- FindStat

Your data matches 116 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001199

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ .

Matching statistic: St001232
(load all 2 compositions to match this statistic)

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 100%

Values

{{1,2},{3,4}}
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
{{1,3},{2,4}}
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
{{1,4},{2,3}}
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
{{1},{2},{3},{4}}
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 1 + 1
{{1,2,3},{4,5}}
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
{{1,2,4},{3,5}}
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
{{1,2,5},{3,4}}
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
{{1,2},{3,4,5}}
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
{{1,2},{3,4},{5}}
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 1 + 1
{{1,2},{3,5},{4}}
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 1 + 1
{{1,2},{3},{4,5}}
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 1 + 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 1 + 1
{{1,3,4},{2,5}}
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
{{1,3,5},{2,4}}
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
{{1,3},{2,4,5}}
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
{{1,3},{2,4},{5}}
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 1 + 1
{{1,3},{2,5},{4}}
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 1 + 1
{{1,3},{2},{4,5}}
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 1 + 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 1 + 1
{{1,4,5},{2,3}}
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
{{1,4},{2,3,5}}
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
{{1,4},{2,3},{5}}
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 1 + 1
{{1,5},{2,3,4}}
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
{{1,5},{2,3},{4}}
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 1 + 1
{{1},{2,3},{4,5}}
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 1 + 1
{{1},{2,3},{4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 1 + 1
{{1,4},{2,5},{3}}
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 1 + 1
{{1,4},{2},{3,5}}
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 1 + 1
{{1,4},{2},{3},{5}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 1 + 1
{{1,5},{2,4},{3}}
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 1 + 1
{{1},{2,4},{3,5}}
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 1 + 1
{{1},{2,4},{3},{5}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 1 + 1
{{1,5},{2},{3,4}}
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 1 + 1
{{1},{2,5},{3,4}}
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 1 + 1
{{1},{2},{3,4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 1 + 1
{{1,5},{2},{3},{4}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 1 + 1
{{1},{2,5},{3},{4}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 1 + 1
{{1},{2},{3,5},{4}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 1 + 1
{{1},{2},{3},{4,5}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 1 + 1
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? = 1 + 1
{{1,2,3,4},{5,6}}
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
{{1,2,3,5},{4,6}}
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
{{1,2,3,6},{4,5}}
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
{{1,2,3},{4,5,6}}
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
{{1,2,3},{4,5},{6}}
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 1 + 1
{{1,2,3},{4,6},{5}}
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 1 + 1
{{1,2,3},{4},{5,6}}
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 1 + 1
{{1,2,3},{4},{5},{6}}
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 1 + 1
{{1,2,4,5},{3,6}}
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
{{1,2,4,6},{3,5}}
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
{{1,2,4},{3,5,6}}
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
{{1,2,4},{3,5},{6}}
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 1 + 1
{{1,2,4},{3,6},{5}}
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 1 + 1
{{1,2,4},{3},{5,6}}
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 1 + 1
{{1,2,4},{3},{5},{6}}
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 1 + 1
{{1,2,5,6},{3,4}}
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
{{1,2,5},{3,4,6}}
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
{{1,2,5},{3,4},{6}}
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 1 + 1
{{1,2,6},{3,4,5}}
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
{{1,2},{3,4,5,6}}
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
{{1,2},{3,4,5},{6}}
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 1 + 1
{{1,2,6},{3,4},{5}}
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 1 + 1
{{1,2},{3,4,6},{5}}
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 1 + 1
{{1,2},{3,4},{5},{6}}
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? = 2 + 1
{{1,2,5},{3,6},{4}}
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 1 + 1
{{1,2,5},{3},{4,6}}
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 1 + 1
{{1,2,5},{3},{4},{6}}
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 1 + 1
{{1,2,6},{3,5},{4}}
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 1 + 1
{{1,2},{3,5,6},{4}}
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 1 + 1
{{1,2},{3,5},{4},{6}}
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? = 2 + 1
{{1,2,6},{3},{4,5}}
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 1 + 1
{{1,2},{3},{4,5,6}}
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 1 + 1
{{1,2},{3},{4,5},{6}}
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? = 2 + 1
{{1,2,6},{3},{4},{5}}
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 1 + 1
{{1,3,4,5},{2,6}}
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
{{1,3,4,6},{2,5}}
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
{{1,3,4},{2,5,6}}
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
{{1,3,5,6},{2,4}}
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
{{1,3,5},{2,4,6}}
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
{{1,3,6},{2,4,5}}
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
{{1,3},{2,4,5,6}}
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
{{1,4,5,6},{2,3}}
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
{{1,4,5},{2,3,6}}
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
{{1,4,6},{2,3,5}}
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
{{1,4},{2,3,5,6}}
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
{{1,5,6},{2,3,4}}
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
{{1,5},{2,3,4,6}}
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
{{1,6},{2,3,4,5}}
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
{{1,2,3,4,5},{6,7}}
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2 = 1 + 1
{{1,2,3,4,6},{5,7}}
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2 = 1 + 1
{{1,2,3,4,7},{5,6}}
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2 = 1 + 1
{{1,2,3,4},{5,6,7}}
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3 = 2 + 1
{{1,2,3,5,6},{4,7}}
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2 = 1 + 1
{{1,2,3,5,7},{4,6}}
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2 = 1 + 1
{{1,2,3,5},{4,6,7}}
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3 = 2 + 1
{{1,2,3,6,7},{4,5}}
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2 = 1 + 1
{{1,2,3,6},{4,5,7}}
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3 = 2 + 1
{{1,2,3,7},{4,5,6}}
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3 = 2 + 1
{{1,2,3},{4,5,6,7}}
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3 = 2 + 1
{{1,2,4,5,6},{3,7}}
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2 = 1 + 1

