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
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 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.
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