],.],.]=>[1,4,5,3,2,6,7]=>{{1},{2,3,4,5},{6},{7}} [[[[.,.],.],[[.,.],[.,.]]],.]=>[[[[.,.],[[.,[.,.]],.]],.],.]=>[1,4,3,5,2,6,7]=>{{1},{2,4,5},{3},{6},{7}} [[[[.,.],.],[[.,[.,.]],.]],.]=>[[[[.,.],[[.,.],[.,.]]],.],.]=>[1,3,5,4,2,6,7]=>{{1},{2,3,5},{4},{6},{7}} [[[[.,.],.],[[[.,.],.],.]],.]=>[[[[.,.],[[[.,.],.],.]],.],.]=>[1,3,4,5,2,6,7]=>{{1},{2,3,4,5},{6},{7}} [[[.,[.,[.,.]]],[.,[.,.]]],.]=>[[[.,.],[.,[.,.]]],[.,[.,.]]]=>[1,4,3,2,7,6,5]=>{{1},{2,4},{3},{5,7},{6}} [[[.,[.,[.,.]]],[[.,.],.]],.]=>[[[.,.],[[.,.],.]],[.,[.,.]]]=>[1,3,4,2,7,6,5]=>{{1},{2,3,4},{5,7},{6}} [[[.,[[.,.],.]],[.,[.,.]]],.]=>[[[.,.],[.,[.,.]]],[[.,.],.]]=>[1,4,3,2,6,7,5]=>{{1},{2,4},{3},{5,6,7}} [[[.,[[.,.],.]],[[.,.],.]],.]=>[[[.,.],[[.,.],.]],[[.,.],.]]=>[1,3,4,2,6,7,5]=>{{1},{2,3,4},{5,6,7}} [[[[.,.],[.,.]],[.,[.,.]]],.]=>[[[[.,.],[.,[.,.]]],[.,.]],.]=>[1,4,3,2,6,5,7]=>{{1},{2,4},{3},{5,6},{7}} [[[[.,.],[.,.]],[[.,.],.]],.]=>[[[[.,.],[[.,.],.]],[.,.]],.]=>[1,3,4,2,6,5,7]=>{{1},{2,3,4},{5,6},{7}} [[[[.,[.,.]],.],[.,[.,.]]],.]=>[[[[.,.],[.,[.,.]]],.],[.,.]]=>[1,4,3,2,5,7,6]=>{{1},{2,4},{3},{5},{6,7}} [[[[.,[.,.]],.],[[.,.],.]],.]=>[[[[.,.],[[.,.],.]],.],[.,.]]=>[1,3,4,2,5,7,6]=>{{1},{2,3,4},{5},{6,7}} [[[[[.,.],.],.],[.,[.,.]]],.]=>[[[[[.,.],[.,[.,.]]],.],.],.]=>[1,4,3,2,5,6,7]=>{{1},{2,4},{3},{5},{6},{7}} [[[[[.,.],.],.],[[.,.],.]],.]=>[[[[[.,.],[[.,.],.]],.],.],.]=>[1,3,4,2,5,6,7]=>{{1},{2,3,4},{5},{6},{7}} [[[.,[.,[.,[.,.]]]],[.,.]],.]=>[[[.,.],[.,.]],[.,[.,[.,.]]]]=>[1,3,2,7,6,5,4]=>{{1},{2,3},{4,7},{5,6}} [[[.,[.,[[.,.],.]]],[.,.]],.]=>[[[.,.],[.,.]],[.,[[.,.],.]]]=>[1,3,2,6,7,5,4]=>{{1},{2,3},{4,5,6,7}} [[[.,[[.,.],[.,.]]],[.,.]],.]=>[[[.,.],[.,.]],[[.,[.,.]],.]]=>[1,3,2,6,5,7,4]=>{{1},{2,3},{4,6,7},{5}} [[[.,[[.,[.,.]],.]],[.,.]],.]=>[[[.,.],[.,.]],[[.,.],[.,.]]]=>[1,3,2,5,7,6,4]=>{{1},{2,3},{4,5,7},{6}} [[[.,[[[.,.],.],.]],[.,.]],.]=>[[[.,.],[.,.]],[[[.,.],.],.]]=>[1,3,2,5,6,7,4]=>{{1},{2,3},{4,5,6,7}} [[[[.,.],[.,[.,.]]],[.,.]],.]=>[[[[.,.],[.,.]],[.,[.,.]]],.]=>[1,3,2,6,5,4,7]=>{{1},{2,3},{4,6},{5},{7}} [[[[.,.],[[.,.],.]],[.,.]],.]=>[[[[.,.],[.,.]],[[.,.],.]],.]=>[1,3,2,5,6,4,7]=>{{1},{2,3},{4,5,6},{7}} [[[[.,[.,.]],[.,.]],[.,.]],.]=>[[[[.,.],[.,.]],[.,.]],[.,.]]=>[1,3,2,5,4,7,6]=>{{1},{2,3},{4,5},{6,7}} [[[[[.,.],.],[.,.]],[.,.]],.]=>[[[[[.,.],[.,.]],[.,.]],.],.]=>[1,3,2,5,4,6,7]=>{{1},{2,3},{4,5},{6},{7}} [[[[.,[.,[.,.]]],.],[.,.]],.]=>[[[[.,.],[.,.]],.],[.,[.,.]]]=>[1,3,2,4,7,6,5]=>{{1},{2,3},{4},{5,7},{6}} [[[[.,[[.,.],.]],.],[.,.]],.]=>[[[[.,.],[.,.]],.],[[.,.],.]]=>[1,3,2,4,6,7,5]=>{{1},{2,3},{4},{5,6,7}} [[[[[.,.],[.,.]],.],[.,.]],.]=>[[[[[.,.],[.,.]],.],[.,.]],.]=>[1,3,2,4,6,5,7]=>{{1},{2,3},{4},{5,6},{7}} [[[[[.,[.,.]],.],.],[.,.]],.]=>[[[[[.,.],[.,.]],.],.],[.,.]]=>[1,3,2,4,5,7,6]=>{{1},{2,3},{4},{5},{6,7}} [[[[[[.,.],.],.],.],[.,.]],.]=>[[[[[[.,.],[.,.]],.],.],.],.]=>[1,3,2,4,5,6,7]=>{{1},{2,3},{4},{5},{6},{7}} [[[.,[.,[.,[.,[.,.]]]]],.],.]=>[[[.,.],.],[.,[.,[.,[.,.]]]]]=>[1,2,7,6,5,4,3]=>{{1},{2},{3,7},{4,6},{5}} [[[.,[.,[.,[[.,.],.]]]],.],.]=>[[[.,.],.],[.,[.,[[.,.],.]]]]=>[1,2,6,7,5,4,3]=>{{1},{2},{3,4,6,7},{5}} [[[.,[.,[[.,.],[.,.]]]],.],.]=>[[[.,.],.],[.,[[.,[.,.]],.]]]=>[1,2,6,5,7,4,3]=>{{1},{2},{3,4,5,6,7}} [[[.,[.,[[.,[.,.]],.]]],.],.]=>[[[.,.],.],[.,[[.,.],[.,.]]]]=>[1,2,5,7,6,4,3]=>{{1},{2},{3,4,5,6,7}} [[[.,[.,[[[.,.],.],.]]],.],.]=>[[[.,.],.],[.,[[[.,.],.],.]]]=>[1,2,5,6,7,4,3]=>{{1},{2},{3,5,7},{4,6}} [[[.,[[.,.],[.,[.,.]]]],.],.]=>[[[.,.],.],[[.,[.,[.,.]]],.]]=>[1,2,6,5,4,7,3]=>{{1},{2},{3,6,7},{4,5}} [[[.,[[.,.],[[.,.],.]]],.],.]=>[[[.,.],.],[[.,[[.,.],.]],.]]=>[1,2,5,6,4,7,3]=>{{1},{2},{3,4,5,6,7}} [[[.,[[.,[.,.]],[.,.]]],.],.]=>[[[.,.],.],[[.,[.,.]],[.,.]]]=>[1,2,5,4,7,6,3]=>{{1},{2},{3,5,7},{4},{6}} [[[.,[[[.,.],.],[.,.]]],.],.]=>[[[.,.],.],[[[.,[.,.]],.],.]]=>[1,2,5,4,6,7,3]=>{{1},{2},{3,5,6,7},{4}} [[[.,[[.,[.,[.,.]]],.]],.],.]=>[[[.,.],.],[[.,.],[.,[.,.]]]]=>[1,2,4,7,6,5,3]=>{{1},{2},{3,4,7},{5,6}} [[[.,[[.,[[.,.],.]],.]],.],.]=>[[[.,.],.],[[.,.],[[.,.],.]]]=>[1,2,4,6,7,5,3]=>{{1},{2},{3,4,5,6,7}} [[[.,[[[.,.],[.,.]],.]],.],.]=>[[[.,.],.],[[[.,.],[.,.]],.]]