- FindStat

Your data matches 10 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

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

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

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: St000453

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ 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
[1,1] => [2] => [1] => ([],1)
 => 1
[2] => [1] => [1] => ([],1)
 => 1
[1,1,1] => [3] => [1] => ([],1)
 => 1
[3] => [1] => [1] => ([],1)
 => 1
[1,1,1,1] => [4] => [1] => ([],1)
 => 1
[1,1,2] => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[2,1,1] => [1,2] => [1,1] => ([(0,1)],2)
 => 2
[2,2] => [2] => [1] => ([],1)
 => 1
[4] => [1] => [1] => ([],1)
 => 1
[1,1,1,1,1] => [5] => [1] => ([],1)
 => 1
[1,1,1,2] => [3,1] => [1,1] => ([(0,1)],2)
 => 2
[1,1,3] => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[1,2,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[1,2,2] => [1,2] => [1,1] => ([(0,1)],2)
 => 2
[2,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
 => 2
[2,2,1] => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[3,1,1] => [1,2] => [1,1] => ([(0,1)],2)
 => 2
[5] => [1] => [1] => ([],1)
 => 1
[1,1,1,1,1,1] => [6] => [1] => ([],1)
 => 1
[1,1,1,1,2] => [4,1] => [1,1] => ([(0,1)],2)
 => 2
[1,1,1,3] => [3,1] => [1,1] => ([(0,1)],2)
 => 2
[1,1,2,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,1,4] => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[1,2,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[1,2,2,1] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,3,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[2,1,1,1,1] => [1,4] => [1,1] => ([(0,1)],2)
 => 2
[2,1,1,2] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,2,2] => [3] => [1] => ([],1)
 => 1
[3,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
 => 2
[3,3] => [2] => [1] => ([],1)
 => 1
[4,1,1] => [1,2] => [1,1] => ([(0,1)],2)
 => 2
[6] => [1] => [1] => ([],1)
 => 1
[1,1,1,1,1,1,1] => [7] => [1] => ([],1)
 => 1
[1,1,1,1,1,2] => [5,1] => [1,1] => ([(0,1)],2)
 => 2
[1,1,1,1,3] => [4,1] => [1,1] => ([(0,1)],2)
 => 2
[1,1,1,2,1,1] => [3,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,1,1,2,2] => [3,2] => [1,1] => ([(0,1)],2)
 => 2
[1,1,1,4] => [3,1] => [1,1] => ([(0,1)],2)
 => 2
[1,1,2,1,1,1] => [2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,1,2,2,1] => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[1,1,3,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,1,5] => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[1,2,1,1,1,1] => [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[1,2,1,1,2] => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,2,2,2] => [1,3] => [1,1] => ([(0,1)],2)
 => 2
[1,3,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[1,3,3] => [1,2] => [1,1] => ([(0,1)],2)
 => 2
[1,4,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 3

Description

The number of distinct Laplacian eigenvalues of a graph.

Matching statistic: St000777

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ 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
[1,1] => [2] => [1] => ([],1)
 => 1
[2] => [1] => [1] => ([],1)
 => 1
[1,1,1] => [3] => [1] => ([],1)
 => 1
[3] => [1] => [1] => ([],1)
 => 1
[1,1,1,1] => [4] => [1] => ([],1)
 => 1
[1,1,2] => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[2,1,1] => [1,2] => [1,1] => ([(0,1)],2)
 => 2
[2,2] => [2] => [1] => ([],1)
 => 1
[4] => [1] => [1] => ([],1)
 => 1
[1,1,1,1,1] => [5] => [1] => ([],1)
 => 1
[1,1,1,2] => [3,1] => [1,1] => ([(0,1)],2)
 => 2
[1,1,3] => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[1,2,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[1,2,2] => [1,2] => [1,1] => ([(0,1)],2)
 => 2
[2,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
 => 2
[2,2,1] => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[3,1,1] => [1,2] => [1,1] => ([(0,1)],2)
 => 2
[5] => [1] => [1] => ([],1)
 => 1
[1,1,1,1,1,1] => [6] => [1] => ([],1)
 => 1
[1,1,1,1,2] => [4,1] => [1,1] => ([(0,1)],2)
 => 2
[1,1,1,3] => [3,1] => [1,1] => ([(0,1)],2)
 => 2
[1,1,2,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,1,4] => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[1,2,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[1,2,2,1] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,3,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[2,1,1,1,1] => [1,4] => [1,1] => ([(0,1)],2)
 => 2
[2,1,1,2] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,2,2] => [3] => [1] => ([],1)
 => 1
[3,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
 => 2
[3,3] => [2] => [1] => ([],1)
 => 1
[4,1,1] => [1,2] => [1,1] => ([(0,1)],2)
 => 2
[6] => [1] => [1] => ([],1)
 => 1
[1,1,1,1,1,1,1] => [7] => [1] => ([],1)
 => 1
[1,1,1,1,1,2] => [5,1] => [1,1] => ([(0,1)],2)
 => 2
[1,1,1,1,3] => [4,1] => [1,1] => ([(0,1)],2)
 => 2
[1,1,1,2,1,1] => [3,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,1,1,2,2] => [3,2] => [1,1] => ([(0,1)],2)
 => 2
[1,1,1,4] => [3,1] => [1,1] => ([(0,1)],2)
 => 2
[1,1,2,1,1,1] => [2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,1,2,2,1] => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[1,1,3,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,1,5] => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[1,2,1,1,1,1] => [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[1,2,1,1,2] => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,2,2,2] => [1,3] => [1,1] => ([(0,1)],2)
 => 2
[1,3,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[1,3,3] => [1,2] => [1,1] => ([(0,1)],2)
 => 2
[1,4,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 3

