- FindStat

Your data matches 15 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001645

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => [1] => ([],1)
 => ([],1)
 => 1
{{1,2}}
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
{{1},{2}}
 => [1,2] => ([],2)
 => ([],1)
 => 1
{{1,2,3}}
 => [2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2
{{1,3},{2}}
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
{{1},{2},{3}}
 => [1,2,3] => ([],3)
 => ([],1)
 => 1
{{1,2,3,4}}
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
{{1,3},{2,4}}
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2
{{1,4},{2,3}}
 => [4,3,2,1] => ([(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)
 => 4
{{1,4},{2},{3}}
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
{{1},{2},{3},{4}}
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 8
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 8
{{1,5},{2,3,4}}
 => [5,3,4,2,1] => ([(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)
 => 4
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => ([(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)
 => 5
{{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2
{{1,2,3,5},{4,6}}
 => [2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 8
{{1,2,4},{3,5,6}}
 => [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 16
{{1,3,5},{2,4,6}}
 => [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 2
{{1,3},{2,4,5,6}}
 => [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 8
{{1,6},{2,3,4,5}}
 => [6,3,4,5,2,1] => ([(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)
 => 4
{{1,6},{2,3,5},{4}}
 => [6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => 6
{{1,4},{2,5},{3,6}}
 => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,1)],2)
 => 2
{{1,4},{2,6},{3,5}}
 => [4,6,5,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
{{1,5},{2,4},{3,6}}
 => [5,4,6,2,1,3] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
{{1,6},{2,4,5},{3}}
 => [6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => 6
{{1,6},{2,4},{3,5}}
 => [6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
{{1,5},{2,6},{3,4}}
 => [5,6,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
{{1,6},{2,5},{3,4}}
 => [6,5,4,3,2,1] => ([(0,1),(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),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
{{1,5},{2,6},{3},{4}}
 => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
{{1,6},{2,5},{3},{4}}
 => [6,5,3,4,2,1] => ([(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)
 => 5
{{1,6},{2},{3},{4},{5}}
 => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
{{1},{2},{3},{4},{5},{6}}
 => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
{{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1)],2)
 => 2
{{1,2,3,4,6},{5,7}}
 => [2,3,4,6,7,1,5] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,3),(1,2),(2,3)],4)
 => 8
{{1,2,3,5},{4,6,7}}
 => [2,3,5,6,1,7,4] => ([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 16
{{1,2,4,6},{3,5,7}}
 => [2,4,5,6,7,1,3] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,3),(1,2),(2,3)],4)
 => 8
{{1,2,4},{3,5,6,7}}
 => [2,4,5,1,6,7,3] => ([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 16
{{1,2,5},{3,6},{4,7}}
 => [2,5,6,7,1,3,4] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
 => ([(0,3),(1,2),(2,3)],4)
 => 8
{{1,3,5,7},{2,4,6}}
 => [3,4,5,6,7,2,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
{{1,3,5},{2,4,6,7}}
 => [3,4,5,6,1,7,2] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,3),(1,2),(2,3)],4)
 => 8
{{1,3},{2,4,5,6,7}}
 => [3,4,1,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,3),(1,2),(2,3)],4)
 => 8
{{1,3},{2,4,6},{5,7}}
 => [3,4,1,6,7,2,5] => ([(0,4),(0,5),(1,2),(1,3),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 16
{{1,7},{2,3,4,5,6}}
 => [7,3,4,5,6,2,1] => ([(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),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
{{1,7},{2,3,4,6},{5}}
 => [7,3,4,6,5,2,1] => ([(0,4),(0,5),(0,6),(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)
 => ([(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)
 => 6

Description

The pebbling number of a connected graph.

Matching statistic: St001330
(load all 3 compositions to match this statistic)

Mapping

Mp00216: Set partitions —inverse Wachs-White⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 78%

Values

{{1}}
 => {{1}}
 => [1] => ([],1)
 => 1
{{1,2}}
 => {{1,2}}
 => [2,1] => ([(0,1)],2)
 => 2
{{1},{2}}
 => {{1},{2}}
 => [1,2] => ([],2)
 => 1
{{1,2,3}}
 => {{1,2,3}}
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 2
{{1,3},{2}}
 => {{1,3},{2}}
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
{{1},{2},{3}}
 => {{1},{2},{3}}
 => [1,2,3] => ([],3)
 => 1
{{1,2,3,4}}
 => {{1,2,3,4}}
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
{{1,3},{2,4}}
 => {{1,2,4},{3}}
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2
{{1,4},{2,3}}
 => {{1,4},{2,3}}
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
{{1,4},{2},{3}}
 => {{1,4},{2},{3}}
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [1,2,3,4] => ([],4)
 => 1
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
{{1,2,4},{3,5}}
 => {{1,2,4,5},{3}}
 => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 8
{{1,3,5},{2,4}}
 => {{1,4},{2,3,5}}
 => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 3
{{1,3},{2,4,5}}
 => {{1,2,3,5},{4}}
 => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 8
{{1,5},{2,3,4}}
 => {{1,5},{2,3,4}}
 => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
{{1,4},{2,5},{3}}
 => {{1,3},{2,5},{4}}
 => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 3
{{1,5},{2,4},{3}}
 => {{1,5},{2,4},{3}}
 => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
{{1,5},{2},{3},{4}}
 => {{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
{{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => ([],5)
 => 1
{{1,2,3,4,5,6}}
 => {{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2
{{1,2,3,5},{4,6}}
 => {{1,2,4,5,6},{3}}
 => [2,4,3,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 8
{{1,2,4},{3,5,6}}
 => {{1,2,3,5,6},{4}}
 => [2,3,5,4,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 16
{{1,3,5},{2,4,6}}
 => {{1,2,5},{3,4,6}}
 => [2,5,4,6,1,3] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 2
{{1,3},{2,4,5,6}}
 => {{1,2,3,4,6},{5}}
 => [2,3,4,6,5,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 8
{{1,6},{2,3,4,5}}
 => {{1,6},{2,3,4,5}}
 => [6,3,4,5,2,1] => ([(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)
 => ? = 4
{{1,6},{2,3,5},{4}}
 => {{1,6},{2,4,5},{3}}
 => [6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
{{1,4},{2,5},{3,6}}
 => {{1,2,4},{3,6},{5}}
 => [2,4,6,1,5,3] => ([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 2
{{1,4},{2,6},{3,5}}
 => {{1,4},{2,3,6},{5}}
 => [4,3,6,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = 6
{{1,5},{2,4},{3,6}}
 => {{1,2,6},{3,5},{4}}
 => [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
{{1,6},{2,4,5},{3}}
 => {{1,6},{2,4},{3,5}}
 => [6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
{{1,6},{2,4},{3,5}}
 => {{1,6},{2,3,5},{4}}
 => [6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
{{1,5},{2,6},{3,4}}
 => {{1,3,4},{2,6},{5}}
 => [3,6,4,1,5,2] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
{{1,6},{2,5},{3,4}}
 => {{1,6},{2,5},{3,4}}
 => [6,5,4,3,2,1] => ([(0,1),(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)
 => 6
{{1,5},{2,6},{3},{4}}
 => {{1,4},{2,6},{3},{5}}
 => [4,6,3,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 3
{{1,6},{2,5},{3},{4}}
 => {{1,6},{2,5},{3},{4}}
 => [6,5,3,4,2,1] => ([(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)
 => ? = 5
{{1,6},{2},{3},{4},{5}}
 => {{1,6},{2},{3},{4},{5}}
 => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
{{1},{2},{3},{4},{5},{6}}
 => {{1},{2},{3},{4},{5},{6}}
 => [1,2,3,4,5,6] => ([],6)
 => 1
{{1,2,3,4,5,6,7}}
 => {{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2
{{1,2,3,4,6},{5,7}}
 => {{1,2,4,5,6,7},{3}}
 => [2,4,3,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 8
{{1,2,3,5},{4,6,7}}
 => {{1,2,3,5,6,7},{4}}
 => [2,3,5,4,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 16
{{1,2,4,6},{3,5,7}}
 => {{1,2,5},{3,4,6,7}}
 => [2,5,4,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 8
{{1,2,4},{3,5,6,7}}
 => {{1,2,3,4,6,7},{5}}
 => [2,3,4,6,5,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 16
{{1,2,5},{3,6},{4,7}}
 => {{1,2,4},{3,6,7},{5}}
 => [2,4,6,1,5,7,3] => ([(0,6),(1,4),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
 => ? = 8
{{1,3,5,7},{2,4,6}}
 => {{1,4,5,7},{2,3,6}}
 => [4,3,6,5,7,2,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
{{1,3,5},{2,4,6,7}}
 => {{1,2,3,6},{4,5,7}}
 => [2,3,6,5,7,1,4] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 8
{{1,3},{2,4,5,6,7}}
 => {{1,2,3,4,5,7},{6}}
 => [2,3,4,5,7,6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 8
{{1,3},{2,4,6},{5,7}}
 => {{1,2,4,5,7},{3},{6}}
 => [2,4,3,5,7,6,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 16
{{1,7},{2,3,4,5,6}}
 => {{1,7},{2,3,4,5,6}}
 => [7,3,4,5,6,2,1] => ([(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)
 => ? = 4
{{1,7},{2,3,4,6},{5}}
 => {{1,7},{2,4,5,6},{3}}
 => [7,4,3,5,6,2,1] => ([(0,4),(0,5),(0,6),(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)
 => ? = 6
{{1,5},{2,3,7},{4,6}}
 => {{1,4},{2,3,7},{5,6}}
 => [4,3,7,1,6,5,2] => ([(0,1),(0,2),(0,6),(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 7
{{1,6},{2,3,5},{4,7}}
 => {{1,2,7},{3,5,6},{4}}
 => [2,7,5,4,6,3,1] => ([(0,6),(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)
 => ? = 7
{{1,7},{2,3,5,6},{4}}
 => {{1,7},{2,4},{3,5,6}}
 => [7,4,5,2,6,3,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
{{1,7},{2,3,5},{4,6}}
 => {{1,7},{2,3,5,6},{4}}
 => [7,3,5,4,6,2,1] => ([(0,4),(0,5),(0,6),(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)
 => ? = 6
{{1,7},{2,3,6},{4,5}}
 => {{1,7},{2,5,6},{3,4}}
 => [7,5,4,3,6,2,1] => ([(0,4),(0,5),(0,6),(1,2),(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)
 => ? = 7
{{1,7},{2,3,6},{4},{5}}
 => {{1,7},{2,5,6},{3},{4}}
 => [7,5,3,4,6,2,1] => ([(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)
 => ? = 6
{{1,4,5},{2,7},{3,6}}
 => {{1,4,7},{2,3,6},{5}}
 => [4,3,6,7,5,2,1] => ([(0,1),(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)
 => ? = 7
{{1,4,6},{2,5},{3,7}}
 => {{1,2,6},{3,5},{4,7}}
 => [2,6,5,7,3,1,4] => ([(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
{{1,4,7},{2,5},{3,6}}
 => {{1,6},{2,3,5},{4,7}}
 => [6,3,5,7,2,1,4] => ([(0,2),(0,4),(0,5),(1,4),(1,5),(1,6),(2,3),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
{{1,4},{2,5},{3,6,7}}
 => {{1,2,3,5},{4,7},{6}}
 => [2,3,5,7,1,6,4] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6)],7)
 => ? = 8
{{1,4,7},{2,6},{3,5}}
 => {{1,6},{2,5},{3,4,7}}
 => [6,5,4,7,2,1,3] => ([(0,1),(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
{{1,4},{2,7},{3,5,6}}
 => {{1,5},{2,3,4,7},{6}}
 => [5,3,4,7,1,6,2] => ([(0,3),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
 => ? = 7
{{1,4,7},{2,6},{3},{5}}
 => {{1,6},{2,5},{3},{4,7}}
 => [6,5,3,7,2,1,4] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 7
{{1,4},{2,7},{3,6},{5}}
 => {{1,5},{2,4,7},{3},{6}}
 => [5,4,3,7,1,6,2] => ([(0,1),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 7
{{1,5,6},{2,4},{3,7}}
 => {{1,2,5},{3,7},{4,6}}
 => [2,5,7,6,1,4,3] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5)],7)
 => ? = 7
{{1,5},{2,4,7},{3,6}}
 => {{1,4,6},{2,3,7},{5}}
 => [4,3,7,6,5,1,2] => ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 6
{{1,7},{2,6},{3,5},{4}}
 => {{1,7},{2,6},{3,5},{4}}
 => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(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)
 => 7

Description

The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of

$q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number

$HG(G)$ of a graph

$G$ is the largest integer

$q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of

$q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.

Matching statistic: St001889
(load all 2 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001889: Signed permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 56%

Values

{{1}}
 => [1] => [1] => [1] => 1
{{1,2}}
 => [2,1] => [1,2] => [1,2] => 2
{{1},{2}}
 => [1,2] => [2,1] => [2,1] => 1
{{1,2,3}}
 => [2,3,1] => [1,3,2] => [1,3,2] => 2
{{1,3},{2}}
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
{{1},{2},{3}}
 => [1,2,3] => [3,2,1] => [3,2,1] => 1
{{1,2,3,4}}
 => [2,3,4,1] => [1,4,3,2] => [1,4,3,2] => 2
{{1,3},{2,4}}
 => [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2
{{1,4},{2,3}}
 => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 4
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 3
{{1},{2},{3},{4}}
 => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,5,4,3,2] => [1,5,4,3,2] => 2
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [3,1,5,4,2] => [3,1,5,4,2] => ? = 8
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,2,5,4,3] => [1,2,5,4,3] => 3
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 8
{{1,5},{2,3,4}}
 => [5,3,4,2,1] => [1,2,4,3,5] => [1,2,4,3,5] => 4
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 3
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5
{{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => [1,4,3,2,5] => [1,4,3,2,5] => 3
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => ? = 2
{{1,2,3,5},{4,6}}
 => [2,3,5,6,1,4] => [4,1,6,5,3,2] => [4,1,6,5,3,2] => ? = 8
{{1,2,4},{3,5,6}}
 => [2,4,5,1,6,3] => [3,6,1,5,4,2] => [3,6,1,5,4,2] => ? = 16
{{1,3,5},{2,4,6}}
 => [3,4,5,6,1,2] => [2,1,6,5,4,3] => [2,1,6,5,4,3] => ? = 2
{{1,3},{2,4,5,6}}
 => [3,4,1,5,6,2] => [2,6,5,1,4,3] => [2,6,5,1,4,3] => ? = 8
{{1,6},{2,3,4,5}}
 => [6,3,4,5,2,1] => [1,2,5,4,3,6] => [1,2,5,4,3,6] => ? = 4
{{1,6},{2,3,5},{4}}
 => [6,3,5,4,2,1] => [1,2,4,5,3,6] => [1,2,4,5,3,6] => ? = 6
{{1,4},{2,5},{3,6}}
 => [4,5,6,1,2,3] => [3,2,1,6,5,4] => [3,2,1,6,5,4] => ? = 2
{{1,4},{2,6},{3,5}}
 => [4,6,5,1,3,2] => [2,3,1,5,6,4] => [2,3,1,5,6,4] => ? = 6
{{1,5},{2,4},{3,6}}
 => [5,4,6,2,1,3] => [3,1,2,6,4,5] => [3,1,2,6,4,5] => ? = 6
{{1,6},{2,4,5},{3}}
 => [6,4,3,5,2,1] => [1,2,5,3,4,6] => [1,2,5,3,4,6] => ? = 6
{{1,6},{2,4},{3,5}}
 => [6,4,5,2,3,1] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => ? = 4
{{1,5},{2,6},{3,4}}
 => [5,6,4,3,1,2] => [2,1,3,4,6,5] => [2,1,3,4,6,5] => ? = 4
{{1,6},{2,5},{3,4}}
 => [6,5,4,3,2,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 6
{{1,5},{2,6},{3},{4}}
 => [5,6,3,4,1,2] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => ? = 3
{{1,6},{2,5},{3},{4}}
 => [6,5,3,4,2,1] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ? = 5
{{1,6},{2},{3},{4},{5}}
 => [6,2,3,4,5,1] => [1,5,4,3,2,6] => [1,5,4,3,2,6] => ? = 3
{{1},{2},{3},{4},{5},{6}}
 => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ? = 1
{{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 2
{{1,2,3,4,6},{5,7}}
 => [2,3,4,6,7,1,5] => [5,1,7,6,4,3,2] => [5,1,7,6,4,3,2] => ? = 8
{{1,2,3,5},{4,6,7}}
 => [2,3,5,6,1,7,4] => [4,7,1,6,5,3,2] => [4,7,1,6,5,3,2] => ? = 16
{{1,2,4,6},{3,5,7}}
 => [2,4,5,6,7,1,3] => [3,1,7,6,5,4,2] => [3,1,7,6,5,4,2] => ? = 8
{{1,2,4},{3,5,6,7}}
 => [2,4,5,1,6,7,3] => [3,7,6,1,5,4,2] => [3,7,6,1,5,4,2] => ? = 16
{{1,2,5},{3,6},{4,7}}
 => [2,5,6,7,1,3,4] => [4,3,1,7,6,5,2] => [4,3,1,7,6,5,2] => ? = 8
{{1,3,5,7},{2,4,6}}
 => [3,4,5,6,7,2,1] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 3
{{1,3,5},{2,4,6,7}}
 => [3,4,5,6,1,7,2] => [2,7,1,6,5,4,3] => [2,7,1,6,5,4,3] => ? = 8
{{1,3},{2,4,5,6,7}}
 => [3,4,1,5,6,7,2] => [2,7,6,5,1,4,3] => [2,7,6,5,1,4,3] => ? = 8
{{1,3},{2,4,6},{5,7}}
 => [3,4,1,6,7,2,5] => [5,2,7,6,1,4,3] => [5,2,7,6,1,4,3] => ? = 16
{{1,7},{2,3,4,5,6}}
 => [7,3,4,5,6,2,1] => [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => ? = 4
{{1,7},{2,3,4,6},{5}}
 => [7,3,4,6,5,2,1] => [1,2,5,6,4,3,7] => [1,2,5,6,4,3,7] => ? = 6
{{1,5},{2,3,7},{4,6}}
 => [5,3,7,6,1,4,2] => [2,4,1,6,7,3,5] => [2,4,1,6,7,3,5] => ? = 7
{{1,6},{2,3,5},{4,7}}
 => [6,3,5,7,2,1,4] => [4,1,2,7,5,3,6] => [4,1,2,7,5,3,6] => ? = 7
{{1,7},{2,3,5,6},{4}}
 => [7,3,5,4,6,2,1] => [1,2,6,4,5,3,7] => [1,2,6,4,5,3,7] => ? = 6
{{1,7},{2,3,5},{4,6}}
 => [7,3,5,6,2,4,1] => [1,4,2,6,5,3,7] => [1,4,2,6,5,3,7] => ? = 6
{{1,7},{2,3,6},{4,5}}
 => [7,3,6,5,4,2,1] => [1,2,4,5,6,3,7] => [1,2,4,5,6,3,7] => ? = 7
{{1,7},{2,3,6},{4},{5}}
 => [7,3,6,4,5,2,1] => [1,2,5,4,6,3,7] => [1,2,5,4,6,3,7] => ? = 6
{{1,4,5},{2,7},{3,6}}
 => [4,7,6,5,1,3,2] => [2,3,1,5,6,7,4] => [2,3,1,5,6,7,4] => ? = 7
{{1,4,6},{2,5},{3,7}}
 => [4,5,7,6,2,1,3] => [3,1,2,6,7,5,4] => [3,1,2,6,7,5,4] => ? = 6
{{1,4,7},{2,5},{3,6}}
 => [4,5,6,7,2,3,1] => [1,3,2,7,6,5,4] => [1,3,2,7,6,5,4] => ? = 3
{{1,4},{2,5},{3,6,7}}
 => [4,5,6,1,2,7,3] => [3,7,2,1,6,5,4] => [3,7,2,1,6,5,4] => ? = 8
{{1,4,7},{2,6},{3,5}}
 => [4,6,5,7,3,2,1] => [1,2,3,7,5,6,4] => [1,2,3,7,5,6,4] => ? = 6
{{1,4},{2,7},{3,5,6}}
 => [4,7,5,1,6,3,2] => [2,3,6,1,5,7,4] => [2,3,6,1,5,7,4] => ? = 7
{{1,4,7},{2,6},{3},{5}}
 => [4,6,3,7,5,2,1] => [1,2,5,7,3,6,4] => [1,2,5,7,3,6,4] => ? = 7
{{1,4},{2,7},{3,6},{5}}
 => [4,7,6,1,5,3,2] => [2,3,5,1,6,7,4] => [2,3,5,1,6,7,4] => ? = 7
{{1,5,6},{2,4},{3,7}}
 => [5,4,7,2,6,1,3] => [3,1,6,2,7,4,5] => [3,1,6,2,7,4,5] => ? = 7
{{1,5},{2,4,7},{3,6}}
 => [5,4,6,7,1,3,2] => [2,3,1,7,6,4,5] => [2,3,1,7,6,4,5] => ? = 6

Description

The size of the connectivity set of a signed permutation. According to [1], the connectivity set of a signed permutation

$w\in\mathfrak H_n$ is

$n$ minus the number of generators appearing in any reduced word for

$w$ . The connectivity set can be defined for arbitrary Coxeter systems. For permutations, see [[St000234]]. For the number of connected elements in a Coxeter system see [[St001888]].

