TamariBinaryTree from DyckPath
A path D = L U R D as the binary tree whose left and right subtrees are those of L and R: the bijection compatible with the Tamari order.
FindStat's Mp00029; TamariDyckPath is its inverse. BinaryTree(DyckPath) reads the other decomposition, U L D R, so this is it after a reverse-complement and a left-right mirror.
Called as CombinatorialMap(x, "TamariBinaryTree"). The head's page has its examples.
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Each element of DyckPath at a size, and its image, applied in your browser; sizes past 7 are not offered.
Properties
Transport a statistic
Pick a statistic on BinaryTree. Its distribution over the images of DyckPath at each size is computed, and compared with the 171 statistics on DyckPath: those that share it are the ones the map sends it to, as far as the sizes computed show.
FindStat Mp00029
to binary tree: left tree, up step, right tree, down step
Matched by value: our map agrees with FindStat's image on each of the 196 rows of size at most 6 we can read (0 unread; its table has 1200 rows).
BijectionGraded mapas FindStat records them, checked by FindStat's contributors.
| Object | Image |
|---|---|
[1,0] | [.,.] |
[1,0,1,0] | [[.,.],.] |
[1,1,0,0] | [.,[.,.]] |
[1,0,1,0,1,0] | [[[.,.],.],.] |
Data from FindStat, the combinatorial statistic finder; the page at findstat.org/Mp00029 has the rest.