Description

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

Matching statistic: St000455
(load all 2 compositions to match this statistic)

Mapping

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 25%

Values

{{1,2},{3,4}}
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 - 1
{{1,3},{2,4}}
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 - 1
{{1,4},{2,3}}
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 - 1
{{1},{2},{3},{4}}
 => [1,1,1,1] => [4] => ([],4)
 => ? = 1 - 1
{{1,2,3},{4,5}}
 => [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
{{1,2,4},{3,5}}
 => [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
{{1,2,5},{3,4}}
 => [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
{{1,2},{3,4,5}}
 => [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
{{1,2},{3,4},{5}}
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
{{1,2},{3,5},{4}}
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
{{1,2},{3},{4,5}}
 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => [1,4] => ([(3,4)],5)
 => 0 = 1 - 1
{{1,3,4},{2,5}}
 => [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
{{1,3,5},{2,4}}
 => [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
{{1,3},{2,4,5}}
 => [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
{{1,3},{2,4},{5}}
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
{{1,3},{2,5},{4}}
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
{{1,3},{2},{4,5}}
 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1] => [1,4] => ([(3,4)],5)
 => 0 = 1 - 1
{{1,4,5},{2,3}}
 => [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
{{1,4},{2,3,5}}
 => [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
{{1,4},{2,3},{5}}
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
{{1,5},{2,3,4}}
 => [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
{{1,5},{2,3},{4}}
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
{{1},{2,3},{4,5}}
 => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
{{1},{2,3},{4},{5}}
 => [1,2,1,1] => [2,3] => ([(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1,4},{2,5},{3}}
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
{{1,4},{2},{3,5}}
 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
{{1,4},{2},{3},{5}}
 => [2,1,1,1] => [1,4] => ([(3,4)],5)
 => 0 = 1 - 1
{{1,5},{2,4},{3}}
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
{{1},{2,4},{3,5}}
 => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
{{1},{2,4},{3},{5}}
 => [1,2,1,1] => [2,3] => ([(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1,5},{2},{3,4}}
 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
{{1},{2,5},{3,4}}
 => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
{{1},{2},{3,4},{5}}
 => [1,1,2,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1,5},{2},{3},{4}}
 => [2,1,1,1] => [1,4] => ([(3,4)],5)
 => 0 = 1 - 1
{{1},{2,5},{3},{4}}
 => [1,2,1,1] => [2,3] => ([(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1},{2},{3,5},{4}}
 => [1,1,2,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1},{2},{3},{4,5}}
 => [1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1] => [5] => ([],5)
 => ? = 1 - 1
{{1,2,3,4},{5,6}}
 => [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1,2,3,5},{4,6}}
 => [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1,2,3,6},{4,5}}
 => [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1,2,3},{4,5,6}}
 => [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
{{1,2,3},{4,5},{6}}
 => [3,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1,2,3},{4,6},{5}}
 => [3,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1,2,3},{4},{5,6}}
 => [3,1,2] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1,2,3},{4},{5},{6}}
 => [3,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,2,4,5},{3,6}}
 => [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1,2,4,6},{3,5}}
 => [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1,2,4},{3,5,6}}
 => [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
{{1,2,4},{3,5},{6}}
 => [3,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1,2,4},{3,6},{5}}
 => [3,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1,2,4},{3},{5,6}}
 => [3,1,2] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1,2,4},{3},{5},{6}}
 => [3,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,2,5,6},{3,4}}
 => [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1,2,5},{3,4,6}}
 => [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
{{1,2,5},{3,4},{6}}
 => [3,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1,2,6},{3,4,5}}
 => [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
{{1,2},{3,4,5,6}}
 => [2,4] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1,2},{3,4,5},{6}}
 => [2,3,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1,2,6},{3,4},{5}}
 => [3,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1,2,5},{3},{4},{6}}
 => [3,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,2,6},{3},{4},{5}}
 => [3,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,2},{3},{4},{5},{6}}
 => [2,1,1,1,1] => [1,5] => ([(4,5)],6)
 => 0 = 1 - 1
{{1,3,4},{2},{5},{6}}
 => [3,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,3,5},{2},{4},{6}}
 => [3,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,3,6},{2},{4},{5}}
 => [3,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,3},{2},{4},{5},{6}}
 => [2,1,1,1,1] => [1,5] => ([(4,5)],6)
 => 0 = 1 - 1
{{1},{2,3,4},{5},{6}}
 => [1,3,1,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1},{2,3,5},{4},{6}}
 => [1,3,1,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1},{2,3,6},{4},{5}}
 => [1,3,1,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1},{2,3},{4},{5},{6}}
 => [1,2,1,1,1] => [2,4] => ([(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,4,5},{2},{3},{6}}
 => [3,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,4,6},{2},{3},{5}}
 => [3,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,4},{2},{3},{5},{6}}
 => [2,1,1,1,1] => [1,5] => ([(4,5)],6)
 => 0 = 1 - 1
{{1},{2,4,5},{3},{6}}
 => [1,3,1,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1},{2,4,6},{3},{5}}
 => [1,3,1,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1},{2,4},{3},{5},{6}}
 => [1,2,1,1,1] => [2,4] => ([(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1},{2},{3,4,5},{6}}
 => [1,1,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1},{2},{3,4,6},{5}}
 => [1,1,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1},{2},{3,4},{5},{6}}
 => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,5,6},{2},{3},{4}}
 => [3,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,5},{2},{3},{4},{6}}
 => [2,1,1,1,1] => [1,5] => ([(4,5)],6)
 => 0 = 1 - 1
{{1},{2,5,6},{3},{4}}
 => [1,3,1,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1},{2,5},{3},{4},{6}}
 => [1,2,1,1,1] => [2,4] => ([(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1},{2},{3,5,6},{4}}
 => [1,1,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1},{2},{3,5},{4},{6}}
 => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1},{2},{3},{4,5,6}}
 => [1,1,1,3] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1},{2},{3},{4,5},{6}}
 => [1,1,1,2,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,6},{2},{3},{4},{5}}
 => [2,1,1,1,1] => [1,5] => ([(4,5)],6)
 => 0 = 1 - 1
{{1},{2,6},{3},{4},{5}}
 => [1,2,1,1,1] => [2,4] => ([(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1},{2},{3,6},{4},{5}}
 => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1},{2},{3},{4,6},{5}}
 => [1,1,1,2,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1},{2},{3},{4},{5,6}}
 => [1,1,1,1,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,2,3,4},{5},{6},{7}}
 => [4,1,1,1] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
{{1,2,3,5},{4},{6},{7}}
 => [4,1,1,1] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
{{1,2,3,6},{4},{5},{7}}
 => [4,1,1,1] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
{{1,2,3,7},{4},{5},{6}}
 => [4,1,1,1] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
{{1,2,3},{4},{5},{6},{7}}
 => [3,1,1,1,1] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1

