- FindStat

Your data matches 6 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000058
(load all 21 compositions to match this statistic)

Mapping

St000058: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The order of a permutation.

$\operatorname{ord}(\pi)$ is given by the minimial

$k$ for which

$\pi^k$ is the identity permutation.

Matching statistic: St001555
(load all 9 compositions to match this statistic)

Mapping

Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001555: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => {{1}}
 => [1] => [1] => 1
[1,2] => {{1},{2}}
 => [1,2] => [1,2] => 1
[2,1] => {{1,2}}
 => [2,1] => [2,1] => 2
[1,2,3] => {{1},{2},{3}}
 => [1,2,3] => [1,2,3] => 1
[1,3,2] => {{1},{2,3}}
 => [1,3,2] => [1,3,2] => 2
[2,1,3] => {{1,2},{3}}
 => [2,1,3] => [2,1,3] => 2
[2,3,1] => {{1,2,3}}
 => [2,3,1] => [2,3,1] => 3
[3,1,2] => {{1,2,3}}
 => [2,3,1] => [2,3,1] => 3
[3,2,1] => {{1,3},{2}}
 => [3,2,1] => [3,2,1] => 2
[1,2,3,4] => {{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => {{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => 2
[1,3,2,4] => {{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2] => {{1},{2,3,4}}
 => [1,3,4,2] => [1,3,4,2] => 3
[1,4,2,3] => {{1},{2,3,4}}
 => [1,3,4,2] => [1,3,4,2] => 3
[1,4,3,2] => {{1},{2,4},{3}}
 => [1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => {{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => 2
[2,1,4,3] => {{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => {{1,2,3},{4}}
 => [2,3,1,4] => [2,3,1,4] => 3
[2,3,4,1] => {{1,2,3,4}}
 => [2,3,4,1] => [2,3,4,1] => 4
[2,4,1,3] => {{1,2,3,4}}
 => [2,3,4,1] => [2,3,4,1] => 4
[2,4,3,1] => {{1,2,4},{3}}
 => [2,4,3,1] => [2,4,3,1] => 3
[3,1,2,4] => {{1,2,3},{4}}
 => [2,3,1,4] => [2,3,1,4] => 3
[3,1,4,2] => {{1,2,3,4}}
 => [2,3,4,1] => [2,3,4,1] => 4
[3,2,1,4] => {{1,3},{2},{4}}
 => [3,2,1,4] => [3,2,1,4] => 2
[3,2,4,1] => {{1,3,4},{2}}
 => [3,2,4,1] => [3,2,4,1] => 3
[3,4,1,2] => {{1,3},{2,4}}
 => [3,4,1,2] => [3,4,1,2] => 2
[3,4,2,1] => {{1,2,3,4}}
 => [2,3,4,1] => [2,3,4,1] => 4
[4,1,2,3] => {{1,2,3,4}}
 => [2,3,4,1] => [2,3,4,1] => 4
[4,1,3,2] => {{1,2,4},{3}}
 => [2,4,3,1] => [2,4,3,1] => 3
[4,2,1,3] => {{1,3,4},{2}}
 => [3,2,4,1] => [3,2,4,1] => 3
[4,2,3,1] => {{1,4},{2},{3}}
 => [4,2,3,1] => [4,2,3,1] => 2
[4,3,1,2] => {{1,2,3,4}}
 => [2,3,4,1] => [2,3,4,1] => 4
[4,3,2,1] => {{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,2,4,5,3] => {{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [1,2,4,5,3] => 3
[1,2,5,3,4] => {{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [1,2,4,5,3] => 3
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
 => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,3,2,5,4] => {{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [1,3,4,2,5] => 3
[1,3,4,5,2] => {{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,3,4,5,2] => 4
[1,3,5,2,4] => {{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,3,4,5,2] => 4
[1,3,5,4,2] => {{1},{2,3,5},{4}}
 => [1,3,5,4,2] => [1,3,5,4,2] => 3
[1,4,2,3,5] => {{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [1,3,4,2,5] => 3
[1,4,2,5,3] => {{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,3,4,5,2] => 4
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
 => [1,4,3,2,5] => [1,4,3,2,5] => 2
[1,4,3,5,2] => {{1},{2,4,5},{3}}
 => [1,4,3,5,2] => [1,4,3,5,2] => 3
[1,4,5,2,3] => {{1},{2,4},{3,5}}
 => [1,4,5,2,3] => [1,4,5,2,3] => 2

Description

The order of a signed permutation.

Matching statistic: St000668

Mapping

Mp00108: Permutations —cycle type⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%

Values

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

Description

The least common multiple of the parts of the partition.

Matching statistic: St001232

Mapping

Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 83%

Values

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

Description

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

Matching statistic: St001207
(load all 3 compositions to match this statistic)

Mapping

Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 67%

Values

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

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: St001624

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 33%

Values

[1] => [1] => ([],1)
 => ([(0,1)],2)
 => 1
[1,2] => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1
[2,1] => [2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2
[2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[2,3,1] => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 3
[3,1,2] => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 3
[3,2,1] => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 2
[1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 2
[1,3,4,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => ? = 3
[1,4,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => ? = 3
[1,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => ? = 2
[2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
[2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 2
[2,3,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => ? = 3
[2,3,4,1] => [4,2,3,1] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? = 4
[2,4,1,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 4
[2,4,3,1] => [4,3,2,1] => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 3
[3,1,2,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => ? = 3
[3,1,4,2] => [4,2,3,1] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? = 4
[3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => ? = 2
[3,2,4,1] => [4,2,3,1] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? = 3
[3,4,1,2] => [4,3,2,1] => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2
[3,4,2,1] => [4,3,2,1] => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 4
[4,1,2,3] => [4,2,3,1] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? = 4
[4,1,3,2] => [4,2,3,1] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? = 3
[4,2,1,3] => [4,3,2,1] => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 3
[4,2,3,1] => [4,3,2,1] => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2
[4,3,1,2] => [4,3,2,1] => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 4
[4,3,2,1] => [4,3,2,1] => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,2,3,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => 2
[1,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 2
[1,2,4,5,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
 => ? = 3
[1,2,5,3,4] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
 => ? = 3
[1,2,5,4,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
 => ? = 2
[1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => 2
[1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
 => ? = 2
[1,3,4,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
 => ? = 3
[1,3,4,5,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
 => ? = 4
[1,3,5,2,4] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ? = 4
[1,3,5,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 3
[1,4,2,3,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
 => ? = 3
[1,4,2,5,3] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
 => ? = 4
[1,4,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
 => ? = 2
[1,4,3,5,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
 => ? = 3
[1,4,5,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 2
[1,4,5,3,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 4
[1,5,2,3,4] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
 => ? = 4
[1,5,2,4,3] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
 => ? = 3
[1,5,3,2,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 3
[1,5,3,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 2
[1,5,4,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 4
[1,5,4,3,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 2
[2,1,3,4,5] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => 2
[2,1,3,5,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
 => ? = 2
[2,1,4,3,5] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
 => ? = 2
[2,1,4,5,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11)
 => ? = 6
[2,1,5,3,4] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11)
 => ? = 6
[2,1,5,4,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11)
 => ? = 2
[2,3,1,4,5] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
 => ? = 3
[2,3,1,5,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(7,10),(8,10),(9,10),(10,1),(10,2)],11)
 => ? = 6
[2,3,4,1,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
 => ? = 4

Description

The breadth of a lattice. The '''breadth''' of a lattice is the least integer

$b$ such that any join

$x_1\vee x_2\vee\cdots\vee x_n$ , with

$n > b$ , can be expressed as a join over a proper subset of

$\{x_1,x_2,\ldots,x_n\}$ .