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Your data matches 23 different statistics following compositions of up to 3 maps.
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Matching statistic: St000259

Mapping

Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1]
 => 1 => [1,1] => ([(0,1)],2)
 => 1
[1,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[3]
 => 1 => [1,1] => ([(0,1)],2)
 => 1
[2,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[1,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[3,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[2,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,1,1]
 => 1111 => [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)
 => 1
[5]
 => 1 => [1,1] => ([(0,1)],2)
 => 1
[4,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[3,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[2,2,1]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[2,1,1,1]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,1,1,1]
 => 11111 => [1,1,1,1,1,1] => ([(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)
 => 1
[5,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[4,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,3]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[3,2,1]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,1,1,1]
 => 1111 => [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)
 => 1
[2,2,1,1]
 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,1,1,1,1]
 => 01111 => [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)
 => 2
[1,1,1,1,1,1]
 => 111111 => [1,1,1,1,1,1,1] => ([(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)
 => 1
[7]
 => 1 => [1,1] => ([(0,1)],2)
 => 1
[6,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[5,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,3]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[4,2,1]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[4,1,1,1]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,3,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[3,2,1,1]
 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,1,1,1,1]
 => 11111 => [1,1,1,1,1,1] => ([(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)
 => 1
[2,2,2,1]
 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[2,2,1,1,1]
 => 00111 => [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)
 => 2
[2,1,1,1,1,1]
 => 011111 => [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)
 => 2
[7,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[6,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[5,3]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[5,2,1]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[5,1,1,1]
 => 1111 => [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)
 => 1
[4,3,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,2,1,1]
 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[4,1,1,1,1]
 => 01111 => [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)
 => 2
[3,3,1,1]
 => 1111 => [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)
 => 1
[3,2,2,1]
 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,2,1,1,1]
 => 10111 => [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)
 => 2
[3,1,1,1,1,1]
 => 111111 => [1,1,1,1,1,1,1] => ([(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)
 => 1
[2,2,2,1,1]
 => 00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[2,2,1,1,1,1]
 => 001111 => [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)
 => 2
[9]
 => 1 => [1,1] => ([(0,1)],2)
 => 1
[8,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2

Description

The diameter of a connected graph. This is the greatest distance between any pair of vertices.

Matching statistic: St001261

Mapping

Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001261: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1]
 => 1 => [1] => ([],1)
 => 1
[1,1]
 => 11 => [2] => ([],2)
 => 1
[3]
 => 1 => [1] => ([],1)
 => 1
[2,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 2
[1,1,1]
 => 111 => [3] => ([],3)
 => 1
[3,1]
 => 11 => [2] => ([],2)
 => 1
[2,1,1]
 => 011 => [1,2] => ([(1,2)],3)
 => 2
[1,1,1,1]
 => 1111 => [4] => ([],4)
 => 1
[5]
 => 1 => [1] => ([],1)
 => 1
[4,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 2
[3,1,1]
 => 111 => [3] => ([],3)
 => 1
[2,2,1]
 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[2,1,1,1]
 => 0111 => [1,3] => ([(2,3)],4)
 => 2
[1,1,1,1,1]
 => 11111 => [5] => ([],5)
 => 1
[5,1]
 => 11 => [2] => ([],2)
 => 1
[4,1,1]
 => 011 => [1,2] => ([(1,2)],3)
 => 2
[3,3]
 => 11 => [2] => ([],2)
 => 1
[3,2,1]
 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,1,1,1]
 => 1111 => [4] => ([],4)
 => 1
[2,2,1,1]
 => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => 2
[2,1,1,1,1]
 => 01111 => [1,4] => ([(3,4)],5)
 => 2
[1,1,1,1,1,1]
 => 111111 => [6] => ([],6)
 => 1
[7]
 => 1 => [1] => ([],1)
 => 1
[6,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 2
[5,1,1]
 => 111 => [3] => ([],3)
 => 1
[4,3]
 => 01 => [1,1] => ([(0,1)],2)
 => 2
[4,2,1]
 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[4,1,1,1]
 => 0111 => [1,3] => ([(2,3)],4)
 => 2
[3,3,1]
 => 111 => [3] => ([],3)
 => 1
[3,2,1,1]
 => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[3,1,1,1,1]
 => 11111 => [5] => ([],5)
 => 1
[2,2,2,1]
 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[2,2,1,1,1]
 => 00111 => [2,3] => ([(2,4),(3,4)],5)
 => 2
[2,1,1,1,1,1]
 => 011111 => [1,5] => ([(4,5)],6)
 => 2
[7,1]
 => 11 => [2] => ([],2)
 => 1
[6,1,1]
 => 011 => [1,2] => ([(1,2)],3)
 => 2
[5,3]
 => 11 => [2] => ([],2)
 => 1
[5,2,1]
 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[5,1,1,1]
 => 1111 => [4] => ([],4)
 => 1
[4,3,1]
 => 011 => [1,2] => ([(1,2)],3)
 => 2
[4,2,1,1]
 => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => 2
[4,1,1,1,1]
 => 01111 => [1,4] => ([(3,4)],5)
 => 2
[3,3,1,1]
 => 1111 => [4] => ([],4)
 => 1
[3,2,2,1]
 => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,2,1,1,1]
 => 10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2
[3,1,1,1,1,1]
 => 111111 => [6] => ([],6)
 => 1
[2,2,2,1,1]
 => 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
[2,2,1,1,1,1]
 => 001111 => [2,4] => ([(3,5),(4,5)],6)
 => 2
[9]
 => 1 => [1] => ([],1)
 => 1
[8,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 2

Description

The Castelnuovo-Mumford regularity of a graph.

Matching statistic: St000535

Mapping

Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000535: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1]
 => 1 => [1] => ([],1)
 => 0 = 1 - 1
[1,1]
 => 11 => [2] => ([],2)
 => 0 = 1 - 1
[3]
 => 1 => [1] => ([],1)
 => 0 = 1 - 1
[2,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,1]
 => 111 => [3] => ([],3)
 => 0 = 1 - 1
[3,1]
 => 11 => [2] => ([],2)
 => 0 = 1 - 1
[2,1,1]
 => 011 => [1,2] => ([(1,2)],3)
 => 1 = 2 - 1
[1,1,1,1]
 => 1111 => [4] => ([],4)
 => 0 = 1 - 1
[5]
 => 1 => [1] => ([],1)
 => 0 = 1 - 1
[4,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[3,1,1]
 => 111 => [3] => ([],3)
 => 0 = 1 - 1
[2,2,1]
 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[2,1,1,1]
 => 0111 => [1,3] => ([(2,3)],4)
 => 1 = 2 - 1
[1,1,1,1,1]
 => 11111 => [5] => ([],5)
 => 0 = 1 - 1
[5,1]
 => 11 => [2] => ([],2)
 => 0 = 1 - 1
[4,1,1]
 => 011 => [1,2] => ([(1,2)],3)
 => 1 = 2 - 1
[3,3]
 => 11 => [2] => ([],2)
 => 0 = 1 - 1
[3,2,1]
 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
[3,1,1,1]
 => 1111 => [4] => ([],4)
 => 0 = 1 - 1
[2,2,1,1]
 => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => 1 = 2 - 1
[2,1,1,1,1]
 => 01111 => [1,4] => ([(3,4)],5)
 => 1 = 2 - 1
[1,1,1,1,1,1]
 => 111111 => [6] => ([],6)
 => 0 = 1 - 1
[7]
 => 1 => [1] => ([],1)
 => 0 = 1 - 1
[6,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[5,1,1]
 => 111 => [3] => ([],3)
 => 0 = 1 - 1
[4,3]
 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[4,2,1]
 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[4,1,1,1]
 => 0111 => [1,3] => ([(2,3)],4)
 => 1 = 2 - 1
[3,3,1]
 => 111 => [3] => ([],3)
 => 0 = 1 - 1
[3,2,1,1]
 => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[3,1,1,1,1]
 => 11111 => [5] => ([],5)
 => 0 = 1 - 1
[2,2,2,1]
 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 2 - 1
[2,2,1,1,1]
 => 00111 => [2,3] => ([(2,4),(3,4)],5)
 => 1 = 2 - 1
[2,1,1,1,1,1]
 => 011111 => [1,5] => ([(4,5)],6)
 => 1 = 2 - 1
[7,1]
 => 11 => [2] => ([],2)
 => 0 = 1 - 1
[6,1,1]
 => 011 => [1,2] => ([(1,2)],3)
 => 1 = 2 - 1
[5,3]
 => 11 => [2] => ([],2)
 => 0 = 1 - 1
[5,2,1]
 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
[5,1,1,1]
 => 1111 => [4] => ([],4)
 => 0 = 1 - 1
[4,3,1]
 => 011 => [1,2] => ([(1,2)],3)
 => 1 = 2 - 1
[4,2,1,1]
 => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => 1 = 2 - 1
[4,1,1,1,1]
 => 01111 => [1,4] => ([(3,4)],5)
 => 1 = 2 - 1
[3,3,1,1]
 => 1111 => [4] => ([],4)
 => 0 = 1 - 1
[3,2,2,1]
 => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[3,2,1,1,1]
 => 10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[3,1,1,1,1,1]
 => 111111 => [6] => ([],6)
 => 0 = 1 - 1
[2,2,2,1,1]
 => 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
[2,2,1,1,1,1]
 => 001111 => [2,4] => ([(3,5),(4,5)],6)
 => 1 = 2 - 1
[9]
 => 1 => [1] => ([],1)
 => 0 = 1 - 1
[8,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1

Description

The rank-width of a graph.

Matching statistic: St001333

Mapping

Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001333: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1]
 => 1 => [1] => ([],1)
 => 0 = 1 - 1
[1,1]
 => 11 => [2] => ([],2)
 => 0 = 1 - 1
[3]
 => 1 => [1] => ([],1)
 => 0 = 1 - 1
[2,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,1]
 => 111 => [3] => ([],3)
 => 0 = 1 - 1
[3,1]
 => 11 => [2] => ([],2)
 => 0 = 1 - 1
[2,1,1]
 => 011 => [1,2] => ([(1,2)],3)
 => 1 = 2 - 1
[1,1,1,1]
 => 1111 => [4] => ([],4)
 => 0 = 1 - 1
[5]
 => 1 => [1] => ([],1)
 => 0 = 1 - 1
[4,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[3,1,1]
 => 111 => [3] => ([],3)
 => 0 = 1 - 1
[2,2,1]
 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[2,1,1,1]
 => 0111 => [1,3] => ([(2,3)],4)
 => 1 = 2 - 1
[1,1,1,1,1]
 => 11111 => [5] => ([],5)
 => 0 = 1 - 1
[5,1]
 => 11 => [2] => ([],2)
 => 0 = 1 - 1
[4,1,1]
 => 011 => [1,2] => ([(1,2)],3)
 => 1 = 2 - 1
[3,3]
 => 11 => [2] => ([],2)
 => 0 = 1 - 1
[3,2,1]
 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
[3,1,1,1]
 => 1111 => [4] => ([],4)
 => 0 = 1 - 1
[2,2,1,1]
 => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => 1 = 2 - 1
[2,1,1,1,1]
 => 01111 => [1,4] => ([(3,4)],5)
 => 1 = 2 - 1
[1,1,1,1,1,1]
 => 111111 => [6] => ([],6)
 => 0 = 1 - 1
[7]
 => 1 => [1] => ([],1)
 => 0 = 1 - 1
[6,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[5,1,1]
 => 111 => [3] => ([],3)
 => 0 = 1 - 1
[4,3]
 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[4,2,1]
 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[4,1,1,1]
 => 0111 => [1,3] => ([(2,3)],4)
 => 1 = 2 - 1
[3,3,1]
 => 111 => [3] => ([],3)
 => 0 = 1 - 1
[3,2,1,1]
 => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[3,1,1,1,1]
 => 11111 => [5] => ([],5)
 => 0 = 1 - 1
[2,2,2,1]
 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 2 - 1
[2,2,1,1,1]
 => 00111 => [2,3] => ([(2,4),(3,4)],5)
 => 1 = 2 - 1
[2,1,1,1,1,1]
 => 011111 => [1,5] => ([(4,5)],6)
 => 1 = 2 - 1
[7,1]
 => 11 => [2] => ([],2)
 => 0 = 1 - 1
[6,1,1]
 => 011 => [1,2] => ([(1,2)],3)
 => 1 = 2 - 1
[5,3]
 => 11 => [2] => ([],2)
 => 0 = 1 - 1
[5,2,1]
 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
[5,1,1,1]
 => 1111 => [4] => ([],4)
 => 0 = 1 - 1
[4,3,1]
 => 011 => [1,2] => ([(1,2)],3)
 => 1 = 2 - 1
[4,2,1,1]
 => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => 1 = 2 - 1
[4,1,1,1,1]
 => 01111 => [1,4] => ([(3,4)],5)
 => 1 = 2 - 1
[3,3,1,1]
 => 1111 => [4] => ([],4)
 => 0 = 1 - 1
[3,2,2,1]
 => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[3,2,1,1,1]
 => 10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[3,1,1,1,1,1]
 => 111111 => [6] => ([],6)
 => 0 = 1 - 1
[2,2,2,1,1]
 => 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
[2,2,1,1,1,1]
 => 001111 => [2,4] => ([(3,5),(4,5)],6)
 => 1 = 2 - 1
[9]
 => 1 => [1] => ([],1)
 => 0 = 1 - 1
[8,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1

Description

The cardinality of a minimal edge-isolating set of a graph. Let

$\mathcal F$ be a set of graphs. A set of vertices

$S$ is

$\mathcal F$ -isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of

$S$ does not contain any graph in

$\mathcal F$ . This statistic returns the cardinality of the smallest isolating set when

$\mathcal F$ contains only the graph with one edge.

Matching statistic: St001393

Mapping

Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001393: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1]
 => 1 => [1] => ([],1)
 => 0 = 1 - 1
[1,1]
 => 11 => [2] => ([],2)
 => 0 = 1 - 1
[3]
 => 1 => [1] => ([],1)
 => 0 = 1 - 1
[2,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,1]
 => 111 => [3] => ([],3)
 => 0 = 1 - 1
[3,1]
 => 11 => [2] => ([],2)
 => 0 = 1 - 1
[2,1,1]
 => 011 => [1,2] => ([(1,2)],3)
 => 1 = 2 - 1
[1,1,1,1]
 => 1111 => [4] => ([],4)
 => 0 = 1 - 1
[5]
 => 1 => [1] => ([],1)
 => 0 = 1 - 1
[4,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[3,1,1]
 => 111 => [3] => ([],3)
 => 0 = 1 - 1
[2,2,1]
 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[2,1,1,1]
 => 0111 => [1,3] => ([(2,3)],4)
 => 1 = 2 - 1
[1,1,1,1,1]
 => 11111 => [5] => ([],5)
 => 0 = 1 - 1
[5,1]
 => 11 => [2] => ([],2)
 => 0 = 1 - 1
[4,1,1]
 => 011 => [1,2] => ([(1,2)],3)
 => 1 = 2 - 1
[3,3]
 => 11 => [2] => ([],2)
 => 0 = 1 - 1
[3,2,1]
 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
[3,1,1,1]
 => 1111 => [4] => ([],4)
 => 0 = 1 - 1
[2,2,1,1]
 => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => 1 = 2 - 1
[2,1,1,1,1]
 => 01111 => [1,4] => ([(3,4)],5)
 => 1 = 2 - 1
[1,1,1,1,1,1]
 => 111111 => [6] => ([],6)
 => 0 = 1 - 1
[7]
 => 1 => [1] => ([],1)
 => 0 = 1 - 1
[6,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[5,1,1]
 => 111 => [3] => ([],3)
 => 0 = 1 - 1
[4,3]
 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[4,2,1]
 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[4,1,1,1]
 => 0111 => [1,3] => ([(2,3)],4)
 => 1 = 2 - 1
[3,3,1]
 => 111 => [3] => ([],3)
 => 0 = 1 - 1
[3,2,1,1]
 => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[3,1,1,1,1]
 => 11111 => [5] => ([],5)
 => 0 = 1 - 1
[2,2,2,1]
 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 2 - 1
[2,2,1,1,1]
 => 00111 => [2,3] => ([(2,4),(3,4)],5)
 => 1 = 2 - 1
[2,1,1,1,1,1]
 => 011111 => [1,5] => ([(4,5)],6)
 => 1 = 2 - 1
[7,1]
 => 11 => [2] => ([],2)
 => 0 = 1 - 1
[6,1,1]
 => 011 => [1,2] => ([(1,2)],3)
 => 1 = 2 - 1
[5,3]
 => 11 => [2] => ([],2)
 => 0 = 1 - 1
[5,2,1]
 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
[5,1,1,1]
 => 1111 => [4] => ([],4)
 => 0 = 1 - 1
[4,3,1]
 => 011 => [1,2] => ([(1,2)],3)
 => 1 = 2 - 1
[4,2,1,1]
 => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => 1 = 2 - 1
[4,1,1,1,1]
 => 01111 => [1,4] => ([(3,4)],5)
 => 1 = 2 - 1
[3,3,1,1]
 => 1111 => [4] => ([],4)
 => 0 = 1 - 1
[3,2,2,1]
 => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[3,2,1,1,1]
 => 10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[3,1,1,1,1,1]
 => 111111 => [6] => ([],6)
 => 0 = 1 - 1
[2,2,2,1,1]
 => 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
[2,2,1,1,1,1]
 => 001111 => [2,4] => ([(3,5),(4,5)],6)
 => 1 = 2 - 1
[9]
 => 1 => [1] => ([],1)
 => 0 = 1 - 1
[8,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1

Description

The induced matching number of a graph. An induced matching of a graph is a set of independent edges which is an induced subgraph. This statistic records the maximal number of edges in an induced matching.

Matching statistic: St001093

Mapping

Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001093: Graphs ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%

Values

[1]
 => 1 => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[1,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[3]
 => 1 => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[2,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[2,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,1,1,1]
 => 1111 => [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)
 => 2 = 1 + 1
[5]
 => 1 => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[4,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 3 = 2 + 1
[3,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,2,1]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,1,1,1]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,1,1,1]
 => 11111 => [1,1,1,1,1,1] => ([(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)
 => 2 = 1 + 1
[5,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[4,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,3]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[3,2,1]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,1,1,1]
 => 1111 => [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)
 => 2 = 1 + 1
[2,2,1,1]
 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[2,1,1,1,1]
 => 01111 => [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)
 => 3 = 2 + 1
[1,1,1,1,1,1]
 => 111111 => [1,1,1,1,1,1,1] => ([(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)
 => 2 = 1 + 1
[7]
 => 1 => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[6,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 3 = 2 + 1
[5,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[4,3]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 3 = 2 + 1
[4,2,1]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,1,1,1]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[3,3,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,2,1,1]
 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[3,1,1,1,1]
 => 11111 => [1,1,1,1,1,1] => ([(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)
 => 2 = 1 + 1
[2,2,2,1]
 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[2,2,1,1,1]
 => 00111 => [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)
 => 3 = 2 + 1
[2,1,1,1,1,1]
 => 011111 => [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)
 => 3 = 2 + 1
[7,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[6,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[5,3]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[5,2,1]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[5,1,1,1]
 => 1111 => [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)
 => 2 = 1 + 1
[4,3,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,2,1,1]
 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[4,1,1,1,1]
 => 01111 => [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)
 => 3 = 2 + 1
[3,3,1,1]
 => 1111 => [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)
 => 2 = 1 + 1
[3,2,2,1]
 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[3,2,1,1,1]
 => 10111 => [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)
 => 3 = 2 + 1
[3,1,1,1,1,1]
 => 111111 => [1,1,1,1,1,1,1] => ([(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)
 => 2 = 1 + 1
[2,2,2,1,1]
 => 00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
[2,2,1,1,1,1]
 => 001111 => [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)
 => 3 = 2 + 1
[9]
 => 1 => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[8,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 3 = 2 + 1
[3,3,2,2,1,1]
 => 110011 => [1,1,3,1,1] => ([(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),(5,6)],7)
 => ? = 2 + 1
[5,3,2,2,1,1]
 => 110011 => [1,1,3,1,1] => ([(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),(5,6)],7)
 => ? = 2 + 1
[7,3,2,2,1,1]
 => 110011 => [1,1,3,1,1] => ([(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),(5,6)],7)
 => ? = 2 + 1
[5,5,2,2,1,1]
 => 110011 => [1,1,3,1,1] => ([(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),(5,6)],7)
 => ? = 2 + 1

Description

The detour number of a graph. This is the number of vertices in a longest induced path in a graph. Note that [1] defines the detour number as the number of edges in a longest induced path, which is unsuitable for the empty graph.

Matching statistic: St001734

Mapping

Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001734: Graphs ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%

Values

[1]
 => 1 => [1,1] => ([(0,1)],2)
 => 1
[1,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[3]
 => 1 => [1,1] => ([(0,1)],2)
 => 1
[2,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[1,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[3,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[2,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,1,1]
 => 1111 => [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)
 => 1
[5]
 => 1 => [1,1] => ([(0,1)],2)
 => 1
[4,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[3,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[2,2,1]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[2,1,1,1]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,1,1,1]
 => 11111 => [1,1,1,1,1,1] => ([(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)
 => 1
[5,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[4,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,3]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[3,2,1]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,1,1,1]
 => 1111 => [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)
 => 1
[2,2,1,1]
 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,1,1,1,1]
 => 01111 => [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)
 => 2
[1,1,1,1,1,1]
 => 111111 => [1,1,1,1,1,1,1] => ([(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)
 => ? = 1
[7]
 => 1 => [1,1] => ([(0,1)],2)
 => 1
[6,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[5,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,3]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[4,2,1]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[4,1,1,1]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,3,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[3,2,1,1]
 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,1,1,1,1]
 => 11111 => [1,1,1,1,1,1] => ([(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)
 => 1
[2,2,2,1]
 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[2,2,1,1,1]
 => 00111 => [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)
 => 2
[2,1,1,1,1,1]
 => 011111 => [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)
 => ? = 2
[7,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[6,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[5,3]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[5,2,1]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[5,1,1,1]
 => 1111 => [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)
 => 1
[4,3,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,2,1,1]
 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[4,1,1,1,1]
 => 01111 => [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)
 => 2
[3,3,1,1]
 => 1111 => [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)
 => 1
[3,2,2,1]
 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,2,1,1,1]
 => 10111 => [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)
 => 2
[3,1,1,1,1,1]
 => 111111 => [1,1,1,1,1,1,1] => ([(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)
 => ? = 1
[2,2,2,1,1]
 => 00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[2,2,1,1,1,1]
 => 001111 => [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)
 => ? = 2
[9]
 => 1 => [1,1] => ([(0,1)],2)
 => 1
[8,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[7,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[6,3]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[6,2,1]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[6,1,1,1]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[4,1,1,1,1,1]
 => 011111 => [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)
 => ? = 2
[3,2,1,1,1,1]
 => 101111 => [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)
 => ? = 2
[2,2,2,1,1,1]
 => 000111 => [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)
 => ? = 2
[5,1,1,1,1,1]
 => 111111 => [1,1,1,1,1,1,1] => ([(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)
 => ? = 1
[4,2,1,1,1,1]
 => 001111 => [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)
 => ? = 2
[3,3,1,1,1,1]
 => 111111 => [1,1,1,1,1,1,1] => ([(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)
 => ? = 1
[3,2,2,1,1,1]
 => 100111 => [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)
 => ? = 2
[2,2,2,2,1,1]
 => 000011 => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[6,1,1,1,1,1]
 => 011111 => [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)
 => ? = 2
[5,2,1,1,1,1]
 => 101111 => [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)
 => ? = 2
[4,3,1,1,1,1]
 => 011111 => [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)
 => ? = 2
[4,2,2,1,1,1]
 => 000111 => [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)
 => ? = 2
[3,3,2,1,1,1]
 => 110111 => [1,1,2,1,1,1] => ([(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)
 => ? = 2
[3,2,2,2,1,1]
 => 100011 => [1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[2,2,2,2,2,1]
 => 000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 2
[7,1,1,1,1,1]
 => 111111 => [1,1,1,1,1,1,1] => ([(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)
 => ? = 1
[6,2,1,1,1,1]
 => 001111 => [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)
 => ? = 2
[5,3,1,1,1,1]
 => 111111 => [1,1,1,1,1,1,1] => ([(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)
 => ? = 1
[5,2,2,1,1,1]
 => 100111 => [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)
 => ? = 2
[4,4,1,1,1,1]
 => 001111 => [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)
 => ? = 2
[4,3,2,1,1,1]
 => 010111 => [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)
 => ? = 2
[4,2,2,2,1,1]
 => 000011 => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[3,3,3,1,1,1]
 => 111111 => [1,1,1,1,1,1,1] => ([(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)
 => ? = 1
[3,3,2,2,1,1]
 => 110011 => [1,1,3,1,1] => ([(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),(5,6)],7)
 => ? = 2
[3,2,2,2,2,1]
 => 100001 => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[8,1,1,1,1,1]
 => 011111 => [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)
 => ? = 2
[7,2,1,1,1,1]
 => 101111 => [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)
 => ? = 2
[6,3,1,1,1,1]
 => 011111 => [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)
 => ? = 2
[6,2,2,1,1,1]
 => 000111 => [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)
 => ? = 2
[5,4,1,1,1,1]
 => 101111 => [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)
 => ? = 2
[5,3,2,1,1,1]
 => 110111 => [1,1,2,1,1,1] => ([(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)
 => ? = 2
[5,2,2,2,1,1]
 => 100011 => [1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[4,4,2,1,1,1]
 => 000111 => [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)
 => ? = 2
[4,3,3,1,1,1]
 => 011111 => [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)
 => ? = 2
[4,3,2,2,1,1]
 => 010011 => [2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[4,2,2,2,2,1]
 => 000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 2
[3,3,3,2,1,1]
 => 111011 => [1,1,1,2,1,1] => ([(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)
 => ? = 2
[3,3,2,2,2,1]
 => 110001 => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[9,1,1,1,1,1]
 => 111111 => [1,1,1,1,1,1,1] => ([(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)
 => ? = 1
[8,2,1,1,1,1]
 => 001111 => [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)
 => ? = 2
[7,3,1,1,1,1]
 => 111111 => [1,1,1,1,1,1,1] => ([(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)
 => ? = 1
[7,2,2,1,1,1]
 => 100111 => [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)
 => ? = 2
[6,4,1,1,1,1]
 => 001111 => [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)
 => ? = 2
[6,3,2,1,1,1]
 => 010111 => [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)
 => ? = 2
[6,2,2,2,1,1]
 => 000011 => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[5,5,1,1,1,1]
 => 111111 => [1,1,1,1,1,1,1] => ([(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)
 => ? = 1

Description

The lettericity of a graph. Let

$D$ be a digraph on

$k$ vertices, possibly with loops and let

$w$ be a word of length

$n$ whose letters are vertices of

$D$ . The letter graph corresponding to

$D$ and

$w$ is the graph with vertex set

$\{1,\dots,n\}$ whose edges are the pairs

$(i,j)$ with

$i < j$ sucht that

$(w_i, w_j)$ is a (directed) edge of

$D$ .

Matching statistic: St000455

Mapping

Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 100%

Values

[1]
 => 1 => [1,1] => ([(0,1)],2)
 => -1 = 1 - 2
[1,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 1 - 2
[3]
 => 1 => [1,1] => ([(0,1)],2)
 => -1 = 1 - 2
[2,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 0 = 2 - 2
[1,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => -1 = 1 - 2
[3,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 1 - 2
[2,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 2 - 2
[1,1,1,1]
 => 1111 => [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)
 => -1 = 1 - 2
[5]
 => 1 => [1,1] => ([(0,1)],2)
 => -1 = 1 - 2
[4,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 0 = 2 - 2
[3,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => -1 = 1 - 2
[2,2,1]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 2 - 2
[2,1,1,1]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 2 - 2
[1,1,1,1,1]
 => 11111 => [1,1,1,1,1,1] => ([(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)
 => -1 = 1 - 2
[5,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 1 - 2
[4,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 2 - 2
[3,3]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 1 - 2
[3,2,1]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 - 2
[3,1,1,1]
 => 1111 => [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)
 => -1 = 1 - 2
[2,2,1,1]
 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 2 - 2
[2,1,1,1,1]
 => 01111 => [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 - 2
[1,1,1,1,1,1]
 => 111111 => [1,1,1,1,1,1,1] => ([(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)
 => -1 = 1 - 2
[7]
 => 1 => [1,1] => ([(0,1)],2)
 => -1 = 1 - 2
[6,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 0 = 2 - 2
[5,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => -1 = 1 - 2
[4,3]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 0 = 2 - 2
[4,2,1]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 2 - 2
[4,1,1,1]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 2 - 2
[3,3,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => -1 = 1 - 2
[3,2,1,1]
 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
[3,1,1,1,1]
 => 11111 => [1,1,1,1,1,1] => ([(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)
 => -1 = 1 - 2
[2,2,2,1]
 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 2 - 2
[2,2,1,1,1]
 => 00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 2 - 2
[2,1,1,1,1,1]
 => 011111 => [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 - 2
[7,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 1 - 2
[6,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 2 - 2
[5,3]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 1 - 2
[5,2,1]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 - 2
[5,1,1,1]
 => 1111 => [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)
 => -1 = 1 - 2
[4,3,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 2 - 2
[4,2,1,1]
 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 2 - 2
[4,1,1,1,1]
 => 01111 => [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 - 2
[3,3,1,1]
 => 1111 => [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)
 => -1 = 1 - 2
[3,2,2,1]
 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
[3,2,1,1,1]
 => 10111 => [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)
 => ? = 2 - 2
[3,1,1,1,1,1]
 => 111111 => [1,1,1,1,1,1,1] => ([(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)
 => -1 = 1 - 2
[2,2,2,1,1]
 => 00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 2 - 2
[2,2,1,1,1,1]
 => 001111 => [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 = 2 - 2
[9]
 => 1 => [1,1] => ([(0,1)],2)
 => -1 = 1 - 2
[8,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 0 = 2 - 2
[7,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => -1 = 1 - 2
[6,3]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 0 = 2 - 2
[6,2,1]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 2 - 2
[6,1,1,1]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 2 - 2
[5,3,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => -1 = 1 - 2
[5,2,1,1]
 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
[3,3,2,1]
 => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
[3,2,2,1,1]
 => 10011 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 2
[3,2,1,1,1,1]
 => 101111 => [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)
 => ? = 2 - 2
[7,2,1]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 - 2
[5,4,1]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 - 2
[5,2,2,1]
 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
[5,2,1,1,1]
 => 10111 => [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)
 => ? = 2 - 2
[4,3,2,1]
 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
[3,3,2,1,1]
 => 11011 => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 2
[3,2,2,2,1]
 => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 2
[3,2,2,1,1,1]
 => 100111 => [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)
 => ? = 2 - 2
[7,2,1,1]
 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
[5,4,1,1]
 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
[5,3,2,1]
 => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
[5,2,2,1,1]
 => 10011 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 2
[5,2,1,1,1,1]
 => 101111 => [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)
 => ? = 2 - 2
[4,3,2,1,1]
 => 01011 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 2
[3,3,2,2,1]
 => 11001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 2
[3,3,2,1,1,1]
 => 110111 => [1,1,2,1,1,1] => ([(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)
 => ? = 2 - 2
[3,2,2,2,1,1]
 => 100011 => [1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 - 2
[9,2,1]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 - 2
[7,4,1]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 - 2
[7,2,2,1]
 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
[7,2,1,1,1]
 => 10111 => [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)
 => ? = 2 - 2
[6,3,2,1]
 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
[5,4,3]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 - 2
[5,4,2,1]
 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
[5,4,1,1,1]
 => 10111 => [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)
 => ? = 2 - 2
[5,3,2,1,1]
 => 11011 => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 2
[5,2,2,2,1]
 => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 2
[5,2,2,1,1,1]
 => 100111 => [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)
 => ? = 2 - 2
[4,3,2,2,1]
 => 01001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 2
[4,3,2,1,1,1]
 => 010111 => [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)
 => ? = 2 - 2
[3,3,3,2,1]
 => 11101 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 2
[3,3,2,2,1,1]
 => 110011 => [1,1,3,1,1] => ([(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),(5,6)],7)
 => ? = 2 - 2
[3,2,2,2,2,1]
 => 100001 => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 - 2
[9,2,1,1]
 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
[7,4,1,1]
 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
[7,3,2,1]
 => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
[7,2,2,1,1]
 => 10011 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 2
[7,2,1,1,1,1]
 => 101111 => [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)
 => ? = 2 - 2
[6,3,2,1,1]
 => 01011 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 2
[5,5,2,1]
 => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
[5,4,3,1]
 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2

Description

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

Matching statistic: St000396
(load all 2 compositions to match this statistic)

Mapping

Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000396: Binary trees ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 100%

Values

[1]
 => [1]
 => [1,0,1,0]
 => [.,[.,.]]
 => 1
[1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => 1
[3]
 => [2,1]
 => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => 1
[2,1]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [[[.,.],.],[.,.]]
 => 2
[1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [.,[[[.,.],.],.]]
 => 1
[3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [.,[[.,[.,.]],.]]
 => 1
[2,1,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[.,[.,.]],[.,.]]
 => 2
[1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[[.,.],.],.],.]]
 => 1
[5]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => 1
[4,1]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [[.,.],[.,[.,.]]]
 => 2
[3,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [.,[[[.,[.,.]],.],.]]
 => 1
[2,2,1]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[[[[.,.],.],.],.],[.,.]]
 => 2
[2,1,1,1]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [.,[[.,.],[.,.]]]
 => 2
[1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [.,[[[[[.,.],.],.],.],.]]
 => 1
[5,1]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => 1
[4,1,1]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[.,[[.,.],.]],[.,.]]
 => 2
[3,3]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => 1
[3,2,1]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => 2
[3,1,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [.,[[[[.,[.,.]],.],.],.]]
 => 1
[2,2,1,1]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[[[.,[.,.]],.],.],[.,.]]
 => 2
[2,1,1,1,1]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => 2
[1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [.,[[[[[[.,.],.],.],.],.],.]]
 => 1
[7]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => 1
[6,1]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[.,.],[[.,[.,.]],.]]
 => 2
[5,1,1]
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [.,[[[.,[[.,.],.]],.],.]]
 => 1
[4,3]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[[.,.],.],[.,[.,.]]]
 => 2
[4,2,1]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[[[.,.],[.,.]],.],[.,.]]
 => 2
[4,1,1,1]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [.,[[[.,.],.],[.,.]]]
 => 2
[3,3,1]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [.,[[.,[.,[.,.]]],.]]
 => 1
[3,2,1,1]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => 2
[3,1,1,1,1]
 => [2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [.,[[[[[.,[.,.]],.],.],.],.]]
 => 1
[2,2,2,1]
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [[[[[[[.,.],.],.],.],.],.],[.,.]]
 => ? = 2
[2,2,1,1,1]
 => [5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [[[.,[[.,.],.]],.],[.,.]]
 => 2
[2,1,1,1,1,1]
 => [3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [.,[[[[.,.],[.,.]],.],.]]
 => 2
[7,1]
 => [2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [.,[[.,[[[.,.],.],.]],.]]
 => 1
[6,1,1]
 => [4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [.,[[.,[.,.]],[.,.]]]
 => 2
[5,3]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[[.,[.,.]],.]]]
 => 1
[5,2,1]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => 2
[5,1,1,1]
 => [2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [.,[[[[.,[[.,.],.]],.],.],.]]
 => 1
[4,3,1]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[.,[.,.]],[.,[.,.]]]
 => 2
[4,2,1,1]
 => [6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [[[[.,[[.,.],.]],.],.],[.,.]]
 => 2
[4,1,1,1,1]
 => [4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [.,[[[[.,.],.],[.,.]],.]]
 => 2
[3,3,1,1]
 => [3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [.,[[[.,[.,[.,.]]],.],.]]
 => 1
[3,2,2,1]
 => [5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [[[[.,.],.],[.,.]],[.,.]]
 => 2
[3,2,1,1,1]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => 2
[3,1,1,1,1,1]
 => [2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [.,[[[[[[.,[.,.]],.],.],.],.],.]]
 => 1
[2,2,2,1,1]
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [[[[[[.,[.,.]],.],.],.],.],[.,.]]
 => ? = 2
[2,2,1,1,1,1]
 => [5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [[.,[[[.,.],.],.]],[.,.]]
 => 2
[9]
 => [2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [.,[.,[[[[.,.],.],.],.]]]
 => 1
[8,1]
 => [3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [[.,.],[[[.,[.,.]],.],.]]
 => 2
[7,1,1]
 => [2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [.,[[[.,[[[.,.],.],.]],.],.]]
 => 1
[6,3]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[.,.],[.,[.,[.,.]]]]
 => 2
[4,2,2,1]
 => [7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
 => ? = 2
[2,2,2,2,1]
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]]
 => ? = 2
[2,2,2,1,1,1]
 => [7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [[[[[.,[[.,.],.]],.],.],.],[.,.]]
 => ? = 2
[5,1,1,1,1,1]
 => [2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [.,[[[[[[.,[[.,.],.]],.],.],.],.],.]]
 => ? = 1
[4,2,2,1,1]
 => [8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [[[[[[.,[[.,.],.]],.],.],.],.],[.,.]]
