- FindStat

Your data matches 329 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001309
(load all 14 compositions to match this statistic)

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001309: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of four-cliques in a graph.

Matching statistic: St001329
(load all 9 compositions to match this statistic)

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001329: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. A graph is outerplanar if and only if in any linear ordering of its vertices, there are no four vertices

$a < b < c < d$ such that

$(a,c)$ and

$(b,d)$ are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.

Matching statistic: St001334
(load all 14 compositions to match this statistic)

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001334: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. A graph is

$3$ -colourable if and only if in any linear ordering of its vertices, there are no four vertices

$a < b < c < d$ such that

$(a,b), (b,c)$ and

$(c,d)$ are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.

Matching statistic: St000273
(load all 3 compositions to match this statistic)

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00156: Graphs —line graph⟶ Graphs
St000273: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The domination number of a graph. The domination number of a graph is given by the minimum size of a dominating set of vertices. A dominating set of vertices is a subset of the vertex set of such that every vertex is either in this subset or adjacent to an element of this subset.

Matching statistic: St000544
(load all 3 compositions to match this statistic)

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00156: Graphs —line graph⟶ Graphs
St000544: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The cop number of a graph. This is the minimal number of cops needed to catch the robber. The algorithm is from [2].

Matching statistic: St000755
(load all 2 compositions to match this statistic)

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000755: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. Consider the recurrence

$f(n)=\sum_{p\in\lambda} f(n-p).$ This statistic returns the number of distinct real roots of the associated characteristic polynomial. For example, the partition

$(2,1)$ corresponds to the recurrence

$f(n)=f(n-1)+f(n-2)$ with associated characteristic polynomial

$x^2-x-1$ , which has two real roots.

Matching statistic: St000913
(load all 5 compositions to match this statistic)

Mapping

Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000913: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of ways to refine the partition into singletons. For example there is only one way to refine

$[2,2]$ :

$[2,2] > [2,1,1] > [1,1,1,1]$ . However, there are two ways to refine

$[3,2]$ :

$[3,2] > [2,2,1] > [2,1,1,1] > [1,1,1,1,1$ and

$[3,2] > [3,1,1] > [2,1,1,1] > [1,1,1,1,1]$ . In other words, this is the number of saturated chains in the refinement order from the bottom element to the given partition. The sequence of values on the partitions with only one part is [[A002846]].

Matching statistic: St001031
(load all 30 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001031: 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]
 => 1
[1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
[1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 1
[2,1] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 2
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 2
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 1
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => 2
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => 2
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => 1
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => 1
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => 1
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => 1
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 1
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => 1
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
 => 1
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
 => 1
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => 1
[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => 1
[4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => 1
[5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => 1

Description

The height of the bicoloured Motzkin path associated with the Dyck path.

Matching statistic: St001063
(load all 6 compositions to match this statistic)

Mapping

Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001063: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra.

Matching statistic: St001064
(load all 7 compositions to match this statistic)

Mapping

Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001064: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules.

The following 319 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001322The size of a minimal independent dominating set in a graph. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001829The common independence number of a graph. St000121The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a binary tree. St000142The number of even parts of a partition. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001092The number of distinct even parts of a partition. St001871The number of triconnected components of a graph. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St000048The multinomial of the parts of a partition. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000346The number of coarsenings of a partition. St000390The number of runs of ones in a binary word. St000659The number of rises of length at least 2 of a Dyck path. St000703The number of deficiencies of a permutation. St000781The number of proper colouring schemes of a Ferrers diagram. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000920The logarithmic height of a Dyck path. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001196The global dimension of

$A$ minus the global dimension of

$eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001316The domatic number of a graph. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001413Half the length of the longest even length palindromic prefix of a binary word. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001517The length of a longest pair of twins in a permutation. St001667The maximal size of a pair of weak twins for a permutation. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000052The number of valleys of a Dyck path not on the x-axis. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000143The largest repeated part of a partition. St000150The floored half-sum of the multiplicities of a partition. St000185The weighted size of a partition. St000257The number of distinct parts of a partition that occur at least twice. St000377The dinv defect of an integer partition. St000481The number of upper covers of a partition in dominance order. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000660The number of rises of length at least 3 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000687The dimension of

$Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000931The number of occurrences of the pattern UUU in a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001091The number of parts in an integer partition whose next smaller part has the same size. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001471The magnitude of a Dyck path. St001584The area statistic between a Dyck path and its bounce path. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001928The number of non-overlapping descents in a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000649The number of 3-excedences of a permutation. St000872The number of very big descents of a permutation. St001513The number of nested exceedences of a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001271The competition number of a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n−1}]$ by adding

$c_0$ to

$c_{n−1}$ . St000354The number of recoils of a permutation. St000650The number of 3-rises of a permutation. St000871The number of very big ascents of a permutation. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001193The dimension of

$Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra

$A$ such that

$eA$ is a minimal faithful projective-injective module. St000274The number of perfect matchings of a graph. St000310The minimal degree of a vertex of a graph. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000352The Elizalde-Pak rank of a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001549The number of restricted non-inversions between exceedances. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000379The number of Hamiltonian cycles in a graph. St001728The number of invisible descents of a permutation. St000387The matching number of a graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001286The annihilation number of a graph. St000323The minimal crossing number of a graph. St000370The genus of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St000149The number of cells of the partition whose leg is zero and arm is odd. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000570The Edelman-Greene number of a permutation. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000934The 2-degree of an integer partition. St001114The number of odd descents of a permutation. St001188The number of simple modules

$S$ with grade

$\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra

$A$ corresponding to the Dyck path. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000322The skewness of a graph. St000365The number of double ascents of a permutation. St000488The number of cycles of a permutation of length at most 2. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001345The Hamming dimension of a graph. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001716The 1-improper chromatic number of a graph. St001792The arboricity of a graph. St000454The largest eigenvalue of a graph if it is integral. St001115The number of even descents of a permutation. St000260The radius of a connected graph. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001933The largest multiplicity of a part in an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001176The size of a partition minus its first part. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001961The sum of the greatest common divisors of all pairs of parts. St000259The diameter of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St000956The maximal displacement of a permutation. St000731The number of double exceedences of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000735The last entry on the main diagonal of a standard tableau. St001394The genus of a permutation. St000699The toughness times the least common multiple of 1,. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000782The indicator function of whether a given perfect matching is an L & P matching. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001191Number of simple modules

$S$ with

$Ext_A^i(S,A)=0$ for all

$i=0,1,...,g-1$ in the corresponding Nakayama algebra

$A$ with global dimension

$g$ . St001208The number of connected components of the quiver of

$A/T$ when

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

$A$ of

$K[x]/(x^n)$ . St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001520The number of strict 3-descents. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001556The number of inversions of the third entry of a permutation. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001570The minimal number of edges to add to make a graph Hamiltonian. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001720The minimal length of a chain of small intervals in a lattice. St001846The number of elements which do not have a complement in the lattice. St000264The girth of a graph, which is not a tree. St001498The normalised height of a Nakayama algebra with magnitude 1. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001490The number of connected components of a skew partition. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001645The pebbling number of a connected graph. St000068The number of minimal elements in a poset. St000456The monochromatic index of a connected graph. St001487The number of inner corners of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000486The number of cycles of length at least 3 of a permutation. St000627The exponent of a binary word. St001884The number of borders of a binary word. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001423The number of distinct cubes in a binary word. St001566The length of the longest arithmetic progression in a permutation. 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. St000706The product of the factorials of the multiplicities of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000929The constant term of the character polynomial of an integer partition. St001099The 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 leaf labelled binary trees. St001100The 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 leaf labelled trees. 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. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001527The cyclic permutation representation number of an integer partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001571The Cartan determinant of the integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001943The sum of the squares of the hook lengths of an integer partition. St000145The Dyson rank of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000474Dyson's crank of a partition. St001060The distinguishing index of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St000302The determinant of the distance matrix of a connected graph. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition.