- FindStat

Your data matches 4 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001074

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001074: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of inversions of the cyclic embedding of a permutation. The cyclic embedding of a permutation

$\pi$ of length

$n$ is given by the permutation of length

$n+1$ represented in cycle notation by

$(\pi_1,\ldots,\pi_n,n+1)$ . This reflects in particular the fact that the number of long cycles of length

$n+1$ equals

$n!$ . This statistic counts the number of inversions of this embedding, see [1]. As shown in [2], the sum of this statistic on all permutations of length

$n$ equals

$n!\cdot(3n-1)/12$ .

Matching statistic: St001207

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 26%●distinct values known / distinct values provided: 20%

Values

[-1,-2] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[2,-1] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[-2,1] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[1,-2,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,2,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,-2,3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,-2,-3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 3
[1,3,-2] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[1,-3,2] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[-1,3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,-3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[2,-1,3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[2,-1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-2,1,3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[-2,1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[2,3,-1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[2,-3,1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[-2,3,1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[-2,-3,-1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[3,1,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[3,-1,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[-3,1,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[-3,-1,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[3,2,-1] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[3,-2,-1] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-3,2,1] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[-3,-2,1] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[1,2,-3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[1,-2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[1,-2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[1,-2,-3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 3
[-1,2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,2,-3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 3
[-1,-2,3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,-2,3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 3
[-1,-2,-3,4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 3
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ? = 4
[1,2,4,-3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[1,2,-4,3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[1,-2,4,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[1,-2,-4,3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,2,4,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,2,-4,3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,-2,4,3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,-2,4,-3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4
[-1,-2,-4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4
[-1,-2,-4,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[1,3,-2,4] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[1,3,-2,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[1,-3,2,4] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[1,-3,2,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,3,2,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,3,-2,4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,3,-2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4
[-1,-3,2,4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,-3,2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4
[-1,-3,-2,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[1,3,4,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[1,3,-4,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[1,-3,4,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[1,-3,-4,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[-1,3,4,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[-1,3,-4,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[-1,-3,4,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[-1,-3,-4,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[1,4,2,-3] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[1,4,-2,3] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[1,-4,2,3] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[1,-4,-2,-3] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[-1,4,2,-3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[-1,4,-2,3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[-1,-4,2,3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[-1,-4,-2,-3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[1,4,3,-2] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[1,4,-3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[1,-4,3,2] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[1,-4,-3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,4,3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,4,-3,2] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,4,-3,-2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4
[-1,-4,3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,-4,-3,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4
[-1,-4,-3,-2] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[2,1,-3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[2,-1,3,4] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[2,-1,-3,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4
[-2,1,-3,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4
[2,-1,4,-3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 6
[2,-1,-4,3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 6
[-2,1,4,-3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 6
[-2,1,-4,3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 6
[2,3,-1,4] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[2,3,-1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[2,-3,1,4] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[2,-3,1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[-2,3,1,4] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[-2,3,1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[-2,-3,-1,4] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[-2,-3,-1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4

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

$K[x]/(x^n)$ .

Matching statistic: St001582

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 26%●distinct values known / distinct values provided: 20%

Values

[-1,-2] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[2,-1] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[-2,1] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[1,-2,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,2,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,-2,3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,-2,-3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 3
[1,3,-2] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[1,-3,2] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[-1,3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,-3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[2,-1,3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[2,-1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-2,1,3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[-2,1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[2,3,-1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[2,-3,1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[-2,3,1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[-2,-3,-1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[3,1,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[3,-1,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[-3,1,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[-3,-1,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[3,2,-1] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[3,-2,-1] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-3,2,1] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[-3,-2,1] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[1,2,-3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[1,-2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[1,-2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[1,-2,-3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 3
[-1,2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,2,-3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 3
[-1,-2,3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,-2,3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 3
[-1,-2,-3,4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 3
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ? = 4
[1,2,4,-3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[1,2,-4,3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[1,-2,4,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[1,-2,-4,3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,2,4,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,2,-4,3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,-2,4,3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,-2,4,-3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4
[-1,-2,-4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4
[-1,-2,-4,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[1,3,-2,4] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[1,3,-2,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[1,-3,2,4] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[1,-3,2,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,3,2,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,3,-2,4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,3,-2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4
[-1,-3,2,4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,-3,2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4
[-1,-3,-2,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[1,3,4,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[1,3,-4,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[1,-3,4,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[1,-3,-4,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[-1,3,4,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[-1,3,-4,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[-1,-3,4,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[-1,-3,-4,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[1,4,2,-3] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[1,4,-2,3] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[1,-4,2,3] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[1,-4,-2,-3] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[-1,4,2,-3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[-1,4,-2,3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[-1,-4,2,3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[-1,-4,-2,-3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[1,4,3,-2] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[1,4,-3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[1,-4,3,2] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[1,-4,-3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,4,3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,4,-3,2] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[-1,4,-3,-2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4
[-1,-4,3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-1,-4,-3,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4
[-1,-4,-3,-2] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[2,1,-3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[2,-1,3,4] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[2,-1,-3,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4
[-2,1,-3,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4
[2,-1,4,-3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 6
[2,-1,-4,3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 6
[-2,1,4,-3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 6
[-2,1,-4,3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 6
[2,3,-1,4] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[2,3,-1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[2,-3,1,4] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[2,-3,1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[-2,3,1,4] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[-2,3,1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4
[-2,-3,-1,4] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3
[-2,-3,-1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4

Description

The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.

Matching statistic: St001171

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001171: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 26%●distinct values known / distinct values provided: 20%

Values

[-1,-2] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[2,-1] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 5 = 2 + 3
[-2,1] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 5 = 2 + 3
[1,-2,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[-1,2,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[-1,-2,3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[-1,-2,-3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 3 + 3
[1,3,-2] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 5 = 2 + 3
[1,-3,2] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 5 = 2 + 3
[-1,3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[-1,-3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[2,-1,3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 5 = 2 + 3
[2,-1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[-2,1,3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 5 = 2 + 3
[-2,1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[2,3,-1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[2,-3,1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[-2,3,1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[-2,-3,-1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[3,1,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[3,-1,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[-3,1,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[-3,-1,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[3,2,-1] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 5 = 2 + 3
[3,-2,-1] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[-3,2,1] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 5 = 2 + 3
[-3,-2,1] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[1,2,-3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[1,-2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[1,-2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[1,-2,-3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 3 + 3
[-1,2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[-1,2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[-1,2,-3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 3 + 3
[-1,-2,3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[-1,-2,3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 3 + 3
[-1,-2,-3,4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 3 + 3
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ? = 4 + 3
[1,2,4,-3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 5 = 2 + 3
[1,2,-4,3] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 5 = 2 + 3
[1,-2,4,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[1,-2,-4,3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[-1,2,4,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[-1,2,-4,3] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[-1,-2,4,3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[-1,-2,4,-3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4 + 3
[-1,-2,-4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4 + 3
[-1,-2,-4,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[1,3,-2,4] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 5 = 2 + 3
[1,3,-2,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[1,-3,2,4] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 5 = 2 + 3
[1,-3,2,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[-1,3,2,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[-1,3,-2,4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[-1,3,-2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4 + 3
[-1,-3,2,4] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[-1,-3,2,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4 + 3
[-1,-3,-2,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[1,3,4,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[1,3,-4,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[1,-3,4,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[1,-3,-4,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[-1,3,4,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 + 3
[-1,3,-4,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 + 3
[-1,-3,4,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 + 3
[-1,-3,-4,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 + 3
[1,4,2,-3] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[1,4,-2,3] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[1,-4,2,3] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[1,-4,-2,-3] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[-1,4,2,-3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 + 3
[-1,4,-2,3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 + 3
[-1,-4,2,3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 + 3
[-1,-4,-2,-3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 + 3
[1,4,3,-2] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 5 = 2 + 3
[1,4,-3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[1,-4,3,2] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 5 = 2 + 3
[1,-4,-3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[-1,4,3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[-1,4,-3,2] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[-1,4,-3,-2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4 + 3
[-1,-4,3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 6 = 3 + 3
[-1,-4,-3,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4 + 3
[-1,-4,-3,-2] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[2,1,-3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 5 = 2 + 3
[2,-1,3,4] => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 5 = 2 + 3
[2,-1,-3,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4 + 3
[-2,1,-3,-4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 4 + 3
[2,-1,4,-3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 6 + 3
[2,-1,-4,3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 6 + 3
[-2,1,4,-3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 6 + 3
[-2,1,-4,3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 6 + 3
[2,3,-1,4] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[2,3,-1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 + 3
[2,-3,1,4] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[2,-3,1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 + 3
[-2,3,1,4] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[-2,3,1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 + 3
[-2,-3,-1,4] => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 3 + 3
[-2,-3,-1,-4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 4 + 3

Description

The vector space dimension of

$Ext_A^1(I_o,A)$ when

$I_o$ is the tilting module corresponding to the permutation

$o$ in the Auslander algebra

$A$ of

$K[x]/(x^n)$ .