- FindStat

Identifier

Mp00256: Decorated permutations —upper permutation⟶ Permutations
St001208: Permutations ⟶ ℤ

Values

[+] => [1] => 1

[-] => [1] => 1

[+,+] => [1,2] => 1

[-,+] => [2,1] => 1

[+,-] => [1,2] => 1

[-,-] => [1,2] => 1

[2,1] => [2,1] => 1

[+,+,+] => [1,2,3] => 1

[-,+,+] => [2,3,1] => 1

[+,-,+] => [1,3,2] => 1

[+,+,-] => [1,2,3] => 1

[-,-,+] => [3,1,2] => 1

[-,+,-] => [2,1,3] => 1

[+,-,-] => [1,2,3] => 1

[-,-,-] => [1,2,3] => 1

[+,3,2] => [1,3,2] => 1

[-,3,2] => [3,1,2] => 1

[2,1,+] => [2,3,1] => 1

[2,1,-] => [2,1,3] => 1

[2,3,1] => [3,1,2] => 1

[3,1,2] => [2,3,1] => 1

[3,+,1] => [2,3,1] => 1

[3,-,1] => [3,1,2] => 1

[+,+,+,+] => [1,2,3,4] => 1

[-,+,+,+] => [2,3,4,1] => 1

[+,-,+,+] => [1,3,4,2] => 1

[+,+,-,+] => [1,2,4,3] => 1

[+,+,+,-] => [1,2,3,4] => 1

[-,-,+,+] => [3,4,1,2] => 1

[-,+,-,+] => [2,4,1,3] => 1

[-,+,+,-] => [2,3,1,4] => 1

[+,-,-,+] => [1,4,2,3] => 1

[+,-,+,-] => [1,3,2,4] => 1

[+,+,-,-] => [1,2,3,4] => 1

[-,-,-,+] => [4,1,2,3] => 1

[-,-,+,-] => [3,1,2,4] => 1

[-,+,-,-] => [2,1,3,4] => 1

[+,-,-,-] => [1,2,3,4] => 1

[-,-,-,-] => [1,2,3,4] => 1

[+,+,4,3] => [1,2,4,3] => 1

[-,+,4,3] => [2,4,1,3] => 1

[+,-,4,3] => [1,4,2,3] => 1

[-,-,4,3] => [4,1,2,3] => 1

[+,3,2,+] => [1,3,4,2] => 1

[-,3,2,+] => [3,4,1,2] => 1

[+,3,2,-] => [1,3,2,4] => 1

[-,3,2,-] => [3,1,2,4] => 1

[+,3,4,2] => [1,4,2,3] => 1

[-,3,4,2] => [4,1,2,3] => 1

[+,4,2,3] => [1,3,4,2] => 1

[-,4,2,3] => [3,4,1,2] => 1

[+,4,+,2] => [1,3,4,2] => 1

[-,4,+,2] => [3,4,1,2] => 1

[+,4,-,2] => [1,4,2,3] => 1

[-,4,-,2] => [4,1,2,3] => 1

[2,1,+,+] => [2,3,4,1] => 1

[2,1,-,+] => [2,4,1,3] => 1

[2,1,+,-] => [2,3,1,4] => 1

[2,1,-,-] => [2,1,3,4] => 1

[2,1,4,3] => [2,4,1,3] => 1

[2,3,1,+] => [3,4,1,2] => 1

[2,3,1,-] => [3,1,2,4] => 1

[2,3,4,1] => [4,1,2,3] => 1

[2,4,1,3] => [3,4,1,2] => 1

[2,4,+,1] => [3,4,1,2] => 1

[2,4,-,1] => [4,1,2,3] => 1

[3,1,2,+] => [2,3,4,1] => 1

[3,1,2,-] => [2,3,1,4] => 1

[3,1,4,2] => [2,4,1,3] => 1

[3,+,1,+] => [2,3,4,1] => 1

[3,-,1,+] => [3,4,1,2] => 1

[3,+,1,-] => [2,3,1,4] => 1

[3,-,1,-] => [3,1,2,4] => 1

[3,+,4,1] => [2,4,1,3] => 1

[3,-,4,1] => [4,1,2,3] => 1

[3,4,1,2] => [3,4,1,2] => 1

[3,4,2,1] => [3,4,1,2] => 1

[4,1,2,3] => [2,3,4,1] => 1

[4,1,+,2] => [2,3,4,1] => 1

[4,1,-,2] => [2,4,1,3] => 1

[4,+,1,3] => [2,3,4,1] => 1

[4,-,1,3] => [3,4,1,2] => 1

[4,+,+,1] => [2,3,4,1] => 1

[4,-,+,1] => [3,4,1,2] => 1

[4,+,-,1] => [2,4,1,3] => 1

[4,-,-,1] => [4,1,2,3] => 1

[4,3,1,2] => [3,4,1,2] => 1

[4,3,2,1] => [3,4,1,2] => 1

[+,+,+,+,+] => [1,2,3,4,5] => 1

[-,+,+,+,+] => [2,3,4,5,1] => 1

[+,-,+,+,+] => [1,3,4,5,2] => 1

[+,+,-,+,+] => [1,2,4,5,3] => 1

[+,+,+,-,+] => [1,2,3,5,4] => 1

[+,+,+,+,-] => [1,2,3,4,5] => 1

[-,-,+,+,+] => [3,4,5,1,2] => 1

[-,+,-,+,+] => [2,4,5,1,3] => 1

[-,+,+,-,+] => [2,3,5,1,4] => 1

[-,+,+,+,-] => [2,3,4,1,5] => 1

[+,-,-,+,+] => [1,4,5,2,3] => 1

[+,-,+,-,+] => [1,3,5,2,4] => 1

[+,-,+,+,-] => [1,3,4,2,5] => 1

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$F_{1} = 2\ q$

$F_{2} = 5\ q$

$F_{3} = 16\ q$

$F_{4} = 65\ q$

$F_{5} = 326\ q$

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

upper permutation

Description

The upper bound in the Grassmann interval corresponding to the decorated permutation.
Let

$I$ be the anti-exceedance set of a decorated permutation

$w$ . Let

$v$ be the

$k$ -Grassmannian permutation determined by

$v[k] = w^{-1}(I)$ and let

$u$ be the permutation satisfying

$u = wv$ . Then

$[u, v]$ is the Grassmann interval corresponding to

$w$ .
This map returns

$v$ .