- FindStat

Your data matches 12 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001645
(load all 9 compositions to match this statistic)

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The pebbling number of a connected graph.

Matching statistic: St000454
(load all 4 compositions to match this statistic)

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%

Values

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

Description

The largest eigenvalue of a graph if it is integral. If a graph is

$d$ -regular, then its largest eigenvalue equals

$d$ . One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.

Matching statistic: St001330
(load all 20 compositions to match this statistic)

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%

Values

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

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: St001880

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 71%

Values

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

Description

The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.

Matching statistic: St001879

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 71%

Values

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

Description

The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.

Matching statistic: St001684

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001684: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 57%

Values

[1] => [1] => [1,0]
 => [2,1] => 1
[2] => [1] => [1,0]
 => [2,1] => 1
[1,2] => [1,1] => [1,0,1,0]
 => [3,1,2] => 2
[2,1] => [1,1] => [1,0,1,0]
 => [3,1,2] => 2
[3] => [1] => [1,0]
 => [2,1] => 1
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[1,3] => [1,1] => [1,0,1,0]
 => [3,1,2] => 2
[2,1,1] => [1,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => 4
[3,1] => [1,1] => [1,0,1,0]
 => [3,1,2] => 2
[4] => [1] => [1,0]
 => [2,1] => 1
[1,2,2] => [1,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => 4
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[1,4] => [1,1] => [1,0,1,0]
 => [3,1,2] => 2
[2,1,2] => [1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[2,3] => [1,1] => [1,0,1,0]
 => [3,1,2] => 2
[3,1,1] => [1,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => 4
[3,2] => [1,1] => [1,0,1,0]
 => [3,1,2] => 2
[4,1] => [1,1] => [1,0,1,0]
 => [3,1,2] => 2
[5] => [1] => [1,0]
 => [2,1] => 1
[1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 4
[1,2,3] => [1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[1,3,2] => [1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[1,4,1] => [1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[1,5] => [1,1] => [1,0,1,0]
 => [3,1,2] => 2
[2,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 4
[2,1,3] => [1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[2,3,1] => [1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[2,4] => [1,1] => [1,0,1,0]
 => [3,1,2] => 2
[3,1,2] => [1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[4,1,1] => [1,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => 4
[4,2] => [1,1] => [1,0,1,0]
 => [3,1,2] => 2
[5,1] => [1,1] => [1,0,1,0]
 => [3,1,2] => 2
[6] => [1] => [1,0]
 => [2,1] => 1
[1,2,1,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => ? = 5
[1,2,1,3] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 4
[1,2,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 4
[1,2,4] => [1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[1,3,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 4
[1,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 4
[1,3,3] => [1,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => 4
[1,4,2] => [1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[1,5,1] => [1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[1,6] => [1,1] => [1,0,1,0]
 => [3,1,2] => 2
[2,1,2,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => ? = 5
[2,1,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 4
