- FindStat

Your data matches 10 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000941

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000941: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of characters of the symmetric group whose value on the partition is even.

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
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 33%

Values

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

Description

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

Matching statistic: St000689

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000689: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 33%

Values

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

Description

The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. The correspondence between LNakayama algebras and Dyck paths is explained in [[St000684]]. A module

$M$ is

$n$ -rigid, if

$\operatorname{Ext}^i(M,M)=0$ for

$1\leq i\leq n$ . This statistic gives the maximal

$n$ such that the minimal generator-cogenerator module

$A \oplus D(A)$ of the LNakayama algebra

$A$ corresponding to a Dyck path is

$n$ -rigid. An application is to check for maximal

$n$ -orthogonal objects in the module category in the sense of [2].

Matching statistic: St001200

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 33%

Values

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

Description

The number of simple modules in

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

$A$ with minimal faithful projective-injective module

$eA$ .

Matching statistic: St000370

Mapping

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000370: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%

Values

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

Description

The genus of a graph. This is the smallest genus of an oriented surface on which the graph can be embedded without crossings. One can indeed compute the genus as the sum of the genuses for the connected components.

Matching statistic: St001728

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001728: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%

Values

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

Description

The number of invisible descents of a permutation. A visible descent of a permutation

$\pi$ is a position

$i$ such that

$\pi(i+1) \leq \min(i, \pi(i))$ . Thus, an invisible descent satisfies

$\pi(i) > \pi(i+1) > i$ .

Matching statistic: St001345

Mapping

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001345: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%

Values

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

Description

The Hamming dimension of a graph. Let

$H(n, k)$ be the graph whose vertices are the subsets of

$\{1,\dots,n\}$ , and

$(u,v)$ being an edge, for

$u\neq v$ , if the symmetric difference of

$u$ and

$v$ has cardinality at most

$k$ . This statistic is the smallest

$n$ such that the graph is an induced subgraph of

$H(n, k)$ for some

$k$ .

Matching statistic: St001716

Mapping

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001716: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%

Values

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

Description

The 1-improper chromatic number of a graph. This is the least number of colours in a vertex-colouring, such that each vertex has at most one neighbour with the same colour.

Matching statistic: St001792

Mapping

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001792: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%

Values

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

Description

The arboricity of a graph. This is the minimum number of forests that covers all edges of the graph.

Matching statistic: St001645

Mapping

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 17%

Values

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

Description

The pebbling number of a connected graph.