- FindStat

Your data matches 26 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001504

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001504: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path.

Matching statistic: St000016
(load all 2 compositions to match this statistic)

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000016: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of attacking pairs of a standard tableau. Note that this is actually a statistic on the underlying partition. A pair of cells

$(c, d)$ of a Young diagram (in English notation) is said to be attacking if one of the following conditions holds: 1.

$c$ and

$d$ lie in the same row with

$c$ strictly to the west of

$d$ . 2.

$c$ is in the row immediately to the south of

$d$ , and

$c$ lies strictly east of

$d$ .

Matching statistic: St000171

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ 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] => [1,1] => ([(0,1)],2)
 => 1
([(0,1)],2)
 => [1,1] => [2] => ([],2)
 => 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] => [3] => ([],3)
 => 0
([(0,3),(1,2)],4)
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,3] => ([(2,3)],4)
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,3] => ([(2,3)],4)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,3] => ([(2,3)],4)
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [4] => ([],4)
 => 0
([(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] => [1,4] => ([(3,4)],5)
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,4] => ([(3,4)],5)
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,4] => ([(3,4)],5)
 => 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)
 => 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
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,4] => ([(3,4)],5)
 => 1
([(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] => [5] => ([],5)
 => 0
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => [2,2,2] => [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),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [2,2,2] => [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),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(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] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [2,2,2] => [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),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [2,2,2] => [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),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [2,2,2] => [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),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [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),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3

Description

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

Matching statistic: St000340

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000340: 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]
 => 1
([(0,1)],2)
 => [1,1] => [1,1] => [1,0,1,0]
 => 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]
 => 0
([(0,3),(1,2)],4)
 => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 3
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 3
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 3
([(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]
 => 1
([(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]
 => 0
([(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]
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
([(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]
 => 1
([(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] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 5
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
 => [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 5
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 3
([(0,1),(0,5),(1,5),(2,3),(2,4),(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]
 => 5
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 3
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 3
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 5
([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(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]
 => 5
([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 3
([(0,3),(0,5),(1,2),(1,4),(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]
 => 5
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 3
([(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] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 5
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 3
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 5
([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(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]
 => 5
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 3
([(0,3),(0,4),(0,5),(1,2),(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]
 => 5
([(0,1),(0,3),(0,5),(1,2),(1,4),(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]
 => 5
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(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]
 => 5
([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 3
([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 3

Description

The number of non-final maximal constant sub-paths of length greater than one. This is the total number of occurrences of the patterns

$110$ and

$001$ .

Matching statistic: St000691
(load all 2 compositions to match this statistic)

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000691: 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] => [2] => 10 => 1
([(0,1)],2)
 => [1,1] => [1,1] => 11 => 0
([(1,2)],3)
 => [2,1] => [1,2] => 110 => 1
([(0,2),(1,2)],3)
 => [2,1] => [1,2] => 110 => 1
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [1,1,1] => 111 => 0
([(0,3),(1,2)],4)
 => [2,2] => [2,2] => 1010 => 3
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [2,2] => 1010 => 3
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,2] => 1110 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,2] => 1110 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [2,2] => 1010 => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,2] => 1110 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,1,1,1] => 1111 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => 11010 => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => 11010 => 3
([(0,3),(0,4),(1,2),(1,4),(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
([(0,1),(0,4),(1,3),(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),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => 11010 => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => 11010 => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,1,2] => 11110 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,1,2] => 11110 => 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] => 11110 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [1,2,2] => 11010 => 3
([(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
([(0,2),(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] => 11110 => 1
([(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] => [1,1,1,1,1] => 11111 => 0
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [2,2,2] => [2,2,2] => 101010 => 5
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
 => [2,2,2] => [2,2,2] => 101010 => 5
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,1,2,2] => 111010 => 3
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [2,2,2] => 101010 => 5
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,1,2,2] => 111010 => 3
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,1,2,2] => 111010 => 3
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => [2,2,2] => [2,2,2] => 101010 => 5
([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [2,2,2] => [2,2,2] => 101010 => 5
([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,1,2,2] => 111010 => 3
([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [2,2,2] => 101010 => 5
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,1,2,2] => 111010 => 3
([(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] => [1,1,2,2] => 111010 => 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [2,2,2] => [2,2,2] => 101010 => 5
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,1,2,2] => 111010 => 3
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [2,2,2] => [2,2,2] => 101010 => 5
([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [2,2,2] => [2,2,2] => 101010 => 5
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,1,2,2] => 111010 => 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [2,2,2] => [2,2,2] => 101010 => 5
([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [2,2,2] => 101010 => 5
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [2,2,2] => 101010 => 5
([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,1,2,2] => 111010 => 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,1,2,2] => 111010 => 3
([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,1,2,2] => 111010 => 3

Description

The number of changes of a binary word. This is the number of indices

$i$ such that

$w_i \neq w_{i+1}$ .

Matching statistic: St000987

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ 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] => [1,1] => ([(0,1)],2)
 => 1
([(0,1)],2)
 => [1,1] => [2] => ([],2)
 => 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] => [3] => ([],3)
 => 0
([(0,3),(1,2)],4)
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,3] => ([(2,3)],4)
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,3] => ([(2,3)],4)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,3] => ([(2,3)],4)
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [4] => ([],4)
 => 0
([(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] => [1,4] => ([(3,4)],5)
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,4] => ([(3,4)],5)
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,4] => ([(3,4)],5)
 => 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)
 => 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
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,4] => ([(3,4)],5)
 => 1
([(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] => [5] => ([],5)
 => 0
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => [2,2,2] => [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),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [2,2,2] => [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),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(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] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [2,2,2] => [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),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [2,2,2] => [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),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [2,2,2] => [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),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [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),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 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: St001120

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001120: Graphs ⟶ 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] => ([(0,1)],2)
 => 1
([(0,1)],2)
 => [1,1] => [2] => ([],2)
 => 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] => [3] => ([],3)
 => 0
([(0,3),(1,2)],4)
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,3] => ([(2,3)],4)
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,3] => ([(2,3)],4)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,3] => ([(2,3)],4)
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [4] => ([],4)
 => 0
([(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] => [1,4] => ([(3,4)],5)
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,4] => ([(3,4)],5)
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,4] => ([(3,4)],5)
 => 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)
 => 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
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,4] => ([(3,4)],5)
 => 1
([(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] => [5] => ([],5)
 => 0
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => [2,2,2] => [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),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [2,2,2] => [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),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(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] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [2,2,2] => [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),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [2,2,2] => [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),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [2,2,2] => [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),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [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),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3

Description

The length of a longest path in a graph.

Matching statistic: St001232

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ 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] => [1,0]
 => 0
([],2)
 => [2] => [1,1] => [1,0,1,0]
 => 1
([(0,1)],2)
 => [1,1] => [2] => [1,1,0,0]
 => 0
([(1,2)],3)
 => [2,1] => [2,1] => [1,1,0,0,1,0]
 => 1
([(0,2),(1,2)],3)
 => [2,1] => [2,1] => [1,1,0,0,1,0]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [3] => [1,1,1,0,0,0]
 => 0
([(0,3),(1,2)],4)
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [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] => [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] => [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] => [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] => [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] => [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] => [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] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [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] => [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] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(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] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
 => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 3
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 3
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [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),(2,5),(3,4),(4,5)],6)
 => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5
([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5
([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 3
([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 3
([(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] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 3
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5
([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5
([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5
([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 3
([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 3

