- FindStat

Identifier

Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000289: Binary words ⟶ ℤ

Values

[.,.] => [1,0] => 10 => 2
[.,[.,.]] => [1,0,1,0] => 1010 => 10
[[.,.],.] => [1,1,0,0] => 1100 => 12
[.,[.,[.,.]]] => [1,0,1,0,1,0] => 101010 => 42
[.,[[.,.],.]] => [1,0,1,1,0,0] => 101100 => 44
[[.,.],[.,.]] => [1,1,0,0,1,0] => 110010 => 50
[[.,[.,.]],.] => [1,1,0,1,0,0] => 110100 => 52
[[[.,.],.],.] => [1,1,1,0,0,0] => 111000 => 56
[.,[.,[.,[.,.]]]] => [1,0,1,0,1,0,1,0] => 10101010 => 170
[.,[.,[[.,.],.]]] => [1,0,1,0,1,1,0,0] => 10101100 => 172
[.,[[.,.],[.,.]]] => [1,0,1,1,0,0,1,0] => 10110010 => 178
[.,[[.,[.,.]],.]] => [1,0,1,1,0,1,0,0] => 10110100 => 180
[.,[[[.,.],.],.]] => [1,0,1,1,1,0,0,0] => 10111000 => 184
[[.,.],[.,[.,.]]] => [1,1,0,0,1,0,1,0] => 11001010 => 202
[[.,.],[[.,.],.]] => [1,1,0,0,1,1,0,0] => 11001100 => 204
[[.,[.,.]],[.,.]] => [1,1,0,1,0,0,1,0] => 11010010 => 210
[[[.,.],.],[.,.]] => [1,1,1,0,0,0,1,0] => 11100010 => 226
[[.,[.,[.,.]]],.] => [1,1,0,1,0,1,0,0] => 11010100 => 212
[[.,[[.,.],.]],.] => [1,1,0,1,1,0,0,0] => 11011000 => 216
[[[.,.],[.,.]],.] => [1,1,1,0,0,1,0,0] => 11100100 => 228
[[[.,[.,.]],.],.] => [1,1,1,0,1,0,0,0] => 11101000 => 232
[[[[.,.],.],.],.] => [1,1,1,1,0,0,0,0] => 11110000 => 240

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$F_{1} = q^{2}$

$F_{2} = q^{10} + q^{12}$

$F_{3} = q^{42} + q^{44} + q^{50} + q^{52} + q^{56}$

$F_{4} = q^{170} + q^{172} + q^{178} + q^{180} + q^{184} + q^{202} + q^{204} + q^{210} + q^{212} + q^{216} + q^{226} + q^{228} + q^{232} + q^{240}$

Description

The decimal representation of a binary word.

Map

to Dyck path: up step, left tree, down step, right tree

Description

Return the associated Dyck path, using the bijection 1L0R.
This is given recursively as follows:

a leaf is associated to the empty Dyck Word
a tree with children $l,r$ is associated with the Dyck path described by 1L0R where $L$ and $R$ are respectively the Dyck words associated with the trees $l$ and $r$ .

Map

to binary word

Description

Return the Dyck word as binary word.