 => ? = 2
[3,2,2,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
 => ? = 2
[2,2,2,2,1,1]
 => [9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [[[[[[[[.,[.,.]],.],.],.],.],.],.],[.,.]]
 => ? = 2
[7,1,1,1,1]
 => [2,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [.,[[[[[.,[[[.,.],.],.]],.],.],.],.]]
 => ? = 1
[6,2,2,1]
 => [7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [[[[[.,.],[[.,.],.]],.],.],[.,.]]
 => ? = 2
[4,4,2,1]
 => [7,4]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
 => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
 => ? = 2
[4,4,1,1,1]
 => [7,2,1,1]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => [[[[.,[[.,[.,.]],.]],.],.],[.,.]]
 => ? = 2
[4,2,2,2,1]
 => [9,2]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
 => [[[[[[[[.,.],[.,.]],.],.],.],.],.],[.,.]]
 => ? = 2
[4,2,2,1,1,1]
 => [8,1,1,1]
 => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
 => [[[[[.,[[[.,.],.],.]],.],.],.],[.,.]]
 => ? = 2
[3,2,2,2,1,1]
 => [7,3,1]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
 => [[[[[.,[.,.]],[.,.]],.],.],[.,.]]
 => ? = 2
[2,2,2,2,2,1]
 => [11]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.],[.,.]]
 => ? = 2
[9,1,1,1]
 => [2,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [.,[[[[.,[[[[.,.],.],.],.]],.],.],.]]
 => ? = 1
[7,1,1,1,1,1]
 => [2,2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
 => [.,[[[[[[.,[[[.,.],.],.]],.],.],.],.],.]]
 => ? = 1
[6,2,2,1,1]
 => [8,2,1,1]
 => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
 => [[[[[.,[[.,[.,.]],.]],.],.],.],[.,.]]
 => ? = 2
[5,3,1,1,1,1]
 => [3,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [.,[[[[[.,[[.,[.,.]],.]],.],.],.],.]]
 => ? = 1
[5,2,2,2,1]
 => [7,3,2]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
 => [[[[[.,.],[.,[.,.]]],.],.],[.,.]]
 => ? = 2
[4,4,2,1,1]
 => [8,3,1]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
 => [[[[[[.,[.,.]],[.,.]],.],.],.],[.,.]]
 => ? = 2
[4,4,1,1,1,1]
 => [7,2,1,1,1]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
 => [[[.,[[[.,[.,.]],.],.]],.],[.,.]]
 => ? = 2
[4,3,2,2,1]
 => [7,5]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
 => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
 => ? = 2
[4,2,2,2,1,1]
 => [10,1,1]
 => [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [[[[[[[[.,[[.,.],.]],.],.],.],.],.],.],[.,.]]
 => ? = 2
[3,2,2,2,2,1]
 => [9,3]
 => [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
 => [[[[[[[[.,.],.],[.,.]],.],.],.],.],[.,.]]
 => ? = 2
[11,1,1]
 => [2,2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => [.,[[[.,[[[[[.,.],.],.],.],.]],.],.]]
 => ? = 1
[9,1,1,1,1]
 => [2,2,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
 => [.,[[[[[.,[[[[.,.],.],.],.]],.],.],.],.]]
 => ? = 1
[8,2,2,1]
 => [7,2,2,2]
 => [1,1,1,1,0,0,1,1,1,0,0,0,0,0,1,0]
 => [[[[.,.],[[[.,.],.],.]],.],[.,.]]
 => ? = 2
[8,1,1,1,1,1]
 => [4,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
 => [.,[[[[[.,[[.,.],[.,.]]],.],.],.],.]]
 => ? = 2
[7,3,1,1,1]
 => [3,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
 => [.,[[[[.,[[[.,[.,.]],.],.]],.],.],.]]
 => ? = 1
[6,4,2,1]
 => [7,4,2]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
 => [[[[[.,.],[.,.]],[.,.]],.],[.,.]]
 => ? = 2
[6,4,1,1,1]
 => [7,2,2,1,1]
 => [1,1,1,0,1,1,0,1,1,0,0,0,0,0,1,0]
 => [[[.,[[.,[[.,.],.]],.]],.],[.,.]]
 => ? = 2
[6,2,2,2,1]
 => [9,2,2]
 => [1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,1,0]
 => ?
 => ? = 2
[6,2,2,1,1,1]
 => [8,2,1,1,1]
 => [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [[[[.,[[[.,[.,.]],.],.]],.],.],[.,.]]
 => ? = 2
[5,2,2,2,1,1]
 => [8,3,1,1]
 => [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,1,0]
 => [[[[[.,[[.,.],[.,.]]],.],.],.],[.,.]]
 => ? = 2
[4,4,4,1]
 => [7,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
 => ? = 2
[4,4,3,1,1]
 => [7,4,1,1]
 => [1,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0]
 => [[[[.,[[.,.],.]],[.,.]],.],[.,.]]
 => ? = 2
[4,4,2,2,1]
 => [9,4]
 => [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
 => [[[[[[[[.,.],.],.],[.,.]],.],.],.],[.,.]]
 => ? = 2
[4,4,2,1,1,1]
 => [9,2,1,1]
 => [1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [[[[[[.,[[.,[.,.]],.]],.],.],.],.],[.,.]]
 => ? = 2
[4,3,2,2,1,1]
 => [7,5,1]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
 => [[[[[.,[.,.]],.],.],[.,.]],[.,.]]
 => ? = 2
[4,2,2,2,2,1]
 => [11,2]
 => [1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0]
 => ?
 => ? = 2
[3,3,2,2,2,1]
 => [7,3,3]
 => [1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
 => [[[[[.,.],.],[[.,.],.]],.],[.,.]]
 => ? = 2
[13,1]
 => [2,2,2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [.,[[.,[[[[[[.,.],.],.],.],.],.]],.]]
 => ? = 1
[11,1,1,1]
 => [2,2,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [.,[[[[.,[[[[[.,.],.],.],.],.]],.],.],.]]
 => ? = 1
[10,1,1,1,1]
 => [4,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
 => ?
 => ? = 2
[9,3,1,1]
 => [3,2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
 => [.,[[[.,[[[[.,[.,.]],.],.],.]],.],.]]
 => ? = 1
[9,1,1,1,1,1]
 => [2,2,2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ?
 => ? = 1
[8,2,2,1,1]
 => [8,2,2,1,1]
 => [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [[[[.,[[.,[[.,.],.]],.]],.],.],[.,.]]
 => ? = 2

Description

The register function (or Horton-Strahler number) of a binary tree. This is different from the dimension of the associated poset for the tree

$[[[.,.],[.,.]],[[.,.],[.,.]]]$ : its register function is 3, whereas the dimension of the associated poset is 2.

Matching statistic: St000298

Mapping

Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000298: Posets ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 100%

Values

[1]
 => 1 => ([(0,1)],2)
 => 1
[1,1]
 => 11 => ([(0,2),(2,1)],3)
 => 1
[3]
 => 1 => ([(0,1)],2)
 => 1
[2,1]
 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,1,1]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1
[3,1]
 => 11 => ([(0,2),(2,1)],3)
 => 1
[2,1,1]
 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[1,1,1,1]
 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[5]
 => 1 => ([(0,1)],2)
 => 1
[4,1]
 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,1,1]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,2,1]
 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[2,1,1,1]
 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[1,1,1,1,1]
 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[5,1]
 => 11 => ([(0,2),(2,1)],3)
 => 1
[4,1,1]
 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[3,3]
 => 11 => ([(0,2),(2,1)],3)
 => 1
[3,2,1]
 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2
[3,1,1,1]
 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,2,1,1]
 => 0011 => ([(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
[2,1,1,1,1]
 => 01111 => ([(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
[1,1,1,1,1,1]
 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[7]
 => 1 => ([(0,1)],2)
 => 1
[6,1]
 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[5,1,1]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1
[4,3]
 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[4,2,1]
 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[4,1,1,1]
 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[3,3,1]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1
[3,2,1,1]
 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 2
[3,1,1,1,1]
 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,2,2,1]
 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[2,2,1,1,1]
 => 00111 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 2
[2,1,1,1,1,1]
 => 011111 => ([(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
[7,1]
 => 11 => ([(0,2),(2,1)],3)
 => 1
[6,1,1]
 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[5,3]
 => 11 => ([(0,2),(2,1)],3)
 => 1
[5,2,1]
 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2
[5,1,1,1]
 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[4,3,1]
 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[4,2,1,1]
 => 0011 => ([(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
[4,1,1,1,1]
 => 01111 => ([(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
[3,3,1,1]
 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[3,2,2,1]
 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 2
[3,2,1,1,1]
 => 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => ? = 2
[3,1,1,1,1,1]
 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[2,2,2,1,1]
 => 00011 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 2
[2,2,1,1,1,1]
 => 001111 => ([(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
[9]
 => 1 => ([(0,1)],2)
 => 1
[8,1]
 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[7,1,1]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1
[6,3]
 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[6,2,1]
 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[6,1,1,1]
 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[5,3,1]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1
[5,2,1,1]
 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 2
[5,1,1,1,1]
 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[4,4,1]
 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[4,3,1,1]
 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[4,2,2,1]
 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[4,2,1,1,1]
 => 00111 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 2
[4,1,1,1,1,1]
 => 011111 => ([(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
[3,3,3]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1
[3,3,2,1]
 => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 2
[3,3,1,1,1]
 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[3,2,2,1,1]
 => 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 2
[3,2,1,1,1,1]
 => 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
 => ? = 2
[2,2,2,2,1]
 => 00001 => ([(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
[2,2,2,1,1,1]
 => 000111 => ([(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
[9,1]
 => 11 => ([(0,2),(2,1)],3)
 => 1
[8,1,1]
 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[7,3]
 => 11 => ([(0,2),(2,1)],3)
 => 1
[7,2,1]
 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2
[7,1,1,1]
 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[6,3,1]
 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[6,2,1,1]
 => 0011 => ([(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
[6,1,1,1,1]
 => 01111 => ([(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
[5,2,2,1]
 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 2
[5,2,1,1,1]
 => 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => ? = 2
[4,4,1,1]
 => 0011 => ([(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
[4,3,2,1]
 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 2
[4,3,1,1,1]
 => 01111 => ([(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
[4,2,2,1,1]
 => 00011 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 2
[4,2,1,1,1,1]
 => 001111 => ([(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
[3,3,2,1,1]
 => 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
 => ? = 2
[3,2,2,2,1]
 => 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
 => ? = 2
[3,2,2,1,1,1]
 => 100111 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
 => ? = 2
[2,2,2,2,1,1]
 => 000011 => ([(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
[8,1,1,1]
 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[7,2,1,1]
 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 2
[6,3,1,1]
 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[6,2,2,1]
 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[6,2,1,1,1]
 => 00111 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 2
[6,1,1,1,1,1]
 => 011111 => ([(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
[5,4,1,1]
 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 2
[5,3,2,1]
 => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 2
[5,2,2,1,1]
 => 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 2
[5,2,1,1,1,1]
 => 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
 => ? = 2
[4,4,2,1]
 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[4,4,1,1,1]
 => 00111 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 2

Description

The order dimension or Dushnik-Miller dimension of a poset. This is the minimal number of linear orderings whose intersection is the given poset.

The following 13 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000264The girth of a graph, which is not a tree. St000640The rank of the largest boolean interval in a poset. St000260The radius of a connected graph. St000836The number of descents of distance 2 of a permutation. St001057The Grundy value of the game of creating an independent set in a graph. St001115The number of even descents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001729The number of visible descents of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ .