Identifier
Mp00018: left border symmetryBinary trees
Mp00017: to 312-avoiding permutationPermutations
Mp00240: Permutations weak exceedance partition
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]=>{{1,3},{2,4}} [.,[[.,.],[.,.]]]=>[.,[[.,[.,.]],.]]=>[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]=>{{1,4},{2,5},{3}} [.,[.,[[.,.],[.,.]]]]=>[.,[.,[[.,[.,.]],.]]]=>[4,3,5,2,1]=>{{1,4},{2,3,5}} [.,[.,[[.,[.,.]],.]]]=>[.,[.,[[.,.],[.,.]]]]=>[3,5,4,2,1]=>{{1,3,4},{2,5}} [.,[.,[[[.,.],.],.]]]=>[.,[.,[[[.,.],.],.]]]=>[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]=>{{1,3},{2,4,5}} [.,[[.,[.,.]],[.,.]]]=>[.,[[.,[.,.]],[.,.]]]=>[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]=>{{1,2,4},{3,5}} [.,[[[.,.],[.,.]],.]]=>[.,[[[.,.],[.,.]],.]]=>[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]=>{{1,3},{2,4},{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},{2,4},{3,5}} [[.,[[.,.],[.,.]]],.]=>[[.,.],[[.,[.,.]],.]]=>[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]=>{{1,5},{2,6},{3,4}} [.,[.,[.,[[.,.],[.,.]]]]]=>[.,[.,[.,[[.,[.,.]],.]]]]=>[5,4,6,3,2,1]=>{{1,5},{2,4},{3,6}} [.,[.,[.,[[.,[.,.]],.]]]]=>[.,[.,[.,[[.,.],[.,.]]]]]=>[4,6,5,3,2,1]=>{{1,4},{2,6},{3,5}} [.,[.,[.,[[[.,.],.],.]]]]=>[.,[.,[.,[[[.,.],.],.]]]]=>[4,5,6,3,2,1]=>{{1,4},{2,5},{3,6}} [.,[.,[[.,.],[.,[.,.]]]]]=>[.,[.,[[.,[.,[.,.]]],.]]]=>[5,4,3,6,2,1]=>{{1,5},{2,4,6},{3}} [.,[.,[[.,.],[[.,.],.]]]]=>[.,[.,[[.,[[.,.],.]],.]]]=>[4,5,3,6,2,1]=>{{1,4,6},{2,5},{3}} [.,[.,[[.,[.,.]],[.,.]]]]=>[.,[.,[[.,[.,.]],[.,.]]]]=>[4,3,6,5,2,1]=>{{1,4,5},{2,3,6}} [.,[.,[[[.,.],.],[.,.]]]]=>[.,[.,[[[.,[.,.]],.],.]]]=>[4,3,5,6,2,1]=>{{1,4,6},{2,3,5}} [.,[.,[[.,[.,[.,.]]],.]]]=>[.,[.,[[.,.],[.,[.,.]]]]]=>[3,6,5,4,2,1]=>{{1,3,5},{2,6},{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]=>{{1,3,5},{2,4,6}} [.,[[.,.],[.,[.,[.,.]]]]]=>[.,[[.,[.,[.,[.,.]]]],.]]=>[5,4,3,2,6,1]=>{{1,5,6},{2,4},{3}} [.,[[.,.],[.,[[.,.],.]]]]=>[.,[[.,[.,[[.,.],.]]],.]]=>[4,5,3,2,6,1]=>{{1,4},{2,5,6},{3}} [.,[[.,.],[[.,.],[.,.]]]]=>[.,[[.,[[.,[.,.]],.]],.]]=>[4,3,5,2,6,1]=>{{1,4},{2,3,5,6}} [.,[[.,.],[[.,[.,.]],.]]]=>[.,[[.,[[.,.],[.,.]]],.]]=>[3,5,4,2,6,1]=>{{1,3,4},{2,5,6}} [.,[[.,.],[[[.,.],.],.]]]=>[.,[[.,[[[.,.],.],.]],.]]=>[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]=>{{1,3},{2,4,6},{5}} [.,[[[.,.],.],[.,[.,.]]]]=>[.,[[[.,[.,[.,.]]],.],.]]=>[4,3,2,5,6,1]=>{{1,4,5,6},{2,3}} [.,[[[.,.],.],[[.,.],.]]]=>[.,[[[.,[[.,.],.]],.],.]]=>[3,4,2,5,6,1]=>{{1,3},{2,4,5,6}} [.,[[.,[.,[.,.]]],[.,.]]]=>[.,[[.,[.,.]],[.,[.,.]]]]=>[3,2,6,5,4,1]=>{{1,3,6},{2},{4,5}} [.,[[.,[[.,.],.]],[.,.]]]=>[.,[[.,[.,.]],[[.,.],.]]]=>[3,2,5,6,4,1]=>{{1,3,5},{2},{4,6}} [.,[[[.,.],[.,.]],[.,.]]]=>[.,[[[.,[.,.]],[.,.]],.]]=>[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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[[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]],.]=>[[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]]=>[1,8,9,7,6,5,4,3,2]=>{{1},{2,8},{3,9},{4,7},{5,6}} [[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]]=>[[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]],.]=>[7,8,6,5,4,3,2,1,9]=>{{1,7},{2,8},{3,6},{4,5},{9}} [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]=>[[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.]=>[9,8,7,6,5,4,3,2,1,10]=>{{1,9},{2,8},{3,7},{4,6},{5},{10}} [[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]],.]=>[[.,.],[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]=>[1,11,10,9,8,7,6,5,4,3,2]=>{{1},{2,11},{3,10},{4,9},{5,8},{6,7}} [[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]],.]=>[[.,.],[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]=>[1,9,10,8,7,6,5,4,3,2]=>{{1},{2,9},{3,10},{4,8},{5,7},{6}} [[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]],.]=>[[.,.],[.,[.,[.,[.,[[.,[.,.]],.]]]]]]=>[1,8,7,9,6,5,4,3,2]=>{{1},{2,8},{3,7},{4,9},{5,6}} [[.,.],[.,[.,[.,[.,[[.,.],[.,.]]]]]]]=>[[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]],.]=>[7,6,8,5,4,3,2,1,9]=>{{1,7},{2,6},{3,8},{4,5},{9}} [[.,.],[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]=>[[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]],.]=>[8,9,7,6,5,4,3,2,1,10]=>{{1,8},{2,9},{3,7},{4,6},{5},{10}} [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]=>[[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]],.]=>[10,9,8,7,6,5,4,3,2,1,11]=>{{1,10},{2,9},{3,8},{4,7},{5,6},{11}} [[[.,.],.],[.,[.,[.,[.,[.,[.,.]]]]]]]=>[[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],.]=>[7,6,5,4,3,2,1,8,9]=>{{1,7},{2,6},{3,5},{4},{8},{9}} [[[[[[[.,[.,[.,.]]],.],.],.],.],.],.]=>[[[[[[[.,.],.],.],.],.],.],[.,[.,.]]]=>[1,2,3,4,5,6,9,8,7]=>{{1},{2},{3},{4},{5},{6},{7,9},{8}} [[[[[[[[.,[.,.]],.],.],.],.],.],.],.]=>[[[[[[[[.,.],.],.],.],.],.],.],[.,.]]=>[1,2,3,4,5,6,7,9,8]=>{{1},{2},{3},{4},{5},{6},{7},{8,9}} [[[[[[[[[.,[.,.]],.],.],.],.],.],.],.],.]=>[[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]]=>[1,2,3,4,5,6,7,8,10,9]=>{{1},{2},{3},{4},{5},{6},{7},{8},{9,10}} [.,[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]]=>[.,[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]]=>[8,7,6,9,5,4,3,2,1]=>{{1,8},{2,7},{3,6},{4,9},{5}} [.,[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]]=>[.,[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]]=>[8,7,6,5,9,4,3,2,1]=>{{1,8},{2,7},{3,6},{4,5,9}} [.,[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]]=>[.,[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]]=>[8,7,6,5,4,9,3,2,1]=>{{1,8},{2,7},{3,6,9},{4,5}} [.,[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]]=>[.,[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]]=>[8,7,6,5,4,3,9,2,1]=>{{1,8},{2,7,9},{3,6},{4,5}} [.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]=>[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]=>[8,7,6,5,4,3,2,9,1]=>{{1,8,9},{2,7},{3,6},{4,5}} [.,[.,[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]]]=>[.,[.,[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]]]=>[9,8,10,7,6,5,4,3,2,1]=>{{1,9},{2,8},{3,10},{4,7},{5,6}} [.,[.,[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]]]=>[.,[.,[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]]]=>[9,8,7,10,6,5,4,3,2,1]=>{{1,9},{2,8},{3,7},{4,10},{5,6}} [.,[.,[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]]]=>[.,[.,[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]]]=>[9,8,7,6,10,5,4,3,2,1]=>{{1,9},{2,8},{3,7},{4,6},{5,10}} [.,[.,[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]]]=>[.,[.,[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]]]=>[9,8,7,6,5,10,4,3,2,1]=>{{1,9},{2,8},{3,7},{4,6,10},{5}} [.,[.,[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]]]=>[.,[.,[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]]]=>[9,8,7,6,5,4,10,3,2,1]=>{{1,9},{2,8},{3,7,10},{4,6},{5}} [.,[.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]]=>[.,[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]]=>[9,8,7,6,5,4,3,10,2,1]=>{{1,9},{2,8,10},{3,7},{4,6},{5}} [.,[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]=>[.,[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]]=>[9,8,7,6,5,4,3,2,10,1]=>{{1,9,10},{2,8},{3,7},{4,6},{5}} [.,[[[[[[[.,.],.],.],.],.],.],[.,.]]]=>[.,[[[[[[[.,[.,.]],.],.],.],.],.],.]]=>[3,2,4,5,6,7,8,9,1]=>{{1,3,4,5,6,7,8,9},{2}} [[[[[[[[[[.,[.,.]],.],.],.],.],.],.],.],.],.]=>[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]=>[1,2,3,4,5,6,7,8,9,11,10]=>{{1},{2},{3},{4},{5},{6},{7},{8},{9},{10,11}} [.,[[[[[[.,.],.],.],.],.],[[.,.],.]]]=>[.,[[[[[[.,[[.,.],.]],.],.],.],.],.]]=>[3,4,2,5,6,7,8,9,1]=>{{1,3},{2,4,5,6,7,8,9}} [.,[[[[[[[[.,.],.],.],.],.],.],.],[.,.]]]=>[.,[[[[[[[[.,[.,.]],.],.],.],.],.],.],.]]=>[3,2,4,5,6,7,8,9,10,1]=>{{1,3,4,5,6,7,8,9,10},{2}} [[.,[[[[[[[.,.],.],.],.],.],.],.]],.]=>[[.,.],[[[[[[[.,.],.],.],.],.],.],.]]=>[1,3,4,5,6,7,8,9,2]=>{{1},{2,3,4,5,6,7,8,9}} [[[[[[[.,[[.,.],.]],.],.],.],.],.],.]=>[[[[[[[.,.],.],.],.],.],.],[[.,.],.]]=>[1,2,3,4,5,6,8,9,7]=>{{1},{2},{3},{4},{5},{6},{7,8,9}} [[[[[[.,[[[.,.],.],.]],.],.],.],.],.]=>[[[[[[.,.],.],.],.],.],[[[.,.],.],.]]=>[1,2,3,4,5,7,8,9,6]=>{{1},{2},{3},{4},{5},{6,7,8,9}} [[[[[.,[[[[.,.],.],.],.]],.],.],.],.]=>[[[[[.,.],.],.],.],[[[[.,.],.],.],.]]=>[1,2,3,4,6,7,8,9,5]=>{{1},{2},{3},{4},{5,6,7,8,9}} [[[[.,[[[[[.,.],.],.],.],.]],.],.],.]=>[[[[.,.],.],.],[[[[[.,.],.],.],.],.]]=>[1,2,3,5,6,7,8,9,4]=>{{1},{2},{3},{4,5,6,7,8,9}} [[[.,[[[[[[.,.],.],.],.],.],.]],.],.]=>[[[.,.],.],[[[[[[.,.],.],.],.],.],.]]=>[1,2,4,5,6,7,8,9,3]=>{{1},{2},{3,4,5,6,7,8,9}} [[[[[[[[.,[[.,.],.]],.],.],.],.],.],.],.]=>[[[[[[[[.,.],.],.],.],.],.],.],[[.,.],.]]=>[1,2,3,4,5,6,7,9,10,8]=>{{1},{2},{3},{4},{5},{6},{7},{8,9,10}} [[[[[[.,[[.,.],[.,.]]],.],.],.],.],.]=>[[[[[[.,.],.],.],.],.],[[.,[.,.]],.]]=>[1,2,3,4,5,8,7,9,6]=>{{1},{2},{3},{4},{5},{6,8,9},{7}} [[[[[[[.,[[[.,.],.],.]],.],.],.],.],.],.]=>[[[[[[[.,.],.],.],.],.],.],[[[.,.],.],.]]=>[1,2,3,4,5,6,8,9,10,7]=>{{1},{2},{3},{4},{5},{6},{7,8,9,10}} [[[[[.,[[[.,.],.],[.,.]]],.],.],.],.]=>[[[[[.,.],.],.],.],[[[.,[.,.]],.],.]]=>[1,2,3,4,7,6,8,9,5]=>{{1},{2},{3},{4},{5,7,8,9},{6}} [[[[[[.,[[[[.,.],.],.],.]],.],.],.],.],.]=>[[[[[[.,.],.],.],.],.],[[[[.,.],.],.],.]]=>[1,2,3,4,5,7,8,9,10,6]=>{{1},{2},{3},{4},{5},{6,7,8,9,10}} [[[[.,[[[[.,.],.],.],[.,.]]],.],.],.]=>[[[[.,.],.],.],[[[[.,[.,.]],.],.],.]]=>[1,2,3,6,5,7,8,9,4]=>{{1},{2},{3},{4,6,7,8,9},{5}} [[[[[.,[[[[[.,.],.],.],.],.]],.],.],.],.]=>[[[[[.,.],.],.],.],[[[[[.,.],.],.],.],.]]=>[1,2,3,4,6,7,8,9,10,5]=>{{1},{2},{3},{4},{5,6,7,8,9,10}} [[[.,[[[[[.,.],.],.],.],[.,.]]],.],.]=>[[[.,.],.],[[[[[.,[.,.]],.],.],.],.]]=>[1,2,5,4,6,7,8,9,3]=>{{1},{2},{3,5,6,7,8,9},{4}} [[[[.,[[[[[[.,.],.],.],.],.],.]],.],.],.]=>[[[[.,.],.],.],[[[[[[.,.],.],.],.],.],.]]=>[1,2,3,5,6,7,8,9,10,4]=>{{1},{2},{3},{4,5,6,7,8,9,10}} [[.,[[[[[[.,.],.],.],.],.],[.,.]]],.]=>[[.,.],[[[[[[.,[.,.]],.],.],.],.],.]]=>[1,4,3,5,6,7,8,9,2]=>{{1},{2,4,5,6,7,8,9},{3}} [[[.,[[[[[[[.,.],.],.],.],.],.],.]],.],.]=>[[[.,.],.],[[[[[[[.,.],.],.],.],.],.],.]]=>[1,2,4,5,6,7,8,9,10,3]=>{{1},{2},{3,4,5,6,7,8,9,10}} [[.,[[[[[[[[.,.],.],.],.],.],.],.],.]],.]=>[[.,.],[[[[[[[[.,.],.],.],.],.],.],.],.]]=>[1,3,4,5,6,7,8,9,10,2]=>{{1},{2,3,4,5,6,7,8,9,10}} [[[.,.],.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]=>[[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.],.]=>[8,7,6,5,4,3,2,1,9,10]=>{{1,8},{2,7},{3,6},{4,5},{9},{10}} [[[[[[[[.,.],.],.],.],.],.],.],[.,[.,.]]]=>[[[[[[[[.,[.,[.,.]]],.],.],.],.],.],.],.]=>[3,2,1,4,5,6,7,8,9,10]=>{{1,3},{2},{4},{5},{6},{7},{8},{9},{10}} [[[[[[.,.],.],.],.],.],[.,[.,[.,.]]]]=>[[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.]=>[4,3,2,1,5,6,7,8,9]=>{{1,4},{2,3},{5},{6},{7},{8},{9}} [[[[[.,.],.],.],.],[.,[.,[.,[.,.]]]]]=>[[[[[.,[.,[.,[.,[.,.]]]]],.],.],.],.]=>[5,4,3,2,1,6,7,8,9]=>{{1,5},{2,4},{3},{6},{7},{8},{9}} [[[[.,.],.],.],[.,[.,[.,[.,[.,.]]]]]]=>[[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.]=>[6,5,4,3,2,1,7,8,9]=>{{1,6},{2,5},{3,4},{7},{8},{9}} [[[[[[[.,.],.],.],.],.],.],[.,[.,[.,.]]]]=>[[[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.],.]=>[4,3,2,1,5,6,7,8,9,10]=>{{1,4},{2,3},{5},{6},{7},{8},{9},{10}} [[[[[[.,.],.],.],.],.],[.,[.,[.,[.,.]]]]]=>[[[[[[.,[.,[.,[.,[.,.]]]]],.],.],.],.],.]=>[5,4,3,2,1,6,7,8,9,10]=>{{1,5},{2,4},{3},{6},{7},{8},{9},{10}} [[[[[.,.],.],.],.],[.,[.,[.,[.,[.,.]]]]]]=>[[[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.],.]=>[6,5,4,3,2,1,7,8,9,10]=>{{1,6},{2,5},{3,4},{7},{8},{9},{10}} [[[[.,.],.],.],[.,[.,[.,[.,[.,[.,.]]]]]]]=>[[[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],.],.]=>[7,6,5,4,3,2,1,8,9,10]=>{{1,7},{2,6},{3,5},{4},{8},{9},{10}}
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
weak exceedance partition
Description
The set partition induced by the weak exceedances of a permutation.
This is the coarsest set partition that contains all arcs $(i, \pi(i))$ with $i\leq\pi(i)$.