=>[1,2,4,6,5,7,3]=>{{1},{2},{3,4,6,7},{5}} [[[.,[[[.,[.,.]],.],.]],.],.]=>[[[.,.],.],[[[.,.],.],[.,.]]]=>[1,2,4,5,7,6,3]=>{{1},{2},{3,4,5,7},{6}} [[[.,[[[[.,.],.],.],.]],.],.]=>[[[.,.],.],[[[[.,.],.],.],.]]=>[1,2,4,5,6,7,3]=>{{1},{2},{3,4,5,6,7}} [[[[.,.],[.,[.,[.,.]]]],.],.]=>[[[[.,.],.],[.,[.,[.,.]]]],.]=>[1,2,6,5,4,3,7]=>{{1},{2},{3,6},{4,5},{7}} [[[[.,.],[.,[[.,.],.]]],.],.]=>[[[[.,.],.],[.,[[.,.],.]]],.]=>[1,2,5,6,4,3,7]=>{{1},{2},{3,4,5,6},{7}} [[[[.,.],[[.,.],[.,.]]],.],.]=>[[[[.,.],.],[[.,[.,.]],.]],.]=>[1,2,5,4,6,3,7]=>{{1},{2},{3,5,6},{4},{7}} [[[[.,.],[[.,[.,.]],.]],.],.]=>[[[[.,.],.],[[.,.],[.,.]]],.]=>[1,2,4,6,5,3,7]=>{{1},{2},{3,4,6},{5},{7}} [[[[.,.],[[[.,.],.],.]],.],.]=>[[[[.,.],.],[[[.,.],.],.]],.]=>[1,2,4,5,6,3,7]=>{{1},{2},{3,4,5,6},{7}} [[[[.,[.,.]],[.,[.,.]]],.],.]=>[[[[.,.],.],[.,[.,.]]],[.,.]]=>[1,2,5,4,3,7,6]=>{{1},{2},{3,5},{4},{6,7}} [[[[.,[.,.]],[[.,.],.]],.],.]=>[[[[.,.],.],[[.,.],.]],[.,.]]=>[1,2,4,5,3,7,6]=>{{1},{2},{3,4,5},{6,7}} [[[[[.,.],.],[.,[.,.]]],.],.]=>[[[[[.,.],.],[.,[.,.]]],.],.]=>[1,2,5,4,3,6,7]=>{{1},{2},{3,5},{4},{6},{7}} [[[[[.,.],.],[[.,.],.]],.],.]=>[[[[[.,.],.],[[.,.],.]],.],.]=>[1,2,4,5,3,6,7]=>{{1},{2},{3,4,5},{6},{7}} [[[[.,[.,[.,.]]],[.,.]],.],.]=>[[[[.,.],.],[.,.]],[.,[.,.]]]=>[1,2,4,3,7,6,5]=>{{1},{2},{3,4},{5,7},{6}} [[[[.,[[.,.],.]],[.,.]],.],.]=>[[[[.,.],.],[.,.]],[[.,.],.]]=>[1,2,4,3,6,7,5]=>{{1},{2},{3,4},{5,6,7}} [[[[[.,.],[.,.]],[.,.]],.],.]=>[[[[[.,.],.],[.,.]],[.,.]],.]=>[1,2,4,3,6,5,7]=>{{1},{2},{3,4},{5,6},{7}} [[[[[.,[.,.]],.],[.,.]],.],.]=>[[[[[.,.],.],[.,.]],.],[.,.]]=>[1,2,4,3,5,7,6]=>{{1},{2},{3,4},{5},{6,7}} [[[[[[.,.],.],.],[.,.]],.],.]=>[[[[[[.,.],.],[.,.]],.],.],.]=>[1,2,4,3,5,6,7]=>{{1},{2},{3,4},{5},{6},{7}} [[[[.,[.,[.,[.,.]]]],.],.],.]=>[[[[.,.],.],.],[.,[.,[.,.]]]]=>[1,2,3,7,6,5,4]=>{{1},{2},{3},{4,7},{5,6}} [[[[.,[.,[[.,.],.]]],.],.],.]=>[[[[.,.],.],.],[.,[[.,.],.]]]=>[1,2,3,6,7,5,4]=>{{1},{2},{3},{4,5,6,7}} [[[[.,[[.,.],[.,.]]],.],.],.]=>[[[[.,.],.],.],[[.,[.,.]],.]]=>[1,2,3,6,5,7,4]=>{{1},{2},{3},{4,6,7},{5}} [[[[.,[[.,[.,.]],.]],.],.],.]