[2,1,4] => [1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[2,3,2] => [1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[2,4,1] => [1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[2,5] => [1,1] => [1,0,1,0]
 => [3,1,2] => 2
[3,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 4
[3,1,3] => [1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[1,1,2,1,2,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,7,1,3,4,5,6] => ? = 6
[1,2,1,2,1,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,1,2,3,4,7,5] => ? = 6
[1,2,1,2,2] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => ? = 5
[1,2,1,3,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => ? = 5
[1,2,3,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => ? = 5
[1,3,1,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => ? = 5
[1,3,2,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => ? = 5
[2,1,2,1,1,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => ? = 6
[2,1,2,1,2] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => ? = 5
[2,1,3,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => ? = 5
[3,1,2,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => ? = 5
[1,1,1,2,1,2,1] => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [2,3,8,1,4,5,6,7] => ? = 7
[1,1,2,1,2,1,1] => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [2,7,1,3,4,5,8,6] => ? = 7
[1,1,2,1,3,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,7,1,3,4,5,6] => ? = 6
[1,1,3,1,2,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,7,1,3,4,5,6] => ? = 6
[1,2,1,2,1,1,1] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,1,2,3,4,7,8,5] => ? = 7
[1,2,1,2,1,2] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6
[1,2,1,2,3] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => ? = 5
[1,2,1,3,1,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,1,2,3,4,7,5] => ? = 6
[1,2,1,3,2] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => ? = 5
[1,2,1,4,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => ? = 5
[1,2,3,1,1,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => ? = 6
[1,2,3,1,2] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => ? = 5
[1,2,3,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => ? = 5
[1,2,4,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => ? = 5
[1,3,1,2,1,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,1,2,3,4,7,5] => ? = 6
[1,3,1,2,2] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => ? = 5
[1,3,1,3,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => ? = 5
[1,3,2,1,1,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => ? = 6
[1,3,2,1,2] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => ? = 5
[1,4,1,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => ? = 5
[1,4,2,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => ? = 5
[2,1,2,1,1,1,1] => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [5,1,2,3,6,7,8,4] => ? = 7
[2,1,2,1,2,1] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6
[2,1,2,1,3] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => ? = 5
[2,1,2,3,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => ? = 5
[2,1,3,1,1,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => ? = 6
[2,1,3,1,2] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => ? = 5
[2,1,3,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => ? = 5
[2,1,4,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => ? = 5
[2,3,1,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => ? = 5
[2,3,2,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => ? = 5
[3,1,2,1,1,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => ? = 6
[3,1,2,1,2] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => ? = 5
[3,1,3,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => ? = 5
[3,2,1,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => ? = 5
[4,1,2,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => ? = 5
[1,1,1,2,1,3,1] => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [2,3,8,1,4,5,6,7] => ? = 7

Description

The reduced word complexity of a permutation. For a permutation

$\pi$ , this is the smallest length of a word in simple transpositions that contains all reduced expressions of

$\pi$ . For example, the permutation

$[3,2,1] = (12)(23)(12) = (23)(12)(23)$ and the reduced word complexity is

$4$ since the smallest words containing those two reduced words as subwords are

$(12),(23),(12),(23)$ and also

$(23),(12),(23),(12)$ . This statistic appears in [1, Question 6.1].

Matching statistic: St000393

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 57%

Values

[1] => [1] => [1,0]
 => 10 => 2 = 1 + 1
[2] => [1] => [1,0]
 => 10 => 2 = 1 + 1
[1,2] => [1,1] => [1,0,1,0]
 => 1010 => 3 = 2 + 1
[2,1] => [1,1] => [1,0,1,0]
 => 1010 => 3 = 2 + 1
[3] => [1] => [1,0]
 => 10 => 2 = 1 + 1
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => 101010 => 4 = 3 + 1
[1,3] => [1,1] => [1,0,1,0]
 => 1010 => 3 = 2 + 1
[2,1,1] => [1,2] => [1,0,1,1,0,0]
 => 101100 => 5 = 4 + 1
[3,1] => [1,1] => [1,0,1,0]
 => 1010 => 3 = 2 + 1
[4] => [1] => [1,0]
 => 10 => 2 = 1 + 1
[1,2,2] => [1,2] => [1,0,1,1,0,0]
 => 101100 => 5 = 4 + 1
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
 => 101010 => 4 = 3 + 1
[1,4] => [1,1] => [1,0,1,0]
 => 1010 => 3 = 2 + 1
[2,1,2] => [1,1,1] => [1,0,1,0,1,0]
 => 101010 => 4 = 3 + 1
[2,3] => [1,1] => [1,0,1,0]
 => 1010 => 3 = 2 + 1
[3,1,1] => [1,2] => [1,0,1,1,0,0]
 => 101100 => 5 = 4 + 1
[3,2] => [1,1] => [1,0,1,0]
 => 1010 => 3 = 2 + 1
[4,1] => [1,1] => [1,0,1,0]
 => 1010 => 3 = 2 + 1
[5] => [1] => [1,0]
 => 10 => 2 = 1 + 1
[1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 10101010 => 5 = 4 + 1
[1,2,3] => [1,1,1] => [1,0,1,0,1,0]
 => 101010 => 4 = 3 + 1
[1,3,2] => [1,1,1] => [1,0,1,0,1,0]
 => 101010 => 4 = 3 + 1
[1,4,1] => [1,1,1] => [1,0,1,0,1,0]
 => 101010 => 4 = 3 + 1
[1,5] => [1,1] => [1,0,1,0]
 => 1010 => 3 = 2 + 1
[2,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 10101010 => 5 = 4 + 1
[2,1,3] => [1,1,1] => [1,0,1,0,1,0]
 => 101010 => 4 = 3 + 1
[2,3,1] => [1,1,1] => [1,0,1,0,1,0]
 => 101010 => 4 = 3 + 1
[2,4] => [1,1] => [1,0,1,0]
 => 1010 => 3 = 2 + 1
[3,1,2] => [1,1,1] => [1,0,1,0,1,0]
 => 101010 => 4 = 3 + 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => 101010 => 4 = 3 + 1
[4,1,1] => [1,2] => [1,0,1,1,0,0]
 => 101100 => 5 = 4 + 1
[4,2] => [1,1] => [1,0,1,0]
 => 1010 => 3 = 2 + 1
[5,1] => [1,1] => [1,0,1,0]
 => 1010 => 3 = 2 + 1
[6] => [1] => [1,0]
 => 10 => 2 = 1 + 1
[1,2,1,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => ? = 5 + 1
[1,2,1,3] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 10101010 => 5 = 4 + 1
[1,2,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 10101010 => 5 = 4 + 1
[1,2,4] => [1,1,1] => [1,0,1,0,1,0]
 => 101010 => 4 = 3 + 1
[1,3,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 10101010 => 5 = 4 + 1
[1,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 10101010 => 5 = 4 + 1
[1,3,3] => [1,2] => [1,0,1,1,0,0]
 => 101100 => 5 = 4 + 1
[1,4,2] => [1,1,1] => [1,0,1,0,1,0]
 => 101010 => 4 = 3 + 1
[1,5,1] => [1,1,1] => [1,0,1,0,1,0]
 => 101010 => 4 = 3 + 1
[1,6] => [1,1] => [1,0,1,0]
 => 1010 => 3 = 2 + 1
[2,1,2,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? = 5 + 1
[2,1,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 10101010 => 5 = 4 + 1
[2,1,4] => [1,1,1] => [1,0,1,0,1,0]
 => 101010 => 4 = 3 + 1
[2,3,2] => [1,1,1] => [1,0,1,0,1,0]
 => 101010 => 4 = 3 + 1
[2,4,1] => [1,1,1] => [1,0,1,0,1,0]
 => 101010 => 4 = 3 + 1
[2,5] => [1,1] => [1,0,1,0]
 => 1010 => 3 = 2 + 1
[3,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 10101010 => 5 = 4 + 1
[3,1,3] => [1,1,1] => [1,0,1,0,1,0]
 => 101010 => 4 = 3 + 1
[1,1,2,1,2,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 110010101010 => ? = 6 + 1
[1,2,1,2,1,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 101010101100 => ? = 6 + 1
[1,2,1,2,2] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? = 5 + 1
[1,2,1,3,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => ? = 5 + 1
[1,2,3,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? = 5 + 1
[1,3,1,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => ? = 5 + 1
[1,3,2,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? = 5 + 1
[2,1,2,1,1,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 101010111000 => ? = 6 + 1
[2,1,2,1,2] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => ? = 5 + 1
[2,1,3,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? = 5 + 1
[3,1,2,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? = 5 + 1
[1,1,1,2,1,2,1] => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => 11100010101010 => ? = 7 + 1
[1,1,2,1,2,1,1] => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => 11001010101100 => ? = 7 + 1
[1,1,2,1,3,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 110010101010 => ? = 6 + 1
[1,1,3,1,2,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 110010101010 => ? = 6 + 1
[1,2,1,2,1,1,1] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => 10101010111000 => ? = 7 + 1
[1,2,1,2,1,2] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 101010101010 => ? = 6 + 1
[1,2,1,2,3] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => ? = 5 + 1
[1,2,1,3,1,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 101010101100 => ? = 6 + 1
[1,2,1,3,2] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => ? = 5 + 1
[1,2,1,4,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => ? = 5 + 1
[1,2,3,1,1,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 101010111000 => ? = 6 + 1
[1,2,3,1,2] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => ? = 5 + 1
[1,2,3,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => ? = 5 + 1
[1,2,4,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? = 5 + 1
[1,3,1,2,1,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 101010101100 => ? = 6 + 1
[1,3,1,2,2] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? = 5 + 1
[1,3,1,3,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => ? = 5 + 1
[1,3,2,1,1,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 101010111000 => ? = 6 + 1
[1,3,2,1,2] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => ? = 5 + 1
[1,4,1,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => ? = 5 + 1
[1,4,2,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? = 5 + 1
[2,1,2,1,1,1,1] => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => 10101011110000 => ? = 7 + 1
[2,1,2,1,2,1] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 101010101010 => ? = 6 + 1
[2,1,2,1,3] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => ? = 5 + 1
[2,1,2,3,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => ? = 5 + 1
[2,1,3,1,1,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 101010111000 => ? = 6 + 1
[2,1,3,1,2] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => ? = 5 + 1
[2,1,3,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => ? = 5 + 1
[2,1,4,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? = 5 + 1
[2,3,1,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => ? = 5 + 1
[2,3,2,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? = 5 + 1
[3,1,2,1,1,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 101010111000 => ? = 6 + 1
[3,1,2,1,2] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => ? = 5 + 1
[3,1,3,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? = 5 + 1
[3,2,1,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => ? = 5 + 1
[4,1,2,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? = 5 + 1
[1,1,1,2,1,3,1] => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => 11100010101010 => ? = 7 + 1

Description

The number of strictly increasing runs in a binary word.

Matching statistic: St001003

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001003: Dyck paths ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 57%

Values

[1] => [1] => [1,0]
 => [1,1,0,0]
 => 6 = 1 + 5
[2] => [1] => [1,0]
 => [1,1,0,0]
 => 6 = 1 + 5
[1,2] => [1,1] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 7 = 2 + 5
[2,1] => [1,1] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 7 = 2 + 5
[3] => [1] => [1,0]
 => [1,1,0,0]
 => 6 = 1 + 5
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 8 = 3 + 5
[1,3] => [1,1] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 7 = 2 + 5
[2,1,1] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 9 = 4 + 5
[3,1] => [1,1] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 7 = 2 + 5
[4] => [1] => [1,0]
 => [1,1,0,0]
 => 6 = 1 + 5
[1,2,2] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 9 = 4 + 5
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 8 = 3 + 5
[1,4] => [1,1] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 7 = 2 + 5
[2,1,2] => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 8 = 3 + 5
[2,3] => [1,1] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 7 = 2 + 5
[3,1,1] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 9 = 4 + 5
[3,2] => [1,1] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 7 = 2 + 5
[4,1] => [1,1] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 7 = 2 + 5
[5] => [1] => [1,0]
 => [1,1,0,0]
 => 6 = 1 + 5
[1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 9 = 4 + 5
[1,2,3] => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 8 = 3 + 5
[1,3,2] => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 8 = 3 + 5
[1,4,1] => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 8 = 3 + 5
[1,5] => [1,1] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 7 = 2 + 5
[2,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 9 = 4 + 5
[2,1,3] => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 8 = 3 + 5
[2,3,1] => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 8 = 3 + 5
[2,4] => [1,1] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 7 = 2 + 5
[3,1,2] => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 8 = 3 + 5
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 8 = 3 + 5
[4,1,1] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 9 = 4 + 5
[4,2] => [1,1] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 7 = 2 + 5
[5,1] => [1,1] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 7 = 2 + 5
[6] => [1] => [1,0]
 => [1,1,0,0]
 => 6 = 1 + 5
[1,2,1,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 5 + 5
[1,2,1,3] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 9 = 4 + 5
[1,2,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 9 = 4 + 5
[1,2,4] => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 8 = 3 + 5
[1,3,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 9 = 4 + 5
[1,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 9 = 4 + 5
[1,3,3] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 9 = 4 + 5
[1,4,2] => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 8 = 3 + 5
[1,5,1] => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 8 = 3 + 5
[1,6] => [1,1] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 7 = 2 + 5
[2,1,2,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 5 + 5
[2,1,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 9 = 4 + 5
[2,1,4] => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 8 = 3 + 5
[2,3,2] => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 8 = 3 + 5
[2,4,1] => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 8 = 3 + 5
[2,5] => [1,1] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 7 = 2 + 5
[3,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 9 = 4 + 5
[3,1,3] => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 8 = 3 + 5
[1,1,2,1,2,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 + 5
[1,2,1,2,1,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 6 + 5
[1,2,1,2,2] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 5 + 5
[1,2,1,3,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 5 + 5
[1,2,3,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 5 + 5
[1,3,1,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 5 + 5
[1,3,2,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 5 + 5
[2,1,2,1,1,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 6 + 5
[2,1,2,1,2] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 5 + 5
[2,1,3,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 5 + 5
[3,1,2,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 5 + 5
[1,1,1,2,1,2,1] => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 5
[1,1,2,1,2,1,1] => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 7 + 5
[1,1,2,1,3,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 + 5
[1,1,3,1,2,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 + 5
[1,2,1,2,1,1,1] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 7 + 5
[1,2,1,2,1,2] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 + 5
[1,2,1,2,3] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 5 + 5
[1,2,1,3,1,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 6 + 5
[1,2,1,3,2] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 5 + 5
[1,2,1,4,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 5 + 5
[1,2,3,1,1,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 6 + 5
[1,2,3,1,2] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 5 + 5
[1,2,3,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 5 + 5
[1,2,4,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 5 + 5
[1,3,1,2,1,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 6 + 5
[1,3,1,2,2] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 5 + 5
[1,3,1,3,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 5 + 5
[1,3,2,1,1,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 6 + 5
[1,3,2,1,2] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 5 + 5
[1,4,1,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 5 + 5
[1,4,2,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 5 + 5
[2,1,2,1,1,1,1] => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 7 + 5
[2,1,2,1,2,1] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 + 5
[2,1,2,1,3] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 5 + 5
[2,1,2,3,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 5 + 5
[2,1,3,1,1,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 6 + 5
[2,1,3,1,2] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 5 + 5
[2,1,3,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 5 + 5
[2,1,4,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 5 + 5
[2,3,1,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 5 + 5
[2,3,2,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 5 + 5
[3,1,2,1,1,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 6 + 5
[3,1,2,1,2] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 5 + 5
[3,1,3,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 5 + 5
[3,2,1,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 5 + 5
[4,1,2,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 5 + 5
[1,1,1,2,1,3,1] => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 5

Description

The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St000264

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 14%

Values

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

Description

The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.

Matching statistic: St001651

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001651: Lattices ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 14%

Values

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

Description

The Frankl number of a lattice. For a lattice

$L$ on at least two elements, this is

$\max_x(|L|-2|[x, 1]|),$ where we maximize over all join irreducible elements and

$[x, 1]$ denotes the interval from

$x$ to the top element. Frankl's conjecture asserts that this number is non-negative, and zero if and only if

$L$ is a Boolean lattice.

The following 2 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001060The distinguishing index of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.