Matching statistic: St001812

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001812: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 44%

Values

{{1}}
 => [1] => ([],1)
 => ([(0,1)],2)
 => 1
{{1,2}}
 => [2,1] => ([(0,1)],2)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
{{1},{2}}
 => [1,2] => ([],2)
 => ([(0,2),(1,2)],3)
 => 1
{{1,2,3}}
 => [2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
{{1,3},{2}}
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
{{1},{2},{3}}
 => [1,2,3] => ([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
{{1,2,3,4}}
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
{{1,3},{2,4}}
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2
{{1,4},{2,3}}
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
{{1,4},{2},{3}}
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
{{1},{2},{3},{4}}
 => [1,2,3,4] => ([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 8
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => ? = 3
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 8
{{1,5},{2,3,4}}
 => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => ? = 4
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => ([(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),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
{{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => ? = 3
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
{{1,2,3,5},{4,6}}
 => [2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 8
{{1,2,4},{3,5,6}}
 => [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 16
{{1,3,5},{2,4,6}}
 => [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 2
{{1,3},{2,4,5,6}}
 => [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 8
{{1,6},{2,3,4,5}}
 => [6,3,4,5,2,1] => ([(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)
 => ([(0,3),(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)
 => ? = 4
{{1,6},{2,3,5},{4}}
 => [6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(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)
 => ? = 6
{{1,4},{2,5},{3,6}}
 => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 2
{{1,4},{2,6},{3,5}}
 => [4,6,5,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
{{1,5},{2,4},{3,6}}
 => [5,4,6,2,1,3] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
{{1,6},{2,4,5},{3}}
 => [6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(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)
 => ? = 6
{{1,6},{2,4},{3,5}}
 => [6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(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)
 => ? = 4
{{1,5},{2,6},{3,4}}
 => [5,6,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(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)
 => ? = 4
{{1,6},{2,5},{3,4}}
 => [6,5,4,3,2,1] => ([(0,1),(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),(0,5),(0,6),(1,2),(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)
 => ? = 6
{{1,5},{2,6},{3},{4}}
 => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 3
{{1,6},{2,5},{3},{4}}
 => [6,5,3,4,2,1] => ([(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)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(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)
 => ? = 5
{{1,6},{2},{3},{4},{5}}
 => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => ? = 3
{{1},{2},{3},{4},{5},{6}}
 => [1,2,3,4,5,6] => ([],6)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 1
{{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
{{1,2,3,4,6},{5,7}}
 => [2,3,4,6,7,1,5] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 8
{{1,2,3,5},{4,6,7}}
 => [2,3,5,6,1,7,4] => ([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 16
{{1,2,4,6},{3,5,7}}
 => [2,4,5,6,7,1,3] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 8
{{1,2,4},{3,5,6,7}}
 => [2,4,5,1,6,7,3] => ([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 16
{{1,2,5},{3,6},{4,7}}
 => [2,5,6,7,1,3,4] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(1,3),(1,4),(1,5),(1,7),(2,3),(2,4),(2,5),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8
{{1,3,5,7},{2,4,6}}
 => [3,4,5,6,7,2,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
{{1,3,5},{2,4,6,7}}
 => [3,4,5,6,1,7,2] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 8
{{1,3},{2,4,5,6,7}}
 => [3,4,1,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 8
{{1,3},{2,4,6},{5,7}}
 => [3,4,1,6,7,2,5] => ([(0,4),(0,5),(1,2),(1,3),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,7),(1,2),(1,3),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 16
{{1,7},{2,3,4,5,6}}
 => [7,3,4,5,6,2,1] => ([(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)
 => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
{{1,7},{2,3,4,6},{5}}
 => [7,3,4,6,5,2,1] => ([(0,4),(0,5),(0,6),(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)
 => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6
{{1,5},{2,3,7},{4,6}}
 => [5,3,7,6,1,4,2] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7
{{1,6},{2,3,5},{4,7}}
 => [6,3,5,7,2,1,4] => ([(0,2),(0,4),(0,5),(1,4),(1,5),(1,6),(2,3),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,2),(0,4),(0,5),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7
{{1,7},{2,3,5,6},{4}}
 => [7,3,5,4,6,2,1] => ([(0,4),(0,5),(0,6),(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)
 => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6
{{1,7},{2,3,5},{4,6}}
 => [7,3,5,6,2,4,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6
{{1,7},{2,3,6},{4,5}}
 => [7,3,6,5,4,2,1] => ([(0,4),(0,5),(0,6),(1,2),(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)
 => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7
{{1,7},{2,3,6},{4},{5}}
 => [7,3,6,4,5,2,1] => ([(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)
 => ([(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6
{{1,4,5},{2,7},{3,6}}
 => [4,7,6,5,1,3,2] => ([(0,1),(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7
{{1,4,6},{2,5},{3,7}}
 => [4,5,7,6,2,1,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6
{{1,4,7},{2,5},{3,6}}
 => [4,5,6,7,2,3,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
{{1,4},{2,5},{3,6,7}}
 => [4,5,6,1,2,7,3] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(1,3),(1,4),(1,5),(1,7),(2,3),(2,4),(2,5),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8
{{1,4,7},{2,6},{3,5}}
 => [4,6,5,7,3,2,1] => ([(0,4),(0,5),(0,6),(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)
 => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6
{{1,4},{2,7},{3,5,6}}
 => [4,7,5,1,6,3,2] => ([(0,2),(0,4),(0,5),(1,4),(1,5),(1,6),(2,3),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,2),(0,4),(0,5),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7
{{1,4,7},{2,6},{3},{5}}
 => [4,6,3,7,5,2,1] => ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(0,7),(1,3),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7

Description

The biclique partition number of a graph. The biclique partition number of a graph is the minimum number of pairwise edge disjoint complete bipartite subgraphs so that each edge belongs to exactly one of them. A theorem of Graham and Pollak [1] asserts that the complete graph

$K_n$ has biclique partition number

$n - 1$ .

Matching statistic: St001896
(load all 9 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001896: Signed permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 44%

Values

{{1}}
 => [1] => [1] => [1] => 0 = 1 - 1
{{1,2}}
 => [2,1] => [2,1] => [2,1] => 1 = 2 - 1
{{1},{2}}
 => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
{{1,2,3}}
 => [2,3,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => [3,2,1] => 2 = 3 - 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
{{1,2,3,4}}
 => [2,3,4,1] => [2,3,4,1] => [2,3,4,1] => 1 = 2 - 1
{{1,3},{2,4}}
 => [3,4,1,2] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3 = 4 - 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,4,3,1] => [2,4,3,1] => 2 = 3 - 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 2 - 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 8 - 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 3 - 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 8 - 1
{{1,5},{2,3,4}}
 => [5,3,4,2,1] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 4 - 1
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => [1,4,5,3,2] => [1,4,5,3,2] => 2 = 3 - 1
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 5 - 1
{{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 3 - 1
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => [2,3,4,5,6,1] => [2,3,4,5,6,1] => ? = 2 - 1
{{1,2,3,5},{4,6}}
 => [2,3,5,6,1,4] => [2,3,1,5,6,4] => [2,3,1,5,6,4] => ? = 8 - 1
{{1,2,4},{3,5,6}}
 => [2,4,5,1,6,3] => [4,2,5,6,1,3] => [4,2,5,6,1,3] => ? = 16 - 1
{{1,3,5},{2,4,6}}
 => [3,4,5,6,1,2] => [1,3,4,5,6,2] => [1,3,4,5,6,2] => ? = 2 - 1
{{1,3},{2,4,5,6}}
 => [3,4,1,5,6,2] => [3,4,5,1,6,2] => [3,4,5,1,6,2] => ? = 8 - 1
{{1,6},{2,3,4,5}}
 => [6,3,4,5,2,1] => [3,4,6,5,2,1] => [3,4,6,5,2,1] => ? = 4 - 1
{{1,6},{2,3,5},{4}}
 => [6,3,5,4,2,1] => [6,3,5,4,2,1] => [6,3,5,4,2,1] => ? = 6 - 1
{{1,4},{2,5},{3,6}}
 => [4,5,6,1,2,3] => [1,2,4,5,6,3] => [1,2,4,5,6,3] => ? = 2 - 1
{{1,4},{2,6},{3,5}}
 => [4,6,5,1,3,2] => [6,1,4,5,3,2] => [6,1,4,5,3,2] => ? = 6 - 1
{{1,5},{2,4},{3,6}}
 => [5,4,6,2,1,3] => [2,5,4,6,1,3] => [2,5,4,6,1,3] => ? = 6 - 1
{{1,6},{2,4,5},{3}}
 => [6,4,3,5,2,1] => [4,6,3,5,2,1] => [4,6,3,5,2,1] => ? = 6 - 1
{{1,6},{2,4},{3,5}}
 => [6,4,5,2,3,1] => [2,4,6,5,3,1] => [2,4,6,5,3,1] => ? = 4 - 1
{{1,5},{2,6},{3,4}}
 => [5,6,4,3,1,2] => [1,5,6,4,3,2] => [1,5,6,4,3,2] => ? = 4 - 1
{{1,6},{2,5},{3,4}}
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ? = 6 - 1
{{1,5},{2,6},{3},{4}}
 => [5,6,3,4,1,2] => [1,3,5,6,4,2] => [1,3,5,6,4,2] => ? = 3 - 1
{{1,6},{2,5},{3},{4}}
 => [6,5,3,4,2,1] => [3,6,5,4,2,1] => [3,6,5,4,2,1] => ? = 5 - 1
{{1,6},{2},{3},{4},{5}}
 => [6,2,3,4,5,1] => [2,3,4,6,5,1] => [2,3,4,6,5,1] => ? = 3 - 1
{{1},{2},{3},{4},{5},{6}}
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
{{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => [2,3,4,5,6,7,1] => [2,3,4,5,6,7,1] => ? = 2 - 1
{{1,2,3,4,6},{5,7}}
 => [2,3,4,6,7,1,5] => [2,3,4,1,6,7,5] => [2,3,4,1,6,7,5] => ? = 8 - 1
{{1,2,3,5},{4,6,7}}
 => [2,3,5,6,1,7,4] => [5,2,3,6,7,1,4] => [5,2,3,6,7,1,4] => ? = 16 - 1
{{1,2,4,6},{3,5,7}}
 => [2,4,5,6,7,1,3] => [2,1,4,5,6,7,3] => [2,1,4,5,6,7,3] => ? = 8 - 1
{{1,2,4},{3,5,6,7}}
 => [2,4,5,1,6,7,3] => [4,2,5,6,1,7,3] => [4,2,5,6,1,7,3] => ? = 16 - 1
{{1,2,5},{3,6},{4,7}}
 => [2,5,6,7,1,3,4] => [2,1,3,5,6,7,4] => [2,1,3,5,6,7,4] => ? = 8 - 1
{{1,3,5,7},{2,4,6}}
 => [3,4,5,6,7,2,1] => [3,4,5,6,7,2,1] => [3,4,5,6,7,2,1] => ? = 3 - 1
{{1,3,5},{2,4,6,7}}
 => [3,4,5,6,1,7,2] => [3,4,5,6,7,1,2] => [3,4,5,6,7,1,2] => ? = 8 - 1
{{1,3},{2,4,5,6,7}}
 => [3,4,1,5,6,7,2] => [3,4,5,1,6,7,2] => [3,4,5,1,6,7,2] => ? = 8 - 1
{{1,3},{2,4,6},{5,7}}
 => [3,4,1,6,7,2,5] => [3,4,1,6,2,7,5] => [3,4,1,6,2,7,5] => ? = 16 - 1
{{1,7},{2,3,4,5,6}}
 => [7,3,4,5,6,2,1] => [3,4,5,7,6,2,1] => [3,4,5,7,6,2,1] => ? = 4 - 1
{{1,7},{2,3,4,6},{5}}
 => [7,3,4,6,5,2,1] => [7,3,4,6,5,2,1] => [7,3,4,6,5,2,1] => ? = 6 - 1
{{1,5},{2,3,7},{4,6}}
 => [5,3,7,6,1,4,2] => [3,7,5,6,1,4,2] => [3,7,5,6,1,4,2] => ? = 7 - 1
{{1,6},{2,3,5},{4,7}}
 => [6,3,5,7,2,1,4] => [3,2,6,5,7,1,4] => [3,2,6,5,7,1,4] => ? = 7 - 1
{{1,7},{2,3,5,6},{4}}
 => [7,3,5,4,6,2,1] => [3,7,5,4,6,2,1] => [3,7,5,4,6,2,1] => ? = 6 - 1
{{1,7},{2,3,5},{4,6}}
 => [7,3,5,6,2,4,1] => [3,2,5,7,6,4,1] => [3,2,5,7,6,4,1] => ? = 6 - 1
{{1,7},{2,3,6},{4,5}}
 => [7,3,6,5,4,2,1] => [7,6,3,5,4,2,1] => [7,6,3,5,4,2,1] => ? = 7 - 1
{{1,7},{2,3,6},{4},{5}}
 => [7,3,6,4,5,2,1] => [3,7,4,6,5,2,1] => [3,7,4,6,5,2,1] => ? = 6 - 1
{{1,4,5},{2,7},{3,6}}
 => [4,7,6,5,1,3,2] => [7,1,6,4,5,3,2] => [7,1,6,4,5,3,2] => ? = 7 - 1
{{1,4,6},{2,5},{3,7}}
 => [4,5,7,6,2,1,3] => [2,7,4,5,6,1,3] => [2,7,4,5,6,1,3] => ? = 6 - 1
{{1,4,7},{2,5},{3,6}}
 => [4,5,6,7,2,3,1] => [2,4,5,6,7,3,1] => [2,4,5,6,7,3,1] => ? = 3 - 1
{{1,4},{2,5},{3,6,7}}
 => [4,5,6,1,2,7,3] => [4,1,5,6,7,2,3] => [4,1,5,6,7,2,3] => ? = 8 - 1
{{1,4,7},{2,6},{3,5}}
 => [4,6,5,7,3,2,1] => [6,4,5,7,3,2,1] => [6,4,5,7,3,2,1] => ? = 6 - 1
{{1,4},{2,7},{3,5,6}}
 => [4,7,5,1,6,3,2] => [4,7,5,6,3,1,2] => [4,7,5,6,3,1,2] => ? = 7 - 1
{{1,4,7},{2,6},{3},{5}}
 => [4,6,3,7,5,2,1] => [6,4,7,3,5,2,1] => [6,4,7,3,5,2,1] => ? = 7 - 1

Description

The number of right descents of a signed permutations. An index is a right descent if it is a left descent of the inverse signed permutation.

Matching statistic: St001613

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001613: Lattices ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 33%

Values

{{1}}
 => [1] => ([],1)
 => ([(0,1)],2)
 => 1
{{1,2}}
 => [2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
{{1},{2}}
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1
{{1,2,3}}
 => [2,3,1] => ([(1,2)],3)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
{{1,3},{2}}
 => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 3
{{1},{2},{3}}
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
{{1,2,3,4}}
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => 2
{{1,3},{2,4}}
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => 2
{{1,4},{2,3}}
 => [4,3,2,1] => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 4
{{1,4},{2},{3}}
 => [4,2,3,1] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? = 3
{{1},{2},{3},{4}}
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 2
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 8
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
 => ? = 3
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
 => ? = 8
{{1,5},{2,3,4}}
 => [5,3,4,2,1] => ([(3,4)],5)
 => ?
 => ? = 4
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => ([(0,3),(0,4),(0,5),(1,8),(1,10),(2,7),(2,9),(3,11),(3,12),(4,2),(4,11),(4,13),(5,1),(5,12),(5,13),(6,17),(7,15),(8,16),(9,6),(9,15),(10,6),(10,16),(11,7),(11,14),(12,8),(12,14),(13,9),(13,10),(13,14),(14,15),(14,16),(15,17),(16,17)],18)
 => ? = 3
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => ([],5)
 => ?
 => ? = 5
{{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
 => ? = 3
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => ([(1,5),(3,4),(4,2),(5,3)],6)
 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
 => ? = 2
{{1,2,3,5},{4,6}}
 => [2,3,5,6,1,4] => ([(0,5),(1,4),(3,2),(4,3),(4,5)],6)
 => ([(0,3),(0,6),(1,9),(2,10),(3,7),(4,5),(4,12),(5,1),(5,8),(6,4),(6,7),(7,12),(8,9),(8,10),(9,11),(10,11),(12,2),(12,8)],13)
 => ? = 8
{{1,2,4},{3,5,6}}
 => [2,4,5,1,6,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(3,5)],6)
 => ([(0,4),(0,6),(1,12),(2,8),(3,10),(4,11),(5,1),(5,7),(6,5),(6,11),(7,10),(7,12),(9,8),(10,9),(11,3),(11,7),(12,2),(12,9)],13)
 => ? = 16
{{1,3,5},{2,4,6}}
 => [3,4,5,6,1,2] => ([(0,5),(1,3),(4,2),(5,4)],6)
 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
 => ? = 2
{{1,3},{2,4,5,6}}
 => [3,4,1,5,6,2] => ([(0,4),(1,3),(1,5),(4,5),(5,2)],6)
 => ([(0,5),(0,6),(1,8),(2,9),(3,10),(4,1),(4,11),(5,2),(5,7),(6,3),(6,7),(7,9),(7,10),(9,12),(10,4),(10,12),(11,8),(12,11)],13)
 => ? = 8
{{1,6},{2,3,4,5}}
 => [6,3,4,5,2,1] => ([(3,4),(4,5)],6)
 => ?
 => ? = 4
{{1,6},{2,3,5},{4}}
 => [6,3,5,4,2,1] => ([(3,4),(3,5)],6)
 => ?
 => ? = 6
{{1,4},{2,5},{3,6}}
 => [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
 => ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
 => ? = 2
{{1,4},{2,6},{3,5}}
 => [4,6,5,1,3,2] => ([(0,4),(0,5),(1,2),(1,3)],6)
 => ?
 => ? = 6
{{1,5},{2,4},{3,6}}
 => [5,4,6,2,1,3] => ([(0,5),(1,5),(2,4),(3,4)],6)
 => ?
 => ? = 6
{{1,6},{2,4,5},{3}}
 => [6,4,3,5,2,1] => ([(3,5),(4,5)],6)
 => ?
 => ? = 6
{{1,6},{2,4},{3,5}}
 => [6,4,5,2,3,1] => ([(2,5),(3,4)],6)
 => ?
 => ? = 4
{{1,5},{2,6},{3,4}}
 => [5,6,4,3,1,2] => ([(2,5),(3,4)],6)
 => ?
 => ? = 4
{{1,6},{2,5},{3,4}}
 => [6,5,4,3,2,1] => ([],6)
 => ?
 => ? = 6
{{1,5},{2,6},{3},{4}}
 => [5,6,3,4,1,2] => ([(0,5),(1,4),(2,3)],6)
 => ?
 => ? = 3
{{1,6},{2,5},{3},{4}}
 => [6,5,3,4,2,1] => ([(4,5)],6)
 => ?
 => ? = 5
{{1,6},{2},{3},{4},{5}}
 => [6,2,3,4,5,1] => ([(2,3),(3,5),(5,4)],6)
 => ([(0,4),(0,5),(0,6),(1,3),(1,12),(1,13),(2,8),(2,9),(3,2),(3,14),(3,15),(4,7),(4,11),(5,7),(5,10),(6,1),(6,10),(6,11),(7,16),(8,17),(9,17),(10,12),(10,16),(11,13),(11,16),(12,14),(12,18),(13,15),(13,18),(14,8),(14,19),(15,9),(15,19),(16,18),(18,19),(19,17)],20)
 => ? = 3
{{1},{2},{3},{4},{5},{6}}
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
{{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
 => ([(0,2),(0,7),(1,9),(2,8),(3,4),(3,11),(4,6),(4,10),(5,3),(5,13),(6,1),(6,12),(7,5),(7,8),(8,13),(10,12),(11,10),(12,9),(13,11)],14)
 => ? = 2
{{1,2,3,4,6},{5,7}}
 => [2,3,4,6,7,1,5] => ([(0,6),(1,4),(3,2),(4,5),(5,3),(5,6)],7)
 => ([(0,3),(0,7),(1,10),(2,9),(3,8),(4,6),(4,12),(5,4),(5,11),(6,1),(6,13),(7,5),(7,8),(8,11),(9,14),(10,14),(11,12),(12,2),(12,13),(13,9),(13,10)],15)
 => ? = 8
{{1,2,3,5},{4,6,7}}
 => [2,3,5,6,1,7,4] => ([(0,5),(0,6),(1,4),(2,6),(3,2),(4,3),(4,5)],7)
 => ([(0,4),(0,7),(1,14),(2,10),(3,12),(4,8),(5,6),(5,13),(6,1),(6,9),(7,5),(7,8),(8,13),(9,12),(9,14),(11,10),(12,11),(13,3),(13,9),(14,2),(14,11)],15)
 => ? = 16
{{1,2,4,6},{3,5,7}}
 => [2,4,5,6,7,1,3] => ([(0,6),(1,5),(1,6),(3,4),(4,2),(5,3)],7)
 => ([(0,6),(0,7),(1,9),(2,5),(2,14),(3,13),(4,3),(4,16),(5,4),(5,15),(6,10),(7,2),(7,10),(9,11),(10,1),(10,14),(11,12),(12,8),(13,8),(14,9),(14,15),(15,11),(15,16),(16,12),(16,13)],17)
 => ? = 8
{{1,2,4},{3,5,6,7}}
 => [2,4,5,1,6,7,3] => ([(0,5),(0,6),(1,4),(1,5),(3,6),(4,3),(6,2)],7)
 => ([(0,4),(0,7),(1,9),(2,10),(3,11),(4,14),(5,3),(5,8),(6,1),(6,12),(7,5),(7,14),(8,10),(8,11),(10,13),(11,6),(11,13),(12,9),(13,12),(14,2),(14,8)],15)
 => ? = 16
{{1,2,5},{3,6},{4,7}}
 => [2,5,6,7,1,3,4] => ([(0,6),(1,5),(1,6),(4,2),(5,4),(6,3)],7)
 => ([(0,6),(0,7),(1,5),(1,14),(2,3),(2,16),(3,13),(4,12),(5,4),(5,17),(6,9),(7,1),(7,9),(9,2),(9,14),(10,8),(11,8),(12,11),(13,10),(14,16),(14,17),(15,10),(15,11),(16,13),(16,15),(17,12),(17,15)],18)
 => ? = 8
{{1,3,5,7},{2,4,6}}
 => [3,4,5,6,7,2,1] => ([(2,6),(4,5),(5,3),(6,4)],7)
 => ?
 => ? = 3
{{1,3,5},{2,4,6,7}}
 => [3,4,5,6,1,7,2] => ([(0,5),(1,3),(1,6),(2,6),(4,2),(5,4)],7)
 => ([(0,6),(0,7),(1,5),(1,16),(2,13),(3,8),(4,12),(5,4),(5,14),(6,1),(6,15),(7,2),(7,15),(9,10),(10,11),(11,8),(12,3),(12,11),(13,9),(14,10),(14,12),(15,13),(15,16),(16,9),(16,14)],17)
 => ? = 8
{{1,3},{2,4,5,6,7}}
 => [3,4,1,5,6,7,2] => ([(0,4),(1,3),(1,6),(4,6),(5,2),(6,5)],7)
 => ([(0,6),(0,7),(1,14),(2,13),(3,9),(4,5),(4,10),(5,3),(5,11),(6,1),(6,8),(7,2),(7,8),(8,13),(8,14),(10,11),(11,9),(12,10),(13,12),(14,4),(14,12)],15)
 => ? = 8
{{1,3},{2,4,6},{5,7}}
 => [3,4,1,6,7,2,5] => ([(0,4),(1,3),(1,6),(3,5),(4,5),(4,6),(6,2)],7)
 => ([(0,6),(0,7),(1,12),(2,5),(2,15),(3,11),(4,13),(5,9),(6,4),(6,14),(7,3),(7,14),(9,8),(10,1),(10,15),(11,10),(12,8),(13,2),(13,10),(14,11),(14,13),(15,9),(15,12)],16)
 => ? = 16
{{1,7},{2,3,4,5,6}}
 => [7,3,4,5,6,2,1] => ([(3,4),(4,6),(6,5)],7)
 => ?
 => ? = 4
{{1,7},{2,3,4,6},{5}}
 => [7,3,4,6,5,2,1] => ([(3,4),(4,5),(4,6)],7)
 => ?
 => ? = 6
{{1,5},{2,3,7},{4,6}}
 => [5,3,7,6,1,4,2] => ([(0,4),(0,5),(1,4),(1,5),(1,6),(2,3),(2,6)],7)
 => ?
 => ? = 7
{{1,6},{2,3,5},{4,7}}
 => [6,3,5,7,2,1,4] => ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5)],7)
 => ?
 => ? = 7
{{1,7},{2,3,5,6},{4}}
 => [7,3,5,4,6,2,1] => ([(3,4),(3,5),(4,6),(5,6)],7)
 => ?
 => ? = 6
{{1,7},{2,3,5},{4,6}}
 => [7,3,5,6,2,4,1] => ([(2,6),(3,4),(3,6),(4,5)],7)
 => ?
 => ? = 6
{{1,7},{2,3,6},{4,5}}
 => [7,3,6,5,4,2,1] => ([(3,4),(3,5),(3,6)],7)
 => ?
 => ? = 7
{{1,7},{2,3,6},{4},{5}}
 => [7,3,6,4,5,2,1] => ([(3,4),(3,5),(5,6)],7)
 => ?
 => ? = 6
{{1,4,5},{2,7},{3,6}}
 => [4,7,6,5,1,3,2] => ([(0,5),(0,6),(1,2),(1,3),(1,4)],7)
 => ?
 => ? = 7
{{1,4,6},{2,5},{3,7}}
 => [4,5,7,6,2,1,3] => ([(0,3),(1,6),(2,6),(3,4),(3,5)],7)
 => ?
 => ? = 6
{{1,4,7},{2,5},{3,6}}
 => [4,5,6,7,2,3,1] => ([(1,6),(2,4),(5,3),(6,5)],7)
 => ?
 => ? = 3
{{1,4},{2,5},{3,6,7}}
 => [4,5,6,1,2,7,3] => ([(0,5),(1,4),(3,6),(4,3),(5,2),(5,6)],7)
 => ([(0,6),(0,7),(1,4),(1,16),(2,5),(2,17),(3,8),(4,9),(5,10),(6,1),(6,15),(7,2),(7,15),(9,11),(10,13),(11,12),(12,8),(13,3),(13,12),(14,11),(14,13),(15,16),(15,17),(16,9),(16,14),(17,10),(17,14)],18)
 => ? = 8
{{1,4,7},{2,6},{3,5}}
 => [4,6,5,7,3,2,1] => ([(3,4),(3,5),(4,6),(5,6)],7)
 => ?
 => ? = 6
{{1},{2},{3},{4},{5},{6},{7}}
 => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => 1

Description

The binary logarithm of the size of the center of a lattice. An element of a lattice is central if it is neutral and has a complement. The subposet induced by central elements is a Boolean lattice.

Matching statistic: St001881

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001881: Lattices ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 33%

Values

{{1}}
 => [1] => ([],1)
 => ([(0,1)],2)
 => 1
{{1,2}}
 => [2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
{{1},{2}}
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1
{{1,2,3}}
 => [2,3,1] => ([(1,2)],3)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
{{1,3},{2}}
 => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 3
{{1},{2},{3}}
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
{{1,2,3,4}}
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => 2
{{1,3},{2,4}}
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => 2
{{1,4},{2,3}}
 => [4,3,2,1] => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 4
{{1,4},{2},{3}}
 => [4,2,3,1] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? = 3
{{1},{2},{3},{4}}
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 2
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 8
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
 => ? = 3
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
 => ? = 8
{{1,5},{2,3,4}}
 => [5,3,4,2,1] => ([(3,4)],5)
 => ?
 => ? = 4
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => ([(0,3),(0,4),(0,5),(1,8),(1,10),(2,7),(2,9),(3,11),(3,12),(4,2),(4,11),(4,13),(5,1),(5,12),(5,13),(6,17),(7,15),(8,16),(9,6),(9,15),(10,6),(10,16),(11,7),(11,14),(12,8),(12,14),(13,9),(13,10),(13,14),(14,15),(14,16),(15,17),(16,17)],18)
 => ? = 3
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => ([],5)
 => ?
 => ? = 5
{{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
 => ? = 3
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => ([(1,5),(3,4),(4,2),(5,3)],6)
 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
 => ? = 2
{{1,2,3,5},{4,6}}
 => [2,3,5,6,1,4] => ([(0,5),(1,4),(3,2),(4,3),(4,5)],6)
 => ([(0,3),(0,6),(1,9),(2,10),(3,7),(4,5),(4,12),(5,1),(5,8),(6,4),(6,7),(7,12),(8,9),(8,10),(9,11),(10,11),(12,2),(12,8)],13)
 => ? = 8
{{1,2,4},{3,5,6}}
 => [2,4,5,1,6,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(3,5)],6)
 => ([(0,4),(0,6),(1,12),(2,8),(3,10),(4,11),(5,1),(5,7),(6,5),(6,11),(7,10),(7,12),(9,8),(10,9),(11,3),(11,7),(12,2),(12,9)],13)
 => ? = 16
{{1,3,5},{2,4,6}}
 => [3,4,5,6,1,2] => ([(0,5),(1,3),(4,2),(5,4)],6)
 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
 => ? = 2
{{1,3},{2,4,5,6}}
 => [3,4,1,5,6,2] => ([(0,4),(1,3),(1,5),(4,5),(5,2)],6)
 => ([(0,5),(0,6),(1,8),(2,9),(3,10),(4,1),(4,11),(5,2),(5,7),(6,3),(6,7),(7,9),(7,10),(9,12),(10,4),(10,12),(11,8),(12,11)],13)
 => ? = 8
{{1,6},{2,3,4,5}}
 => [6,3,4,5,2,1] => ([(3,4),(4,5)],6)
 => ?
 => ? = 4
{{1,6},{2,3,5},{4}}
 => [6,3,5,4,2,1] => ([(3,4),(3,5)],6)
 => ?
 => ? = 6
{{1,4},{2,5},{3,6}}
 => [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
 => ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
 => ? = 2
{{1,4},{2,6},{3,5}}
 => [4,6,5,1,3,2] => ([(0,4),(0,5),(1,2),(1,3)],6)
 => ?
 => ? = 6
{{1,5},{2,4},{3,6}}
 => [5,4,6,2,1,3] => ([(0,5),(1,5),(2,4),(3,4)],6)
 => ?
 => ? = 6
{{1,6},{2,4,5},{3}}
 => [6,4,3,5,2,1] => ([(3,5),(4,5)],6)
 => ?
 => ? = 6
{{1,6},{2,4},{3,5}}
 => [6,4,5,2,3,1] => ([(2,5),(3,4)],6)
 => ?
 => ? = 4
{{1,5},{2,6},{3,4}}
 => [5,6,4,3,1,2] => ([(2,5),(3,4)],6)
 => ?
 => ? = 4
{{1,6},{2,5},{3,4}}
 => [6,5,4,3,2,1] => ([],6)
 => ?
 => ? = 6
{{1,5},{2,6},{3},{4}}
 => [5,6,3,4,1,2] => ([(0,5),(1,4),(2,3)],6)
 => ?
 => ? = 3
{{1,6},{2,5},{3},{4}}
 => [6,5,3,4,2,1] => ([(4,5)],6)
 => ?
 => ? = 5
{{1,6},{2},{3},{4},{5}}
 => [6,2,3,4,5,1] => ([(2,3),(3,5),(5,4)],6)
 => ([(0,4),(0,5),(0,6),(1,3),(1,12),(1,13),(2,8),(2,9),(3,2),(3,14),(3,15),(4,7),(4,11),(5,7),(5,10),(6,1),(6,10),(6,11),(7,16),(8,17),(9,17),(10,12),(10,16),(11,13),(11,16),(12,14),(12,18),(13,15),(13,18),(14,8),(14,19),(15,9),(15,19),(16,18),(18,19),(19,17)],20)
 => ? = 3
{{1},{2},{3},{4},{5},{6}}
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
{{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
 => ([(0,2),(0,7),(1,9),(2,8),(3,4),(3,11),(4,6),(4,10),(5,3),(5,13),(6,1),(6,12),(7,5),(7,8),(8,13),(10,12),(11,10),(12,9),(13,11)],14)
 => ? = 2
{{1,2,3,4,6},{5,7}}
 => [2,3,4,6,7,1,5] => ([(0,6),(1,4),(3,2),(4,5),(5,3),(5,6)],7)
 => ([(0,3),(0,7),(1,10),(2,9),(3,8),(4,6),(4,12),(5,4),(5,11),(6,1),(6,13),(7,5),(7,8),(8,11),(9,14),(10,14),(11,12),(12,2),(12,13),(13,9),(13,10)],15)
 => ? = 8
{{1,2,3,5},{4,6,7}}
 => [2,3,5,6,1,7,4] => ([(0,5),(0,6),(1,4),(2,6),(3,2),(4,3),(4,5)],7)
 => ([(0,4),(0,7),(1,14),(2,10),(3,12),(4,8),(5,6),(5,13),(6,1),(6,9),(7,5),(7,8),(8,13),(9,12),(9,14),(11,10),(12,11),(13,3),(13,9),(14,2),(14,11)],15)
 => ? = 16
{{1,2,4,6},{3,5,7}}
 => [2,4,5,6,7,1,3] => ([(0,6),(1,5),(1,6),(3,4),(4,2),(5,3)],7)
 => ([(0,6),(0,7),(1,9),(2,5),(2,14),(3,13),(4,3),(4,16),(5,4),(5,15),(6,10),(7,2),(7,10),(9,11),(10,1),(10,14),(11,12),(12,8),(13,8),(14,9),(14,15),(15,11),(15,16),(16,12),(16,13)],17)
 => ? = 8
{{1,2,4},{3,5,6,7}}
 => [2,4,5,1,6,7,3] => ([(0,5),(0,6),(1,4),(1,5),(3,6),(4,3),(6,2)],7)
 => ([(0,4),(0,7),(1,9),(2,10),(3,11),(4,14),(5,3),(5,8),(6,1),(6,12),(7,5),(7,14),(8,10),(8,11),(10,13),(11,6),(11,13),(12,9),(13,12),(14,2),(14,8)],15)
 => ? = 16
{{1,2,5},{3,6},{4,7}}
 => [2,5,6,7,1,3,4] => ([(0,6),(1,5),(1,6),(4,2),(5,4),(6,3)],7)
 => ([(0,6),(0,7),(1,5),(1,14),(2,3),(2,16),(3,13),(4,12),(5,4),(5,17),(6,9),(7,1),(7,9),(9,2),(9,14),(10,8),(11,8),(12,11),(13,10),(14,16),(14,17),(15,10),(15,11),(16,13),(16,15),(17,12),(17,15)],18)
 => ? = 8
{{1,3,5,7},{2,4,6}}
 => [3,4,5,6,7,2,1] => ([(2,6),(4,5),(5,3),(6,4)],7)
 => ?
 => ? = 3
{{1,3,5},{2,4,6,7}}
 => [3,4,5,6,1,7,2] => ([(0,5),(1,3),(1,6),(2,6),(4,2),(5,4)],7)
 => ([(0,6),(0,7),(1,5),(1,16),(2,13),(3,8),(4,12),(5,4),(5,14),(6,1),(6,15),(7,2),(7,15),(9,10),(10,11),(11,8),(12,3),(12,11),(13,9),(14,10),(14,12),(15,13),(15,16),(16,9),(16,14)],17)
 => ? = 8
{{1,3},{2,4,5,6,7}}
 => [3,4,1,5,6,7,2] => ([(0,4),(1,3),(1,6),(4,6),(5,2),(6,5)],7)
 => ([(0,6),(0,7),(1,14),(2,13),(3,9),(4,5),(4,10),(5,3),(5,11),(6,1),(6,8),(7,2),(7,8),(8,13),(8,14),(10,11),(11,9),(12,10),(13,12),(14,4),(14,12)],15)
 => ? = 8
{{1,3},{2,4,6},{5,7}}
 => [3,4,1,6,7,2,5] => ([(0,4),(1,3),(1,6),(3,5),(4,5),(4,6),(6,2)],7)
 => ([(0,6),(0,7),(1,12),(2,5),(2,15),(3,11),(4,13),(5,9),(6,4),(6,14),(7,3),(7,14),(9,8),(10,1),(10,15),(11,10),(12,8),(13,2),(13,10),(14,11),(14,13),(15,9),(15,12)],16)
 => ? = 16
{{1,7},{2,3,4,5,6}}
 => [7,3,4,5,6,2,1] => ([(3,4),(4,6),(6,5)],7)
 => ?
 => ? = 4
{{1,7},{2,3,4,6},{5}}
 => [7,3,4,6,5,2,1] => ([(3,4),(4,5),(4,6)],7)
 => ?
 => ? = 6
{{1,5},{2,3,7},{4,6}}
 => [5,3,7,6,1,4,2] => ([(0,4),(0,5),(1,4),(1,5),(1,6),(2,3),(2,6)],7)
 => ?
 => ? = 7
{{1,6},{2,3,5},{4,7}}
 => [6,3,5,7,2,1,4] => ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5)],7)
 => ?
 => ? = 7
{{1,7},{2,3,5,6},{4}}
 => [7,3,5,4,6,2,1] => ([(3,4),(3,5),(4,6),(5,6)],7)
 => ?
 => ? = 6
{{1,7},{2,3,5},{4,6}}
 => [7,3,5,6,2,4,1] => ([(2,6),(3,4),(3,6),(4,5)],7)
 => ?
 => ? = 6
{{1,7},{2,3,6},{4,5}}
 => [7,3,6,5,4,2,1] => ([(3,4),(3,5),(3,6)],7)
 => ?
 => ? = 7
{{1,7},{2,3,6},{4},{5}}
 => [7,3,6,4,5,2,1] => ([(3,4),(3,5),(5,6)],7)
 => ?
 => ? = 6
{{1,4,5},{2,7},{3,6}}
 => [4,7,6,5,1,3,2] => ([(0,5),(0,6),(1,2),(1,3),(1,4)],7)
 => ?
 => ? = 7
{{1,4,6},{2,5},{3,7}}
 => [4,5,7,6,2,1,3] => ([(0,3),(1,6),(2,6),(3,4),(3,5)],7)
 => ?
 => ? = 6
{{1,4,7},{2,5},{3,6}}
 => [4,5,6,7,2,3,1] => ([(1,6),(2,4),(5,3),(6,5)],7)
 => ?
 => ? = 3
{{1,4},{2,5},{3,6,7}}
 => [4,5,6,1,2,7,3] => ([(0,5),(1,4),(3,6),(4,3),(5,2),(5,6)],7)
 => ([(0,6),(0,7),(1,4),(1,16),(2,5),(2,17),(3,8),(4,9),(5,10),(6,1),(6,15),(7,2),(7,15),(9,11),(10,13),(11,12),(12,8),(13,3),(13,12),(14,11),(14,13),(15,16),(15,17),(16,9),(16,14),(17,10),(17,14)],18)
 => ? = 8
{{1,4,7},{2,6},{3,5}}
 => [4,6,5,7,3,2,1] => ([(3,4),(3,5),(4,6),(5,6)],7)
 => ?
 => ? = 6
{{1},{2},{3},{4},{5},{6},{7}}
 => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => 1

Description

The number of factors of a lattice as a Cartesian product of lattices. Since the cardinality of a lattice is the product of the cardinalities of its factors, this statistic is one whenever the cardinality of the lattice is prime.

Matching statistic: St001864
(load all 3 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001864: Signed permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 44%

Values

{{1}}
 => [1] => [1] => [1] => 0 = 1 - 1
{{1,2}}
 => [2,1] => [2,1] => [2,1] => 1 = 2 - 1
{{1},{2}}
 => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
{{1,2,3}}
 => [2,3,1] => [3,2,1] => [3,2,1] => 1 = 2 - 1
{{1,3},{2}}
 => [3,2,1] => [2,3,1] => [2,3,1] => 2 = 3 - 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
{{1,2,3,4}}
 => [2,3,4,1] => [4,2,3,1] => [4,2,3,1] => 1 = 2 - 1
{{1,3},{2,4}}
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1 = 2 - 1
{{1,4},{2,3}}
 => [4,3,2,1] => [2,3,4,1] => [2,3,4,1] => 3 = 4 - 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [3,4,2,1] => [3,4,2,1] => 2 = 3 - 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 2 - 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 8 - 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [2,5,3,4,1] => [2,5,3,4,1] => ? = 3 - 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [4,5,3,1,2] => [4,5,3,1,2] => ? = 8 - 1
{{1,5},{2,3,4}}
 => [5,3,4,2,1] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 4 - 1
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => [3,1,5,4,2] => [3,1,5,4,2] => ? = 3 - 1
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 5 - 1
{{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => [4,5,2,3,1] => [4,5,2,3,1] => ? = 3 - 1
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => [6,2,3,4,5,1] => [6,2,3,4,5,1] => ? = 2 - 1
{{1,2,3,5},{4,6}}
 => [2,3,5,6,1,4] => [6,2,3,1,5,4] => [6,2,3,1,5,4] => ? = 8 - 1
{{1,2,4},{3,5,6}}
 => [2,4,5,1,6,3] => [5,2,6,4,1,3] => [5,2,6,4,1,3] => ? = 16 - 1
{{1,3,5},{2,4,6}}
 => [3,4,5,6,1,2] => [6,1,3,4,5,2] => [6,1,3,4,5,2] => ? = 2 - 1
{{1,3},{2,4,5,6}}
 => [3,4,1,5,6,2] => [4,6,3,1,5,2] => [4,6,3,1,5,2] => ? = 8 - 1
{{1,6},{2,3,4,5}}
 => [6,3,4,5,2,1] => [2,5,6,3,4,1] => [2,5,6,3,4,1] => ? = 4 - 1
{{1,6},{2,3,5},{4}}
 => [6,3,5,4,2,1] => [2,4,5,6,1,3] => [2,4,5,6,1,3] => ? = 6 - 1
{{1,4},{2,5},{3,6}}
 => [4,5,6,1,2,3] => [6,1,2,4,5,3] => [6,1,2,4,5,3] => ? = 2 - 1
{{1,4},{2,6},{3,5}}
 => [4,6,5,1,3,2] => [5,3,1,4,6,2] => [5,3,1,4,6,2] => ? = 6 - 1
{{1,5},{2,4},{3,6}}
 => [5,4,6,2,1,3] => [2,6,1,5,4,3] => [2,6,1,5,4,3] => ? = 6 - 1
{{1,6},{2,4,5},{3}}
 => [6,4,3,5,2,1] => [2,5,4,6,3,1] => [2,5,4,6,3,1] => ? = 6 - 1
{{1,6},{2,4},{3,5}}
 => [6,4,5,2,3,1] => [3,5,2,6,4,1] => [3,5,2,6,4,1] => ? = 4 - 1
{{1,5},{2,6},{3,4}}
 => [5,6,4,3,1,2] => [3,1,4,6,5,2] => [3,1,4,6,5,2] => ? = 4 - 1
{{1,6},{2,5},{3,4}}
 => [6,5,4,3,2,1] => [2,3,4,5,6,1] => [2,3,4,5,6,1] => ? = 6 - 1
{{1,5},{2,6},{3},{4}}
 => [5,6,3,4,1,2] => [4,1,6,3,5,2] => [4,1,6,3,5,2] => ? = 3 - 1
{{1,6},{2,5},{3},{4}}
 => [6,5,3,4,2,1] => [2,4,5,3,6,1] => [2,4,5,3,6,1] => ? = 5 - 1
{{1,6},{2},{3},{4},{5}}
 => [6,2,3,4,5,1] => [5,6,2,3,4,1] => [5,6,2,3,4,1] => ? = 3 - 1
{{1},{2},{3},{4},{5},{6}}
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
{{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => [7,2,3,4,5,6,1] => [7,2,3,4,5,6,1] => ? = 2 - 1
{{1,2,3,4,6},{5,7}}
 => [2,3,4,6,7,1,5] => [7,2,3,4,1,6,5] => [7,2,3,4,1,6,5] => ? = 8 - 1
{{1,2,3,5},{4,6,7}}
 => [2,3,5,6,1,7,4] => [6,2,3,7,5,1,4] => [6,2,3,7,5,1,4] => ? = 16 - 1
{{1,2,4,6},{3,5,7}}
 => [2,4,5,6,7,1,3] => [7,2,1,4,5,6,3] => [7,2,1,4,5,6,3] => ? = 8 - 1
{{1,2,4},{3,5,6,7}}
 => [2,4,5,1,6,7,3] => [5,2,7,4,1,6,3] => [5,2,7,4,1,6,3] => ? = 16 - 1
{{1,2,5},{3,6},{4,7}}
 => [2,5,6,7,1,3,4] => [7,2,1,3,5,6,4] => [7,2,1,3,5,6,4] => ? = 8 - 1
{{1,3,5,7},{2,4,6}}
 => [3,4,5,6,7,2,1] => [2,7,3,4,5,6,1] => [2,7,3,4,5,6,1] => ? = 3 - 1
{{1,3,5},{2,4,6,7}}
 => [3,4,5,6,1,7,2] => [6,7,3,4,5,1,2] => [6,7,3,4,5,1,2] => ? = 8 - 1
{{1,3},{2,4,5,6,7}}
 => [3,4,1,5,6,7,2] => [4,7,3,1,5,6,2] => [4,7,3,1,5,6,2] => ? = 8 - 1
{{1,3},{2,4,6},{5,7}}
 => [3,4,1,6,7,2,5] => [4,7,3,1,2,6,5] => [4,7,3,1,2,6,5] => ? = 16 - 1
{{1,7},{2,3,4,5,6}}
 => [7,3,4,5,6,2,1] => [2,6,7,3,4,5,1] => [2,6,7,3,4,5,1] => ? = 4 - 1
{{1,7},{2,3,4,6},{5}}
 => [7,3,4,6,5,2,1] => [2,5,6,1,7,4,3] => [2,5,6,1,7,4,3] => ? = 6 - 1
{{1,5},{2,3,7},{4,6}}
 => [5,3,7,6,1,4,2] => [6,4,5,1,3,7,2] => [6,4,5,1,3,7,2] => ? = 7 - 1
{{1,6},{2,3,5},{4,7}}
 => [6,3,5,7,2,1,4] => [2,7,6,1,3,5,4] => [2,7,6,1,3,5,4] => ? = 7 - 1
{{1,7},{2,3,5,6},{4}}
 => [7,3,5,4,6,2,1] => [2,6,7,5,3,4,1] => [2,6,7,5,3,4,1] => ? = 6 - 1
{{1,7},{2,3,5},{4,6}}
 => [7,3,5,6,2,4,1] => [4,6,7,2,3,5,1] => [4,6,7,2,3,5,1] => ? = 6 - 1
{{1,7},{2,3,6},{4,5}}
 => [7,3,6,5,4,2,1] => [2,4,5,6,7,3,1] => [2,4,5,6,7,3,1] => ? = 7 - 1
{{1,7},{2,3,6},{4},{5}}
 => [7,3,6,4,5,2,1] => [2,5,7,6,4,3,1] => [2,5,7,6,4,3,1] => ? = 6 - 1
{{1,4,5},{2,7},{3,6}}
 => [4,7,6,5,1,3,2] => [5,3,1,4,6,7,2] => [5,3,1,4,6,7,2] => ? = 7 - 1
{{1,4,6},{2,5},{3,7}}
 => [4,5,7,6,2,1,3] => [2,6,1,4,5,7,3] => [2,6,1,4,5,7,3] => ? = 6 - 1
{{1,4,7},{2,5},{3,6}}
 => [4,5,6,7,2,3,1] => [3,7,2,4,5,6,1] => [3,7,2,4,5,6,1] => ? = 3 - 1
{{1,4},{2,5},{3,6,7}}
 => [4,5,6,1,2,7,3] => [6,1,7,4,5,2,3] => [6,1,7,4,5,2,3] => ? = 8 - 1
{{1,4,7},{2,6},{3,5}}
 => [4,6,5,7,3,2,1] => [2,3,7,4,6,5,1] => [2,3,7,4,6,5,1] => ? = 6 - 1
{{1,4},{2,7},{3,5,6}}
 => [4,7,5,1,6,3,2] => [5,3,7,4,6,2,1] => [5,3,7,4,6,2,1] => ? = 7 - 1

Description

The number of excedances of a signed permutation. For a signed permutation

$\pi\in\mathfrak H_n$ , this is

$\lvert\{i\in[n] \mid \pi(i) > i\}\rvert$ .

Matching statistic: St001946
(load all 6 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001946: Parking functions ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 44%

Values

{{1}}
 => [1] => [1] => 0 = 1 - 1
{{1,2}}
 => [2,1] => [2,1] => 1 = 2 - 1
{{1},{2}}
 => [1,2] => [1,2] => 0 = 1 - 1
{{1,2,3}}
 => [2,3,1] => [2,3,1] => 1 = 2 - 1
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => 0 = 1 - 1
{{1,2,3,4}}
 => [2,3,4,1] => [2,3,4,1] => 1 = 2 - 1
{{1,3},{2,4}}
 => [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => 3 = 4 - 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [4,2,3,1] => 2 = 3 - 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [2,3,4,5,1] => ? = 2 - 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [2,4,5,1,3] => ? = 8 - 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [3,4,5,2,1] => ? = 3 - 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,4,1,5,2] => ? = 8 - 1
{{1,5},{2,3,4}}
 => [5,3,4,2,1] => [5,3,4,2,1] => ? = 4 - 1
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => [4,5,3,1,2] => ? = 3 - 1
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [5,4,3,2,1] => ? = 5 - 1
{{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => [5,2,3,4,1] => ? = 3 - 1
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,2,3,4,5] => ? = 1 - 1
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => [2,3,4,5,6,1] => ? = 2 - 1
{{1,2,3,5},{4,6}}
 => [2,3,5,6,1,4] => [2,3,5,6,1,4] => ? = 8 - 1
{{1,2,4},{3,5,6}}
 => [2,4,5,1,6,3] => [2,4,5,1,6,3] => ? = 16 - 1
{{1,3,5},{2,4,6}}
 => [3,4,5,6,1,2] => [3,4,5,6,1,2] => ? = 2 - 1
{{1,3},{2,4,5,6}}
 => [3,4,1,5,6,2] => [3,4,1,5,6,2] => ? = 8 - 1
{{1,6},{2,3,4,5}}
 => [6,3,4,5,2,1] => [6,3,4,5,2,1] => ? = 4 - 1
{{1,6},{2,3,5},{4}}
 => [6,3,5,4,2,1] => [6,3,5,4,2,1] => ? = 6 - 1
{{1,4},{2,5},{3,6}}
 => [4,5,6,1,2,3] => [4,5,6,1,2,3] => ? = 2 - 1
{{1,4},{2,6},{3,5}}
 => [4,6,5,1,3,2] => [4,6,5,1,3,2] => ? = 6 - 1
{{1,5},{2,4},{3,6}}
 => [5,4,6,2,1,3] => [5,4,6,2,1,3] => ? = 6 - 1
{{1,6},{2,4,5},{3}}
 => [6,4,3,5,2,1] => [6,4,3,5,2,1] => ? = 6 - 1
{{1,6},{2,4},{3,5}}
 => [6,4,5,2,3,1] => [6,4,5,2,3,1] => ? = 4 - 1
{{1,5},{2,6},{3,4}}
 => [5,6,4,3,1,2] => [5,6,4,3,1,2] => ? = 4 - 1
{{1,6},{2,5},{3,4}}
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ? = 6 - 1
{{1,5},{2,6},{3},{4}}
 => [5,6,3,4,1,2] => [5,6,3,4,1,2] => ? = 3 - 1
{{1,6},{2,5},{3},{4}}
 => [6,5,3,4,2,1] => [6,5,3,4,2,1] => ? = 5 - 1
{{1,6},{2},{3},{4},{5}}
 => [6,2,3,4,5,1] => [6,2,3,4,5,1] => ? = 3 - 1
{{1},{2},{3},{4},{5},{6}}
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
{{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => [2,3,4,5,6,7,1] => ? = 2 - 1
{{1,2,3,4,6},{5,7}}
 => [2,3,4,6,7,1,5] => [2,3,4,6,7,1,5] => ? = 8 - 1
{{1,2,3,5},{4,6,7}}
 => [2,3,5,6,1,7,4] => [2,3,5,6,1,7,4] => ? = 16 - 1
{{1,2,4,6},{3,5,7}}
 => [2,4,5,6,7,1,3] => [2,4,5,6,7,1,3] => ? = 8 - 1
{{1,2,4},{3,5,6,7}}
 => [2,4,5,1,6,7,3] => [2,4,5,1,6,7,3] => ? = 16 - 1
{{1,2,5},{3,6},{4,7}}
 => [2,5,6,7,1,3,4] => [2,5,6,7,1,3,4] => ? = 8 - 1
{{1,3,5,7},{2,4,6}}
 => [3,4,5,6,7,2,1] => [3,4,5,6,7,2,1] => ? = 3 - 1
{{1,3,5},{2,4,6,7}}
 => [3,4,5,6,1,7,2] => [3,4,5,6,1,7,2] => ? = 8 - 1
{{1,3},{2,4,5,6,7}}
 => [3,4,1,5,6,7,2] => [3,4,1,5,6,7,2] => ? = 8 - 1
{{1,3},{2,4,6},{5,7}}
 => [3,4,1,6,7,2,5] => [3,4,1,6,7,2,5] => ? = 16 - 1
{{1,7},{2,3,4,5,6}}
 => [7,3,4,5,6,2,1] => [7,3,4,5,6,2,1] => ? = 4 - 1
{{1,7},{2,3,4,6},{5}}
 => [7,3,4,6,5,2,1] => [7,3,4,6,5,2,1] => ? = 6 - 1
{{1,5},{2,3,7},{4,6}}
 => [5,3,7,6,1,4,2] => [5,3,7,6,1,4,2] => ? = 7 - 1
{{1,6},{2,3,5},{4,7}}
 => [6,3,5,7,2,1,4] => [6,3,5,7,2,1,4] => ? = 7 - 1
{{1,7},{2,3,5,6},{4}}
 => [7,3,5,4,6,2,1] => [7,3,5,4,6,2,1] => ? = 6 - 1
{{1,7},{2,3,5},{4,6}}
 => [7,3,5,6,2,4,1] => [7,3,5,6,2,4,1] => ? = 6 - 1
{{1,7},{2,3,6},{4,5}}
 => [7,3,6,5,4,2,1] => [7,3,6,5,4,2,1] => ? = 7 - 1
{{1,7},{2,3,6},{4},{5}}
 => [7,3,6,4,5,2,1] => [7,3,6,4,5,2,1] => ? = 6 - 1
{{1,4,5},{2,7},{3,6}}
 => [4,7,6,5,1,3,2] => [4,7,6,5,1,3,2] => ? = 7 - 1
{{1,4,6},{2,5},{3,7}}
 => [4,5,7,6,2,1,3] => [4,5,7,6,2,1,3] => ? = 6 - 1
{{1,4,7},{2,5},{3,6}}
 => [4,5,6,7,2,3,1] => [4,5,6,7,2,3,1] => ? = 3 - 1
{{1,4},{2,5},{3,6,7}}
 => [4,5,6,1,2,7,3] => [4,5,6,1,2,7,3] => ? = 8 - 1
{{1,4,7},{2,6},{3,5}}
 => [4,6,5,7,3,2,1] => [4,6,5,7,3,2,1] => ? = 6 - 1

Description

The number of descents in a parking function. This is the number of indices

$i$ such that

$p_i > p_{i+1}$ .

Matching statistic: St000942
(load all 2 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St000942: Parking functions ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 44%

Values

{{1}}
 => [1] => [1] => [1] => 1
{{1,2}}
 => [2,1] => [1,2] => [1,2] => 2
{{1},{2}}
 => [1,2] => [2,1] => [2,1] => 1
{{1,2,3}}
 => [2,3,1] => [2,1,3] => [2,1,3] => 2
{{1,3},{2}}
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
{{1},{2},{3}}
 => [1,2,3] => [3,2,1] => [3,2,1] => 1
{{1,2,3,4}}
 => [2,3,4,1] => [3,2,1,4] => [3,2,1,4] => 2
{{1,3},{2,4}}
 => [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2
{{1,4},{2,3}}
 => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 4
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 3
{{1},{2},{3},{4}}
 => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 2
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,2,1,5,3] => [4,2,1,5,3] => ? = 8
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 3
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 8
{{1,5},{2,3,4}}
 => [5,3,4,2,1] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 4
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 3
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 5
{{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => [1,4,3,2,5] => [1,4,3,2,5] => ? = 3
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => ? = 2
{{1,2,3,5},{4,6}}
 => [2,3,5,6,1,4] => [5,4,2,1,6,3] => [5,4,2,1,6,3] => ? = 8
{{1,2,4},{3,5,6}}
 => [2,4,5,1,6,3] => [5,3,2,6,1,4] => [5,3,2,6,1,4] => ? = 16
{{1,3,5},{2,4,6}}
 => [3,4,5,6,1,2] => [4,3,2,1,6,5] => [4,3,2,1,6,5] => ? = 2
{{1,3},{2,4,5,6}}
 => [3,4,1,5,6,2] => [4,3,6,2,1,5] => [4,3,6,2,1,5] => ? = 8
{{1,6},{2,3,4,5}}
 => [6,3,4,5,2,1] => [1,4,3,2,5,6] => [1,4,3,2,5,6] => ? = 4
{{1,6},{2,3,5},{4}}
 => [6,3,5,4,2,1] => [1,4,2,3,5,6] => [1,4,2,3,5,6] => ? = 6
{{1,4},{2,5},{3,6}}
 => [4,5,6,1,2,3] => [3,2,1,6,5,4] => [3,2,1,6,5,4] => ? = 2
{{1,4},{2,6},{3,5}}
 => [4,6,5,1,3,2] => [3,1,2,6,4,5] => [3,1,2,6,4,5] => ? = 6
{{1,5},{2,4},{3,6}}
 => [5,4,6,2,1,3] => [2,3,1,5,6,4] => [2,3,1,5,6,4] => ? = 6
{{1,6},{2,4,5},{3}}
 => [6,4,3,5,2,1] => [1,3,4,2,5,6] => [1,3,4,2,5,6] => ? = 6
{{1,6},{2,4},{3,5}}
 => [6,4,5,2,3,1] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => ? = 4
{{1,5},{2,6},{3,4}}
 => [5,6,4,3,1,2] => [2,1,3,4,6,5] => [2,1,3,4,6,5] => ? = 4
{{1,6},{2,5},{3,4}}
 => [6,5,4,3,2,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 6
{{1,5},{2,6},{3},{4}}
 => [5,6,3,4,1,2] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => ? = 3
{{1,6},{2,5},{3},{4}}
 => [6,5,3,4,2,1] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ? = 5
{{1,6},{2},{3},{4},{5}}
 => [6,2,3,4,5,1] => [1,5,4,3,2,6] => [1,5,4,3,2,6] => ? = 3
{{1},{2},{3},{4},{5},{6}}
 => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ? = 1
{{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => ? = 2
{{1,2,3,4,6},{5,7}}
 => [2,3,4,6,7,1,5] => [6,5,4,2,1,7,3] => [6,5,4,2,1,7,3] => ? = 8
{{1,2,3,5},{4,6,7}}
 => [2,3,5,6,1,7,4] => [6,5,3,2,7,1,4] => [6,5,3,2,7,1,4] => ? = 16
{{1,2,4,6},{3,5,7}}
 => [2,4,5,6,7,1,3] => [6,4,3,2,1,7,5] => [6,4,3,2,1,7,5] => ? = 8
{{1,2,4},{3,5,6,7}}
 => [2,4,5,1,6,7,3] => [6,4,3,7,2,1,5] => [6,4,3,7,2,1,5] => ? = 16
{{1,2,5},{3,6},{4,7}}
 => [2,5,6,7,1,3,4] => [6,3,2,1,7,5,4] => [6,3,2,1,7,5,4] => ? = 8
{{1,3,5,7},{2,4,6}}
 => [3,4,5,6,7,2,1] => [5,4,3,2,1,6,7] => [5,4,3,2,1,6,7] => ? = 3
{{1,3,5},{2,4,6,7}}
 => [3,4,5,6,1,7,2] => [5,4,3,2,7,1,6] => [5,4,3,2,7,1,6] => ? = 8
{{1,3},{2,4,5,6,7}}
 => [3,4,1,5,6,7,2] => [5,4,7,3,2,1,6] => [5,4,7,3,2,1,6] => ? = 8
{{1,3},{2,4,6},{5,7}}
 => [3,4,1,6,7,2,5] => [5,4,7,2,1,6,3] => [5,4,7,2,1,6,3] => ? = 16
{{1,7},{2,3,4,5,6}}
 => [7,3,4,5,6,2,1] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => ? = 4
{{1,7},{2,3,4,6},{5}}
 => [7,3,4,6,5,2,1] => [1,5,4,2,3,6,7] => [1,5,4,2,3,6,7] => ? = 6
{{1,5},{2,3,7},{4,6}}
 => [5,3,7,6,1,4,2] => [3,5,1,2,7,4,6] => [3,5,1,2,7,4,6] => ? = 7
{{1,6},{2,3,5},{4,7}}
 => [6,3,5,7,2,1,4] => [2,5,3,1,6,7,4] => [2,5,3,1,6,7,4] => ? = 7
{{1,7},{2,3,5,6},{4}}
 => [7,3,5,4,6,2,1] => [1,5,3,4,2,6,7] => [1,5,3,4,2,6,7] => ? = 6
{{1,7},{2,3,5},{4,6}}
 => [7,3,5,6,2,4,1] => [1,5,3,2,6,4,7] => [1,5,3,2,6,4,7] => ? = 6
{{1,7},{2,3,6},{4,5}}
 => [7,3,6,5,4,2,1] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? = 7
{{1,7},{2,3,6},{4},{5}}
 => [7,3,6,4,5,2,1] => [1,5,2,4,3,6,7] => [1,5,2,4,3,6,7] => ? = 6
{{1,4,5},{2,7},{3,6}}
 => [4,7,6,5,1,3,2] => [4,1,2,3,7,5,6] => [4,1,2,3,7,5,6] => ? = 7
{{1,4,6},{2,5},{3,7}}
 => [4,5,7,6,2,1,3] => [4,3,1,2,6,7,5] => [4,3,1,2,6,7,5] => ? = 6
{{1,4,7},{2,5},{3,6}}
 => [4,5,6,7,2,3,1] => [4,3,2,1,6,5,7] => [4,3,2,1,6,5,7] => ? = 3
{{1,4},{2,5},{3,6,7}}
 => [4,5,6,1,2,7,3] => [4,3,2,7,6,1,5] => [4,3,2,7,6,1,5] => ? = 8
{{1,4,7},{2,6},{3,5}}
 => [4,6,5,7,3,2,1] => [4,2,3,1,5,6,7] => [4,2,3,1,5,6,7] => ? = 6

Description

The number of critical left to right maxima of the parking functions. An entry

$p$ in a parking function is critical, if there are exactly

$p-1$ entries smaller than

$p$ and

$n-p$ entries larger than

$p$ . It is a left to right maximum, if there are no larger entries before it. This statistic allows the computation of the Tutte polynomial of the complete graph

$K_{n+1}$ , via

$\sum_{P} x^{st(P)}y^{\binom{n+1}{2}-\sum P},$ where the sum is over all parking functions of length

$n$ , see [1, thm.13.5.16].

The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001937The size of the center of a parking function. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St000455The second largest eigenvalue of a graph if it is integral.