- FindStat

Identifier

Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
St000920: Dyck paths ⟶ ℤ

Values

[.,.] => [1,0] => 1

[.,[.,.]] => [1,0,1,0] => 1

[[.,.],.] => [1,1,0,0] => 1

[.,[.,[.,.]]] => [1,0,1,0,1,0] => 1

[.,[[.,.],.]] => [1,0,1,1,0,0] => 1

[[.,.],[.,.]] => [1,1,1,0,0,0] => 2

[[.,[.,.]],.] => [1,1,0,0,1,0] => 1

[[[.,.],.],.] => [1,1,0,1,0,0] => 1

[.,[.,[.,[.,.]]]] => [1,0,1,0,1,0,1,0] => 1

[.,[.,[[.,.],.]]] => [1,0,1,0,1,1,0,0] => 1

[.,[[.,.],[.,.]]] => [1,0,1,1,1,0,0,0] => 2

[.,[[.,[.,.]],.]] => [1,0,1,1,0,0,1,0] => 1

[.,[[[.,.],.],.]] => [1,0,1,1,0,1,0,0] => 1

[[.,.],[.,[.,.]]] => [1,1,1,0,0,0,1,0] => 2

[[.,.],[[.,.],.]] => [1,1,1,0,0,1,0,0] => 2

[[.,[.,.]],[.,.]] => [1,1,1,0,1,0,0,0] => 2

[[[.,.],.],[.,.]] => [1,1,1,1,0,0,0,0] => 2

[[.,[.,[.,.]]],.] => [1,1,0,0,1,0,1,0] => 1

[[.,[[.,.],.]],.] => [1,1,0,0,1,1,0,0] => 1

[[[.,.],[.,.]],.] => [1,1,0,1,1,0,0,0] => 2

[[[.,[.,.]],.],.] => [1,1,0,1,0,0,1,0] => 1

[[[[.,.],.],.],.] => [1,1,0,1,0,1,0,0] => 1

[.,[.,[.,[.,[.,.]]]]] => [1,0,1,0,1,0,1,0,1,0] => 1

[.,[.,[.,[[.,.],.]]]] => [1,0,1,0,1,0,1,1,0,0] => 1

[.,[.,[[.,.],[.,.]]]] => [1,0,1,0,1,1,1,0,0,0] => 2

[.,[.,[[.,[.,.]],.]]] => [1,0,1,0,1,1,0,0,1,0] => 1

[.,[.,[[[.,.],.],.]]] => [1,0,1,0,1,1,0,1,0,0] => 1

[.,[[.,.],[.,[.,.]]]] => [1,0,1,1,1,0,0,0,1,0] => 2

[.,[[.,.],[[.,.],.]]] => [1,0,1,1,1,0,0,1,0,0] => 2

[.,[[.,[.,.]],[.,.]]] => [1,0,1,1,1,0,1,0,0,0] => 2

[.,[[[.,.],.],[.,.]]] => [1,0,1,1,1,1,0,0,0,0] => 2

[.,[[.,[.,[.,.]]],.]] => [1,0,1,1,0,0,1,0,1,0] => 1

[.,[[.,[[.,.],.]],.]] => [1,0,1,1,0,0,1,1,0,0] => 1

[.,[[[.,.],[.,.]],.]] => [1,0,1,1,0,1,1,0,0,0] => 2

[.,[[[.,[.,.]],.],.]] => [1,0,1,1,0,1,0,0,1,0] => 1

[.,[[[[.,.],.],.],.]] => [1,0,1,1,0,1,0,1,0,0] => 1

[[.,.],[.,[.,[.,.]]]] => [1,1,1,0,0,0,1,0,1,0] => 2

[[.,.],[.,[[.,.],.]]] => [1,1,1,0,0,0,1,1,0,0] => 2

[[.,.],[[.,.],[.,.]]] => [1,1,1,0,0,1,1,0,0,0] => 2

[[.,.],[[.,[.,.]],.]] => [1,1,1,0,0,1,0,0,1,0] => 2

[[.,.],[[[.,.],.],.]] => [1,1,1,0,0,1,0,1,0,0] => 2

[[.,[.,.]],[.,[.,.]]] => [1,1,1,0,1,0,0,0,1,0] => 2

[[.,[.,.]],[[.,.],.]] => [1,1,1,0,1,0,0,1,0,0] => 2

