- FindStat

Your data matches 178 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001486
(load all 4 compositions to match this statistic)

Mapping

St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of corners of the ribbon associated with an integer composition. We associate a ribbon shape to a composition

$c=(c_1,\dots,c_n)$ with

$c_i$ cells in the

$i$ -th row from bottom to top, such that the cells in two rows overlap in precisely one cell. This statistic records the total number of corners of the ribbon shape.

Matching statistic: St001036
(load all 33 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of inner corners of the parallelogram polyomino associated with the Dyck path.

Matching statistic: St000388

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000388: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of orbits of vertices of a graph under automorphisms.

Matching statistic: St001352

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001352: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of internal nodes in the modular decomposition of a graph.

Matching statistic: St001949

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001949: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The rigidity index of a graph. A base of a permutation group is a set

$B$ such that the pointwise stabilizer of

$B$ is trivial. For example, a base of the symmetric group on

$n$ letters must contain all but one letter. This statistic yields the minimal size of a base for the automorphism group of a graph.

Matching statistic: St001951

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001951: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. The disjoint direct product decomposition of a permutation group factors the group corresponding to the product

$(G, X) \ast (H, Y) = (G\times H, Z)$ , where

$Z$ is the disjoint union of

$X$ and

$Y$ . In particular, for an asymmetric graph, i.e., with trivial automorphism group, this statistic equals the number of vertices, because the trivial action factors completely.

Matching statistic: St001315
(load all 3 compositions to match this statistic)

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001315: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The dissociation number of a graph.

Matching statistic: St000483
(load all 13 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000483: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of times a permutation switches from increasing to decreasing or decreasing to increasing. This is the same as the number of inner peaks plus the number of inner valleys and called alternating runs in [2]

Matching statistic: St000691
(load all 9 compositions to match this statistic)

Mapping

Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,1] => 11 => 00 => 0
[2] => 10 => 11 => 0
[1,1,1] => 111 => 000 => 0
[1,2] => 110 => 111 => 0
[2,1] => 101 => 110 => 1
[3] => 100 => 011 => 1
[1,1,1,1] => 1111 => 0000 => 0
[1,1,2] => 1110 => 1111 => 0
[1,2,1] => 1101 => 1110 => 1
[1,3] => 1100 => 0111 => 1
[2,1,1] => 1011 => 1100 => 1
[2,2] => 1010 => 1101 => 2
[3,1] => 1001 => 0110 => 2
[4] => 1000 => 0011 => 1
[1,1,1,1,1] => 11111 => 00000 => 0
[1,1,1,2] => 11110 => 11111 => 0
[1,1,2,1] => 11101 => 11110 => 1
[1,1,3] => 11100 => 01111 => 1
[1,2,1,1] => 11011 => 11100 => 1
[1,2,2] => 11010 => 11101 => 2
[1,3,1] => 11001 => 01110 => 2
[1,4] => 11000 => 00111 => 1
[2,1,1,1] => 10111 => 11000 => 1
[2,1,2] => 10110 => 11011 => 2
[2,2,1] => 10101 => 11010 => 3
[2,3] => 10100 => 11001 => 2
[3,1,1] => 10011 => 01100 => 2
[3,2] => 10010 => 01101 => 3
[4,1] => 10001 => 00110 => 2
[5] => 10000 => 00011 => 1
[1,1,1,1,1,1] => 111111 => 000000 => 0
[1,1,1,1,2] => 111110 => 111111 => 0
[1,1,1,2,1] => 111101 => 111110 => 1
[1,1,1,3] => 111100 => 011111 => 1
[1,1,2,1,1] => 111011 => 111100 => 1
[1,1,2,2] => 111010 => 111101 => 2
[1,1,3,1] => 111001 => 011110 => 2
[1,1,4] => 111000 => 001111 => 1
[1,2,1,1,1] => 110111 => 111000 => 1
[1,2,1,2] => 110110 => 111011 => 2
[1,2,2,1] => 110101 => 111010 => 3
[1,2,3] => 110100 => 111001 => 2
[1,3,1,1] => 110011 => 011100 => 2
[1,3,2] => 110010 => 011101 => 3
[1,4,1] => 110001 => 001110 => 2
[1,5] => 110000 => 000111 => 1
[2,1,1,1,1] => 101111 => 110000 => 1
[2,1,1,2] => 101110 => 110111 => 2
[2,1,2,1] => 101101 => 110110 => 3
[2,1,3] => 101100 => 110011 => 2

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: St001035
(load all 2 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001035: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The convexity degree of the parallelogram polyomino associated with the Dyck path. A parallelogram polyomino is

$k$ -convex if

$k$ is the maximal number of turns an axis-parallel path must take to connect two cells of the polyomino. For example, any rotation of a Ferrers shape has convexity degree at most one. The (bivariate) generating function is given in Theorem 2 of [1].

The following 168 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000071The number of maximal chains in a poset. St000340The number of non-final maximal constant sub-paths of length greater than one. St000482The (zero)-forcing number of a graph. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000288The number of ones in a binary word. St000142The number of even parts of a partition. St000204The number of internal nodes of a binary tree. St000242The number of indices that are not cyclical small weak excedances. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000646The number of big ascents of a permutation. St000648The number of 2-excedences of a permutation. St000711The number of big exceedences of a permutation. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St000871The number of very big ascents of a permutation. St001388The number of non-attacking neighbors of a permutation. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000390The number of runs of ones in a binary word. St000552The number of cut vertices of a graph. St000619The number of cyclic descents of a permutation. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001304The number of maximally independent sets of vertices of a graph. St001405The number of bonds in a permutation. St001692The number of vertices with higher degree than the average degree in a graph. St000203The number of external nodes of a binary tree. St000236The number of cyclical small weak excedances. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000259The diameter of a connected graph. St000312The number of leaves in a graph. St000636The hull number of a graph. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St001120The length of a longest path in a graph. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001883The mutual visibility number of a graph. St001093The detour number of a graph. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000159The number of distinct parts of the integer partition. St001280The number of parts of an integer partition that are at least two. St000741The Colin de Verdière graph invariant. St000335The difference of lower and upper interactions. St000444The length of the maximal rise of a Dyck path. St000982The length of the longest constant subword. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000681The Grundy value of Chomp on Ferrers diagrams. St001083The number of boxed occurrences of 132 in a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001638The book thickness of a graph. St000365The number of double ascents of a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000366The number of double descents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000742The number of big ascents of a permutation after prepending zero. St000670The reversal length of a permutation. St000675The number of centered multitunnels of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001902The number of potential covers of a poset. St000120The number of left tunnels of a Dyck path. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001488The number of corners of a skew partition. St001537The number of cyclic crossings of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000831The number of indices that are either descents or recoils. St001557The number of inversions of the second entry of a permutation. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000956The maximal displacement of a permutation. St001516The number of cyclic bonds of a permutation. St001649The length of a longest trail in a graph. St001180Number of indecomposable injective modules with projective dimension at most 1. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000993The multiplicity of the largest part of an integer partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001720The minimal length of a chain of small intervals in a lattice. St001090The number of pop-stack-sorts needed to sort a permutation. St000455The second largest eigenvalue of a graph if it is integral. St001118The acyclic chromatic index of a graph. St001624The breadth of a lattice. St000454The largest eigenvalue of a graph if it is integral. St000392The length of the longest run of ones in a binary word. St000628The balance of a binary word. St001626The number of maximal proper sublattices of a lattice. St000456The monochromatic index of a connected graph. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000667The greatest common divisor of the parts of the partition. St000668The least common multiple of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001128The exponens consonantiae of a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001389The number of partitions of the same length below the given integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001527The cyclic permutation representation number of an integer partition. St001564The value of the forgotten symmetric functions when all variables set to 1. St001571The Cartan determinant of the integer partition. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001933The largest multiplicity of a part in an integer partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001637The number of (upper) dissectors of a poset. St000735The last entry on the main diagonal of a standard tableau. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000939The number of characters of the symmetric group whose value on the partition is positive. St001568The smallest positive integer that does not appear twice in the partition. St000834The number of right outer peaks of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001095The number of non-isomorphic posets with precisely one further covering relation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000717The number of ordinal summands of a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000260The radius of a connected graph. St000264The girth of a graph, which is not a tree. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000567The sum of the products of all pairs of parts. St000706The product of the factorials of the multiplicities of an integer partition. St000934The 2-degree of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000075The orbit size of a standard tableau under promotion. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St000089The absolute variation of a composition. St000650The number of 3-rises of a permutation. St001060The distinguishing index of a graph. St001822The number of alignments of a signed permutation. St000023The number of inner peaks of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000779The tier of a permutation. St001469The holeyness of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001768The number of reduced words of a signed permutation. St000099The number of valleys of a permutation, including the boundary. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix.