- FindStat

Your data matches 14 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000362
(load all 7 compositions to match this statistic)

Mapping

St000362: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The size of a minimal vertex cover of a graph. A '''vertex cover''' of a graph

$G$ is a subset

$S$ of the vertices of

$G$ such that each edge of

$G$ contains at least one vertex of

$S$ . Finding a minimal vertex cover is an NP-hard optimization problem.

Matching statistic: St001176

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The size of a partition minus its first part. This is the number of boxes in its diagram that are not in the first row.

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

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
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%

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

Description

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

Matching statistic: St000228

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The size of a partition. This statistic is the constant statistic of the level sets.

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

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
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%

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

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: St000738

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The first entry in the last row of a standard tableau. For the last entry in the first row, see [[St000734]].

Matching statistic: St000476

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
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%

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

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

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001725: Graphs ⟶ ℤResult quality: 86% ●values known / values provided: 95%●distinct values known / distinct values provided: 86%

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

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: St001232

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 68% ●values known / values provided: 68%●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)
 => [2,1] => [1,1,0,0,1,0]
 => 1
([(0,2),(1,2)],3)
 => [2,1] => [1,1,0,0,1,0]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [1,0,1,0,1,0]
 => ? = 2
([],4)
 => [4] => [1,1,1,1,0,0,0,0]
 => 0
([(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
([(1,3),(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
([(0,3),(1,2)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => ? = 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => ? = 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => ? = 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => ? = 3
([],5)
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
([(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(2,4),(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(1,4),(2,3)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
([(1,4),(2,3),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => ? = 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => ? = 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => ? = 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => ? = 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => ? = 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => ? = 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => ? = 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3
([(0,3),(1,2),(1,4),(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),(2,4),(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
([(0,1),(0,4),(1,3),(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),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],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)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => ? = 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => ? = 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => ? = 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3
([(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
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => ? = 3
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 4
([],6)
 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
([(4,5)],6)
 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(3,5),(4,5)],6)
 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(2,5),(3,5),(4,5)],6)
 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(2,5),(3,4)],6)
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2
([(2,5),(3,4),(4,5)],6)
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2
([(1,2),(3,5),(4,5)],6)
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2
([(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2
([(1,5),(2,5),(3,4),(4,5)],6)
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2
([(0,5),(1,5),(2,4),(3,4)],6)
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 3
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 3
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 3
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 3
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 3
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 3
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 3
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 3
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => ? = 3
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 3
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 3
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 4
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 4
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 4
([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 4
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 4
([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 4

Description

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

Matching statistic: St001227

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 17%●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)
 => [2,1] => [1,1,0,0,1,0]
 => 1
([(0,2),(1,2)],3)
 => [2,1] => [1,1,0,0,1,0]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [1,0,1,0,1,0]
 => 2
([],4)
 => [4] => [1,1,1,1,0,0,0,0]
 => 0
([(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
([(1,3),(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
([(0,3),(1,2)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
([],5)
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
([(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(2,4),(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(1,4),(2,3)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
([(1,4),(2,3),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3
([(0,3),(1,2),(1,4),(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),(2,4),(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
([(0,1),(0,4),(1,3),(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),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],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)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3
([(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
([],7)
 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
([(5,6)],7)
 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
([(4,6),(5,6)],7)
 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
([(3,6),(4,6),(5,6)],7)
 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
([(2,6),(3,6),(4,6),(5,6)],7)
 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
([(3,6),(4,5)],7)
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 2
([(3,6),(4,5),(5,6)],7)
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 2
([(2,3),(4,6),(5,6)],7)
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 2
([(4,5),(4,6),(5,6)],7)
 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 2
([(2,6),(3,6),(4,5),(5,6)],7)
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 2
([(1,2),(3,6),(4,6),(5,6)],7)
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 2
([(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 2
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 2
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 2
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 2
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 2
([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 2
([(3,5),(3,6),(4,5),(4,6)],7)
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 2
([(1,6),(2,6),(3,5),(4,5)],7)
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 2
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 2
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 2
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 2
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 2
([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 2
([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 2
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 2
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 2
([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 2
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 2
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 2
([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 2
([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 2
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 2
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 2
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 2
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 2
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 2
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 2
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 2
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 2
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 2
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 2
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 2
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 2
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 2
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 2

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

St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in 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. St000454The largest eigenvalue of a graph if it is integral.