=>[[[[.,.],.],.],[[.,.],[.,.]]]=>[1,2,3,5,7,6,4]=>{{1},{2},{3},{4,5,7},{6}} [[[[.,[[[.,.],.],.]],.],.],.]=>[[[[.,.],.],.],[[[.,.],.],.]]=>[1,2,3,5,6,7,4]=>{{1},{2},{3},{4,5,6,7}} [[[[[.,.],[.,[.,.]]],.],.],.]=>[[[[[.,.],.],.],[.,[.,.]]],.]=>[1,2,3,6,5,4,7]=>{{1},{2},{3},{4,6},{5},{7}} [[[[[.,.],[[.,.],.]],.],.],.]=>[[[[[.,.],.],.],[[.,.],.]],.]=>[1,2,3,5,6,4,7]=>{{1},{2},{3},{4,5,6},{7}} [[[[[.,[.,.]],[.,.]],.],.],.]=>[[[[[.,.],.],.],[.,.]],[.,.]]=>[1,2,3,5,4,7,6]=>{{1},{2},{3},{4,5},{6,7}} [[[[[[.,.],.],[.,.]],.],.],.]=>[[[[[[.,.],.],.],[.,.]],.],.]=>[1,2,3,5,4,6,7]=>{{1},{2},{3},{4,5},{6},{7}} [[[[[.,[.,[.,.]]],.],.],.],.]=>[[[[[.,.],.],.],.],[.,[.,.]]]=>[1,2,3,4,7,6,5]=>{{1},{2},{3},{4},{5,7},{6}} [[[[[.,[[.,.],.]],.],.],.],.]=>[[[[[.,.],.],.],.],[[.,.],.]]=>[1,2,3,4,6,7,5]=>{{1},{2},{3},{4},{5,6,7}} [[[[[[.,.],[.,.]],.],.],.],.]=>[[[[[[.,.],.],.],.],[.,.]],.]=>[1,2,3,4,6,5,7]=>{{1},{2},{3},{4},{5,6},{7}} [[[[[[.,[.,.]],.],.],.],.],.]=>[[[[[[.,.],.],.],.],.],[.,.]]=>[1,2,3,4,5,7,6]=>{{1},{2},{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]=>{{1,8},{2,7},{3,6},{4,5}} [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]=>[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]=>[6,5,4,3,2,1,8,7]=>{{1,6},{2,5},{3,4},{7,8}} [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]=>[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]=>[4,3,2,1,8,7,6,5]=>{{1,4},{2,3},{5,8},{6,7}} [[.,[.,[[.,.],.]]],[.,[[.,.],.]]]=>[[.,[.,[[.,.],.]]],[.,[[.,.],.]]]=>[3,4,2,1,7,8,6,5]=>{{1,2,3,4},{5,6,7,8}} [[.,[[.,.],[.,.]]],[[.,[.,.]],.]]=>[[.,[[.,.],[.,.]]],[[.,[.,.]],.]]=>[2,4,3,1,7,6,8,5]=>{{1,2,4},{3},{5,7,8},{6}} [[.,[[.,[.,.]],.]],[[.,.],[.,.]]]=>[[.,[[.,[.,.]],.]],[[.,.],[.,.]]]=>[3,2,4,1,6,8,7,5]=>{{1,3,4},{2},{5,6,8},{7}} [[.,[[[.,.],.],.]],[[[.,.],.],.]]=>[[.,[[[.,.],.],.]],[[[.,.],.],.]]