- FindStat

Your data matches 16 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001232
(load all 3 compositions to match this statistic)

Mapping

Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
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%

Values

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

Description

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

Matching statistic: St000171
(load all 2 compositions to match this statistic)

Mapping

Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ 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%

Values

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

Description

The degree of the graph. This is the maximal vertex degree of a graph.

Matching statistic: St000987
(load all 2 compositions to match this statistic)

Mapping

Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ 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%

Values

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

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: St001721

Mapping

Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
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%

Values

([],1)
 => [1] => [1] => 1 => 0
([],2)
 => [2] => [1,1] => 11 => 0
([(0,1)],2)
 => [1,1] => [2] => 10 => 1
([],3)
 => [3] => [1,1,1] => 111 => 0
([(1,2)],3)
 => [1,2] => [2,1] => 101 => 2
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [1,2] => 110 => 1
([],4)
 => [4] => [1,1,1,1] => 1111 => 0
([(2,3)],4)
 => [1,3] => [2,1,1] => 1011 => 3
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [2,2] => 1010 => 3
([(0,3),(1,2)],4)
 => [2,2] => [1,2,1] => 1101 => 2
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [1,2,1] => 1101 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [2,2] => 1010 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [1,1,2] => 1110 => 1
([],5)
 => [5] => [1,1,1,1,1] => 11111 => 0
([(3,4)],5)
 => [1,4] => [2,1,1,1] => 10111 => 4
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [2,2,1] => 10101 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [2,1,2] => 10110 => 4
([(1,4),(2,3)],5)
 => [2,3] => [1,2,1,1] => 11011 => 3
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [1,2,1,1] => 11011 => 3
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => [2,2,1] => 10101 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => 11010 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,2,2] => 11010 => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [1,1,2,1] => 11101 => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => 11010 => 3
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => [1,1,1,2] => 11110 => 1
([],6)
 => [6] => [1,1,1,1,1,1] => 111111 => 0
([(4,5)],6)
 => [1,5] => [2,1,1,1,1] => 101111 => 5
([(2,5),(3,5),(4,5)],6)
 => [1,2,3] => [2,2,1,1] => 101011 => 5
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,3,2] => [2,1,2,1] => 101101 => 5
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,4,1] => [2,1,1,2] => 101110 => 5
([(2,5),(3,4)],6)
 => [2,4] => [1,2,1,1,1] => 110111 => 4
([(3,4),(3,5),(4,5)],6)
 => [2,4] => [1,2,1,1,1] => 110111 => 4
([(2,4),(2,5),(3,4),(3,5)],6)
 => [1,2,3] => [2,2,1,1] => 101011 => 5
([(0,5),(1,5),(2,4),(3,4)],6)
 => [2,2,2] => [1,2,2,1] => 110101 => 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [1,2,2,1] => 110101 => 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,3,1] => [1,2,1,2] => 110110 => 4
([(0,5),(1,4),(2,3)],6)
 => [3,3] => [1,1,2,1,1] => 111011 => 3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,2,2,1] => [2,2,2] => 101010 => 5
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [2,2,2] => [1,2,2,1] => 110101 => 4
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,3,2] => [2,1,2,1] => 101101 => 5
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [1,2,2,1] => 110101 => 4
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3] => [1,1,2,1,1] => 111011 => 3
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,2,2,1] => [2,2,2] => 101010 => 5
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,2,2,1] => [2,2,2] => 101010 => 5
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [2,2,2] => [1,2,2,1] => 110101 => 4
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => [1,2,2,1] => [2,2,2] => 101010 => 5
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => [1,4,1] => [2,1,1,2] => 101110 => 5
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,2,1] => [1,1,2,2] => 111010 => 3
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [4,2] => [1,1,1,2,1] => 111101 => 2
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,2,2,1] => [2,2,2] => 101010 => 5

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

Mapping

Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
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%

Values

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

Description

The position of the first return of a Dyck path.

Matching statistic: St000476
(load all 2 compositions to match this statistic)

Mapping

Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%

Values

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

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: St001118

Mapping

Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001118: Graphs ⟶ ℤResult quality: 86% ●values known / values provided: 94%●distinct values known / distinct values provided: 86%

Values

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

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: St001725
(load all 2 compositions to match this statistic)

Mapping

Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001725: Graphs ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%

Values

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

Description

The harmonious chromatic number of a graph. A harmonious colouring is a proper vertex colouring such that any pair of colours appears at most once on adjacent vertices.

Matching statistic: St000010

Mapping

Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the partition.

Matching statistic: St001227
(load all 2 compositions to match this statistic)

Mapping

Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 86%

Values

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

Description

The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.

The following 6 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000653The last descent of a permutation. 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. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001480The number of simple summands of the module J^2/J^3. St001645The pebbling number of a connected graph.