Identifier
Values
[1,0,1,0] => [1,1,0,0] => [[0,1],[1,0]] => [2,1] => 1
[1,1,0,0] => [1,0,1,0] => [[1,0],[0,1]] => [1,2] => 0
[1,0,1,0,1,0] => [1,1,1,0,0,0] => [[0,0,1],[0,1,0],[1,0,0]] => [3,2,1] => 2
[1,0,1,1,0,0] => [1,1,0,1,0,0] => [[0,1,0],[1,-1,1],[0,1,0]] => [1,3,2] => 1
[1,1,0,0,1,0] => [1,1,0,0,1,0] => [[0,1,0],[1,0,0],[0,0,1]] => [2,1,3] => 1
[1,1,0,1,0,0] => [1,0,1,1,0,0] => [[1,0,0],[0,0,1],[0,1,0]] => [1,3,2] => 1
[1,1,1,0,0,0] => [1,0,1,0,1,0] => [[1,0,0],[0,1,0],[0,0,1]] => [1,2,3] => 0
[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]] => [4,3,2,1] => 3
[1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]] => [1,4,3,2] => 2
[1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]] => [2,1,4,3] => 1
[1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]] => [1,4,3,2] => 2
[1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]] => [1,2,4,3] => 1
[1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]] => [3,2,1,4] => 2
[1,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]] => [1,3,2,4] => 1
[1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]] => [2,1,4,3] => 1
[1,1,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]] => [1,4,3,2] => 2
[1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0] => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]] => [1,2,4,3] => 1
[1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]] => [2,1,3,4] => 1
[1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]] => [1,3,2,4] => 1
[1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,0] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]] => [1,2,4,3] => 1
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]] => [1,2,3,4] => 0
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Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Map
to left key permutation
Description
Return the permutation of the left key of an alternating sign matrix.
This was originally defined by Lascoux and then further studied by Aval [1].
Map
to symmetric ASM
Description
The diagonally symmetric alternating sign matrix corresponding to a Dyck path.
Map
decomposition reverse
Description
This map is recursively defined as follows.
The unique empty path of semilength $0$ is sent to itself.
Let $D$ be a Dyck path of semilength $n > 0$ and decompose it into $1 D_1 0 D_2$ with Dyck paths $D_1, D_2$ of respective semilengths $n_1$ and $n_2$ such that $n_1$ is minimal. One then has $n_1+n_2 = n-1$.
Now let $\tilde D_1$ and $\tilde D_2$ be the recursively defined respective images of $D_1$ and $D_2$ under this map. The image of $D$ is then defined as $1 \tilde D_2 0 \tilde D_1$.