Description

The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.

Matching statistic: St001195
(load all 6 compositions to match this statistic)

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%

Values

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

Description

The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ .

Matching statistic: St001208
(load all 156 compositions to match this statistic)

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%

Values

{{1,2},{3,4}}
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 1
{{1,3},{2,4}}
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 1
{{1,4},{2,3}}
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 1
{{1},{2},{3},{4}}
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ? = 1
{{1,2,3},{4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1
{{1,2,4},{3,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1
{{1,2,5},{3,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1
{{1,2},{3,4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1
{{1,2},{3,4},{5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1
{{1,2},{3,5},{4}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1
{{1,2},{3},{4,5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ? = 1
{{1,3,4},{2,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1
{{1,3,5},{2,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1
{{1,3},{2,4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1
{{1,3},{2,4},{5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1
{{1,3},{2,5},{4}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1
{{1,3},{2},{4,5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ? = 1
{{1,4,5},{2,3}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1
{{1,4},{2,3,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1
{{1,4},{2,3},{5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1
{{1,5},{2,3,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1
{{1,5},{2,3},{4}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1
{{1},{2,3},{4,5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1
{{1},{2,3},{4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ? = 1
{{1,4},{2,5},{3}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1
{{1,4},{2},{3,5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1
{{1,4},{2},{3},{5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ? = 1
{{1,5},{2,4},{3}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1
{{1},{2,4},{3,5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1
{{1},{2,4},{3},{5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ? = 1
{{1,5},{2},{3,4}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1
{{1},{2,5},{3,4}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1
{{1},{2},{3,4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ? = 1
{{1,5},{2},{3},{4}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ? = 1
{{1},{2,5},{3},{4}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ? = 1
{{1},{2},{3,5},{4}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ? = 1
{{1},{2},{3},{4,5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ? = 1
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 1
{{1,2,3,4},{5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 1
{{1,2,3,5},{4,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 1
{{1,2,3,6},{4,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 1
{{1,2,3},{4,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 2
{{1,2,3},{4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
{{1,2,3},{4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
{{1,2,3},{4},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
{{1,2,3},{4},{5},{6}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => ? = 1
{{1,2,4,5},{3,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 1
{{1,2,4,6},{3,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 1
{{1,2,4},{3,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 2
{{1,2,4},{3,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
{{1,2,4},{3,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
{{1,2,4},{3},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
{{1,2,4},{3},{5},{6}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => ? = 1
{{1,2,5,6},{3,4}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 1
{{1,2,5},{3,4,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 2
{{1,2,5},{3,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
{{1,2,6},{3,4,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 2
{{1,2},{3,4,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 1
{{1,2},{3,4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
{{1,2,6},{3,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
{{1,2},{3,4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
{{1,2},{3,4},{5},{6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ? = 2
{{1,2,5},{3,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
{{1,2,5},{3},{4,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
{{1,2,5},{3},{4},{6}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => ? = 1
{{1,2,6},{3,5},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
{{1,2},{3,5,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
{{1,2},{3,5},{4},{6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ? = 2
{{1,2,6},{3},{4,5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
{{1,2},{3},{4,5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
{{1,2},{3},{4,5},{6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ? = 2
{{1,2,6},{3},{4},{5}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => ? = 1
{{1,2},{3,6},{4},{5}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ? = 2
{{1,2},{3},{4,6},{5}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ? = 2
{{1,2},{3},{4},{5,6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ? = 2
{{1,2},{3},{4},{5},{6}}
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [7,1,4,5,6,2,3] => ? = 1
{{1,3,4,5},{2,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 1
{{1,3,4,6},{2,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 1
{{1,3,4},{2,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 2
{{1,3,4},{2,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
{{1,3,4},{2,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
{{1,3,4},{2},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
{{1,3,4},{2},{5},{6}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => ? = 1
{{1,3,5,6},{2,4}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 1
{{1,3,5},{2,4,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 2
{{1,3,5},{2,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
{{1,3,6},{2,4,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 2
{{1,3},{2,4,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 1
{{1,3},{2,4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
{{1,3,6},{2,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
{{1,3},{2,4},{5},{6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ? = 2
{{1,3,5},{2},{4},{6}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => ? = 1
{{1,3},{2,5},{4},{6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ? = 2
{{1,3},{2},{4,5},{6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ? = 2
{{1,3,6},{2},{4},{5}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => ? = 1
{{1,3},{2,6},{4},{5}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ? = 2
{{1,3},{2},{4,6},{5}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ? = 2
{{1,3},{2},{4},{5,6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ? = 2

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

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

Matching statistic: St001001
(load all 3 compositions to match this statistic)

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%

Values

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

Description

The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001371

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%

Values

{{1,2},{3,4}}
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => 0 = 1 - 1
{{1,3},{2,4}}
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => 0 = 1 - 1
{{1,4},{2,3}}
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => 0 = 1 - 1
{{1},{2},{3},{4}}
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => ? = 1 - 1
{{1,2,3},{4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0 = 1 - 1
{{1,2,4},{3,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0 = 1 - 1
{{1,2,5},{3,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0 = 1 - 1
{{1,2},{3,4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0 = 1 - 1
{{1,2},{3,4},{5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
{{1,2},{3,5},{4}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
{{1,2},{3},{4,5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 1 - 1
{{1,3,4},{2,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0 = 1 - 1
{{1,3,5},{2,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0 = 1 - 1
{{1,3},{2,4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0 = 1 - 1
{{1,3},{2,4},{5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
{{1,3},{2,5},{4}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
{{1,3},{2},{4,5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 1 - 1
{{1,4,5},{2,3}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0 = 1 - 1
{{1,4},{2,3,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0 = 1 - 1
{{1,4},{2,3},{5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
{{1,5},{2,3,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0 = 1 - 1
{{1,5},{2,3},{4}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
{{1},{2,3},{4,5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
{{1},{2,3},{4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 1 - 1
{{1,4},{2,5},{3}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
{{1,4},{2},{3,5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
{{1,4},{2},{3},{5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 1 - 1
{{1,5},{2,4},{3}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
{{1},{2,4},{3,5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
{{1},{2,4},{3},{5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 1 - 1
{{1,5},{2},{3,4}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
{{1},{2,5},{3,4}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
{{1},{2},{3,4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 1 - 1
{{1,5},{2},{3},{4}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 1 - 1
{{1},{2,5},{3},{4}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 1 - 1
{{1},{2},{3,5},{4}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 1 - 1
{{1},{2},{3},{4,5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 1 - 1
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 101111100000 => ? = 1 - 1
{{1,2,3,4},{5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 1 - 1
{{1,2,3,5},{4,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 1 - 1
{{1,2,3,6},{4,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 1 - 1
{{1,2,3},{4,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 2 - 1
{{1,2,3},{4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,2,3},{4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,2,3},{4},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,2,3},{4},{5},{6}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 1 - 1
{{1,2,4,5},{3,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 1 - 1
{{1,2,4,6},{3,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 1 - 1
{{1,2,4},{3,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 2 - 1
{{1,2,4},{3,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,2,4},{3,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,2,4},{3},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,2,4},{3},{5},{6}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 1 - 1
{{1,2,5,6},{3,4}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 1 - 1
{{1,2,5},{3,4,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 2 - 1
{{1,2,5},{3,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,2,6},{3,4,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 2 - 1
{{1,2},{3,4,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 1 - 1
{{1,2},{3,4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,2,6},{3,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,2},{3,4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,2},{3,4},{5},{6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? = 2 - 1
{{1,2,5},{3,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,2,5},{3},{4,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,2,5},{3},{4},{6}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 1 - 1
{{1,2,6},{3,5},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,2},{3,5,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,2},{3,5},{4},{6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? = 2 - 1
{{1,2,6},{3},{4,5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,2},{3},{4,5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,2},{3},{4,5},{6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? = 2 - 1
{{1,2,6},{3},{4},{5}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 1 - 1
{{1,2},{3,6},{4},{5}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? = 2 - 1
{{1,2},{3},{4,6},{5}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? = 2 - 1
{{1,2},{3},{4},{5,6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? = 2 - 1
{{1,2},{3},{4},{5},{6}}
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => ? = 1 - 1
{{1,3,4,5},{2,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 1 - 1
{{1,3,4,6},{2,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 1 - 1
{{1,3,4},{2,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 2 - 1
{{1,3,4},{2,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,3,4},{2,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,3,4},{2},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,3,4},{2},{5},{6}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 1 - 1
{{1,3,5,6},{2,4}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 1 - 1
{{1,3,5},{2,4,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 2 - 1
{{1,3,5},{2,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,3,6},{2,4,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 2 - 1
{{1,3},{2,4,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 1 - 1
{{1,3},{2,4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,3,6},{2,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,3},{2,4},{5},{6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? = 2 - 1
{{1,3,5},{2},{4},{6}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 1 - 1
{{1,3},{2,5},{4},{6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? = 2 - 1
{{1,3},{2},{4,5},{6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? = 2 - 1
{{1,3,6},{2},{4},{5}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 1 - 1
{{1,3},{2,6},{4},{5}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? = 2 - 1
{{1,3},{2},{4,6},{5}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? = 2 - 1
{{1,3},{2},{4},{5,6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? = 2 - 1

Description

The length of the longest Yamanouchi prefix of a binary word. This is the largest index

$i$ such that in each of the prefixes

$w_1$ ,

$w_1w_2$ ,

$w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.

