- FindStat

Identifier

Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St001554: Binary trees ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 12881 entries. <<<

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$F_{1} = q$

$F_{2} = 2\ q^{2}$

$F_{3} = 2\ q^{2} + 4\ q^{3}$

$F_{4} = 16\ q^{3} + 8\ q^{4}$

$F_{5} = 28\ q^{3} + 76\ q^{4} + 16\ q^{5}$

$F_{6} = 376\ q^{4} + 312\ q^{5} + 32\ q^{6}$

$F_{7} = 80\ q^{3} + 776\ q^{4} + 2904\ q^{5} + 1216\ q^{6} + 64\ q^{7}$

Description

The number of distinct nonempty subtrees of a binary tree.

Map

binary search tree: left to right

Description

Return the shape of the binary search tree of the permutation as a non labelled binary tree.