- FindStat

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

Matching statistic: St000925
(load all 16 compositions to match this statistic)

Mapping

Mp00151: Permutations —to cycle type⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of topologically connected components of a set partition. For example, the set partition

$\{\{1,5\},\{2,3\},\{4,6\}\}$ has the two connected components

$\{1,4,5,6\}$ and

$\{2,3\}$ . The number of set partitions with only one block is [[oeis:A099947]].

Matching statistic: St001232

Mapping

Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 100%

Values

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

Description

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

Matching statistic: St001461
(load all 31 compositions to match this statistic)

Mapping

Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St001461: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 100%

Values

[1,2] => {{1},{2}}
 => [1,2] => 2
[2,1] => {{1,2}}
 => [2,1] => 1
[1,2,3] => {{1},{2},{3}}
 => [1,2,3] => 3
[1,3,2] => {{1},{2,3}}
 => [1,3,2] => 2
[2,1,3] => {{1,2},{3}}
 => [2,1,3] => 2
[2,3,1] => {{1,2,3}}
 => [2,3,1] => 1
[3,1,2] => {{1,2,3}}
 => [2,3,1] => 1
[3,2,1] => {{1,3},{2}}
 => [3,2,1] => 2
[1,2,3,4] => {{1},{2},{3},{4}}
 => [1,2,3,4] => 4
[1,2,4,3] => {{1},{2},{3,4}}
 => [1,2,4,3] => 3
[1,3,2,4] => {{1},{2,3},{4}}
 => [1,3,2,4] => 3
[1,3,4,2] => {{1},{2,3,4}}
 => [1,3,4,2] => 2
[1,4,2,3] => {{1},{2,3,4}}
 => [1,3,4,2] => 2
[1,4,3,2] => {{1},{2,4},{3}}
 => [1,4,3,2] => 3
[2,1,3,4] => {{1,2},{3},{4}}
 => [2,1,3,4] => 3
[2,1,4,3] => {{1,2},{3,4}}
 => [2,1,4,3] => 2
[2,3,1,4] => {{1,2,3},{4}}
 => [2,3,1,4] => 2
[2,3,4,1] => {{1,2,3,4}}
 => [2,3,4,1] => 1
[2,4,1,3] => {{1,2,3,4}}
 => [2,3,4,1] => 1
[2,4,3,1] => {{1,2,4},{3}}
 => [2,4,3,1] => 2
[3,1,2,4] => {{1,2,3},{4}}
 => [2,3,1,4] => 2
[3,1,4,2] => {{1,2,3,4}}
 => [2,3,4,1] => 1
[3,2,1,4] => {{1,3},{2},{4}}
 => [3,2,1,4] => 3
[3,2,4,1] => {{1,3,4},{2}}
 => [3,2,4,1] => 2
[3,4,1,2] => {{1,3},{2,4}}
 => [3,4,1,2] => 1
[3,4,2,1] => {{1,2,3,4}}
 => [2,3,4,1] => 1
[4,1,2,3] => {{1,2,3,4}}
 => [2,3,4,1] => 1
[4,1,3,2] => {{1,2,4},{3}}
 => [2,4,3,1] => 2
[4,2,1,3] => {{1,3,4},{2}}
 => [3,2,4,1] => 2
[4,2,3,1] => {{1,4},{2},{3}}
 => [4,2,3,1] => 3
[4,3,1,2] => {{1,2,3,4}}
 => [2,3,4,1] => 1
[4,3,2,1] => {{1,4},{2,3}}
 => [4,3,2,1] => 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => 5
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => 4
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => 4
[1,2,4,5,3] => {{1},{2},{3,4,5}}
 => [1,2,4,5,3] => 3
[1,2,5,3,4] => {{1},{2},{3,4,5}}
 => [1,2,4,5,3] => 3
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
 => [1,2,5,4,3] => 4
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => 4
[1,3,2,5,4] => {{1},{2,3},{4,5}}
 => [1,3,2,5,4] => 3
[1,3,4,2,5] => {{1},{2,3,4},{5}}
 => [1,3,4,2,5] => 3
[1,3,4,5,2] => {{1},{2,3,4,5}}
 => [1,3,4,5,2] => 2
[1,3,5,2,4] => {{1},{2,3,4,5}}
 => [1,3,4,5,2] => 2
[1,3,5,4,2] => {{1},{2,3,5},{4}}
 => [1,3,5,4,2] => 3
[1,4,2,3,5] => {{1},{2,3,4},{5}}
 => [1,3,4,2,5] => 3
[1,4,2,5,3] => {{1},{2,3,4,5}}
 => [1,3,4,5,2] => 2
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
 => [1,4,3,2,5] => 4
[1,4,3,5,2] => {{1},{2,4,5},{3}}
 => [1,4,3,5,2] => 3
[1,4,5,2,3] => {{1},{2,4},{3,5}}
 => [1,4,5,2,3] => 2
[1,4,5,3,2] => {{1},{2,3,4,5}}
 => [1,3,4,5,2] => 2
[1,4,6,5,7,3,2] => {{1},{2,4,5,7},{3,6}}
 => [1,4,6,5,7,3,2] => ? = 2
[1,4,6,7,2,3,5] => {{1},{2,4,5,7},{3,6}}
 => [1,4,6,5,7,3,2] => ? = 2
[1,4,6,7,5,3,2] => {{1},{2,4,7},{3,6},{5}}
 => [1,4,6,7,5,3,2] => ? = 3
[1,4,7,2,5,6,3] => {{1},{2,4},{3,7},{5},{6}}
 => [1,4,7,2,5,6,3] => ? = 4
[1,4,7,2,6,5,3] => {{1},{2,4},{3,7},{5,6}}
 => [1,4,7,2,6,5,3] => ? = 3
[1,4,7,5,2,6,3] => {{1},{2,4,5},{3,7},{6}}
 => [1,4,7,5,2,6,3] => ? = 3
[1,4,7,5,6,2,3] => {{1},{2,4,5,6},{3,7}}
 => [1,4,7,5,6,2,3] => ? = 2
[1,4,7,6,2,5,3] => {{1},{2,4,5,6},{3,7}}
 => [1,4,7,5,6,2,3] => ? = 2
[1,4,7,6,5,2,3] => {{1},{2,4,6},{3,7},{5}}
 => [1,4,7,6,5,2,3] => ? = 3
[1,5,3,4,2,6,7] => {{1},{2,5},{3},{4},{6},{7}}
 => [1,5,3,4,2,6,7] => ? = 6
[1,5,3,4,2,7,6] => {{1},{2,5},{3},{4},{6,7}}
 => [1,5,3,4,2,7,6] => ? = 5
[1,5,3,4,6,2,7] => {{1},{2,5,6},{3},{4},{7}}
 => [1,5,3,4,6,2,7] => ? = 5
[1,5,3,4,6,7,2] => {{1},{2,5,6,7},{3},{4}}
 => [1,5,3,4,6,7,2] => ? = 4
[1,5,3,4,7,2,6] => {{1},{2,5,6,7},{3},{4}}
 => [1,5,3,4,6,7,2] => ? = 4
[1,5,3,4,7,6,2] => {{1},{2,5,7},{3},{4},{6}}
 => [1,5,3,4,7,6,2] => ? = 5
[1,5,3,6,2,4,7] => {{1},{2,5},{3},{4,6},{7}}
 => [1,5,3,6,2,4,7] => ? = 4
[1,5,3,6,2,7,4] => {{1},{2,5},{3},{4,6,7}}
 => [1,5,3,6,2,7,4] => ? = 3
[1,5,3,6,7,4,2] => {{1},{2,5,7},{3},{4,6}}
 => [1,5,3,6,7,4,2] => ? = 3
[1,5,3,7,2,4,6] => {{1},{2,5},{3},{4,6,7}}
 => [1,5,3,6,2,7,4] => ? = 3
[1,5,3,7,2,6,4] => {{1},{2,5},{3},{4,7},{6}}
 => [1,5,3,7,2,6,4] => ? = 4
[1,5,3,7,6,2,4] => {{1},{2,5,6},{3},{4,7}}
 => [1,5,3,7,6,2,4] => ? = 3
[1,5,4,3,2,6,7] => {{1},{2,5},{3,4},{6},{7}}
 => [1,5,4,3,2,6,7] => ? = 5
[1,5,4,3,2,7,6] => {{1},{2,5},{3,4},{6,7}}
 => [1,5,4,3,2,7,6] => ? = 4
[1,5,4,3,6,2,7] => {{1},{2,5,6},{3,4},{7}}
 => [1,5,4,3,6,2,7] => ? = 4
[1,5,4,3,6,7,2] => {{1},{2,5,6,7},{3,4}}
 => [1,5,4,3,6,7,2] => ? = 3
[1,5,4,3,7,2,6] => {{1},{2,5,6,7},{3,4}}
 => [1,5,4,3,6,7,2] => ? = 3
[1,5,4,3,7,6,2] => {{1},{2,5,7},{3,4},{6}}
 => [1,5,4,3,7,6,2] => ? = 4
[1,5,4,6,2,3,7] => {{1},{2,5},{3,4,6},{7}}
 => [1,5,4,6,2,3,7] => ? = 3
[1,5,4,6,2,7,3] => {{1},{2,5},{3,4,6,7}}
 => [1,5,4,6,2,7,3] => ? = 2
[1,5,4,6,7,3,2] => {{1},{2,5,7},{3,4,6}}
 => [1,5,4,6,7,3,2] => ? = 2
[1,5,4,7,2,3,6] => {{1},{2,5},{3,4,6,7}}
 => [1,5,4,6,2,7,3] => ? = 2
[1,5,4,7,2,6,3] => {{1},{2,5},{3,4,7},{6}}
 => [1,5,4,7,2,6,3] => ? = 3
[1,5,4,7,6,2,3] => {{1},{2,5,6},{3,4,7}}
 => [1,5,4,7,6,2,3] => ? = 2
[1,5,6,2,7,3,4] => {{1},{2,4,5,7},{3,6}}
 => [1,4,6,5,7,3,2] => ? = 2
[1,5,6,3,2,4,7] => {{1},{2,5},{3,4,6},{7}}
 => [1,5,4,6,2,3,7] => ? = 3
[1,5,6,3,2,7,4] => {{1},{2,5},{3,4,6,7}}
 => [1,5,4,6,2,7,3] => ? = 2
[1,5,6,3,7,4,2] => {{1},{2,5,7},{3,4,6}}
 => [1,5,4,6,7,3,2] => ? = 2
[1,5,6,4,2,3,7] => {{1},{2,5},{3,6},{4},{7}}
 => [1,5,6,4,2,3,7] => ? = 4
[1,5,6,4,2,7,3] => {{1},{2,5},{3,6,7},{4}}
 => [1,5,6,4,2,7,3] => ? = 3
[1,5,6,4,7,3,2] => {{1},{2,5,7},{3,6},{4}}
 => [1,5,6,4,7,3,2] => ? = 3
[1,5,6,7,2,3,4] => {{1},{2,5},{3,6},{4,7}}
 => [1,5,6,7,2,3,4] => ? = 2
[1,5,6,7,2,4,3] => {{1},{2,5},{3,4,6,7}}
 => [1,5,4,6,2,7,3] => ? = 2
[1,5,6,7,4,3,2] => {{1},{2,4,5,7},{3,6}}
 => [1,4,6,5,7,3,2] => ? = 2
[1,5,7,2,4,6,3] => {{1},{2,4,5},{3,7},{6}}
 => [1,4,7,5,2,6,3] => ? = 3
[1,5,7,2,6,4,3] => {{1},{2,4,5,6},{3,7}}
 => [1,4,7,5,6,2,3] => ? = 2
[1,5,7,3,2,4,6] => {{1},{2,5},{3,4,6,7}}
 => [1,5,4,6,2,7,3] => ? = 2
[1,5,7,3,2,6,4] => {{1},{2,5},{3,4,7},{6}}
 => [1,5,4,7,2,6,3] => ? = 3
[1,5,7,3,6,2,4] => {{1},{2,5,6},{3,4,7}}
 => [1,5,4,7,6,2,3] => ? = 2
[1,5,7,4,2,3,6] => {{1},{2,5},{3,6,7},{4}}
 => [1,5,6,4,2,7,3] => ? = 3
[1,5,7,4,2,6,3] => {{1},{2,5},{3,7},{4},{6}}
 => [1,5,7,4,2,6,3] => ? = 4

Description

The number of topologically connected components of the chord diagram of a permutation. The chord diagram of a permutation

$\pi\in\mathfrak S_n$ is obtained by placing labels

$1,\dots,n$ in cyclic order on a cycle and drawing a (straight) arc from

$i$ to

$\pi(i)$ for every label

$i$ . This statistic records the number of topologically connected components in the chord diagram. In particular, if two arcs cross, all four labels connected by the two arcs are in the same component. The permutation

$\pi\in\mathfrak S_n$ stabilizes an interval

$I=\{a,a+1,\dots,b\}$ if

$\pi(I)=I$ . It is stabilized-interval-free, if the only interval

$\pi$ stablizes is

$\{1,\dots,n\}$ . Thus, this statistic is

$1$ if

$\pi$ is stabilized-interval-free.