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Identifier

Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St001207: Permutations ⟶ ℤ

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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$F_{2} = 1 + q$

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

$F_{4} = 1 + 8\ q + 4\ q^{2} + q^{3}$

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

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

$1 \tilde D_2 0 \tilde D_1$ .

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.