Description

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

Matching statistic: St001419

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St001419: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => [1] => 1 => 0 => 0
([],2)
 => [2] => 10 => 01 => 1
([(0,1)],2)
 => [1,1] => 11 => 00 => 0
([(1,2)],3)
 => [2,1] => 101 => 010 => 1
([(0,2),(1,2)],3)
 => [2,1] => 101 => 010 => 1
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => 111 => 000 => 0
([(0,3),(1,2)],4)
 => [2,2] => 1010 => 0101 => 3
([(0,3),(1,2),(2,3)],4)
 => [2,2] => 1010 => 0101 => 3
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 1011 => 0100 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 1011 => 0100 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => 1010 => 0101 => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 1011 => 0100 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => 1111 => 0000 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => 10101 => 01010 => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => 10101 => 01010 => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => 10101 => 01010 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => 10101 => 01010 => 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => 10101 => 01010 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => 10101 => 01010 => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => 10101 => 01010 => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => 10111 => 01000 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => 10111 => 01000 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => 10111 => 01000 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => 10101 => 01010 => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => 10101 => 01010 => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => 10111 => 01000 => 1
([(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] => 11111 => 00000 => 0
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [2,2,2] => 101010 => 010101 => 5
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
 => [2,2,2] => 101010 => 010101 => 5
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => 101011 => 010100 => 3
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => 101010 => 010101 => 5
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => 101011 => 010100 => 3
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => 101011 => 010100 => 3
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => [2,2,2] => 101010 => 010101 => 5
([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [2,2,2] => 101010 => 010101 => 5
([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => 101011 => 010100 => 3
([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => 101010 => 010101 => 5
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => 101011 => 010100 => 3
([(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] => 101011 => 010100 => 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [2,2,2] => 101010 => 010101 => 5
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => 101011 => 010100 => 3
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [2,2,2] => 101010 => 010101 => 5
([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [2,2,2] => 101010 => 010101 => 5
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [2,2,1,1] => 101011 => 010100 => 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [2,2,2] => 101010 => 010101 => 5
([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => 101010 => 010101 => 5
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => 101010 => 010101 => 5
([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => 101011 => 010100 => 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => 101011 => 010100 => 3
([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => 101011 => 010100 => 3

Description

The length of the longest palindromic factor beginning with a one of a binary word.

Matching statistic: St001436

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St001436: 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] => 10 => 10 => 1
([(0,1)],2)
 => [1,1] => 11 => 11 => 0
([(1,2)],3)
 => [2,1] => 101 => 101 => 1
([(0,2),(1,2)],3)
 => [2,1] => 101 => 101 => 1
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => 111 => 111 => 0
([(0,3),(1,2)],4)
 => [2,2] => 1010 => 1100 => 3
([(0,3),(1,2),(2,3)],4)
 => [2,2] => 1010 => 1100 => 3
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 1011 => 1011 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 1011 => 1011 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => 1010 => 1100 => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 1011 => 1011 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => 1111 => 1111 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => 10101 => 11001 => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => 10101 => 11001 => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => 10101 => 11001 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => 10101 => 11001 => 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => 10101 => 11001 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => 10101 => 11001 => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => 10101 => 11001 => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => 10111 => 10111 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => 10111 => 10111 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => 10111 => 10111 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => 10101 => 11001 => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => 10101 => 11001 => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => 10111 => 10111 => 1
([(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] => 11111 => 11111 => 0
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [2,2,2] => 101010 => 111000 => 5
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
 => [2,2,2] => 101010 => 111000 => 5
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => 101011 => 110011 => 3
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => 101010 => 111000 => 5
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => 101011 => 110011 => 3
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => 101011 => 110011 => 3
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => [2,2,2] => 101010 => 111000 => 5
([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [2,2,2] => 101010 => 111000 => 5
([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => 101011 => 110011 => 3
([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => 101010 => 111000 => 5
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => 101011 => 110011 => 3
([(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] => 101011 => 110011 => 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [2,2,2] => 101010 => 111000 => 5
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => 101011 => 110011 => 3
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [2,2,2] => 101010 => 111000 => 5
([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [2,2,2] => 101010 => 111000 => 5
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [2,2,1,1] => 101011 => 110011 => 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [2,2,2] => 101010 => 111000 => 5
([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => 101010 => 111000 => 5
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => 101010 => 111000 => 5
([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => 101011 => 110011 => 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => 101011 => 110011 => 3
([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => 101011 => 110011 => 3

Description

The index of a given binary word in the lex-order among all its cyclic shifts.

The following 16 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001721The degree of a binary word. St000453The number of distinct Laplacian eigenvalues of a graph. St000722The number of different neighbourhoods in a graph. St000983The length of the longest alternating subword. St001725The harmonious chromatic number of a graph. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St001500The global dimension of magnitude 1 Nakayama algebras. St001118The acyclic chromatic index of a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001486The number of corners of the ribbon associated with an integer composition. St001117The game chromatic index of a graph. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. 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. St001330The hat guessing number of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph.