searching the database
Your data matches 171 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001232
(load all 25 compositions to match this statistic)
(load all 25 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [1,0,1,0]
=> 1
[1,2] => [1,0,1,1,0,0]
=> 2
[2,1] => [1,1,0,0,1,0]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[1,3] => [1,0,1,1,1,0,0,0]
=> 3
[2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 4
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 6
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> 6
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> 6
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> 6
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> 6
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> 6
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> 5
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> 5
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> 5
[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> 5
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> 5
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> 4
[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> 4
[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 4
[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> 3
[4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> 3
[5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 2
[6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000171
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000171: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000171: 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
[1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,1] => [1,2] => ([(1,2)],3)
=> 1
[1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,1] => [1,3] => ([(2,3)],4)
=> 1
[1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[4,1] => [1,4] => ([(3,4)],5)
=> 1
[1,2,2,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,2,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,3,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,4,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,5] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[2,3,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[3,2,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 3
[4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
[5,1] => [1,5] => ([(4,5)],6)
=> 1
[1,2,2,2] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,2,3,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,2,4] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,3,2,1] => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,3,3] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,4,2] => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,5,1] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,6] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6
[2,2,2,1] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[2,2,3] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[2,3,2] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[2,4,1] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[2,5] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
[3,2,2] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[3,3,1] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 4
[4,2,1] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[4,3] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 3
[5,2] => [2,5] => ([(4,6),(5,6)],7)
=> 2
[6,1] => [1,6] => ([(5,6)],7)
=> 1
Description
The degree of the graph.
This is the maximal vertex degree of a graph.
Matching statistic: St000271
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000271: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000271: 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
[1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,1] => [1,2] => ([(1,2)],3)
=> 1
[1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,1] => [1,3] => ([(2,3)],4)
=> 1
[1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[4,1] => [1,4] => ([(3,4)],5)
=> 1
[1,2,2,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,2,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,3,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,4,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,5] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[2,3,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[3,2,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 3
[4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
[5,1] => [1,5] => ([(4,5)],6)
=> 1
[1,2,2,2] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,2,3,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,2,4] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,3,2,1] => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,3,3] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,4,2] => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,5,1] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,6] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6
[2,2,2,1] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[2,2,3] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[2,3,2] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[2,4,1] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[2,5] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
[3,2,2] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[3,3,1] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 4
[4,2,1] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[4,3] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 3
[5,2] => [2,5] => ([(4,6),(5,6)],7)
=> 2
[6,1] => [1,6] => ([(5,6)],7)
=> 1
Description
The chromatic index of a graph.
This is the minimal number of colours needed such that no two adjacent edges have the same colour.
Matching statistic: St000476
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [1,1] => [1,0,1,0]
=> 1
[1,2] => [2,1] => [1,1,0,0,1,0]
=> 2
[2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 4
[1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 4
[1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
[2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,2,2,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
[1,2,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 5
[1,3,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 5
[1,4,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5
[1,5] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
[2,3,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 4
[2,4] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[3,2,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 3
[3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[4,2] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[5,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,2,2] => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> 6
[1,2,3,1] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> 6
[1,2,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> 6
[1,3,2,1] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> 6
[1,3,3] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> 6
[1,4,2] => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> 6
[1,5,1] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[1,6] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 6
[2,2,2,1] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 5
[2,2,3] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> 5
[2,3,2] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> 5
[2,4,1] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> 5
[2,5] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 5
[3,2,2] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> 4
[3,3,1] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> 4
[3,4] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> 4
[4,2,1] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> 3
[4,3] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 3
[5,2] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> 2
[6,1] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
Description
The sum of the semi-lengths of tunnels before a valley of a Dyck path.
For each valley $v$ in a Dyck path $D$ there is a corresponding tunnel, which
is the factor $T_v = s_i\dots s_j$ of $D$ where $s_i$ is the step after the first intersection of $D$ with the line $y = ht(v)$ to the left of $s_j$. This statistic is
$$
\sum_v (j_v-i_v)/2.
$$
Matching statistic: St000987
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000987: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000987: 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
[1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,1] => [1,2] => ([(1,2)],3)
=> 1
[1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,1] => [1,3] => ([(2,3)],4)
=> 1
[1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[4,1] => [1,4] => ([(3,4)],5)
=> 1
[1,2,2,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,2,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,3,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,4,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,5] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[2,3,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[3,2,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 3
[4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
[5,1] => [1,5] => ([(4,5)],6)
=> 1
[1,2,2,2] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,2,3,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,2,4] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,3,2,1] => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,3,3] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,4,2] => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,5,1] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,6] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6
[2,2,2,1] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[2,2,3] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[2,3,2] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[2,4,1] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[2,5] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
[3,2,2] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[3,3,1] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 4
[4,2,1] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[4,3] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 3
[5,2] => [2,5] => ([(4,6),(5,6)],7)
=> 2
[6,1] => [1,6] => ([(5,6)],7)
=> 1
Description
The number of positive eigenvalues of the Laplacian matrix of the graph.
This is the number of vertices minus the number of connected components of the graph.
Matching statistic: St001118
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001118: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001118: 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
[1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,1] => [1,2] => ([(1,2)],3)
=> 1
[1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,1] => [1,3] => ([(2,3)],4)
=> 1
[1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[4,1] => [1,4] => ([(3,4)],5)
=> 1
[1,2,2,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,2,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,3,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,4,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,5] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[2,3,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[3,2,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 3
[4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
[5,1] => [1,5] => ([(4,5)],6)
=> 1
[1,2,2,2] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,2,3,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,2,4] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,3,2,1] => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,3,3] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,4,2] => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,5,1] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,6] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6
[2,2,2,1] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[2,2,3] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[2,3,2] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[2,4,1] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[2,5] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
[3,2,2] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[3,3,1] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 4
[4,2,1] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[4,3] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 3
[5,2] => [2,5] => ([(4,6),(5,6)],7)
=> 2
[6,1] => [1,6] => ([(5,6)],7)
=> 1
Description
The acyclic chromatic index of a graph.
An acyclic edge coloring of a graph is a proper colouring of the edges of a graph such that the union of the edges colored with any two given colours is a forest.
The smallest number of colours such that such a colouring exists is the acyclic chromatic index.
Matching statistic: St001721
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St001721: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
St001721: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [2] => 10 => 1
[1,2] => [2,1] => 101 => 2
[2,1] => [1,2] => 110 => 1
[1,2,1] => [2,2] => 1010 => 3
[1,3] => [2,1,1] => 1011 => 3
[2,2] => [1,2,1] => 1101 => 2
[3,1] => [1,1,2] => 1110 => 1
[1,2,2] => [2,2,1] => 10101 => 4
[1,3,1] => [2,1,2] => 10110 => 4
[1,4] => [2,1,1,1] => 10111 => 4
[2,2,1] => [1,2,2] => 11010 => 3
[2,3] => [1,2,1,1] => 11011 => 3
[3,2] => [1,1,2,1] => 11101 => 2
[4,1] => [1,1,1,2] => 11110 => 1
[1,2,2,1] => [2,2,2] => 101010 => 5
[1,2,3] => [2,2,1,1] => 101011 => 5
[1,3,2] => [2,1,2,1] => 101101 => 5
[1,4,1] => [2,1,1,2] => 101110 => 5
[1,5] => [2,1,1,1,1] => 101111 => 5
[2,2,2] => [1,2,2,1] => 110101 => 4
[2,3,1] => [1,2,1,2] => 110110 => 4
[2,4] => [1,2,1,1,1] => 110111 => 4
[3,2,1] => [1,1,2,2] => 111010 => 3
[3,3] => [1,1,2,1,1] => 111011 => 3
[4,2] => [1,1,1,2,1] => 111101 => 2
[5,1] => [1,1,1,1,2] => 111110 => 1
[1,2,2,2] => [2,2,2,1] => 1010101 => 6
[1,2,3,1] => [2,2,1,2] => 1010110 => 6
[1,2,4] => [2,2,1,1,1] => 1010111 => 6
[1,3,2,1] => [2,1,2,2] => 1011010 => 6
[1,3,3] => [2,1,2,1,1] => 1011011 => 6
[1,4,2] => [2,1,1,2,1] => 1011101 => 6
[1,5,1] => [2,1,1,1,2] => 1011110 => 6
[1,6] => [2,1,1,1,1,1] => 1011111 => 6
[2,2,2,1] => [1,2,2,2] => 1101010 => 5
[2,2,3] => [1,2,2,1,1] => 1101011 => 5
[2,3,2] => [1,2,1,2,1] => 1101101 => 5
[2,4,1] => [1,2,1,1,2] => 1101110 => 5
[2,5] => [1,2,1,1,1,1] => 1101111 => 5
[3,2,2] => [1,1,2,2,1] => 1110101 => 4
[3,3,1] => [1,1,2,1,2] => 1110110 => 4
[3,4] => [1,1,2,1,1,1] => 1110111 => 4
[4,2,1] => [1,1,1,2,2] => 1111010 => 3
[4,3] => [1,1,1,2,1,1] => 1111011 => 3
[5,2] => [1,1,1,1,2,1] => 1111101 => 2
[6,1] => [1,1,1,1,1,2] => 1111110 => 1
Description
The degree of a binary word.
A valley in a binary word is a letter $0$ which is not immediately followed by a $1$. A peak is a letter $1$ which is not immediately followed by a $0$.
Let $f$ be the map that replaces every valley with a peak. The degree of a binary word $w$ is the number of times $f$ has to be applied to obtain a binary word without zeros.
Matching statistic: St000026
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2 = 1 + 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 5 = 4 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 5 = 4 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 5 = 4 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 3 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> 6 = 5 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 6 = 5 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 6 = 5 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 6 = 5 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 6 = 5 + 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 5 = 4 + 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> 5 = 4 + 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 5 = 4 + 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> 7 = 6 + 1
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> 7 = 6 + 1
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> 7 = 6 + 1
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> 7 = 6 + 1
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> 7 = 6 + 1
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> 7 = 6 + 1
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 7 = 6 + 1
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 7 = 6 + 1
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> 6 = 5 + 1
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> 6 = 5 + 1
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> 6 = 5 + 1
[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> 6 = 5 + 1
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> 6 = 5 + 1
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
Description
The position of the first return of a Dyck path.
Matching statistic: St000718
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000718: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000718: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,1] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 4 = 3 + 1
[2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,1] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,3,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5 = 4 + 1
[2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,1] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,2,2,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,2,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,3,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,4,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,5] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 6 = 5 + 1
[2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
[2,3,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
[2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 5 = 4 + 1
[3,2,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 4 = 3 + 1
[4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
[5,1] => [1,5] => ([(4,5)],6)
=> 2 = 1 + 1
[1,2,2,2] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[1,2,3,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[1,2,4] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[1,3,2,1] => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[1,3,3] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[1,4,2] => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[1,5,1] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[1,6] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 7 = 6 + 1
[2,2,2,1] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 5 + 1
[2,2,3] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 5 + 1
[2,3,2] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 5 + 1
[2,4,1] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 5 + 1
[2,5] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6 = 5 + 1
[3,2,2] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 4 + 1
[3,3,1] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 4 + 1
[3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 5 = 4 + 1
[4,2,1] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[4,3] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[5,2] => [2,5] => ([(4,6),(5,6)],7)
=> 3 = 2 + 1
[6,1] => [1,6] => ([(5,6)],7)
=> 2 = 1 + 1
Description
The largest Laplacian eigenvalue of a graph if it is integral.
This statistic is undefined if the largest Laplacian eigenvalue of the graph is not integral.
Various results are collected in Section 3.9 of [1]
Matching statistic: St001723
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001723: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001723: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [1,1] => ([(0,1)],2)
=> 0 = 1 - 1
[1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,1] => [1,2] => ([(1,2)],3)
=> 0 = 1 - 1
[1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,1] => [1,3] => ([(2,3)],4)
=> 0 = 1 - 1
[1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,3,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[4,1] => [1,4] => ([(3,4)],5)
=> 0 = 1 - 1
[1,2,2,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[1,2,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[1,3,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[1,4,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[1,5] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 5 - 1
[2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[2,3,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 3 = 4 - 1
[3,2,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 1 = 2 - 1
[5,1] => [1,5] => ([(4,5)],6)
=> 0 = 1 - 1
[1,2,2,2] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 6 - 1
[1,2,3,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 6 - 1
[1,2,4] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 6 - 1
[1,3,2,1] => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 6 - 1
[1,3,3] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 6 - 1
[1,4,2] => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 6 - 1
[1,5,1] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 6 - 1
[1,6] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5 = 6 - 1
[2,2,2,1] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[2,2,3] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[2,3,2] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[2,4,1] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[2,5] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 4 = 5 - 1
[3,2,2] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[3,3,1] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 3 = 4 - 1
[4,2,1] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[4,3] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[5,2] => [2,5] => ([(4,6),(5,6)],7)
=> 1 = 2 - 1
[6,1] => [1,6] => ([(5,6)],7)
=> 0 = 1 - 1
Description
The differential of a graph.
The external neighbourhood (or boundary) of a set of vertices $S\subseteq V(G)$ is the set of vertices not in $S$ which are adjacent to a vertex in $S$.
The differential of a set of vertices $S\subseteq V(G)$ is the difference of the size of the external neighbourhood of $S$ and the size of $S$.
The differential of a graph is the maximal differential of a set of vertices.
The following 161 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001724The 2-packing differential of a graph. St000272The treewidth of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000362The size of a minimal vertex cover of a graph. St000454The largest eigenvalue of a graph if it is integral. St000503The maximal difference between two elements in a common block. St000536The pathwidth of a graph. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000728The dimension of a set partition. St000730The maximal arc length of a set partition. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St001120The length of a longest path in a graph. St001644The dimension of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001971The number of negative eigenvalues of the adjacency matrix of the graph. St000010The length of the partition. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000147The largest part of an integer partition. St000172The Grundy number of a graph. St000363The number of minimal vertex covers of a graph. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000459The hook length of the base cell of a partition. St000505The biggest entry in the block containing the 1. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000738The first entry in the last row of a standard tableau. St000808The number of up steps of the associated bargraph. St000839The largest opener of a set partition. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001279The sum of the parts of an integer partition that are at least two. St001389The number of partitions of the same length below the given integer partition. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001883The mutual visibility number of a graph. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000653The last descent of a permutation. St001725The harmonious chromatic number of a graph. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000007The number of saliances of the permutation. St000356The number of occurrences of the pattern 13-2. St000463The number of admissible inversions of a permutation. St001330The hat guessing number of a graph. St000019The cardinality of the support of a permutation. St000209Maximum difference of elements in cycles. St000740The last entry of a permutation. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000028The number of stack-sorts needed to sort a permutation. St000141The maximum drop size of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St000054The first entry of the permutation. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001962The proper pathwidth of a graph. St000822The Hadwiger number of the graph. St000673The number of non-fixed points of a permutation. St000741The Colin de Verdière graph invariant. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001368The number of vertices of maximal degree in a graph. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001963The tree-depth of a graph. St000288The number of ones in a binary word. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001497The position of the largest weak excedence of a permutation. St000733The row containing the largest entry of a standard tableau. St000157The number of descents of a standard tableau. St001812The biclique partition number of a graph. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001117The game chromatic index of a graph. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000030The sum of the descent differences of a permutations. St000051The size of the left subtree of a binary tree. St000067The inversion number of the alternating sign matrix. St000133The "bounce" of a permutation. St000204The number of internal nodes of a binary tree. St000304The load of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St001584The area statistic between a Dyck path and its bounce path. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000143The largest repeated part of a partition. St000235The number of indices that are not cyclical small weak excedances. St000240The number of indices that are not small excedances. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000501The size of the first part in the decomposition of a permutation. St000692Babson and Steingrímsson's statistic of a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001727The number of invisible inversions of a permutation. St001468The smallest fixpoint of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000651The maximal size of a rise in a permutation. St001645The pebbling number of a connected graph. St001622The number of join-irreducible elements of a lattice. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001411The number of patterns 321 or 3412 in a permutation. St001613The binary logarithm of the size of the center of a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St000119The number of occurrences of the pattern 321 in a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000840The number of closers smaller than the largest opener in a perfect matching. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001742The difference of the maximal and the minimal degree in a graph. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000652The maximal difference between successive positions of a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001863The number of weak excedances of a signed permutation. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001267The length of the Lyndon factorization of the binary word. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St000422The energy of a graph, if it is integral. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001879The 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. St000455The second largest eigenvalue of a graph if it is integral. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001087The number of occurrences of the vincular pattern |12-3 in a permutation.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!