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Identifier

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
St001431: Dyck paths ⟶ ℤ

Values

[[1,2]] => [1,2] => [.,[.,.]] => [1,0,1,0] => 0
[[1],[2]] => [2,1] => [[.,.],.] => [1,1,0,0] => 1
[[1,2,3]] => [1,2,3] => [.,[.,[.,.]]] => [1,0,1,0,1,0] => 0
[[1,3],[2]] => [2,1,3] => [[.,.],[.,.]] => [1,1,1,0,0,0] => 1
[[1,2],[3]] => [3,1,2] => [[.,.],[.,.]] => [1,1,1,0,0,0] => 1
[[1],[2],[3]] => [3,2,1] => [[[.,.],.],.] => [1,1,0,1,0,0] => 2
[[1,2,3,4]] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => [1,0,1,0,1,0,1,0] => 0
[[1,3,4],[2]] => [2,1,3,4] => [[.,.],[.,[.,.]]] => [1,1,1,0,0,0,1,0] => 1
[[1,2,4],[3]] => [3,1,2,4] => [[.,.],[.,[.,.]]] => [1,1,1,0,0,0,1,0] => 1
[[1,2,3],[4]] => [4,1,2,3] => [[.,.],[.,[.,.]]] => [1,1,1,0,0,0,1,0] => 1
[[1,3],[2,4]] => [2,4,1,3] => [[.,[.,.]],[.,.]] => [1,1,1,0,1,0,0,0] => 1
[[1,2],[3,4]] => [3,4,1,2] => [[.,[.,.]],[.,.]] => [1,1,1,0,1,0,0,0] => 1
[[1,4],[2],[3]] => [3,2,1,4] => [[[.,.],.],[.,.]] => [1,1,1,1,0,0,0,0] => 1
[[1,3],[2],[4]] => [4,2,1,3] => [[[.,.],.],[.,.]] => [1,1,1,1,0,0,0,0] => 1
[[1,2],[3],[4]] => [4,3,1,2] => [[[.,.],.],[.,.]] => [1,1,1,1,0,0,0,0] => 1
[[1],[2],[3],[4]] => [4,3,2,1] => [[[[.,.],.],.],.] => [1,1,0,1,0,1,0,0] => 3
[[1,2,3,4,5]] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => [1,0,1,0,1,0,1,0,1,0] => 0
[[1,3,4,5],[2]] => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]] => [1,1,1,0,0,0,1,0,1,0] => 1
[[1,2,4,5],[3]] => [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]] => [1,1,1,0,0,0,1,0,1,0] => 1
[[1,2,3,5],[4]] => [4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]] => [1,1,1,0,0,0,1,0,1,0] => 1
[[1,2,3,4],[5]] => [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]] => [1,1,1,0,0,0,1,0,1,0] => 1
[[1,3,5],[2,4]] => [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]] => [1,1,1,0,1,0,0,0,1,0] => 1
[[1,2,5],[3,4]] => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]] => [1,1,1,0,1,0,0,0,1,0] => 1
[[1,3,4],[2,5]] => [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]] => [1,1,1,0,1,0,0,0,1,0] => 1
[[1,2,4],[3,5]] => [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]] => [1,1,1,0,1,0,0,0,1,0] => 1
[[1,2,3],[4,5]] => [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]] => [1,1,1,0,1,0,0,0,1,0] => 1
[[1,4,5],[2],[3]] => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]] => [1,1,1,1,0,0,0,0,1,0] => 1
[[1,3,5],[2],[4]] => [4,2,1,3,5] => [[[.,.],.],[.,[.,.]]] => [1,1,1,1,0,0,0,0,1,0] => 1
[[1,2,5],[3],[4]] => [4,3,1,2,5] => [[[.,.],.],[.,[.,.]]] => [1,1,1,1,0,0,0,0,1,0] => 1
[[1,3,4],[2],[5]] => [5,2,1,3,4] => [[[.,.],.],[.,[.,.]]] => [1,1,1,1,0,0,0,0,1,0] => 1
[[1,2,4],[3],[5]] => [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]] => [1,1,1,1,0,0,0,0,1,0] => 1
[[1,2,3],[4],[5]] => [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]] => [1,1,1,1,0,0,0,0,1,0] => 1
[[1,4],[2,5],[3]] => [3,2,5,1,4] => [[[.,.],[.,.]],[.,.]] => [1,1,1,1,1,0,0,0,0,0] => 1
[[1,3],[2,5],[4]] => [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]] => [1,1,1,1,1,0,0,0,0,0] => 1
[[1,2],[3,5],[4]] => [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]] => [1,1,1,1,1,0,0,0,0,0] => 1
[[1,3],[2,4],[5]] => [5,2,4,1,3] => [[[.,.],[.,.]],[.,.]] => [1,1,1,1,1,0,0,0,0,0] => 1
[[1,2],[3,4],[5]] => [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]] => [1,1,1,1,1,0,0,0,0,0] => 1
[[1,5],[2],[3],[4]] => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]] => [1,1,1,1,0,1,0,0,0,0] => 1
[[1,4],[2],[3],[5]] => [5,3,2,1,4] => [[[[.,.],.],.],[.,.]] => [1,1,1,1,0,1,0,0,0,0] => 1
[[1,3],[2],[4],[5]] => [5,4,2,1,3] => [[[[.,.],.],.],[.,.]] => [1,1,1,1,0,1,0,0,0,0] => 1
[[1,2],[3],[4],[5]] => [5,4,3,1,2] => [[[[.,.],.],.],[.,.]] => [1,1,1,1,0,1,0,0,0,0] => 1
[[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.] => [1,1,0,1,0,1,0,1,0,0] => 4
[[1,2,3,4,5,6]] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0

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$F_{2} = 1 + q$

$F_{3} = 1 + 2\ q + q^{2}$

$F_{4} = 1 + 8\ q + q^{3}$

$F_{5} = 1 + 24\ q + q^{4}$

Description

Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.
The modified algebra B is obtained from the stable Auslander algebra kQ/I by deleting all relations which contain walks of length at least three (conjectural this step of deletion is not necessary as the stable higher Auslander algebras might be quadratic) and taking as B then the algebra kQ^(op)/J when J is the quadratic perp of the ideal I.
See www.findstat.org/DyckPaths/NakayamaAlgebras for the definition of Loewy length and Nakayama algebras associated to Dyck paths.

Map

pruning number to logarithmic height

Description

Francon's map from binary trees to Dyck paths.
This bijection sends the pruning number of the binary tree, St000396The register function (or Horton-Strahler number) of a binary tree., to the logarithmic height of the Dyck path, St000920The logarithmic height of a Dyck path.. The implementation is a literal translation of Knuth's [2].

Map

reading word permutation

Description

Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.

Map

to increasing tree

Description

Sends a permutation to its associated increasing tree.
This tree is recursively obtained by sending the unique permutation of length

$0$ to the empty tree, and sending a permutation

$\sigma$ of length

$n \geq 1$ to a root node with two subtrees

$L$ and

$R$ by splitting

$\sigma$ at the index

$\sigma^{-1}(1)$ , normalizing both sides again to permutations and sending the permutations on the left and on the right of

$\sigma^{-1}(1)$ to the trees

$L$ and

$R$ , respectively.