Description

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

Matching statistic: St001035

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001035: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 97%●distinct values known / distinct values provided: 80%

Values

[1] => [1] => [1] => [1,0]
 => ? = 1 - 2
[1,1] => [2] => [1] => [1,0]
 => ? = 1 - 2
[2] => [1] => [1] => [1,0]
 => ? = 1 - 2
[1,1,1] => [3] => [1] => [1,0]
 => ? = 1 - 2
[3] => [1] => [1] => [1,0]
 => ? = 1 - 2
[1,1,1,1] => [4] => [1] => [1,0]
 => ? = 1 - 2
[1,1,2] => [2,1] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
[2,1,1] => [1,2] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
[2,2] => [2] => [1] => [1,0]
 => ? = 1 - 2
[4] => [1] => [1] => [1,0]
 => ? = 1 - 2
[1,1,1,1,1] => [5] => [1] => [1,0]
 => ? = 1 - 2
[1,1,1,2] => [3,1] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
[1,1,3] => [2,1] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
[1,2,1,1] => [1,1,2] => [2,1] => [1,1,0,0,1,0]
 => 1 = 3 - 2
[1,2,2] => [1,2] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
[2,1,1,1] => [1,3] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
[2,2,1] => [2,1] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
[3,1,1] => [1,2] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
[5] => [1] => [1] => [1,0]
 => ? = 1 - 2
[1,1,1,1,1,1] => [6] => [1] => [1,0]
 => ? = 1 - 2
[1,1,1,1,2] => [4,1] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
[1,1,1,3] => [3,1] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
[1,1,2,1,1] => [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 2 - 2
[1,1,4] => [2,1] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
[1,2,1,1,1] => [1,1,3] => [2,1] => [1,1,0,0,1,0]
 => 1 = 3 - 2
[1,2,2,1] => [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 2 - 2
[1,3,1,1] => [1,1,2] => [2,1] => [1,1,0,0,1,0]
 => 1 = 3 - 2
[2,1,1,1,1] => [1,4] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
[2,1,1,2] => [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 2 - 2
[2,2,2] => [3] => [1] => [1,0]
 => ? = 1 - 2
[3,1,1,1] => [1,3] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
[3,3] => [2] => [1] => [1,0]
 => ? = 1 - 2
[4,1,1] => [1,2] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
[6] => [1] => [1] => [1,0]
 => ? = 1 - 2
[1,1,1,1,1,1,1] => [7] => [1] => [1,0]
 => ? = 1 - 2
[1,1,1,1,1,2] => [5,1] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
[1,1,1,1,3] => [4,1] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
[1,1,1,2,1,1] => [3,1,2] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 2 - 2
[1,1,1,2,2] => [3,2] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
[1,1,1,4] => [3,1] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
[1,1,2,1,1,1] => [2,1,3] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 2 - 2
[1,1,2,2,1] => [2,2,1] => [2,1] => [1,1,0,0,1,0]
 => 1 = 3 - 2
[1,1,3,1,1] => [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 2 - 2
[1,1,5] => [2,1] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
[1,2,1,1,1,1] => [1,1,4] => [2,1] => [1,1,0,0,1,0]
 => 1 = 3 - 2
[1,2,1,1,2] => [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 3 - 2
[1,2,2,2] => [1,3] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
[1,3,1,1,1] => [1,1,3] => [2,1] => [1,1,0,0,1,0]
 => 1 = 3 - 2
[1,3,3] => [1,2] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
[1,4,1,1] => [1,1,2] => [2,1] => [1,1,0,0,1,0]
 => 1 = 3 - 2
[2,1,1,1,1,1] => [1,5] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
[2,1,1,1,2] => [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 2 - 2
[2,1,1,3] => [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 2 - 2
[2,1,2,1,1] => [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 3 - 2
[2,1,2,2] => [1,1,2] => [2,1] => [1,1,0,0,1,0]
 => 1 = 3 - 2
[2,2,1,1,1] => [2,3] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
[2,2,2,1] => [3,1] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
[2,2,3] => [2,1] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
[2,3,1,1] => [1,1,2] => [2,1] => [1,1,0,0,1,0]
 => 1 = 3 - 2
[3,1,1,1,1] => [1,4] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
[3,1,1,2] => [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 2 - 2
[3,2,1,1] => [1,1,2] => [2,1] => [1,1,0,0,1,0]
 => 1 = 3 - 2
[3,2,2] => [1,2] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
[3,3,1] => [2,1] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
[4,1,1,1] => [1,3] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
[7] => [1] => [1] => [1,0]
 => ? = 1 - 2
[1,1,1,1,1,1,1,1] => [8] => [1] => [1,0]
 => ? = 1 - 2
[2,2,2,2] => [4] => [1] => [1,0]
 => ? = 1 - 2
[4,4] => [2] => [1] => [1,0]
 => ? = 1 - 2
[8] => [1] => [1] => [1,0]
 => ? = 1 - 2
[1,1,1,1,1,1,1,1,1] => [9] => [1] => [1,0]
 => ? = 1 - 2
[3,3,3] => [3] => [1] => [1,0]
 => ? = 1 - 2
[9] => [1] => [1] => [1,0]
 => ? = 1 - 2
[1,1,1,1,1,1,1,1,1,1] => [10] => [1] => [1,0]
 => ? = 1 - 2
[2,2,2,2,2] => [5] => [1] => [1,0]
 => ? = 1 - 2
[5,5] => [2] => [1] => [1,0]
 => ? = 1 - 2
[10] => [1] => [1] => [1,0]
 => ? = 1 - 2
[1,1,1,1,1,1,1,1,1,1,1,1] => [12] => [1] => [1,0]
 => ? = 1 - 2
[2,2,2,2,2,2] => [6] => [1] => [1,0]
 => ? = 1 - 2
[3,3,3,3] => [4] => [1] => [1,0]
 => ? = 1 - 2
[6,6] => [2] => [1] => [1,0]
 => ? = 1 - 2
[4,4,4] => [3] => [1] => [1,0]
 => ? = 1 - 2
[12] => [1] => [1] => [1,0]
 => ? = 1 - 2
[11] => [1] => [1] => [1,0]
 => ? = 1 - 2
[1,1,1,1,1,1,1,1,1,1,1] => [11] => [1] => [1,0]
 => ? = 1 - 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: St001488

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001488: Skew partitions ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1] => [[1],[]]
 => 1
[1,1] => [2] => [1] => [[1],[]]
 => 1
[2] => [1] => [1] => [[1],[]]
 => 1
[1,1,1] => [3] => [1] => [[1],[]]
 => 1
[3] => [1] => [1] => [[1],[]]
 => 1
[1,1,1,1] => [4] => [1] => [[1],[]]
 => 1
[1,1,2] => [2,1] => [1,1] => [[1,1],[]]
 => 2
[2,1,1] => [1,2] => [1,1] => [[1,1],[]]
 => 2
[2,2] => [2] => [1] => [[1],[]]
 => 1
[4] => [1] => [1] => [[1],[]]
 => 1
[1,1,1,1,1] => [5] => [1] => [[1],[]]
 => 1
[1,1,1,2] => [3,1] => [1,1] => [[1,1],[]]
 => 2
[1,1,3] => [2,1] => [1,1] => [[1,1],[]]
 => 2
[1,2,1,1] => [1,1,2] => [2,1] => [[2,2],[1]]
 => 3
[1,2,2] => [1,2] => [1,1] => [[1,1],[]]
 => 2
[2,1,1,1] => [1,3] => [1,1] => [[1,1],[]]
 => 2
[2,2,1] => [2,1] => [1,1] => [[1,1],[]]
 => 2
[3,1,1] => [1,2] => [1,1] => [[1,1],[]]
 => 2
[5] => [1] => [1] => [[1],[]]
 => 1
[1,1,1,1,1,1] => [6] => [1] => [[1],[]]
 => 1
[1,1,1,1,2] => [4,1] => [1,1] => [[1,1],[]]
 => 2
[1,1,1,3] => [3,1] => [1,1] => [[1,1],[]]
 => 2
[1,1,2,1,1] => [2,1,2] => [1,1,1] => [[1,1,1],[]]
 => 2
[1,1,4] => [2,1] => [1,1] => [[1,1],[]]
 => 2
[1,2,1,1,1] => [1,1,3] => [2,1] => [[2,2],[1]]
 => 3
[1,2,2,1] => [1,2,1] => [1,1,1] => [[1,1,1],[]]
 => 2
[1,3,1,1] => [1,1,2] => [2,1] => [[2,2],[1]]
 => 3
[2,1,1,1,1] => [1,4] => [1,1] => [[1,1],[]]
 => 2
[2,1,1,2] => [1,2,1] => [1,1,1] => [[1,1,1],[]]
 => 2
[2,2,2] => [3] => [1] => [[1],[]]
 => 1
[3,1,1,1] => [1,3] => [1,1] => [[1,1],[]]
 => 2
[3,3] => [2] => [1] => [[1],[]]
 => 1
[4,1,1] => [1,2] => [1,1] => [[1,1],[]]
 => 2
[6] => [1] => [1] => [[1],[]]
 => 1
[1,1,1,1,1,1,1] => [7] => [1] => [[1],[]]
 => 1
[1,1,1,1,1,2] => [5,1] => [1,1] => [[1,1],[]]
 => 2
[1,1,1,1,3] => [4,1] => [1,1] => [[1,1],[]]
 => 2
[1,1,1,2,1,1] => [3,1,2] => [1,1,1] => [[1,1,1],[]]
 => 2
[1,1,1,2,2] => [3,2] => [1,1] => [[1,1],[]]
 => 2
[1,1,1,4] => [3,1] => [1,1] => [[1,1],[]]
 => 2
[1,1,2,1,1,1] => [2,1,3] => [1,1,1] => [[1,1,1],[]]
 => 2
[1,1,2,2,1] => [2,2,1] => [2,1] => [[2,2],[1]]
 => 3
[1,1,3,1,1] => [2,1,2] => [1,1,1] => [[1,1,1],[]]
 => 2
[1,1,5] => [2,1] => [1,1] => [[1,1],[]]
 => 2
[1,2,1,1,1,1] => [1,1,4] => [2,1] => [[2,2],[1]]
 => 3
[1,2,1,1,2] => [1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
 => 3
[1,2,2,2] => [1,3] => [1,1] => [[1,1],[]]
 => 2
[1,3,1,1,1] => [1,1,3] => [2,1] => [[2,2],[1]]
 => 3
[1,3,3] => [1,2] => [1,1] => [[1,1],[]]
 => 2
[1,4,1,1] => [1,1,2] => [2,1] => [[2,2],[1]]
 => 3
[1,2,1,2,1,1,2] => [1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
 => ? = 3
[2,1,2,1,2,1,1] => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
[1,1,2,1,2,3,1,1] => [2,1,1,1,1,2] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
[1,1,2,1,2,1,1,3] => [2,1,1,1,2,1] => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => ? = 4
[1,1,3,2,1,2,1,1] => [2,1,1,1,1,2] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
[2,1,2,2,1,2,1,1] => [1,1,2,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ? = 5
[2,1,2,1,1,3,1,1] => [1,1,1,2,1,2] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => ? = 3
[3,1,1,2,1,2,1,1] => [1,2,1,1,1,2] => [1,1,3,1] => [[3,3,1,1],[2]]
 => ? = 4
[1,2,1,3,1,1,4] => [1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
 => ? = 3
[1,2,1,2,3,2,2] => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
[1,2,1,2,1,3,3] => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
[1,2,1,2,1,1,5] => [1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
 => ? = 3
[1,3,1,2,1,1,3] => [1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
 => ? = 3
[3,1,2,1,2,2,1] => [1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
 => ? = 3
[1,3,2,1,2,2,1] => [1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
 => ? = 3
[1,2,4,1,2,1,1] => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
[1,4,2,1,2,1,1] => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
[1,1,2,1,2,1,2,1,1] => [2,1,1,1,1,1,2] => [1,5,1] => [[5,5,1],[4]]
 => ? = 4
[1,3,1,2,3,1,1] => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
[2,1,2,1,1,2,2,1] => [1,1,1,2,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
[2,1,1,2,1,2,2,1] => [1,2,1,1,2,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ? = 4
[1,2,1,2,1,2,2,1] => [1,1,1,1,1,2,1] => [5,1,1] => [[5,5,5],[4,4]]
 => ? = 3
[1,2,1,1,2,1,2,2] => [1,1,2,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ? = 5
[1,1,2,1,2,1,2,2] => [2,1,1,1,1,2] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
[3,1,3,1,2,2,1] => [1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
 => ? = 3
[2,3,2,1,2,2,1] => [1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
 => ? = 3
[4,1,2,1,2,2,1] => [1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
 => ? = 3
[2,3,1,2,3,1,1] => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
[3,2,1,2,3,1,1] => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
[3,1,3,1,3,1,1] => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
[2,3,2,1,3,1,1] => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
[4,1,2,1,3,1,1] => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
[3,1,2,3,2,1,1] => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
[2,3,1,3,2,1,1] => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
[3,2,1,3,2,1,1] => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
[3,1,4,1,2,1,1] => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
[3,2,3,1,2,1,1] => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
[4,1,3,1,2,1,1] => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
[2,4,2,1,2,1,1] => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
[5,1,2,1,2,1,1] => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
[2,2,1,2,1,2,1,1] => [2,1,1,1,1,2] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
[2,1,2,1,1,1,2,1,1] => [1,1,1,3,1,2] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => ? = 3
[2,1,2,1,2,1,1,2] => [1,1,1,1,1,2,1] => [5,1,1] => [[5,5,5],[4,4]]
 => ? = 3
[2,1,2,1,1,2,1,1,1] => [1,1,1,2,1,3] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => ? = 3
[2,1,1,2,1,1,2,1,1] => [1,2,1,2,1,2] => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => ? = 2
[2,1,1,2,1,2,1,1,1] => [1,2,1,1,1,3] => [1,1,3,1] => [[3,3,1,1],[2]]
 => ? = 4
[2,1,1,1,2,1,2,1,1] => [1,3,1,1,1,2] => [1,1,3,1] => [[3,3,1,1],[2]]
 => ? = 4
[1,2,1,2,1,1,2,1,1] => [1,1,1,1,2,1,2] => [4,1,1,1] => [[4,4,4,4],[3,3,3]]
 => ? = 3
[1,2,1,1,2,1,2,1,1] => [1,1,2,1,1,1,2] => [2,1,3,1] => [[4,4,2,2],[3,1,1]]
 => ? = 5
[1,2,1,1,1,2,1,1,2] => [1,1,3,1,2,1] => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => ? = 3

Description

The number of corners of a skew partition. This is also known as the number of removable cells of the skew partition.

Matching statistic: St000455
(load all 2 compositions to match this statistic)

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 84%●distinct values known / distinct values provided: 40%

Values

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

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: St001431

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001431: Dyck paths ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 60%

Values

[1] => [1] => [1,0]
 => [1,1,0,0]
 => 1
[1,1] => [2] => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 1
[2] => [1] => [1,0]
 => [1,1,0,0]
 => 1
[1,1,1] => [3] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[3] => [1] => [1,0]
 => [1,1,0,0]
 => 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,2] => [2,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[2,1,1] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[2,2] => [2] => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 1
[4] => [1] => [1,0]
 => [1,1,0,0]
 => 1
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,1,3] => [2,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3
[1,2,2] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2
[2,2,1] => [2,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[3,1,1] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[5] => [1] => [1,0]
 => [1,1,0,0]
 => 1
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 2
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => ? = 2
[1,1,4] => [2,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 3
[1,2,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2
[1,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3
[2,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 2
[2,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2
[2,2,2] => [3] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[3,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2
[3,3] => [2] => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 1
[4,1,1] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[6] => [1] => [1,0]
 => [1,1,0,0]
 => 1
[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
[1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 2
[1,1,1,1,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 2
[1,1,1,2,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
 => ? = 2
[1,1,1,2,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 2
[1,1,1,4] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,1,2,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
 => ? = 2
[1,1,2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => ? = 3
[1,1,3,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => ? = 2
[1,1,5] => [2,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,2,1,1,1,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 3
[1,2,1,1,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => ? = 3
[1,2,2,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2
[1,3,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 3
[1,3,3] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[1,4,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3
[2,1,1,1,1,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[2,1,1,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => ? = 2
[2,1,1,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2
[2,1,2,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 3
[2,1,2,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3
[2,2,1,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 2
[2,2,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[2,2,3] => [2,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[2,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3
[3,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 2
[3,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2
[3,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3
[3,2,2] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[3,3,1] => [2,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[4,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2
[5,1,1] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[7] => [1] => [1,0]
 => [1,1,0,0]
 => 1
[1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 1
[1,1,1,1,1,1,2] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => ? = 2
[1,1,1,1,1,3] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 2
[1,1,1,1,2,1,1] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,1,0,0,0]
 => ? = 2
[1,1,1,1,2,2] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => ? = 2
[1,1,1,1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 2
[1,1,1,2,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
 => ? = 2
[1,1,1,2,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => ? = 2
[1,1,1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
 => ? = 2
[1,1,1,5] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,1,2,1,1,1,1] => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 2
[1,1,2,1,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
 => ? = 2
[1,1,2,2,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 2
[1,1,3,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
 => ? = 2
[1,1,4,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => ? = 2
[1,1,6] => [2,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,2,1,1,1,1,1] => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 3
[1,2,1,1,1,2] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => ? = 3
[1,2,1,1,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => ? = 3
[1,2,1,2,1,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 3
[1,2,1,2,2] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 3
[1,2,2,1,1,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 2
[1,2,2,2,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => ? = 2
[1,2,2,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2
[1,2,3,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 3
[1,3,1,1,1,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 3
[1,3,1,1,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => ? = 3
[1,3,2,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 3
[1,3,2,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3
[1,3,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2
[1,4,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 3
[2,1,1,1,1,1,1] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 2
[2,1,1,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 2

Description

Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. The modified algebra B is obtained from the stable Auslander algebra kQ/I by deleting all relations which contain walks of length at least three (conjectural this step of deletion is not necessary as the stable higher Auslander algebras might be quadratic) and taking as B then the algebra kQ^(op)/J when J is the quadratic perp of the ideal I. See http://www.findstat.org/DyckPaths/NakayamaAlgebras for the definition of Loewy length and Nakayama algebras associated to Dyck paths.

Matching statistic: St001207

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 60%

Values

[1] => [1] => [1,0]
 => [1] => ? = 1 - 1
[1,1] => [2] => [1,1,0,0]
 => [1,2] => 0 = 1 - 1
[2] => [1] => [1,0]
 => [1] => ? = 1 - 1
[1,1,1] => [3] => [1,1,1,0,0,0]
 => [1,2,3] => 0 = 1 - 1
[3] => [1] => [1,0]
 => [1] => ? = 1 - 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0 = 1 - 1
[1,1,2] => [2,1] => [1,1,0,0,1,0]
 => [1,3,2] => 1 = 2 - 1
[2,1,1] => [1,2] => [1,0,1,1,0,0]
 => [2,1,3] => 1 = 2 - 1
[2,2] => [2] => [1,1,0,0]
 => [1,2] => 0 = 1 - 1
[4] => [1] => [1,0]
 => [1] => ? = 1 - 1
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ? = 1 - 1
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 1 = 2 - 1
[1,1,3] => [2,1] => [1,1,0,0,1,0]
 => [1,3,2] => 1 = 2 - 1
[1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 2 = 3 - 1
[1,2,2] => [1,2] => [1,0,1,1,0,0]
 => [2,1,3] => 1 = 2 - 1
[2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 1 = 2 - 1
[2,2,1] => [2,1] => [1,1,0,0,1,0]
 => [1,3,2] => 1 = 2 - 1
[3,1,1] => [1,2] => [1,0,1,1,0,0]
 => [2,1,3] => 1 = 2 - 1
[5] => [1] => [1,0]
 => [1] => ? = 1 - 1
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => ? = 1 - 1
[1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => ? = 2 - 1
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 1 = 2 - 1
[1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => ? = 2 - 1
[1,1,4] => [2,1] => [1,1,0,0,1,0]
 => [1,3,2] => 1 = 2 - 1
[1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => ? = 3 - 1
[1,2,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 1 = 2 - 1
[1,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 2 = 3 - 1
[2,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => ? = 2 - 1
[2,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 1 = 2 - 1
[2,2,2] => [3] => [1,1,1,0,0,0]
 => [1,2,3] => 0 = 1 - 1
[3,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 1 = 2 - 1
[3,3] => [2] => [1,1,0,0]
 => [1,2] => 0 = 1 - 1
[4,1,1] => [1,2] => [1,0,1,1,0,0]
 => [2,1,3] => 1 = 2 - 1
[6] => [1] => [1,0]
 => [1] => ? = 1 - 1
[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7] => ? = 1 - 1
[1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,2,3,4,6,5] => ? = 2 - 1
[1,1,1,1,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => ? = 2 - 1
[1,1,1,2,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,2,4,5,3,6] => ? = 2 - 1
[1,1,1,2,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => ? = 2 - 1
[1,1,1,4] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 1 = 2 - 1
[1,1,2,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,3,4,2,5,6] => ? = 2 - 1
[1,1,2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => ? = 3 - 1
[1,1,3,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => ? = 2 - 1
[1,1,5] => [2,1] => [1,1,0,0,1,0]
 => [1,3,2] => 1 = 2 - 1
[1,2,1,1,1,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,3,1,4,5,6] => ? = 3 - 1
[1,2,1,1,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => ? = 3 - 1
[1,2,2,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 1 = 2 - 1
[1,3,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => ? = 3 - 1
[1,3,3] => [1,2] => [1,0,1,1,0,0]
 => [2,1,3] => 1 = 2 - 1
[1,4,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 2 = 3 - 1
[2,1,1,1,1,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,3,4,5,6] => ? = 2 - 1
[2,1,1,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => ? = 2 - 1
[2,1,1,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 1 = 2 - 1
[2,1,2,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => ? = 3 - 1
[2,1,2,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 2 = 3 - 1
[2,2,1,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => ? = 2 - 1
[2,2,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 1 = 2 - 1
[2,2,3] => [2,1] => [1,1,0,0,1,0]
 => [1,3,2] => 1 = 2 - 1
[2,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 2 = 3 - 1
[3,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => ? = 2 - 1
[3,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 1 = 2 - 1
[3,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 2 = 3 - 1
[3,2,2] => [1,2] => [1,0,1,1,0,0]
 => [2,1,3] => 1 = 2 - 1
[3,3,1] => [2,1] => [1,1,0,0,1,0]
 => [1,3,2] => 1 = 2 - 1
[4,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 1 = 2 - 1
[5,1,1] => [1,2] => [1,0,1,1,0,0]
 => [2,1,3] => 1 = 2 - 1
[7] => [1] => [1,0]
 => [1] => ? = 1 - 1
[1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7,8] => ? = 1 - 1
[1,1,1,1,1,1,2] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,7,6] => ? = 2 - 1
[1,1,1,1,1,3] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,2,3,4,6,5] => ? = 2 - 1
[1,1,1,1,2,1,1] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [1,2,3,5,6,4,7] => ? = 2 - 1
[1,1,1,1,2,2] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,2,3,5,4,6] => ? = 2 - 1
[1,1,1,1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => ? = 2 - 1
[1,1,1,2,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [1,2,4,5,3,6,7] => ? = 2 - 1
[1,1,1,2,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,2,4,3,6,5] => ? = 2 - 1
[1,1,1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,2,4,5,3,6] => ? = 2 - 1
[1,1,1,5] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 1 = 2 - 1
[1,1,2,1,1,1,1] => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,3,4,2,5,6,7] => ? = 2 - 1
[1,1,2,1,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,3,4,2,6,5] => ? = 2 - 1
[1,1,2,2,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => ? = 2 - 1
[1,1,3,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,3,4,2,5,6] => ? = 2 - 1
[1,1,4,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => ? = 2 - 1
[1,1,6] => [2,1] => [1,1,0,0,1,0]
 => [1,3,2] => 1 = 2 - 1
[1,2,1,1,1,1,1] => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,1,4,5,6,7] => ? = 3 - 1
[1,2,1,1,1,2] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [2,3,1,4,6,5] => ? = 3 - 1
[1,2,1,1,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => ? = 3 - 1
[1,2,1,2,1,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,5,1,6] => ? = 3 - 1
[1,2,1,2,2] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => ? = 3 - 1
[1,2,2,1,1,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [2,1,4,3,5,6] => ? = 2 - 1
[1,2,2,2,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => ? = 2 - 1
[1,2,2,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 1 = 2 - 1
[1,3,2,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 2 = 3 - 1
[1,3,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 1 = 2 - 1
[1,5,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 2 = 3 - 1
[2,1,1,4] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 1 = 2 - 1
[2,2,2,2] => [4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0 = 1 - 1
[2,2,4] => [2,1] => [1,1,0,0,1,0]
 => [1,3,2] => 1 = 2 - 1
[2,3,3] => [1,2] => [1,0,1,1,0,0]
 => [2,1,3] => 1 = 2 - 1
[2,4,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 2 = 3 - 1
[3,1,1,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 1 = 2 - 1

Description

The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ .

Matching statistic: St001330
(load all 3 compositions to match this statistic)

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 40%

Values

[1] => [1] => [1] => ([],1)
 => 1
[1,1] => [2] => [2] => ([],2)
 => 1
[2] => [1] => [1] => ([],1)
 => 1
[1,1,1] => [3] => [3] => ([],3)
 => 1
[3] => [1] => [1] => ([],1)
 => 1
[1,1,1,1] => [4] => [4] => ([],4)
 => 1
[1,1,2] => [2,1] => [1,2] => ([(1,2)],3)
 => 2
[2,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 2
[2,2] => [2] => [2] => ([],2)
 => 1
[4] => [1] => [1] => ([],1)
 => 1
[1,1,1,1,1] => [5] => [5] => ([],5)
 => 1
[1,1,1,2] => [3,1] => [1,3] => ([(2,3)],4)
 => 2
[1,1,3] => [2,1] => [1,2] => ([(1,2)],3)
 => 2
[1,2,1,1] => [1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3
[1,2,2] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 2
[2,1,1,1] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[2,2,1] => [2,1] => [1,2] => ([(1,2)],3)
 => 2
[3,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 2
[5] => [1] => [1] => ([],1)
 => 1
[1,1,1,1,1,1] => [6] => [6] => ([],6)
 => 1
[1,1,1,1,2] => [4,1] => [1,4] => ([(3,4)],5)
 => 2
[1,1,1,3] => [3,1] => [1,3] => ([(2,3)],4)
 => 2
[1,1,2,1,1] => [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
[1,1,4] => [2,1] => [1,2] => ([(1,2)],3)
 => 2
[1,2,1,1,1] => [1,1,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,2,2,1] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2
[1,3,1,1] => [1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3
[2,1,1,1,1] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[2,1,1,2] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2
[2,2,2] => [3] => [3] => ([],3)
 => 1
[3,1,1,1] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[3,3] => [2] => [2] => ([],2)
 => 1
[4,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 2
[6] => [1] => [1] => ([],1)
 => 1
[1,1,1,1,1,1,1] => [7] => [7] => ([],7)
 => ? = 1
[1,1,1,1,1,2] => [5,1] => [1,5] => ([(4,5)],6)
 => 2
[1,1,1,1,3] => [4,1] => [1,4] => ([(3,4)],5)
 => 2
[1,1,1,2,1,1] => [3,1,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[1,1,1,2,2] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2
[1,1,1,4] => [3,1] => [1,3] => ([(2,3)],4)
 => 2
[1,1,2,1,1,1] => [2,1,3] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[1,1,2,2,1] => [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,1,3,1,1] => [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
[1,1,5] => [2,1] => [1,2] => ([(1,2)],3)
 => 2
[1,2,1,1,1,1] => [1,1,4] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
[1,2,1,1,2] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,2,2,2] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[1,3,1,1,1] => [1,1,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,3,3] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 2
[1,4,1,1] => [1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3
[2,1,1,1,1,1] => [1,5] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[2,1,1,1,2] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
[2,1,1,3] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2
[2,1,2,1,1] => [1,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[2,1,2,2] => [1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3
[2,2,1,1,1] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
[2,2,2,1] => [3,1] => [1,3] => ([(2,3)],4)
 => 2
[2,2,3] => [2,1] => [1,2] => ([(1,2)],3)
 => 2
[2,3,1,1] => [1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3
[3,1,1,1,1] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[3,1,1,2] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2
[3,2,1,1] => [1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3
[3,2,2] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 2
[3,3,1] => [2,1] => [1,2] => ([(1,2)],3)
 => 2
[4,1,1,1] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[5,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 2
[7] => [1] => [1] => ([],1)
 => 1
[1,1,1,1,1,1,1,1] => [8] => [8] => ([],8)
 => ? = 1
[1,1,1,1,1,1,2] => [6,1] => [1,6] => ([(5,6)],7)
 => ? = 2
[1,1,1,1,1,3] => [5,1] => [1,5] => ([(4,5)],6)
 => 2
[1,1,1,1,2,1,1] => [4,1,2] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[1,1,1,1,2,2] => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => 2
[1,1,1,1,4] => [4,1] => [1,4] => ([(3,4)],5)
 => 2
[1,1,1,2,1,1,1] => [3,1,3] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[1,1,1,2,2,1] => [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[1,1,1,3,1,1] => [3,1,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[1,1,1,5] => [3,1] => [1,3] => ([(2,3)],4)
 => 2
