************************************************************************ * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2013 The FindStatCrew * * * * This information is distributed in the hope that it will be * * useful, but WITHOUT ANY WARRANTY; without even the implied * * warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. * ************************************************************************ ------------------------------------------------------------------------ Map identifier: Mp00328 ------------------------------------------------------------------------ Map name: DeBruijn-Morselt plane tree automorphism ------------------------------------------------------------------------ Domain: Ordered trees ------------------------------------------------------------------------ Codomain: Ordered trees ------------------------------------------------------------------------ Description: This automorphism on the set of plane trees with a given number of vertices is the combination of the two "canonical" bijections between plane trees and binary trees, using either the left or right branches. In [3] Shapiro essentially attributes this automorphism to DeBruijn and Morselt [1], hence the name I chose. Shapiro shows in that paper that for a large subset of plane trees (those corresponding to compositions), the sixth power of the automorphism acts as the identity on this subset. However, its behavior in general is quite chaotic. In [2] Donaghey studies this automorphism further and shows that its behavior on all trees can be reduced to its behavior on certain "primitive" trees. ------------------------------------------------------------------------ References: [1] de Bruijn, N. G. and Morselt, B. J. M. "A note on plane trees." J. Combinatorial Theory 2 (1967), 27–34. [[https://mathscinet.ams.org/mathscinet/article?mr=205872|MR0205872]] [2] Donaghey, Robert. "Automorphisms on Catalan trees and bracketings." J. Combin. Theory Ser. B 29 (1980), no. 1, 75–90. [[https://mathscinet.ams.org/mathscinet/article?mr=584162|MR0584162]] [3] Shapiro, Louis W. "The cycle of six." Fibonacci Quart. 17 (1979), no. 3, 253–259. [[https://mathscinet.ams.org/mathscinet/article?mr=549808|MR0549808]] ------------------------------------------------------------------------ Code: def mapping(T): return T.to_binary_tree_right_branch().to_ordered_tree_left_branch() ------------------------------------------------------------------------ Map images: [] => [] [[]] => [[]] [[],[]] => [[[]]] [[[]]] => [[],[]] [[],[],[]] => [[[[]]]] [[],[[]]] => [[[],[]]] [[[]],[]] => [[],[[]]] [[[],[]]] => [[[]],[]] [[[[]]]] => [[],[],[]] [[],[],[],[]] => [[[[[]]]]] [[],[],[[]]] => [[[[],[]]]] [[],[[]],[]] => [[[],[[]]]] [[],[[],[]]] => [[[[]],[]]] [[],[[[]]]] => [[[],[],[]]] [[[]],[],[]] => [[],[[[]]]] [[[]],[[]]] => [[],[[],[]]] [[[],[]],[]] => [[[]],[[]]] [[[[]]],[]] => [[],[],[[]]] [[[],[],[]]] => [[[[]]],[]] [[[],[[]]]] => [[[],[]],[]] [[[[]],[]]] => [[],[[]],[]] [[[[],[]]]] => [[[]],[],[]] [[[[[]]]]] => [[],[],[],[]] [[],[],[],[],[]] => [[[[[[]]]]]] [[],[],[],[[]]] => [[[[[],[]]]]] [[],[],[[]],[]] => [[[[],[[]]]]] [[],[],[[],[]]] 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[[[[]],[]],[[[]]]] => [[],[[]],[[],[],[]]] [[[[],[]]],[[],[]]] => [[[]],[],[[[]],[]]] [[[[],[]]],[[[]]]] => [[[]],[],[[],[],[]]] [[[[[]]]],[[],[]]] => [[],[],[],[[[]],[]]] [[[[[]]]],[[[]]]] => [[],[],[],[[],[],[]]] [[[],[],[],[]],[],[]] => [[[[[]]]],[[[]]]] [[[],[],[[]]],[],[]] => [[[[],[]]],[[[]]]] [[[],[[]],[]],[],[]] => [[[],[[]]],[[[]]]] [[[],[[],[]]],[],[]] => [[[[]],[]],[[[]]]] [[[],[[[]]]],[],[]] => [[[],[],[]],[[[]]]] [[[[]],[],[]],[],[]] => [[],[[[]]],[[[]]]] [[[[]],[[]]],[],[]] => [[],[[],[]],[[[]]]] [[[[],[]],[]],[],[]] => [[[]],[[]],[[[]]]] [[[[[]]],[]],[],[]] => [[],[],[[]],[[[]]]] [[[[],[],[]]],[],[]] => [[[[]]],[],[[[]]]] [[[[],[[]]]],[],[]] => [[[],[]],[],[[[]]]] [[[[[]],[]]],[],[]] => [[],[[]],[],[[[]]]] [[[[[],[]]]],[],[]] => [[[]],[],[],[[[]]]] [[[[[[]]]]],[],[]] => [[],[],[],[],[[[]]]] [[[],[],[],[]],[[]]] => [[[[[]]]],[[],[]]] [[[],[],[[]]],[[]]] => [[[[],[]]],[[],[]]] [[[],[[]],[]],[[]]] => [[[],[[]]],[[],[]]] [[[],[[],[]]],[[]]] => [[[[]],[]],[[],[]]] [[[],[[[]]]],[[]]] => [[[],[],[]],[[],[]]] [[[[]],[],[]],[[]]] => [[],[[[]]],[[],[]]] [[[[]],[[]]],[[]]] => [[],[[],[]],[[],[]]] [[[[],[]],[]],[[]]] => [[[]],[[]],[[],[]]] [[[[[]]],[]],[[]]] => [[],[],[[]],[[],[]]] [[[[],[],[]]],[[]]] => [[[[]]],[],[[],[]]] [[[[],[[]]]],[[]]] => [[[],[]],[],[[],[]]] [[[[[]],[]]],[[]]] => [[],[[]],[],[[],[]]] [[[[[],[]]]],[[]]] => [[[]],[],[],[[],[]]] [[[[[[]]]]],[[]]] => [[],[],[],[],[[],[]]] [[[],[],[],[],[]],[]] => [[[[[[]]]]],[[]]] [[[],[],[],[[]]],[]] => [[[[[],[]]]],[[]]] [[[],[],[[]],[]],[]] => [[[[],[[]]]],[[]]] [[[],[],[[],[]]],[]] => [[[[[]],[]]],[[]]] [[[],[],[[[]]]],[]] => [[[[],[],[]]],[[]]] [[[],[[]],[],[]],[]] => [[[],[[[]]]],[[]]] [[[],[[]],[[]]],[]] => [[[],[[],[]]],[[]]] [[[],[[],[]],[]],[]] => [[[[]],[[]]],[[]]] [[[],[[[]]],[]],[]] => [[[],[],[[]]],[[]]] [[[],[[],[],[]]],[]] => [[[[[]]],[]],[[]]] [[[],[[],[[]]]],[]] => [[[[],[]],[]],[[]]] [[[],[[[]],[]]],[]] => [[[],[[]],[]],[[]]] [[[],[[[],[]]]],[]] => [[[[]],[],[]],[[]]] 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[[[[[[]]],[]]],[]] => [[],[],[[]],[],[[]]] [[[[[],[],[]]]],[]] => [[[[]]],[],[],[[]]] [[[[[],[[]]]]],[]] => [[[],[]],[],[],[[]]] [[[[[[]],[]]]],[]] => [[],[[]],[],[],[[]]] [[[[[[],[]]]]],[]] => [[[]],[],[],[],[[]]] [[[[[[[]]]]]],[]] => [[],[],[],[],[],[[]]] [[[],[],[],[],[],[]]] => [[[[[[[]]]]]],[]] [[[],[],[],[],[[]]]] => [[[[[[],[]]]]],[]] [[[],[],[],[[]],[]]] => [[[[[],[[]]]]],[]] [[[],[],[],[[],[]]]] => [[[[[[]],[]]]],[]] [[[],[],[],[[[]]]]] => [[[[[],[],[]]]],[]] [[[],[],[[]],[],[]]] => [[[[],[[[]]]]],[]] [[[],[],[[]],[[]]]] => [[[[],[[],[]]]],[]] [[[],[],[[],[]],[]]] => [[[[[]],[[]]]],[]] [[[],[],[[[]]],[]]] => [[[[],[],[[]]]],[]] [[[],[],[[],[],[]]]] => [[[[[[]]],[]]],[]] [[[],[],[[],[[]]]]] => [[[[[],[]],[]]],[]] [[[],[],[[[]],[]]]] => [[[[],[[]],[]]],[]] [[[],[],[[[],[]]]]] => [[[[[]],[],[]]],[]] [[[],[],[[[[]]]]]] => [[[[],[],[],[]]],[]] [[[],[[]],[],[],[]]] => [[[],[[[[]]]]],[]] [[[],[[]],[],[[]]]] => [[[],[[[],[]]]],[]] [[[],[[]],[[]],[]]] => [[[],[[],[[]]]],[]] [[[],[[]],[[],[]]]] => [[[],[[[]],[]]],[]] [[[],[[]],[[[]]]]] => [[[],[[],[],[]]],[]] [[[],[[],[]],[],[]]] => [[[[]],[[[]]]],[]] [[[],[[[]]],[],[]]] => [[[],[],[[[]]]],[]] [[[],[[],[]],[[]]]] => [[[[]],[[],[]]],[]] [[[],[[[]]],[[]]]] => [[[],[],[[],[]]],[]] [[[],[[],[],[]],[]]] => [[[[[]]],[[]]],[]] [[[],[[],[[]]],[]]] => [[[[],[]],[[]]],[]] [[[],[[[]],[]],[]]] => [[[],[[]],[[]]],[]] [[[],[[[],[]]],[]]] => [[[[]],[],[[]]],[]] [[[],[[[[]]]],[]]] => [[[],[],[],[[]]],[]] [[[],[[],[],[],[]]]] => [[[[[[]]]],[]],[]] [[[],[[],[],[[]]]]] => [[[[[],[]]],[]],[]] [[[],[[],[[]],[]]]] => [[[[],[[]]],[]],[]] [[[],[[],[[],[]]]]] => [[[[[]],[]],[]],[]] [[[],[[],[[[]]]]]] => [[[[],[],[]],[]],[]] [[[],[[[]],[],[]]]] => [[[],[[[]]],[]],[]] [[[],[[[]],[[]]]]] => [[[],[[],[]],[]],[]] [[[],[[[],[]],[]]]] => [[[[]],[[]],[]],[]] [[[],[[[[]]],[]]]] => [[[],[],[[]],[]],[]] [[[],[[[],[],[]]]]] => [[[[[]]],[],[]],[]] [[[],[[[],[[]]]]]] => [[[[],[]],[],[]],[]] [[[],[[[[]],[]]]]] => [[[],[[]],[],[]],[]] [[[],[[[[],[]]]]]] => [[[[]],[],[],[]],[]] [[[],[[[[[]]]]]]] => [[[],[],[],[],[]],[]] [[[[]],[],[],[],[]]] => [[],[[[[[]]]]],[]] [[[[]],[],[],[[]]]] => [[],[[[[],[]]]],[]] [[[[]],[],[[]],[]]] => [[],[[[],[[]]]],[]] [[[[]],[],[[],[]]]] => [[],[[[[]],[]]],[]] [[[[]],[],[[[]]]]] => [[],[[[],[],[]]],[]] [[[[]],[[]],[],[]]] => [[],[[],[[[]]]],[]] [[[[]],[[]],[[]]]] => [[],[[],[[],[]]],[]] [[[[]],[[],[]],[]]] => [[],[[[]],[[]]],[]] [[[[]],[[[]]],[]]] => [[],[[],[],[[]]],[]] [[[[]],[[],[],[]]]] => [[],[[[[]]],[]],[]] [[[[]],[[],[[]]]]] => [[],[[[],[]],[]],[]] [[[[]],[[[]],[]]]] => [[],[[],[[]],[]],[]] [[[[]],[[[],[]]]]] => [[],[[[]],[],[]],[]] [[[[]],[[[[]]]]]] => [[],[[],[],[],[]],[]] [[[[],[]],[],[],[]]] => [[[]],[[[[]]]],[]] [[[[[]]],[],[],[]]] => [[],[],[[[[]]]],[]] [[[[],[]],[],[[]]]] => [[[]],[[[],[]]],[]] [[[[[]]],[],[[]]]] => [[],[],[[[],[]]],[]] [[[[],[]],[[]],[]]] => [[[]],[[],[[]]],[]] [[[[[]]],[[]],[]]] => [[],[],[[],[[]]],[]] [[[[],[]],[[],[]]]] => [[[]],[[[]],[]],[]] [[[[],[]],[[[]]]]] => [[[]],[[],[],[]],[]] [[[[[]]],[[],[]]]] => [[],[],[[[]],[]],[]] [[[[[]]],[[[]]]]] => [[],[],[[],[],[]],[]] [[[[],[],[]],[],[]]] => [[[[]]],[[[]]],[]] [[[[],[[]]],[],[]]] => [[[],[]],[[[]]],[]] [[[[[]],[]],[],[]]] => [[],[[]],[[[]]],[]] [[[[[],[]]],[],[]]] => [[[]],[],[[[]]],[]] [[[[[[]]]],[],[]]] => [[],[],[],[[[]]],[]] [[[[],[],[]],[[]]]] => [[[[]]],[[],[]],[]] [[[[],[[]]],[[]]]] => [[[],[]],[[],[]],[]] [[[[[]],[]],[[]]]] => [[],[[]],[[],[]],[]] [[[[[],[]]],[[]]]] => [[[]],[],[[],[]],[]] [[[[[[]]]],[[]]]] => [[],[],[],[[],[]],[]] [[[[],[],[],[]],[]]] => [[[[[]]]],[[]],[]] [[[[],[],[[]]],[]]] => [[[[],[]]],[[]],[]] [[[[],[[]],[]],[]]] => [[[],[[]]],[[]],[]] [[[[],[[],[]]],[]]] => [[[[]],[]],[[]],[]] [[[[],[[[]]]],[]]] => [[[],[],[]],[[]],[]] [[[[[]],[],[]],[]]] => [[],[[[]]],[[]],[]] [[[[[]],[[]]],[]]] => [[],[[],[]],[[]],[]] [[[[[],[]],[]],[]]] => [[[]],[[]],[[]],[]] [[[[[[]]],[]],[]]] => [[],[],[[]],[[]],[]] [[[[[],[],[]]],[]]] => [[[[]]],[],[[]],[]] [[[[[],[[]]]],[]]] => [[[],[]],[],[[]],[]] [[[[[[]],[]]],[]]] => [[],[[]],[],[[]],[]] [[[[[[],[]]]],[]]] => [[[]],[],[],[[]],[]] [[[[[[[]]]]],[]]] => [[],[],[],[],[[]],[]] [[[[],[],[],[],[]]]] => [[[[[[]]]]],[],[]] [[[[],[],[],[[]]]]] => [[[[[],[]]]],[],[]] [[[[],[],[[]],[]]]] => [[[[],[[]]]],[],[]] [[[[],[],[[],[]]]]] => [[[[[]],[]]],[],[]] [[[[],[],[[[]]]]]] => [[[[],[],[]]],[],[]] [[[[],[[]],[],[]]]] => [[[],[[[]]]],[],[]] [[[[],[[]],[[]]]]] => [[[],[[],[]]],[],[]] [[[[],[[],[]],[]]]] => [[[[]],[[]]],[],[]] [[[[],[[[]]],[]]]] => [[[],[],[[]]],[],[]] [[[[],[[],[],[]]]]] => [[[[[]]],[]],[],[]] [[[[],[[],[[]]]]]] => [[[[],[]],[]],[],[]] [[[[],[[[]],[]]]]] => [[[],[[]],[]],[],[]] [[[[],[[[],[]]]]]] => [[[[]],[],[]],[],[]] [[[[],[[[[]]]]]]] => [[[],[],[],[]],[],[]] [[[[[]],[],[],[]]]] => [[],[[[[]]]],[],[]] [[[[[]],[],[[]]]]] => [[],[[[],[]]],[],[]] [[[[[]],[[]],[]]]] => [[],[[],[[]]],[],[]] [[[[[]],[[],[]]]]] => [[],[[[]],[]],[],[]] [[[[[]],[[[]]]]]] => [[],[[],[],[]],[],[]] [[[[[],[]],[],[]]]] => [[[]],[[[]]],[],[]] [[[[[[]]],[],[]]]] => [[],[],[[[]]],[],[]] [[[[[],[]],[[]]]]] => [[[]],[[],[]],[],[]] [[[[[[]]],[[]]]]] => [[],[],[[],[]],[],[]] [[[[[],[],[]],[]]]] => [[[[]]],[[]],[],[]] [[[[[],[[]]],[]]]] => [[[],[]],[[]],[],[]] [[[[[[]],[]],[]]]] => [[],[[]],[[]],[],[]] [[[[[[],[]]],[]]]] => [[[]],[],[[]],[],[]] [[[[[[[]]]],[]]]] => [[],[],[],[[]],[],[]] [[[[[],[],[],[]]]]] => [[[[[]]]],[],[],[]] [[[[[],[],[[]]]]]] => [[[[],[]]],[],[],[]] [[[[[],[[]],[]]]]] => [[[],[[]]],[],[],[]] [[[[[],[[],[]]]]]] => [[[[]],[]],[],[],[]] [[[[[],[[[]]]]]]] => [[[],[],[]],[],[],[]] [[[[[[]],[],[]]]]] => [[],[[[]]],[],[],[]] [[[[[[]],[[]]]]]] => [[],[[],[]],[],[],[]] [[[[[[],[]],[]]]]] => [[[]],[[]],[],[],[]] [[[[[[[]]],[]]]]] => [[],[],[[]],[],[],[]] [[[[[[],[],[]]]]]] => [[[[]]],[],[],[],[]] [[[[[[],[[]]]]]]] => [[[],[]],[],[],[],[]] [[[[[[[]],[]]]]]] => [[],[[]],[],[],[],[]] [[[[[[[],[]]]]]]] => [[[]],[],[],[],[],[]] [[[[[[[[]]]]]]]] => [[],[],[],[],[],[],[]] [[],[],[],[],[],[],[],[]] => [[[[[[[[[]]]]]]]]] [[],[],[],[],[],[],[[]]] => [[[[[[[[],[]]]]]]]] [[],[],[],[],[],[[]],[]] => [[[[[[[],[[]]]]]]]] [[],[],[],[],[],[[],[]]] => [[[[[[[[]],[]]]]]]] [[],[],[],[],[],[[[]]]] => [[[[[[[],[],[]]]]]]] [[],[],[],[],[[]],[],[]] => [[[[[[],[[[]]]]]]]] [[],[],[],[],[[]],[[]]] => [[[[[[],[[],[]]]]]]] [[],[],[],[],[[],[]],[]] => [[[[[[[]],[[]]]]]]] [[],[],[],[],[[[]]],[]] => [[[[[[],[],[[]]]]]]] [[],[],[],[],[[],[],[]]] => [[[[[[[[]]],[]]]]]] [[],[],[],[],[[],[[]]]] => [[[[[[[],[]],[]]]]]] [[],[],[],[],[[[]],[]]] => [[[[[[],[[]],[]]]]]] [[],[],[],[],[[[],[]]]] => [[[[[[[]],[],[]]]]]] [[],[],[],[],[[[[]]]]] => [[[[[[],[],[],[]]]]]] [[],[],[],[[]],[],[],[]] => [[[[[],[[[[]]]]]]]] [[],[],[],[[]],[],[[]]] => [[[[[],[[[],[]]]]]]] [[],[],[],[[]],[[]],[]] => [[[[[],[[],[[]]]]]]] [[],[],[],[[]],[[],[]]] => [[[[[],[[[]],[]]]]]] [[],[],[],[[]],[[[]]]] => [[[[[],[[],[],[]]]]]] [[],[],[],[[],[]],[],[]] => [[[[[[]],[[[]]]]]]] [[],[],[],[[[]]],[],[]] => [[[[[],[],[[[]]]]]]] [[],[],[],[[],[]],[[]]] => [[[[[[]],[[],[]]]]]] [[],[],[],[[[]]],[[]]] => [[[[[],[],[[],[]]]]]] [[],[],[],[[],[],[]],[]] => [[[[[[[]]],[[]]]]]] [[],[],[],[[],[[]]],[]] => [[[[[[],[]],[[]]]]]] [[],[],[],[[[]],[]],[]] => [[[[[],[[]],[[]]]]]] [[],[],[],[[[],[]]],[]] => [[[[[[]],[],[[]]]]]] [[],[],[],[[[[]]]],[]] => [[[[[],[],[],[[]]]]]] [[],[],[],[[],[],[],[]]] => [[[[[[[[]]]],[]]]]] [[],[],[],[[],[],[[]]]] => [[[[[[[],[]]],[]]]]] [[],[],[],[[],[[]],[]]] => [[[[[[],[[]]],[]]]]] [[],[],[],[[],[[],[]]]] => [[[[[[[]],[]],[]]]]] [[],[],[],[[],[[[]]]]] => [[[[[[],[],[]],[]]]]] [[],[],[],[[[]],[],[]]] => [[[[[],[[[]]],[]]]]] [[],[],[],[[[]],[[]]]] => [[[[[],[[],[]],[]]]]] [[],[],[],[[[],[]],[]]] => [[[[[[]],[[]],[]]]]] [[],[],[],[[[[]]],[]]] => [[[[[],[],[[]],[]]]]] [[],[],[],[[[],[],[]]]] => [[[[[[[]]],[],[]]]]] [[],[],[],[[[],[[]]]]] => [[[[[[],[]],[],[]]]]] [[],[],[],[[[[]],[]]]] => [[[[[],[[]],[],[]]]]] [[],[],[],[[[[],[]]]]] => [[[[[[]],[],[],[]]]]] [[],[],[],[[[[[]]]]]] => [[[[[],[],[],[],[]]]]] [[],[],[[]],[],[],[],[]] => [[[[],[[[[[]]]]]]]] [[],[],[[]],[],[],[[]]] => [[[[],[[[[],[]]]]]]] [[],[],[[]],[],[[]],[]] => [[[[],[[[],[[]]]]]]] [[],[],[[]],[],[[],[]]] => [[[[],[[[[]],[]]]]]] [[],[],[[]],[],[[[]]]] => [[[[],[[[],[],[]]]]]] [[],[],[[]],[[]],[],[]] => [[[[],[[],[[[]]]]]]] [[],[],[[]],[[]],[[]]] => [[[[],[[],[[],[]]]]]] [[],[],[[]],[[],[]],[]] => [[[[],[[[]],[[]]]]]] [[],[],[[]],[[[]]],[]] => [[[[],[[],[],[[]]]]]] [[],[],[[]],[[],[],[]]] => [[[[],[[[[]]],[]]]]] [[],[],[[]],[[],[[]]]] => [[[[],[[[],[]],[]]]]] [[],[],[[]],[[[]],[]]] => [[[[],[[],[[]],[]]]]] [[],[],[[]],[[[],[]]]] => [[[[],[[[]],[],[]]]]] [[],[],[[]],[[[[]]]]] => [[[[],[[],[],[],[]]]]] [[],[],[[],[]],[],[],[]] => [[[[[]],[[[[]]]]]]] [[],[],[[[]]],[],[],[]] => [[[[],[],[[[[]]]]]]] [[],[],[[],[]],[],[[]]] => [[[[[]],[[[],[]]]]]] [[],[],[[[]]],[],[[]]] => [[[[],[],[[[],[]]]]]] [[],[],[[],[]],[[]],[]] => [[[[[]],[[],[[]]]]]] [[],[],[[[]]],[[]],[]] => 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[[],[],[[[]],[[]]],[]] => [[[[],[[],[]],[[]]]]] [[],[],[[[],[]],[]],[]] => [[[[[]],[[]],[[]]]]] [[],[],[[[[]]],[]],[]] => [[[[],[],[[]],[[]]]]] [[],[],[[[],[],[]]],[]] => [[[[[[]]],[],[[]]]]] [[],[],[[[],[[]]]],[]] => [[[[[],[]],[],[[]]]]] [[],[],[[[[]],[]]],[]] => [[[[],[[]],[],[[]]]]] [[],[],[[[[],[]]]],[]] => [[[[[]],[],[],[[]]]]] [[],[],[[[[[]]]]],[]] => [[[[],[],[],[],[[]]]]] [[],[],[[],[],[],[],[]]] => [[[[[[[[]]]]],[]]]] [[],[],[[],[],[],[[]]]] => [[[[[[[],[]]]],[]]]] [[],[],[[],[],[[]],[]]] => [[[[[[],[[]]]],[]]]] [[],[],[[],[],[[],[]]]] => [[[[[[[]],[]]],[]]]] [[],[],[[],[],[[[]]]]] => [[[[[[],[],[]]],[]]]] [[],[],[[],[[]],[],[]]] => [[[[[],[[[]]]],[]]]] [[],[],[[],[[]],[[]]]] => [[[[[],[[],[]]],[]]]] [[],[],[[],[[],[]],[]]] => [[[[[[]],[[]]],[]]]] [[],[],[[],[[[]]],[]]] => [[[[[],[],[[]]],[]]]] [[],[],[[],[[],[],[]]]] => [[[[[[[]]],[]],[]]]] [[],[],[[],[[],[[]]]]] => [[[[[[],[]],[]],[]]]] [[],[],[[],[[[]],[]]]] => [[[[[],[[]],[]],[]]]] [[],[],[[],[[[],[]]]]] => [[[[[[]],[],[]],[]]]] [[],[],[[],[[[[]]]]]] => [[[[[],[],[],[]],[]]]] [[],[],[[[]],[],[],[]]] => [[[[],[[[[]]]],[]]]] [[],[],[[[]],[],[[]]]] => [[[[],[[[],[]]],[]]]] [[],[],[[[]],[[]],[]]] => [[[[],[[],[[]]],[]]]] [[],[],[[[]],[[],[]]]] => [[[[],[[[]],[]],[]]]] [[],[],[[[]],[[[]]]]] => [[[[],[[],[],[]],[]]]] [[],[],[[[],[]],[],[]]] => [[[[[]],[[[]]],[]]]] [[],[],[[[[]]],[],[]]] => [[[[],[],[[[]]],[]]]] [[],[],[[[],[]],[[]]]] => [[[[[]],[[],[]],[]]]] [[],[],[[[[]]],[[]]]] => [[[[],[],[[],[]],[]]]] [[],[],[[[],[],[]],[]]] => [[[[[[]]],[[]],[]]]] [[],[],[[[],[[]]],[]]] => [[[[[],[]],[[]],[]]]] [[],[],[[[[]],[]],[]]] => [[[[],[[]],[[]],[]]]] [[],[],[[[[],[]]],[]]] => [[[[[]],[],[[]],[]]]] [[],[],[[[[[]]]],[]]] => [[[[],[],[],[[]],[]]]] [[],[],[[[],[],[],[]]]] => [[[[[[[]]]],[],[]]]] [[],[],[[[],[],[[]]]]] => [[[[[[],[]]],[],[]]]] [[],[],[[[],[[]],[]]]] => [[[[[],[[]]],[],[]]]] [[],[],[[[],[[],[]]]]] => [[[[[[]],[]],[],[]]]] [[],[],[[[],[[[]]]]]] => [[[[[],[],[]],[],[]]]] [[],[],[[[[]],[],[]]]] => [[[[],[[[]]],[],[]]]] [[],[],[[[[]],[[]]]]] => [[[[],[[],[]],[],[]]]] [[],[],[[[[],[]],[]]]] => [[[[[]],[[]],[],[]]]] [[],[],[[[[[]]],[]]]] => [[[[],[],[[]],[],[]]]] [[],[],[[[[],[],[]]]]] => [[[[[[]]],[],[],[]]]] [[],[],[[[[],[[]]]]]] => [[[[[],[]],[],[],[]]]] [[],[],[[[[[]],[]]]]] => [[[[],[[]],[],[],[]]]] [[],[],[[[[[],[]]]]]] => [[[[[]],[],[],[],[]]]] [[],[],[[[[[[]]]]]]] => [[[[],[],[],[],[],[]]]] [[],[[]],[],[],[],[],[]] => [[[],[[[[[[]]]]]]]] [[],[[]],[],[],[],[[]]] => [[[],[[[[[],[]]]]]]] [[],[[]],[],[],[[]],[]] => [[[],[[[[],[[]]]]]]] [[],[[]],[],[],[[],[]]] => [[[],[[[[[]],[]]]]]] [[],[[]],[],[],[[[]]]] => [[[],[[[[],[],[]]]]]] [[],[[]],[],[[]],[],[]] => [[[],[[[],[[[]]]]]]] [[],[[]],[],[[]],[[]]] => [[[],[[[],[[],[]]]]]] [[],[[]],[],[[],[]],[]] => [[[],[[[[]],[[]]]]]] [[],[[]],[],[[[]]],[]] => [[[],[[[],[],[[]]]]]] [[],[[]],[],[[],[],[]]] => [[[],[[[[[]]],[]]]]] [[],[[]],[],[[],[[]]]] => [[[],[[[[],[]],[]]]]] [[],[[]],[],[[[]],[]]] => [[[],[[[],[[]],[]]]]] [[],[[]],[],[[[],[]]]] => [[[],[[[[]],[],[]]]]] [[],[[]],[],[[[[]]]]] => [[[],[[[],[],[],[]]]]] [[],[[]],[[]],[],[],[]] => [[[],[[],[[[[]]]]]]] [[],[[]],[[]],[],[[]]] => [[[],[[],[[[],[]]]]]] [[],[[]],[[]],[[]],[]] => [[[],[[],[[],[[]]]]]] [[],[[]],[[]],[[],[]]] => [[[],[[],[[[]],[]]]]] [[],[[]],[[]],[[[]]]] => [[[],[[],[[],[],[]]]]] [[],[[]],[[],[]],[],[]] => [[[],[[[]],[[[]]]]]] [[],[[]],[[[]]],[],[]] => [[[],[[],[],[[[]]]]]] [[],[[]],[[],[]],[[]]] => [[[],[[[]],[[],[]]]]] [[],[[]],[[[]]],[[]]] => [[[],[[],[],[[],[]]]]] [[],[[]],[[],[],[]],[]] => [[[],[[[[]]],[[]]]]] [[],[[]],[[],[[]]],[]] => [[[],[[[],[]],[[]]]]] [[],[[]],[[[]],[]],[]] => [[[],[[],[[]],[[]]]]] [[],[[]],[[[],[]]],[]] => [[[],[[[]],[],[[]]]]] [[],[[]],[[[[]]]],[]] => [[[],[[],[],[],[[]]]]] [[],[[]],[[],[],[],[]]] => [[[],[[[[[]]]],[]]]] [[],[[]],[[],[],[[]]]] => [[[],[[[[],[]]],[]]]] [[],[[]],[[],[[]],[]]] => [[[],[[[],[[]]],[]]]] [[],[[]],[[],[[],[]]]] => [[[],[[[[]],[]],[]]]] [[],[[]],[[],[[[]]]]] => [[[],[[[],[],[]],[]]]] [[],[[]],[[[]],[],[]]] => [[[],[[],[[[]]],[]]]] [[],[[]],[[[]],[[]]]] => [[[],[[],[[],[]],[]]]] [[],[[]],[[[],[]],[]]] => [[[],[[[]],[[]],[]]]] [[],[[]],[[[[]]],[]]] => [[[],[[],[],[[]],[]]]] [[],[[]],[[[],[],[]]]] => [[[],[[[[]]],[],[]]]] [[],[[]],[[[],[[]]]]] => [[[],[[[],[]],[],[]]]] [[],[[]],[[[[]],[]]]] => [[[],[[],[[]],[],[]]]] [[],[[]],[[[[],[]]]]] => [[[],[[[]],[],[],[]]]] [[],[[]],[[[[[]]]]]] => [[[],[[],[],[],[],[]]]] [[],[[],[]],[],[],[],[]] => [[[[]],[[[[[]]]]]]] [[],[[[]]],[],[],[],[]] => [[[],[],[[[[[]]]]]]] [[],[[],[]],[],[],[[]]] => [[[[]],[[[[],[]]]]]] [[],[[[]]],[],[],[[]]] => [[[],[],[[[[],[]]]]]] [[],[[],[]],[],[[]],[]] => [[[[]],[[[],[[]]]]]] [[],[[[]]],[],[[]],[]] => [[[],[],[[[],[[]]]]]] [[],[[],[]],[],[[],[]]] => [[[[]],[[[[]],[]]]]] [[],[[],[]],[],[[[]]]] => [[[[]],[[[],[],[]]]]] [[],[[[]]],[],[[],[]]] => [[[],[],[[[[]],[]]]]] [[],[[[]]],[],[[[]]]] => [[[],[],[[[],[],[]]]]] [[],[[],[]],[[]],[],[]] => 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[[],[[[],[]]],[],[],[]] => [[[[]],[],[[[[]]]]]] [[],[[[[]]]],[],[],[]] => [[[],[],[],[[[[]]]]]] [[],[[],[],[]],[],[[]]] => [[[[[]]],[[[],[]]]]] [[],[[],[[]]],[],[[]]] => [[[[],[]],[[[],[]]]]] [[],[[[]],[]],[],[[]]] => [[[],[[]],[[[],[]]]]] [[],[[[],[]]],[],[[]]] => [[[[]],[],[[[],[]]]]] [[],[[[[]]]],[],[[]]] => [[[],[],[],[[[],[]]]]] [[],[[],[],[]],[[]],[]] => [[[[[]]],[[],[[]]]]] [[],[[],[[]]],[[]],[]] => [[[[],[]],[[],[[]]]]] [[],[[[]],[]],[[]],[]] => [[[],[[]],[[],[[]]]]] [[],[[[],[]]],[[]],[]] => [[[[]],[],[[],[[]]]]] [[],[[[[]]]],[[]],[]] => [[[],[],[],[[],[[]]]]] [[],[[],[],[]],[[],[]]] => [[[[[]]],[[[]],[]]]] [[],[[],[],[]],[[[]]]] => [[[[[]]],[[],[],[]]]] [[],[[],[[]]],[[],[]]] => [[[[],[]],[[[]],[]]]] [[],[[],[[]]],[[[]]]] => [[[[],[]],[[],[],[]]]] [[],[[[]],[]],[[],[]]] => [[[],[[]],[[[]],[]]]] [[],[[[]],[]],[[[]]]] => [[[],[[]],[[],[],[]]]] [[],[[[],[]]],[[],[]]] => [[[[]],[],[[[]],[]]]] [[],[[[],[]]],[[[]]]] => [[[[]],[],[[],[],[]]]] [[],[[[[]]]],[[],[]]] => 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[[],[[[]],[],[]],[[]]] => [[[],[[[]]],[[],[]]]] [[],[[[]],[[]]],[[]]] => [[[],[[],[]],[[],[]]]] [[],[[[],[]],[]],[[]]] => [[[[]],[[]],[[],[]]]] [[],[[[[]]],[]],[[]]] => [[[],[],[[]],[[],[]]]] [[],[[[],[],[]]],[[]]] => [[[[[]]],[],[[],[]]]] [[],[[[],[[]]]],[[]]] => [[[[],[]],[],[[],[]]]] [[],[[[[]],[]]],[[]]] => [[[],[[]],[],[[],[]]]] [[],[[[[],[]]]],[[]]] => [[[[]],[],[],[[],[]]]] [[],[[[[[]]]]],[[]]] => [[[],[],[],[],[[],[]]]] [[],[[],[],[],[],[]],[]] => [[[[[[[]]]]],[[]]]] [[],[[],[],[],[[]]],[]] => [[[[[[],[]]]],[[]]]] [[],[[],[],[[]],[]],[]] => [[[[[],[[]]]],[[]]]] [[],[[],[],[[],[]]],[]] => [[[[[[]],[]]],[[]]]] [[],[[],[],[[[]]]],[]] => [[[[[],[],[]]],[[]]]] [[],[[],[[]],[],[]],[]] => [[[[],[[[]]]],[[]]]] [[],[[],[[]],[[]]],[]] => [[[[],[[],[]]],[[]]]] [[],[[],[[],[]],[]],[]] => [[[[[]],[[]]],[[]]]] [[],[[],[[[]]],[]],[]] => [[[[],[],[[]]],[[]]]] [[],[[],[[],[],[]]],[]] => [[[[[[]]],[]],[[]]]] [[],[[],[[],[[]]]],[]] => [[[[[],[]],[]],[[]]]] [[],[[],[[[]],[]]],[]] => [[[[],[[]],[]],[[]]]] [[],[[],[[[],[]]]],[]] => [[[[[]],[],[]],[[]]]] [[],[[],[[[[]]]]],[]] => [[[[],[],[],[]],[[]]]] [[],[[[]],[],[],[]],[]] => [[[],[[[[]]]],[[]]]] [[],[[[]],[],[[]]],[]] => [[[],[[[],[]]],[[]]]] [[],[[[]],[[]],[]],[]] => [[[],[[],[[]]],[[]]]] [[],[[[]],[[],[]]],[]] => [[[],[[[]],[]],[[]]]] [[],[[[]],[[[]]]],[]] => [[[],[[],[],[]],[[]]]] [[],[[[],[]],[],[]],[]] => [[[[]],[[[]]],[[]]]] [[],[[[[]]],[],[]],[]] => [[[],[],[[[]]],[[]]]] [[],[[[],[]],[[]]],[]] => [[[[]],[[],[]],[[]]]] [[],[[[[]]],[[]]],[]] => [[[],[],[[],[]],[[]]]] [[],[[[],[],[]],[]],[]] => [[[[[]]],[[]],[[]]]] [[],[[[],[[]]],[]],[]] => [[[[],[]],[[]],[[]]]] [[],[[[[]],[]],[]],[]] => [[[],[[]],[[]],[[]]]] [[],[[[[],[]]],[]],[]] => [[[[]],[],[[]],[[]]]] [[],[[[[[]]]],[]],[]] => [[[],[],[],[[]],[[]]]] [[],[[[],[],[],[]]],[]] => [[[[[[]]]],[],[[]]]] [[],[[[],[],[[]]]],[]] => [[[[[],[]]],[],[[]]]] [[],[[[],[[]],[]]],[]] => [[[[],[[]]],[],[[]]]] [[],[[[],[[],[]]]],[]] => [[[[[]],[]],[],[[]]]] [[],[[[],[[[]]]]],[]] => [[[[],[],[]],[],[[]]]] [[],[[[[]],[],[]]],[]] => [[[],[[[]]],[],[[]]]] [[],[[[[]],[[]]]],[]] => [[[],[[],[]],[],[[]]]] [[],[[[[],[]],[]]],[]] => [[[[]],[[]],[],[[]]]] [[],[[[[[]]],[]]],[]] => [[[],[],[[]],[],[[]]]] [[],[[[[],[],[]]]],[]] => [[[[[]]],[],[],[[]]]] [[],[[[[],[[]]]]],[]] => [[[[],[]],[],[],[[]]]] [[],[[[[[]],[]]]],[]] => [[[],[[]],[],[],[[]]]] [[],[[[[[],[]]]]],[]] => [[[[]],[],[],[],[[]]]] [[],[[[[[[]]]]]],[]] => [[[],[],[],[],[],[[]]]] [[],[[],[],[],[],[],[]]] => [[[[[[[[]]]]]],[]]] [[],[[],[],[],[],[[]]]] => [[[[[[[],[]]]]],[]]] [[],[[],[],[],[[]],[]]] => [[[[[[],[[]]]]],[]]] [[],[[],[],[],[[],[]]]] => [[[[[[[]],[]]]],[]]] [[],[[],[],[],[[[]]]]] => [[[[[[],[],[]]]],[]]] [[],[[],[],[[]],[],[]]] => [[[[[],[[[]]]]],[]]] [[],[[],[],[[]],[[]]]] => [[[[[],[[],[]]]],[]]] [[],[[],[],[[],[]],[]]] => [[[[[[]],[[]]]],[]]] [[],[[],[],[[[]]],[]]] => [[[[[],[],[[]]]],[]]] [[],[[],[],[[],[],[]]]] => [[[[[[[]]],[]]],[]]] [[],[[],[],[[],[[]]]]] => 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[[],[[],[[],[[],[]]]]] => [[[[[[]],[]],[]],[]]] [[],[[],[[],[[[]]]]]] => [[[[[],[],[]],[]],[]]] [[],[[],[[[]],[],[]]]] => [[[[],[[[]]],[]],[]]] [[],[[],[[[]],[[]]]]] => [[[[],[[],[]],[]],[]]] [[],[[],[[[],[]],[]]]] => [[[[[]],[[]],[]],[]]] [[],[[],[[[[]]],[]]]] => [[[[],[],[[]],[]],[]]] [[],[[],[[[],[],[]]]]] => [[[[[[]]],[],[]],[]]] [[],[[],[[[],[[]]]]]] => [[[[[],[]],[],[]],[]]] [[],[[],[[[[]],[]]]]] => [[[[],[[]],[],[]],[]]] [[],[[],[[[[],[]]]]]] => [[[[[]],[],[],[]],[]]] [[],[[],[[[[[]]]]]]] => [[[[],[],[],[],[]],[]]] [[],[[[]],[],[],[],[]]] => [[[],[[[[[]]]]],[]]] [[],[[[]],[],[],[[]]]] => [[[],[[[[],[]]]],[]]] [[],[[[]],[],[[]],[]]] => [[[],[[[],[[]]]],[]]] [[],[[[]],[],[[],[]]]] => [[[],[[[[]],[]]],[]]] [[],[[[]],[],[[[]]]]] => [[[],[[[],[],[]]],[]]] [[],[[[]],[[]],[],[]]] => [[[],[[],[[[]]]],[]]] [[],[[[]],[[]],[[]]]] => [[[],[[],[[],[]]],[]]] [[],[[[]],[[],[]],[]]] => [[[],[[[]],[[]]],[]]] [[],[[[]],[[[]]],[]]] => [[[],[[],[],[[]]],[]]] [[],[[[]],[[],[],[]]]] => 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[[[]],[[],[]],[[[]]]] => [[],[[[]],[[],[],[]]]] [[[]],[[[]]],[[],[]]] => [[],[[],[],[[[]],[]]]] [[[]],[[[]]],[[[]]]] => [[],[[],[],[[],[],[]]]] [[[]],[[],[],[]],[],[]] => [[],[[[[]]],[[[]]]]] [[[]],[[],[[]]],[],[]] => [[],[[[],[]],[[[]]]]] [[[]],[[[]],[]],[],[]] => [[],[[],[[]],[[[]]]]] [[[]],[[[],[]]],[],[]] => [[],[[[]],[],[[[]]]]] [[[]],[[[[]]]],[],[]] => [[],[[],[],[],[[[]]]]] [[[]],[[],[],[]],[[]]] => [[],[[[[]]],[[],[]]]] [[[]],[[],[[]]],[[]]] => [[],[[[],[]],[[],[]]]] [[[]],[[[]],[]],[[]]] => [[],[[],[[]],[[],[]]]] [[[]],[[[],[]]],[[]]] => [[],[[[]],[],[[],[]]]] [[[]],[[[[]]]],[[]]] => [[],[[],[],[],[[],[]]]] [[[]],[[],[],[],[]],[]] => [[],[[[[[]]]],[[]]]] [[[]],[[],[],[[]]],[]] => [[],[[[[],[]]],[[]]]] [[[]],[[],[[]],[]],[]] => [[],[[[],[[]]],[[]]]] [[[]],[[],[[],[]]],[]] => [[],[[[[]],[]],[[]]]] [[[]],[[],[[[]]]],[]] => [[],[[[],[],[]],[[]]]] [[[]],[[[]],[],[]],[]] => [[],[[],[[[]]],[[]]]] [[[]],[[[]],[[]]],[]] => [[],[[],[[],[]],[[]]]] [[[]],[[[],[]],[]],[]] => 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[[],[[],[[],[]],[],[]]] [[[]],[[[[],[]],[]]]] => [[],[[[]],[[]],[],[]]] [[[]],[[[[[]]],[]]]] => [[],[[],[],[[]],[],[]]] [[[]],[[[[],[],[]]]]] => [[],[[[[]]],[],[],[]]] [[[]],[[[[],[[]]]]]] => [[],[[[],[]],[],[],[]]] [[[]],[[[[[]],[]]]]] => [[],[[],[[]],[],[],[]]] [[[]],[[[[[],[]]]]]] => [[],[[[]],[],[],[],[]]] [[[]],[[[[[[]]]]]]] => [[],[[],[],[],[],[],[]]] [[[],[]],[],[],[],[],[]] => [[[]],[[[[[[]]]]]]] [[[[]]],[],[],[],[],[]] => [[],[],[[[[[[]]]]]]] [[[],[]],[],[],[],[[]]] => [[[]],[[[[[],[]]]]]] [[[[]]],[],[],[],[[]]] => [[],[],[[[[[],[]]]]]] [[[],[]],[],[],[[]],[]] => [[[]],[[[[],[[]]]]]] [[[[]]],[],[],[[]],[]] => [[],[],[[[[],[[]]]]]] [[[],[]],[],[],[[],[]]] => [[[]],[[[[[]],[]]]]] [[[],[]],[],[],[[[]]]] => [[[]],[[[[],[],[]]]]] [[[[]]],[],[],[[],[]]] => [[],[],[[[[[]],[]]]]] [[[[]]],[],[],[[[]]]] => [[],[],[[[[],[],[]]]]] [[[],[]],[],[[]],[],[]] => [[[]],[[[],[[[]]]]]] [[[[]]],[],[[]],[],[]] => [[],[],[[[],[[[]]]]]] [[[],[]],[],[[]],[[]]] => [[[]],[[[],[[],[]]]]] ----------------------------------------------------------------------------- Created: Jul 01, 2024 at 02:15 by Sam Hopkins ----------------------------------------------------------------------------- Last Updated: Jul 01, 2024 at 02:15 by Sam Hopkins