=>[2,3,4,1,6,7,8,5]=>{{1,2,3,4},{5,6,7,8}} [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]=>[[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]=>[4,3,2,1,6,5,8,7]=>{{1,4},{2,3},{5,6},{7,8}} [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]=>[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]=>[3,2,1,4,5,8,7,6]=>{{1,3},{2},{4},{5},{6,8},{7}} [[[[.,[[.,.],.]],.],.],[[.,.],.]]=>[[[[.,[[.,.],.]],.],.],[[.,.],.]]=>[2,3,1,4,5,7,8,6]=>{{1,2,3},{4},{5},{6,7,8}} [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]=>[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]=>[2,1,8,7,6,5,4,3]=>{{1,2},{3,8},{4,7},{5,6}} [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]=>[[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]=>[2,1,6,5,4,3,8,7]=>{{1,2},{3,6},{4,5},{7,8}} [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]=>[[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]=>[2,1,4,3,8,7,6,5]=>{{1,2},{3,4},{5,8},{6,7}} [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]=>[[[[.,[.,.]],[.,.]],[.,.]],[.,.]]=>[2,1,4,3,6,5,8,7]=>{{1,2},{3,4},{5,6},{7,8}} [[[[[[.,[.,.]],.],.],.],.],[.,.]]=>[[[[[[.,[.,.]],.],.],.],.],[.,.]]=>[2,1,3,4,5,6,8,7]=>{{1,2},{3},{4},{5},{6},{7,8}} [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]=>[[[[.,.],[.,[.,.]]],[.,[.,.]]],.]=>[1,4,3,2,7,6,5,8]=>{{1},{2,4},{3},{5,7},{6},{8}} [[[[.,.],[[.,.],.]],[[.,.],.]],.]=>[[[[.,.],[[.,.],.]],[[.,.],.]],.]=>[1,3,4,2,6,7,5,8]=>{{1},{2,3,4},{5,6,7},{8}} [[[[[[.,.],[.,.]],.],.],[.,.]],.]=>[[[[[[.,.],[.,.]],.],.],[.,.]],.]=>[1,3,2,4,5,7,6,8]=>{{1},{2,3},{4},{5},{6,7},{8}} [[[[[[.,.],.],[.,.]],[.,.]],.],.]=>[[[[[[.,.],.],[.,.]],[.,.]],.],.]=>[1,2,4,3,6,5,7,8]=>{{1},{2},{3,4},{5,6},{7},{8}} [[[[[[[[.,.],.],.],.],.],.],.],.]=>[[[[[[[[.,.],.],.],.],.],.],.],.]=>[1,2,3,4,5,6,7,8]=>{{1},{2},{3},{4},{5},{6},{7},{8}}
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
to cycle type
Description
Let $\pi=c_1\dots c_r$ a permutation of size $n$ decomposed in its cyclic parts. The associated set partition of $[n]$ then is $S=S_1\cup\dots\cup S_r$ such that $S_i$ is the set of integers in the cycle $c_i$.
A permutation is cyclic [1] if and only if its cycle type is a hook partition [2].