Identifier
-
Mp00081:
Standard tableaux
—reading word permutation⟶
Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St001208: Permutations ⟶ ℤ
Values
[[1]] => [1] => [.,.] => [1] => 1
[[1,2]] => [1,2] => [.,[.,.]] => [2,1] => 1
[[1],[2]] => [2,1] => [[.,.],.] => [1,2] => 1
[[1,2,3]] => [1,2,3] => [.,[.,[.,.]]] => [3,2,1] => 1
[[1,3],[2]] => [2,1,3] => [[.,.],[.,.]] => [1,3,2] => 1
[[1,2],[3]] => [3,1,2] => [[.,.],[.,.]] => [1,3,2] => 1
[[1],[2],[3]] => [3,2,1] => [[[.,.],.],.] => [1,2,3] => 1
[[1,2,3,4]] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => [4,3,2,1] => 1
[[1,3,4],[2]] => [2,1,3,4] => [[.,.],[.,[.,.]]] => [1,4,3,2] => 1
[[1,2,4],[3]] => [3,1,2,4] => [[.,.],[.,[.,.]]] => [1,4,3,2] => 1
[[1,2,3],[4]] => [4,1,2,3] => [[.,.],[.,[.,.]]] => [1,4,3,2] => 1
[[1,3],[2,4]] => [2,4,1,3] => [[.,[.,.]],[.,.]] => [2,1,4,3] => 2
[[1,2],[3,4]] => [3,4,1,2] => [[.,[.,.]],[.,.]] => [2,1,4,3] => 2
[[1,4],[2],[3]] => [3,2,1,4] => [[[.,.],.],[.,.]] => [1,2,4,3] => 1
[[1,3],[2],[4]] => [4,2,1,3] => [[[.,.],.],[.,.]] => [1,2,4,3] => 1
[[1,2],[3],[4]] => [4,3,1,2] => [[[.,.],.],[.,.]] => [1,2,4,3] => 1
[[1],[2],[3],[4]] => [4,3,2,1] => [[[[.,.],.],.],.] => [1,2,3,4] => 1
[[1,2,3,4,5]] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => 1
[[1,3,4,5],[2]] => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => 1
[[1,2,4,5],[3]] => [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => 1
[[1,2,3,5],[4]] => [4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => 1
[[1,2,3,4],[5]] => [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => 1
[[1,3,5],[2,4]] => [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => 2
[[1,2,5],[3,4]] => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => 2
[[1,3,4],[2,5]] => [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => 2
[[1,2,4],[3,5]] => [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => 2
[[1,2,3],[4,5]] => [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => 2
[[1,4,5],[2],[3]] => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => 1
[[1,3,5],[2],[4]] => [4,2,1,3,5] => [[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => 1
[[1,2,5],[3],[4]] => [4,3,1,2,5] => [[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => 1
[[1,3,4],[2],[5]] => [5,2,1,3,4] => [[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => 1
[[1,2,4],[3],[5]] => [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => 1
[[1,2,3],[4],[5]] => [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => 1
[[1,4],[2,5],[3]] => [3,2,5,1,4] => [[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => 2
[[1,3],[2,5],[4]] => [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => 2
[[1,2],[3,5],[4]] => [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => 2
[[1,3],[2,4],[5]] => [5,2,4,1,3] => [[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => 2
[[1,2],[3,4],[5]] => [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => 2
[[1,5],[2],[3],[4]] => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => 1
[[1,4],[2],[3],[5]] => [5,3,2,1,4] => [[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => 1
[[1,3],[2],[4],[5]] => [5,4,2,1,3] => [[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => 1
[[1,2],[3],[4],[5]] => [5,4,3,1,2] => [[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => 1
[[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.] => [1,2,3,4,5] => 1
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Description
The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$.
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.
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.
Map
to 312-avoiding permutation
Description
Return a 312-avoiding permutation corresponding to a binary tree.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the minimal element of this Sylvester class.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the minimal element of this Sylvester class.
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.
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