- FindStat

Identifier

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000289: Binary words ⟶ ℤ

Values

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

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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 Tamari-corresponding Dyck path

Description

Return the Dyck path associated with a binary tree in consistency with the Tamari order on Dyck words and binary trees.
The bijection is defined recursively as follows:

a leaf is associated with an empty Dyck path,
a tree with children $l,r$ is associated with the Dyck word $T(l) 1 T(r) 0$ where $T(l)$ and $T(r)$ are the images of this bijection to $l$ and $r$ .

Map

to binary word

Description

Return the Dyck word as binary word.