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Identifier

Mp00061: Permutations —to increasing tree⟶ Binary trees

Images

[] => .

[1] => [.,.]

[1,2] => [.,[.,.]]

[2,1] => [[.,.],.]

[1,2,3] => [.,[.,[.,.]]]

[1,3,2] => [.,[[.,.],.]]

[2,1,3] => [[.,.],[.,.]]

[2,3,1] => [[.,[.,.]],.]

[3,1,2] => [[.,.],[.,.]]

[3,2,1] => [[[.,.],.],.]

[1,2,3,4] => [.,[.,[.,[.,.]]]]

[1,2,4,3] => [.,[.,[[.,.],.]]]

[1,3,2,4] => [.,[[.,.],[.,.]]]

[1,3,4,2] => [.,[[.,[.,.]],.]]

[1,4,2,3] => [.,[[.,.],[.,.]]]

[1,4,3,2] => [.,[[[.,.],.],.]]

[2,1,3,4] => [[.,.],[.,[.,.]]]

[2,1,4,3] => [[.,.],[[.,.],.]]

[2,3,1,4] => [[.,[.,.]],[.,.]]

[2,3,4,1] => [[.,[.,[.,.]]],.]

[2,4,1,3] => [[.,[.,.]],[.,.]]

[2,4,3,1] => [[.,[[.,.],.]],.]

[3,1,2,4] => [[.,.],[.,[.,.]]]

[3,1,4,2] => [[.,.],[[.,.],.]]

[3,2,1,4] => [[[.,.],.],[.,.]]

[3,2,4,1] => [[[.,.],[.,.]],.]

[3,4,1,2] => [[.,[.,.]],[.,.]]

[3,4,2,1] => [[[.,[.,.]],.],.]

[4,1,2,3] => [[.,.],[.,[.,.]]]

[4,1,3,2] => [[.,.],[[.,.],.]]

[4,2,1,3] => [[[.,.],.],[.,.]]

[4,2,3,1] => [[[.,.],[.,.]],.]

[4,3,1,2] => [[[.,.],.],[.,.]]

[4,3,2,1] => [[[[.,.],.],.],.]

[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]

[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]

[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]

[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]

[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]

[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]

[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]

[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]

[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]

[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]

[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]

[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]

[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]

[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]

[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]

[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]

[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]

[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]

[1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]

[1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]

[1,5,3,2,4] => [.,[[[.,.],.],[.,.]]]

[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]

[1,5,4,2,3] => [.,[[[.,.],.],[.,.]]]

[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]

[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]

[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]

[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]

[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]

[2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]

[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]

[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]

[2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]

[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]

[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]

[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]

[2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]

[2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]

[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]

[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]

[2,4,3,5,1] => [[.,[[.,.],[.,.]]],.]

[2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]

[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.]

[2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]

[2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]

[2,5,3,1,4] => [[.,[[.,.],.]],[.,.]]

[2,5,3,4,1] => [[.,[[.,.],[.,.]]],.]

[2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]

[2,5,4,3,1] => [[.,[[[.,.],.],.]],.]

[3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]

[3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]

[3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]

[3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]

[3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]

[3,1,5,4,2] => [[.,.],[[[.,.],.],.]]

[3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]

[3,2,1,5,4] => [[[.,.],.],[[.,.],.]]

[3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]

[3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]

[3,2,5,1,4] => [[[.,.],[.,.]],[.,.]]

[3,2,5,4,1] => [[[.,.],[[.,.],.]],.]

[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]

[3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]

[3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]

[3,4,2,5,1] => [[[.,[.,.]],[.,.]],.]

[3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]

[3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]

[3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]

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Description

Sends a permutation to its associated increasing tree.
This tree is recursively obtained by sending the unique permutation of length

$0$ to the empty tree, and sending a permutation

$\sigma$ of length

$n \geq 1$ to a root node with two subtrees

$L$ and

$R$ by splitting

$\sigma$ at the index

$\sigma^{-1}(1)$ , normalizing both sides again to permutations and sending the permutations on the left and on the right of

$\sigma^{-1}(1)$ to the trees

$L$ and

$R$ , respectively.

Properties

surjective, graded

Sage code

def mapping(sigma):
    return sigma.increasing_tree_shape()

Weight

Created

Jan 26, 2020 at 12:32 by FindStatCrew

Updated

Jan 26, 2020 at 12:32 by Martin Rubey