Matching statistic: St001556

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001556: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%

Values

{{1,2},{3,4}}
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 0 = 1 - 1
{{1,3},{2,4}}
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 0 = 1 - 1
{{1,4},{2,3}}
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 0 = 1 - 1
{{1},{2},{3},{4}}
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ? = 1 - 1
{{1,2,3},{4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0 = 1 - 1
{{1,2,4},{3,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0 = 1 - 1
{{1,2,5},{3,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0 = 1 - 1
{{1,2},{3,4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0 = 1 - 1
{{1,2},{3,4},{5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 0 = 1 - 1
{{1,2},{3,5},{4}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 0 = 1 - 1
{{1,2},{3},{4,5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 0 = 1 - 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ? = 1 - 1
{{1,3,4},{2,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0 = 1 - 1
{{1,3,5},{2,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0 = 1 - 1
{{1,3},{2,4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0 = 1 - 1
{{1,3},{2,4},{5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 0 = 1 - 1
{{1,3},{2,5},{4}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 0 = 1 - 1
{{1,3},{2},{4,5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 0 = 1 - 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ? = 1 - 1
{{1,4,5},{2,3}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0 = 1 - 1
{{1,4},{2,3,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0 = 1 - 1
{{1,4},{2,3},{5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 0 = 1 - 1
{{1,5},{2,3,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0 = 1 - 1
{{1,5},{2,3},{4}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 0 = 1 - 1
{{1},{2,3},{4,5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 0 = 1 - 1
{{1},{2,3},{4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ? = 1 - 1
{{1,4},{2,5},{3}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 0 = 1 - 1
{{1,4},{2},{3,5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 0 = 1 - 1
{{1,4},{2},{3},{5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ? = 1 - 1
{{1,5},{2,4},{3}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 0 = 1 - 1
{{1},{2,4},{3,5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 0 = 1 - 1
{{1},{2,4},{3},{5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ? = 1 - 1
{{1,5},{2},{3,4}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 0 = 1 - 1
{{1},{2,5},{3,4}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 0 = 1 - 1
{{1},{2},{3,4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ? = 1 - 1
{{1,5},{2},{3},{4}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ? = 1 - 1
{{1},{2,5},{3},{4}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ? = 1 - 1
{{1},{2},{3,5},{4}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ? = 1 - 1
{{1},{2},{3},{4,5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ? = 1 - 1
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 1 - 1
{{1,2,3,4},{5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 1 - 1
{{1,2,3,5},{4,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 1 - 1
{{1,2,3,6},{4,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 1 - 1
{{1,2,3},{4,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 2 - 1
{{1,2,3},{4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
{{1,2,3},{4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
{{1,2,3},{4},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
{{1,2,3},{4},{5},{6}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => ? = 1 - 1
{{1,2,4,5},{3,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 1 - 1
{{1,2,4,6},{3,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 1 - 1
{{1,2,4},{3,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 2 - 1
{{1,2,4},{3,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
{{1,2,4},{3,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
{{1,2,4},{3},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
{{1,2,4},{3},{5},{6}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => ? = 1 - 1
{{1,2,5,6},{3,4}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 1 - 1
{{1,2,5},{3,4,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 2 - 1
{{1,2,5},{3,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
{{1,2,6},{3,4,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 2 - 1
{{1,2},{3,4,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 1 - 1
{{1,2},{3,4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
{{1,2,6},{3,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
{{1,2},{3,4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
{{1,2},{3,4},{5},{6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ? = 2 - 1
{{1,2,5},{3,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
{{1,2,5},{3},{4,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
{{1,2,5},{3},{4},{6}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => ? = 1 - 1
{{1,2,6},{3,5},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
{{1,2},{3,5,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
{{1,2},{3,5},{4},{6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ? = 2 - 1
{{1,2,6},{3},{4,5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
{{1,2},{3},{4,5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
{{1,2},{3},{4,5},{6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ? = 2 - 1
{{1,2,6},{3},{4},{5}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => ? = 1 - 1
{{1,2},{3,6},{4},{5}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ? = 2 - 1
{{1,2},{3},{4,6},{5}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ? = 2 - 1
{{1,2},{3},{4},{5,6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ? = 2 - 1
{{1,2},{3},{4},{5},{6}}
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [7,1,4,5,6,2,3] => ? = 1 - 1
{{1,3,4,5},{2,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 1 - 1
{{1,3,4,6},{2,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 1 - 1
{{1,3,4},{2,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 2 - 1
{{1,3,4},{2,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
{{1,3,4},{2,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
{{1,3,4},{2},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
{{1,3,4},{2},{5},{6}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => ? = 1 - 1
{{1,3,5,6},{2,4}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 1 - 1
{{1,3,5},{2,4,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 2 - 1
{{1,3,5},{2,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
{{1,3,6},{2,4,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 2 - 1
{{1,3},{2,4,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 1 - 1
{{1,3},{2,4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
{{1,3,6},{2,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
{{1,3},{2,4},{5},{6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ? = 2 - 1
{{1,3,5},{2},{4},{6}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => ? = 1 - 1
{{1,3},{2,5},{4},{6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ? = 2 - 1
{{1,3},{2},{4,5},{6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ? = 2 - 1
{{1,3,6},{2},{4},{5}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => ? = 1 - 1
{{1,3},{2,6},{4},{5}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ? = 2 - 1
{{1,3},{2},{4,6},{5}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ? = 2 - 1
{{1,3},{2},{4},{5,6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ? = 2 - 1

Description

The number of inversions of the third entry of a permutation. This is, for a permutation

$\pi$ of length

$n$ ,

$\# \{3 < k \leq n \mid \pi(3) > \pi(k)\}.$ The number of inversions of the first entry is [[St000054]] and the number of inversions of the second entry is [[St001557]]. The sequence of inversions of all the entries define the [[http://www.findstat.org/Permutations#The_Lehmer_code_and_the_major_code_of_a_permutation|Lehmer code]] of a permutation.

Matching statistic: St001730

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%

Values

{{1,2},{3,4}}
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => 0 = 1 - 1
{{1,3},{2,4}}
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => 0 = 1 - 1
{{1,4},{2,3}}
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => 0 = 1 - 1
{{1},{2},{3},{4}}
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => ? = 1 - 1
{{1,2,3},{4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0 = 1 - 1
{{1,2,4},{3,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0 = 1 - 1
{{1,2,5},{3,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0 = 1 - 1
{{1,2},{3,4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0 = 1 - 1
{{1,2},{3,4},{5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
{{1,2},{3,5},{4}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
{{1,2},{3},{4,5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 1 - 1
{{1,3,4},{2,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0 = 1 - 1
{{1,3,5},{2,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0 = 1 - 1
{{1,3},{2,4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0 = 1 - 1
{{1,3},{2,4},{5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
{{1,3},{2,5},{4}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
{{1,3},{2},{4,5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 1 - 1
{{1,4,5},{2,3}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0 = 1 - 1
{{1,4},{2,3,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0 = 1 - 1
{{1,4},{2,3},{5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
{{1,5},{2,3,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0 = 1 - 1
{{1,5},{2,3},{4}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
{{1},{2,3},{4,5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
{{1},{2,3},{4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 1 - 1
{{1,4},{2,5},{3}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
{{1,4},{2},{3,5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
{{1,4},{2},{3},{5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 1 - 1
{{1,5},{2,4},{3}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
{{1},{2,4},{3,5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
{{1},{2,4},{3},{5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 1 - 1
{{1,5},{2},{3,4}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
{{1},{2,5},{3,4}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
{{1},{2},{3,4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 1 - 1
{{1,5},{2},{3},{4}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 1 - 1
{{1},{2,5},{3},{4}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 1 - 1
{{1},{2},{3,5},{4}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 1 - 1
{{1},{2},{3},{4,5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 1 - 1
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 101111100000 => ? = 1 - 1
{{1,2,3,4},{5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 1 - 1
{{1,2,3,5},{4,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 1 - 1
{{1,2,3,6},{4,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 1 - 1
{{1,2,3},{4,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 2 - 1
{{1,2,3},{4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,2,3},{4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,2,3},{4},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,2,3},{4},{5},{6}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 1 - 1
{{1,2,4,5},{3,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 1 - 1
{{1,2,4,6},{3,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 1 - 1
{{1,2,4},{3,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 2 - 1
{{1,2,4},{3,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,2,4},{3,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,2,4},{3},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,2,4},{3},{5},{6}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 1 - 1
{{1,2,5,6},{3,4}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 1 - 1
{{1,2,5},{3,4,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 2 - 1
{{1,2,5},{3,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,2,6},{3,4,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 2 - 1
{{1,2},{3,4,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 1 - 1
{{1,2},{3,4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,2,6},{3,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,2},{3,4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,2},{3,4},{5},{6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? = 2 - 1
{{1,2,5},{3,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,2,5},{3},{4,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,2,5},{3},{4},{6}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 1 - 1
{{1,2,6},{3,5},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,2},{3,5,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,2},{3,5},{4},{6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? = 2 - 1
{{1,2,6},{3},{4,5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,2},{3},{4,5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,2},{3},{4,5},{6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? = 2 - 1
{{1,2,6},{3},{4},{5}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 1 - 1
{{1,2},{3,6},{4},{5}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? = 2 - 1
{{1,2},{3},{4,6},{5}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? = 2 - 1
{{1,2},{3},{4},{5,6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? = 2 - 1
{{1,2},{3},{4},{5},{6}}
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => ? = 1 - 1
{{1,3,4,5},{2,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 1 - 1
{{1,3,4,6},{2,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 1 - 1
{{1,3,4},{2,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 2 - 1
{{1,3,4},{2,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,3,4},{2,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,3,4},{2},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,3,4},{2},{5},{6}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 1 - 1
{{1,3,5,6},{2,4}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 1 - 1
{{1,3,5},{2,4,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 2 - 1
{{1,3,5},{2,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,3,6},{2,4,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 2 - 1
{{1,3},{2,4,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 1 - 1
{{1,3},{2,4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,3,6},{2,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
{{1,3},{2,4},{5},{6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? = 2 - 1
{{1,3,5},{2},{4},{6}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 1 - 1
{{1,3},{2,5},{4},{6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? = 2 - 1
{{1,3},{2},{4,5},{6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? = 2 - 1
{{1,3,6},{2},{4},{5}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 1 - 1
{{1,3},{2,6},{4},{5}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? = 2 - 1
{{1,3},{2},{4,6},{5}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? = 2 - 1
{{1,3},{2},{4},{5,6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? = 2 - 1

Description

The number of times the path corresponding to a binary word crosses the base line. Interpret each

$0$ as a step

$(1,-1)$ and

$1$ as a step

$(1,1)$ . Then this statistic counts the number of times the path crosses the

$x$ -axis.

Matching statistic: St001803

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%

Values

{{1,2},{3,4}}
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => 0 = 1 - 1
{{1,3},{2,4}}
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => 0 = 1 - 1
{{1,4},{2,3}}
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => 0 = 1 - 1
{{1},{2},{3},{4}}
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => ? = 1 - 1
{{1,2,3},{4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => 0 = 1 - 1
{{1,2,4},{3,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => 0 = 1 - 1
{{1,2,5},{3,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => 0 = 1 - 1
{{1,2},{3,4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => 0 = 1 - 1
{{1,2},{3,4},{5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 0 = 1 - 1
{{1,2},{3,5},{4}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 0 = 1 - 1
{{1,2},{3},{4,5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 0 = 1 - 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[1,3,4,5,7],[2,6,8,9,10]]
 => ? = 1 - 1
{{1,3,4},{2,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => 0 = 1 - 1
{{1,3,5},{2,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => 0 = 1 - 1
{{1,3},{2,4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => 0 = 1 - 1
{{1,3},{2,4},{5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 0 = 1 - 1
{{1,3},{2,5},{4}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 0 = 1 - 1
{{1,3},{2},{4,5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 0 = 1 - 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[1,3,4,5,7],[2,6,8,9,10]]
 => ? = 1 - 1
{{1,4,5},{2,3}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => 0 = 1 - 1
{{1,4},{2,3,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => 0 = 1 - 1
{{1,4},{2,3},{5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 0 = 1 - 1
{{1,5},{2,3,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => 0 = 1 - 1
{{1,5},{2,3},{4}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 0 = 1 - 1
{{1},{2,3},{4,5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 0 = 1 - 1
{{1},{2,3},{4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[1,3,4,5,7],[2,6,8,9,10]]
 => ? = 1 - 1
{{1,4},{2,5},{3}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 0 = 1 - 1
{{1,4},{2},{3,5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 0 = 1 - 1
{{1,4},{2},{3},{5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[1,3,4,5,7],[2,6,8,9,10]]
 => ? = 1 - 1
{{1,5},{2,4},{3}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 0 = 1 - 1
{{1},{2,4},{3,5}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 0 = 1 - 1
{{1},{2,4},{3},{5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[1,3,4,5,7],[2,6,8,9,10]]
 => ? = 1 - 1
{{1,5},{2},{3,4}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 0 = 1 - 1
{{1},{2,5},{3,4}}
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 0 = 1 - 1
{{1},{2},{3,4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[1,3,4,5,7],[2,6,8,9,10]]
 => ? = 1 - 1
{{1,5},{2},{3},{4}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[1,3,4,5,7],[2,6,8,9,10]]
 => ? = 1 - 1
{{1},{2,5},{3},{4}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[1,3,4,5,7],[2,6,8,9,10]]
 => ? = 1 - 1
{{1},{2},{3,5},{4}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[1,3,4,5,7],[2,6,8,9,10]]
 => ? = 1 - 1
{{1},{2},{3},{4,5}}
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[1,3,4,5,7],[2,6,8,9,10]]
 => ? = 1 - 1
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,3,4,5,6,7],[2,8,9,10,11,12]]
 => ? = 1 - 1
{{1,2,3,4},{5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[1,2,3,6,9],[4,5,7,8,10]]
 => ? = 1 - 1
{{1,2,3,5},{4,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[1,2,3,6,9],[4,5,7,8,10]]
 => ? = 1 - 1
{{1,2,3,6},{4,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[1,2,3,6,9],[4,5,7,8,10]]
 => ? = 1 - 1
{{1,2,3},{4,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 2 - 1
{{1,2,3},{4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
{{1,2,3},{4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
{{1,2,3},{4},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
{{1,2,3},{4},{5},{6}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 1 - 1
{{1,2,4,5},{3,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[1,2,3,6,9],[4,5,7,8,10]]
 => ? = 1 - 1
{{1,2,4,6},{3,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[1,2,3,6,9],[4,5,7,8,10]]
 => ? = 1 - 1
{{1,2,4},{3,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 2 - 1
{{1,2,4},{3,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
{{1,2,4},{3,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
{{1,2,4},{3},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
{{1,2,4},{3},{5},{6}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 1 - 1
{{1,2,5,6},{3,4}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[1,2,3,6,9],[4,5,7,8,10]]
 => ? = 1 - 1
{{1,2,5},{3,4,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 2 - 1
{{1,2,5},{3,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
{{1,2,6},{3,4,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 2 - 1
{{1,2},{3,4,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[1,2,3,6,9],[4,5,7,8,10]]
 => ? = 1 - 1
{{1,2},{3,4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
{{1,2,6},{3,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
{{1,2},{3,4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
{{1,2},{3,4},{5},{6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[1,3,4,6,7],[2,5,8,9,10]]
 => ? = 2 - 1
{{1,2,5},{3,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
{{1,2,5},{3},{4,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
{{1,2,5},{3},{4},{6}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 1 - 1
{{1,2,6},{3,5},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
{{1,2},{3,5,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
{{1,2},{3,5},{4},{6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[1,3,4,6,7],[2,5,8,9,10]]
 => ? = 2 - 1
{{1,2,6},{3},{4,5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
{{1,2},{3},{4,5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
{{1,2},{3},{4,5},{6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[1,3,4,6,7],[2,5,8,9,10]]
 => ? = 2 - 1
{{1,2,6},{3},{4},{5}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 1 - 1
{{1,2},{3,6},{4},{5}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[1,3,4,6,7],[2,5,8,9,10]]
 => ? = 2 - 1
{{1,2},{3},{4,6},{5}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[1,3,4,6,7],[2,5,8,9,10]]
 => ? = 2 - 1
{{1,2},{3},{4},{5,6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[1,3,4,6,7],[2,5,8,9,10]]
 => ? = 2 - 1
{{1,2},{3},{4},{5},{6}}
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[1,3,4,5,6,8],[2,7,9,10,11,12]]
 => ? = 1 - 1
{{1,3,4,5},{2,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[1,2,3,6,9],[4,5,7,8,10]]
 => ? = 1 - 1
{{1,3,4,6},{2,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[1,2,3,6,9],[4,5,7,8,10]]
 => ? = 1 - 1
{{1,3,4},{2,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 2 - 1
{{1,3,4},{2,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
{{1,3,4},{2,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
{{1,3,4},{2},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
{{1,3,4},{2},{5},{6}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 1 - 1
{{1,3,5,6},{2,4}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[1,2,3,6,9],[4,5,7,8,10]]
 => ? = 1 - 1
{{1,3,5},{2,4,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 2 - 1
{{1,3,5},{2,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
{{1,3,6},{2,4,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 2 - 1
{{1,3},{2,4,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[1,2,3,6,9],[4,5,7,8,10]]
 => ? = 1 - 1
{{1,3},{2,4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
{{1,3,6},{2,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
{{1,3},{2,4},{5},{6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[1,3,4,6,7],[2,5,8,9,10]]
 => ? = 2 - 1
{{1,3,5},{2},{4},{6}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 1 - 1
{{1,3},{2,5},{4},{6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[1,3,4,6,7],[2,5,8,9,10]]
 => ? = 2 - 1
{{1,3},{2},{4,5},{6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[1,3,4,6,7],[2,5,8,9,10]]
 => ? = 2 - 1
{{1,3,6},{2},{4},{5}}
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 1 - 1
{{1,3},{2,6},{4},{5}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[1,3,4,6,7],[2,5,8,9,10]]
 => ? = 2 - 1
{{1,3},{2},{4,6},{5}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[1,3,4,6,7],[2,5,8,9,10]]
 => ? = 2 - 1
{{1,3},{2},{4},{5,6}}
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[1,3,4,6,7],[2,5,8,9,10]]
 => ? = 2 - 1

Description

The maximal overlap of the cylindrical tableau associated with a tableau. A cylindrical tableau associated with a standard Young tableau

$T$ is the skew row-strict tableau obtained by gluing two copies of

$T$ such that the inner shape is a rectangle. The overlap, recorded in this statistic, equals

$\max_C\big(2\ell(T) - \ell(C)\big)$ , where

$\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux. In particular, the statistic equals

$0$ , if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.

The following 106 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000735The last entry on the main diagonal of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000044The number of vertices of the unicellular map given by a perfect matching. St000017The number of inversions of a standard tableau. St001721The degree of a binary word. St000016The number of attacking pairs of a standard tableau. St000958The number of Bruhat factorizations of a permutation. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001429The number of negative entries in a signed permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000056The decomposition (or block) number of a permutation. St000069The number of maximal elements of a poset. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000287The number of connected components of a graph. St000486The number of cycles of length at least 3 of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000694The number of affine bounded permutations that project to a given permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001081The number of minimal length factorizations of a permutation into star transpositions. St001174The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001256Number of simple reflexive modules that are 2-stable reflexive. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001518The number of graphs with the same ordinary spectrum as the given graph. St001590The crossing number of a perfect matching. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001665The number of pure excedances of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001890The maximum magnitude of the Möbius function of a poset. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000221The number of strong fixed points of a permutation. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000315The number of isolated vertices of a graph. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000406The number of occurrences of the pattern 3241 in a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000542The number of left-to-right-minima of a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000666The number of right tethers of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000750The number of occurrences of the pattern 4213 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001192The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ . St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001381The fertility of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001430The number of positive entries in a signed permutation. St001434The number of negative sum pairs of a signed permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001513The number of nested exceedences of a permutation. St001520The number of strict 3-descents. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001557The number of inversions of the second entry of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001569The maximal modular displacement of a permutation. St001577The minimal number of edges to add or remove to make a graph a cograph. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001847The number of occurrences of the pattern 1432 in a permutation. St001850The number of Hecke atoms of a permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001948The number of augmented double ascents of a permutation. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.