[[[.,.],.],[.,[.,.]]] => [1,1,1,1,0,0,0,0,1,0] => 2

[[[.,.],.],[[.,.],.]] => [1,1,1,1,0,0,0,1,0,0] => 2

[[.,[.,[.,.]]],[.,.]] => [1,1,1,0,1,0,1,0,0,0] => 2

[[.,[[.,.],.]],[.,.]] => [1,1,1,0,1,1,0,0,0,0] => 2

[[[.,.],[.,.]],[.,.]] => [1,1,1,1,1,0,0,0,0,0] => 2

[[[.,[.,.]],.],[.,.]] => [1,1,1,1,0,0,1,0,0,0] => 2

[[[[.,.],.],.],[.,.]] => [1,1,1,1,0,1,0,0,0,0] => 2

[[.,[.,[.,[.,.]]]],.] => [1,1,0,0,1,0,1,0,1,0] => 1

[[.,[.,[[.,.],.]]],.] => [1,1,0,0,1,0,1,1,0,0] => 1

[[.,[[.,.],[.,.]]],.] => [1,1,0,0,1,1,1,0,0,0] => 2

[[.,[[.,[.,.]],.]],.] => [1,1,0,0,1,1,0,0,1,0] => 1

[[.,[[[.,.],.],.]],.] => [1,1,0,0,1,1,0,1,0,0] => 1

[[[.,.],[.,[.,.]]],.] => [1,1,0,1,1,0,0,0,1,0] => 2

[[[.,.],[[.,.],.]],.] => [1,1,0,1,1,0,0,1,0,0] => 2

[[[.,[.,.]],[.,.]],.] => [1,1,0,1,1,0,1,0,0,0] => 2

[[[[.,.],.],[.,.]],.] => [1,1,0,1,1,1,0,0,0,0] => 2

[[[.,[.,[.,.]]],.],.] => [1,1,0,1,0,0,1,0,1,0] => 1

[[[.,[[.,.],.]],.],.] => [1,1,0,1,0,0,1,1,0,0] => 1

[[[[.,.],[.,.]],.],.] => [1,1,0,1,0,1,1,0,0,0] => 2

[[[[.,[.,.]],.],.],.] => [1,1,0,1,0,1,0,0,1,0] => 1

[[[[[.,.],.],.],.],.] => [1,1,0,1,0,1,0,1,0,0] => 1

[.,[.,[.,[.,[.,[.,.]]]]]] => [1,0,1,0,1,0,1,0,1,0,1,0] => 1

[.,[.,[.,[.,[[.,.],.]]]]] => [1,0,1,0,1,0,1,0,1,1,0,0] => 1

[.,[.,[.,[[.,.],[.,.]]]]] => [1,0,1,0,1,0,1,1,1,0,0,0] => 2

[.,[.,[.,[[.,[.,.]],.]]]] => [1,0,1,0,1,0,1,1,0,0,1,0] => 1

[.,[.,[.,[[[.,.],.],.]]]] => [1,0,1,0,1,0,1,1,0,1,0,0] => 1

[.,[.,[[.,.],[.,[.,.]]]]] => [1,0,1,0,1,1,1,0,0,0,1,0] => 2

[.,[.,[[.,.],[[.,.],.]]]] => [1,0,1,0,1,1,1,0,0,1,0,0] => 2

[.,[.,[[.,[.,.]],[.,.]]]] => [1,0,1,0,1,1,1,0,1,0,0,0] => 2

[.,[.,[[[.,.],.],[.,.]]]] => [1,0,1,0,1,1,1,1,0,0,0,0] => 2

[.,[.,[[.,[.,[.,.]]],.]]] => [1,0,1,0,1,1,0,0,1,0,1,0] => 1

[.,[.,[[.,[[.,.],.]],.]]] => [1,0,1,0,1,1,0,0,1,1,0,0] => 1

[.,[.,[[[.,.],[.,.]],.]]] => [1,0,1,0,1,1,0,1,1,0,0,0] => 2

[.,[.,[[[.,[.,.]],.],.]]] => [1,0,1,0,1,1,0,1,0,0,1,0] => 1

[.,[.,[[[[.,.],.],.],.]]] => [1,0,1,0,1,1,0,1,0,1,0,0] => 1

[.,[[.,.],[.,[.,[.,.]]]]] => [1,0,1,1,1,0,0,0,1,0,1,0] => 2

[.,[[.,.],[.,[[.,.],.]]]] => [1,0,1,1,1,0,0,0,1,1,0,0] => 2

[.,[[.,.],[[.,.],[.,.]]]] => [1,0,1,1,1,0,0,1,1,0,0,0] => 2

[.,[[.,.],[[.,[.,.]],.]]] => [1,0,1,1,1,0,0,1,0,0,1,0] => 2

[.,[[.,.],[[[.,.],.],.]]] => [1,0,1,1,1,0,0,1,0,1,0,0] => 2

[.,[[.,[.,.]],[.,[.,.]]]] => [1,0,1,1,1,0,1,0,0,0,1,0] => 2

[.,[[.,[.,.]],[[.,.],.]]] => [1,0,1,1,1,0,1,0,0,1,0,0] => 2

[.,[[[.,.],.],[.,[.,.]]]] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2

[.,[[[.,.],.],[[.,.],.]]] => [1,0,1,1,1,1,0,0,0,1,0,0] => 2

[.,[[.,[.,[.,.]]],[.,.]]] => [1,0,1,1,1,0,1,0,1,0,0,0] => 2

[.,[[.,[[.,.],.]],[.,.]]] => [1,0,1,1,1,0,1,1,0,0,0,0] => 2

[.,[[[.,.],[.,.]],[.,.]]] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2

[.,[[[.,[.,.]],.],[.,.]]] => [1,0,1,1,1,1,0,0,1,0,0,0] => 2

[.,[[[[.,.],.],.],[.,.]]] => [1,0,1,1,1,1,0,1,0,0,0,0] => 2

[.,[[.,[.,[.,[.,.]]]],.]] => [1,0,1,1,0,0,1,0,1,0,1,0] => 1

[.,[[.,[.,[[.,.],.]]],.]] => [1,0,1,1,0,0,1,0,1,1,0,0] => 1

[.,[[.,[[.,.],[.,.]]],.]] => [1,0,1,1,0,0,1,1,1,0,0,0] => 2

[.,[[.,[[.,[.,.]],.]],.]] => [1,0,1,1,0,0,1,1,0,0,1,0] => 1

[.,[[.,[[[.,.],.],.]],.]] => [1,0,1,1,0,0,1,1,0,1,0,0] => 1

[.,[[[.,.],[.,[.,.]]],.]] => [1,0,1,1,0,1,1,0,0,0,1,0] => 2

[.,[[[.,.],[[.,.],.]],.]] => [1,0,1,1,0,1,1,0,0,1,0,0] => 2

[.,[[[.,[.,.]],[.,.]],.]] => [1,0,1,1,0,1,1,0,1,0,0,0] => 2

[.,[[[[.,.],.],[.,.]],.]] => [1,0,1,1,0,1,1,1,0,0,0,0] => 2

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$F_{1} = q$

$F_{2} = 2\ q$

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

$F_{4} = 8\ q + 6\ q^{2}$

$F_{5} = 16\ q + 26\ q^{2}$

$F_{6} = 32\ q + 100\ q^{2}$

$F_{7} = 64\ q + 364\ q^{2} + q^{3}$

Description

The logarithmic height of a Dyck path.
This is the floor of the binary logarithm of the usual height increased by one:

$\lfloor\log_2(1+height(D))\rfloor$

Map

pruning number to logarithmic height

Description

Francon's map from binary trees to Dyck paths.
This bijection sends the pruning number of the binary tree, St000396The register function (or Horton-Strahler number) of a binary tree., to the logarithmic height of the Dyck path, St000920The logarithmic height of a Dyck path.. The implementation is a literal translation of Knuth's [2].