Identifier
Values
[.,[.,.]] => [[],[[],[]]] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => 1
[[.,.],.] => [[[],[]],[]] => [1,1,0,1,0,0,1,0] => [2,3,1,4] => 1
[.,[.,[.,.]]] => [[],[[],[[],[]]]] => [1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,2,4] => 4
[.,[[.,.],.]] => [[],[[[],[]],[]]] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,2,6,3] => 4
[[.,.],[.,.]] => [[[],[]],[[],[]]] => [1,1,0,1,0,0,1,1,0,1,0,0] => [2,3,1,5,6,4] => 2
[[.,[.,.]],.] => [[[],[[],[]]],[]] => [1,1,0,1,1,0,1,0,0,0,1,0] => [2,4,5,1,3,6] => 4
[[[.,.],.],.] => [[[[],[]],[]],[]] => [1,1,1,0,1,0,0,1,0,0,1,0] => [3,4,1,5,2,6] => 4
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Description
The number of occurrences of the pattern 231 in a permutation.
Map
to complete tree
Description
Return the same tree seen as an ordered tree. By default, leaves are transformed into actual nodes.
Map
to Dyck path
Description
Return the Dyck path of the corresponding ordered tree induced by the recurrence of the Catalan numbers, see wikipedia:Catalan_number.
This sends the maximal height of the Dyck path to the depth of the tree.
Map
to 321-avoiding permutation (Krattenthaler)
Description
Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength $n$ in an $n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.