- FindStat

Your data matches 30 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001486
(load all 9 compositions to match this statistic)

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

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: St000340
(load all 2 compositions to match this statistic)

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

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

$110$ and

$001$ .

Matching statistic: St000691
(load all 6 compositions to match this statistic)

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => [1] => 1 => 0 = 1 - 1
([],2)
 => [2] => 10 => 1 = 2 - 1
([],3)
 => [3] => 100 => 1 = 2 - 1
([],4)
 => [4] => 1000 => 1 = 2 - 1
([(0,3),(1,2)],4)
 => [2,2] => 1010 => 3 = 4 - 1
([(0,3),(1,2),(2,3)],4)
 => [2,2] => 1010 => 3 = 4 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => 1010 => 3 = 4 - 1
([],5)
 => [5] => 10000 => 1 = 2 - 1
([(1,4),(2,3)],5)
 => [3,2] => 10010 => 3 = 4 - 1
([(1,4),(2,3),(3,4)],5)
 => [3,2] => 10010 => 3 = 4 - 1
([(0,1),(2,4),(3,4)],5)
 => [3,2] => 10010 => 3 = 4 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,2] => 10010 => 3 = 4 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => 10010 => 3 = 4 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2] => 10010 => 3 = 4 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => 10010 => 3 = 4 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [3,2] => 10010 => 3 = 4 - 1
([],6)
 => [6] => 100000 => 1 = 2 - 1
([(2,5),(3,4)],6)
 => [4,2] => 100010 => 3 = 4 - 1
([(2,5),(3,4),(4,5)],6)
 => [4,2] => 100010 => 3 = 4 - 1
([(1,2),(3,5),(4,5)],6)
 => [4,2] => 100010 => 3 = 4 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [4,2] => 100010 => 3 = 4 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2] => 100010 => 3 = 4 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [4,2] => 100010 => 3 = 4 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,2] => 100010 => 3 = 4 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [4,2] => 100010 => 3 = 4 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,2] => 100010 => 3 = 4 - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [4,2] => 100010 => 3 = 4 - 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [4,2] => 100010 => 3 = 4 - 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,2] => 100010 => 3 = 4 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,2] => 100010 => 3 = 4 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,2] => 100010 => 3 = 4 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,2] => 100010 => 3 = 4 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,2] => 100010 => 3 = 4 - 1
([(0,5),(1,4),(2,3)],6)
 => [3,3] => 100100 => 3 = 4 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
 => [4,2] => 100010 => 3 = 4 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => [3,3] => 100100 => 3 = 4 - 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [3,3] => 100100 => 3 = 4 - 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,3] => 100100 => 3 = 4 - 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => [3,3] => 100100 => 3 = 4 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,3] => 100100 => 3 = 4 - 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => [3,3] => 100100 => 3 = 4 - 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,3] => 100100 => 3 = 4 - 1
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => [3,3] => 100100 => 3 = 4 - 1
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => [3,3] => 100100 => 3 = 4 - 1
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,3] => 100100 => 3 = 4 - 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,2] => 100110 => 3 = 4 - 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => [3,3] => 100100 => 3 = 4 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [2,2,2] => 101010 => 5 = 6 - 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
 => [2,2,2] => 101010 => 5 = 6 - 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => 101010 => 5 = 6 - 1

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: St000453
(load all 2 compositions to match this statistic)

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000453: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of distinct Laplacian eigenvalues of a graph.

Matching statistic: St000777
(load all 3 compositions to match this statistic)

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of distinct eigenvalues of the distance Laplacian of a connected graph.

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

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001035: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 100%●distinct values known / distinct values provided: 75%

Values

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

Matching statistic: St000762

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000762: Integer compositions ⟶ ℤResult quality: 50% ●values known / values provided: 89%●distinct values known / distinct values provided: 50%

Values

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

Description

The sum of the positions of the weak records of an integer composition. A weak record is an element

$a_i$ such that

$a_i \geq a_j$ for all

$j < i$ . This statistic is the sum of their positions.

Matching statistic: St000455
(load all 5 compositions to match this statistic)

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 38%●distinct values known / distinct values provided: 25%

Values

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

Description

The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.

Matching statistic: St001596
(load all 3 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001596: Skew partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 50%

Values

([],1)
 => []
 => [[],[]]
 => ? = 1 - 4
([],2)
 => []
 => [[],[]]
 => ? = 2 - 4
([],3)
 => []
 => [[],[]]
 => ? = 2 - 4
([],4)
 => []
 => [[],[]]
 => ? = 2 - 4
([(0,3),(1,2)],4)
 => [1,1]
 => [[1,1],[]]
 => 0 = 4 - 4
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [[1,1,1],[]]
 => 0 = 4 - 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [[4],[]]
 => 0 = 4 - 4
([],5)
 => []
 => [[],[]]
 => ? = 2 - 4
([(1,4),(2,3)],5)
 => [1,1]
 => [[1,1],[]]
 => 0 = 4 - 4
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [[1,1,1],[]]
 => 0 = 4 - 4
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [[1,1,1],[]]
 => 0 = 4 - 4
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 0 = 4 - 4
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [[4],[]]
 => 0 = 4 - 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [[4,1],[]]
 => 0 = 4 - 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [[6],[]]
 => 0 = 4 - 4
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 0 = 4 - 4
([],6)
 => []
 => [[],[]]
 => ? = 2 - 4
([(2,5),(3,4)],6)
 => [1,1]
 => [[1,1],[]]
 => 0 = 4 - 4
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [[1,1,1],[]]
 => 0 = 4 - 4
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [[1,1,1],[]]
 => 0 = 4 - 4
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 0 = 4 - 4
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 0 = 4 - 4
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => 0 = 4 - 4
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => [[4],[]]
 => 0 = 4 - 4
([(0,5),(1,5),(2,4),(3,4)],6)
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 0 = 4 - 4
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1]
 => [[4,1],[]]
 => 0 = 4 - 4
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => 0 = 4 - 4
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => 0 = 4 - 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [[4,1,1],[]]
 => 0 = 4 - 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => [[6],[]]
 => 0 = 4 - 4
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [[4,1,1],[]]
 => 0 = 4 - 4
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,1]
 => [[6,1],[]]
 => 0 = 4 - 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [[8],[]]
 => ? = 4 - 4
([(0,5),(1,4),(2,3)],6)
 => [1,1,1]
 => [[1,1,1],[]]
 => 0 = 4 - 4
([(1,5),(2,4),(3,4),(3,5)],6)
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 0 = 4 - 4
([(0,1),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 0 = 4 - 4
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => 0 = 4 - 4
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [[4,1,1],[]]
 => 0 = 4 - 4
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => [1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => 0 = 4 - 4
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1]
 => [[4,1],[]]
 => 0 = 4 - 4
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [[4,1,1],[]]
 => 0 = 4 - 4
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [6,1]
 => [[6,1],[]]
 => 0 = 4 - 4
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => [6]
 => [[6],[]]
 => 0 = 4 - 4
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => [7]
 => [[7],[]]
 => 0 = 4 - 4
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [[8],[]]
 => ? = 4 - 4
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [9]
 => [[9],[]]
 => ? = 4 - 4
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => [9]
 => [[9],[]]
 => ? = 4 - 4
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [[3,3],[]]
 => 2 = 6 - 4
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
 => [3,3,1]
 => [[3,3,1],[]]
 => 2 = 6 - 4
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [5,3]
 => [[5,3],[]]
 => ? = 6 - 4
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => [8]
 => [[8],[]]
 => ? = 6 - 4
([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [9]
 => [[9],[]]
 => ? = 6 - 4
([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [[9],[]]
 => ? = 6 - 4
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [10]
 => [[10],[]]
 => ? = 6 - 4
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [[8],[]]
 => ? = 6 - 4
([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [9]
 => [[9],[]]
 => ? = 6 - 4
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [9]
 => [[9],[]]
 => ? = 6 - 4
([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [[10],[]]
 => ? = 6 - 4
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [10]
 => [[10],[]]
 => ? = 6 - 4
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [11]
 => [[11],[]]
 => ? = 6 - 4
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [12]
 => [[12],[]]
 => ? = 6 - 4
([],7)
 => []
 => [[],[]]
 => ? = 2 - 4
([(3,6),(4,5)],7)
 => [1,1]
 => [[1,1],[]]
 => 0 = 4 - 4
([(3,6),(4,5),(5,6)],7)
 => [1,1,1]
 => [[1,1,1],[]]
 => 0 = 4 - 4
([(2,3),(4,6),(5,6)],7)
 => [1,1,1]
 => [[1,1,1],[]]
 => 0 = 4 - 4
([(2,6),(3,6),(4,5),(5,6)],7)
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 0 = 4 - 4
([(1,2),(3,6),(4,6),(5,6)],7)
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 0 = 4 - 4
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => [1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => 0 = 4 - 4
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => [1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => 0 = 4 - 4
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => [1,1,1,1,1,1]
 => [[1,1,1,1,1,1],[]]
 => 0 = 4 - 4
([(3,5),(3,6),(4,5),(4,6)],7)
 => [4]
 => [[4],[]]
 => 0 = 4 - 4
([(1,6),(2,6),(3,5),(4,5)],7)
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 0 = 4 - 4
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => [4,1]
 => [[4,1],[]]
 => 0 = 4 - 4
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [6,1,1]
 => [[6,1,1],[]]
 => ? = 4 - 4
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [8]
 => [[8],[]]
 => ? = 4 - 4
([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [6,1,1]
 => [[6,1,1],[]]
 => ? = 4 - 4
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [8,1]
 => [[8,1],[]]
 => ? = 4 - 4
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [10]
 => [[10],[]]
 => ? = 4 - 4
([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [6,1,1]
 => [[6,1,1],[]]
 => ? = 4 - 4
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
 => [6,1,1]
 => [[6,1,1],[]]
 => ? = 4 - 4
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [8,1]
 => [[8,1],[]]
 => ? = 4 - 4
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => [6,1,1]
 => [[6,1,1],[]]
 => ? = 4 - 4
([(0,6),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
 => [7,1]
 => [[7,1],[]]
 => ? = 4 - 4
([(0,4),(0,5),(1,2),(1,3),(2,6),(3,6),(4,6),(5,6)],7)
 => [4,4]
 => [[4,4],[]]
 => ? = 4 - 4
([(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [8]
 => [[8],[]]
 => ? = 4 - 4
([(0,6),(1,5),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [6,1,1]
 => [[6,1,1],[]]
 => ? = 4 - 4
([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => [8,1]
 => [[8,1],[]]
 => ? = 4 - 4
([(0,1),(0,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => [8]
 => [[8],[]]
 => ? = 4 - 4
([(0,1),(0,6),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
 => [9]
 => [[9],[]]
 => ? = 4 - 4
([(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => [9]
 => [[9],[]]
 => ? = 4 - 4
([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => [9,1]
 => [[9,1],[]]
 => ? = 4 - 4
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [10]
 => [[10],[]]
 => ? = 4 - 4
([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => [11]
 => [[11],[]]
 => ? = 4 - 4
([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => [9]
 => [[9],[]]
 => ? = 4 - 4
([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
 => [9,1]
 => [[9,1],[]]
 => ? = 4 - 4
([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => [10]
 => [[10],[]]
 => ? = 4 - 4
([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [11]
 => [[11],[]]
 => ? = 4 - 4
([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => [12]
 => [[12],[]]
 => ? = 4 - 4
([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7)
 => [3,3,1,1]
 => [[3,3,1,1],[]]
 => ? = 6 - 4
([(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => [5,3]
 => [[5,3],[]]
 => ? = 6 - 4

Description

The number of two-by-two squares inside a skew partition. This is, the number of cells

$(i,j)$ in a skew partition for which the box

$(i+1,j+1)$ is also a cell inside the skew partition.

Matching statistic: St001890
(load all 2 compositions to match this statistic)

Mapping

Mp00117: Graphs —Ore closure⟶ Graphs
Mp00243: Graphs —weak duplicate order⟶ Posets
St001890: Posets ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 25%

Values

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

Description

The maximum magnitude of the Möbius function of a poset. The '''Möbius function''' of a poset is the multiplicative inverse of the zeta function in the incidence algebra. The Möbius value

$\mu(x, y)$ is equal to the signed sum of chains from

$x$ to

$y$ , where odd-length chains are counted with a minus sign, so this statistic is bounded above by the total number of chains in the poset.

The following 20 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001651The Frankl number of a lattice. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew 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. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001763The Hurwitz number of an integer partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian.