Identifier
Values
[.,.] => [1,0] => 10 => [1,2] => 2
[.,[.,.]] => [1,0,1,0] => 1010 => [1,2,2] => 2
[[.,.],.] => [1,1,0,0] => 1100 => [1,1,3] => 2
[.,[.,[.,.]]] => [1,0,1,0,1,0] => 101010 => [1,2,2,2] => 2
[.,[[.,.],.]] => [1,0,1,1,0,0] => 101100 => [1,2,1,3] => 3
[[.,.],[.,.]] => [1,1,0,0,1,0] => 110010 => [1,1,3,2] => 2
[[.,[.,.]],.] => [1,1,0,1,0,0] => 110100 => [1,1,2,3] => 3
[[[.,.],.],.] => [1,1,1,0,0,0] => 111000 => [1,1,1,4] => 2
[.,[.,[.,[.,.]]]] => [1,0,1,0,1,0,1,0] => 10101010 => [1,2,2,2,2] => 2
[.,[.,[[.,.],.]]] => [1,0,1,0,1,1,0,0] => 10101100 => [1,2,2,1,3] => 3
[.,[[.,.],[.,.]]] => [1,0,1,1,0,0,1,0] => 10110010 => [1,2,1,3,2] => 3
[.,[[.,[.,.]],.]] => [1,0,1,1,0,1,0,0] => 10110100 => [1,2,1,2,3] => 3
[.,[[[.,.],.],.]] => [1,0,1,1,1,0,0,0] => 10111000 => [1,2,1,1,4] => 3
[[.,.],[.,[.,.]]] => [1,1,0,0,1,0,1,0] => 11001010 => [1,1,3,2,2] => 2
[[.,.],[[.,.],.]] => [1,1,0,0,1,1,0,0] => 11001100 => [1,1,3,1,3] => 3
[[.,[.,.]],[.,.]] => [1,1,0,1,0,0,1,0] => 11010010 => [1,1,2,3,2] => 3
[[[.,.],.],[.,.]] => [1,1,1,0,0,0,1,0] => 11100010 => [1,1,1,4,2] => 2
[[.,[.,[.,.]]],.] => [1,1,0,1,0,1,0,0] => 11010100 => [1,1,2,2,3] => 3
[[.,[[.,.],.]],.] => [1,1,0,1,1,0,0,0] => 11011000 => [1,1,2,1,4] => 3
[[[.,.],[.,.]],.] => [1,1,1,0,0,1,0,0] => 11100100 => [1,1,1,3,3] => 3
[[[.,[.,.]],.],.] => [1,1,1,0,1,0,0,0] => 11101000 => [1,1,1,2,4] => 3
[[[[.,.],.],.],.] => [1,1,1,1,0,0,0,0] => 11110000 => [1,1,1,1,5] => 2
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Description
The length of the longest staircase fitting into an integer composition.
For a given composition $c_1,\dots,c_n$, this is the maximal number $\ell$ such that there are indices $i_1 < \dots < i_\ell$ with $c_{i_k} \geq k$, see [def.3.1, 1]
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.
Map
to composition
Description
The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.