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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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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 $T_1, T_2, T_3$.
Properties
bijective, graded, involutive
Sage code
def mapping(elt): return elt.left_border_symmetry()
Weight
11
Created
Jan 19, 2020 at 07:28 by FindStatCrew
Updated
Jan 19, 2020 at 07:28 by Martin Rubey
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