[1,1,2,1,1,1,1] => [2,1,4] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[1,1,2,1,1,2] => [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[1,1,2,2,2] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
[1,1,3,1,1,1] => [2,1,3] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[1,1,4,1,1] => [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
[1,2,1,1,1,1,1] => [1,1,5] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[1,2,1,1,1,2] => [1,1,3,1] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
[1,2,1,1,3] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,2,1,2,1,1] => [1,1,1,1,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
[1,2,1,2,2] => [1,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,2,2,1,1,1] => [1,2,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[1,2,2,2,1] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
[1,2,2,3] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2
[1,2,3,1,1] => [1,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,3,1,1,1,1] => [1,1,4] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
[1,3,1,1,2] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,3,2,1,1] => [1,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,3,2,2] => [1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3
[1,3,3,1] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2
[1,4,1,1,1] => [1,1,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,5,1,1] => [1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3
[2,1,1,1,1,2] => [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[2,1,1,1,3] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2

Description

The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of

$q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number

$HG(G)$ of a graph

$G$ is the largest integer

$q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of

$q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.

Matching statistic: St001645

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 40%

Values

[1] => [1] => ([],1)
 => ([],1)
 => 1
[1,1] => [2] => ([],2)
 => ([],1)
 => 1
[2] => [1] => ([],1)
 => ([],1)
 => 1
[1,1,1] => [3] => ([],3)
 => ([],1)
 => 1
[3] => [1] => ([],1)
 => ([],1)
 => 1
[1,1,1,1] => [4] => ([],4)
 => ([],1)
 => 1
[1,1,2] => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2
[2,1,1] => [1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 2
[2,2] => [2] => ([],2)
 => ([],1)
 => 1
[4] => [1] => ([],1)
 => ([],1)
 => 1
[1,1,1,1,1] => [5] => ([],5)
 => ([],1)
 => 1
[1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
[1,1,3] => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2
[1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 3
[1,2,2] => [1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 2
[2,1,1,1] => [1,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 2
[2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2
[3,1,1] => [1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 2
[5] => [1] => ([],1)
 => ([],1)
 => 1
[1,1,1,1,1,1] => [6] => ([],6)
 => ([],1)
 => 1
[1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
[1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
[1,1,2,1,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2
[1,1,4] => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2
[1,2,1,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 3
[1,2,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2
[1,3,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 3
[2,1,1,1,1] => [1,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[2,1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2
[2,2,2] => [3] => ([],3)
 => ([],1)
 => 1
[3,1,1,1] => [1,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 2
[3,3] => [2] => ([],2)
 => ([],1)
 => 1
[4,1,1] => [1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 2
[6] => [1] => ([],1)
 => ([],1)
 => 1
[1,1,1,1,1,1,1] => [7] => ([],7)
 => ([],1)
 => 1
[1,1,1,1,1,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2
[1,1,1,1,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
[1,1,1,2,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2
[1,1,1,2,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[1,1,1,4] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
[1,1,2,1,1,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2
[1,1,2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3
[1,1,3,1,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2
[1,1,5] => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2
[1,2,1,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 3
[1,2,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,2,2,2] => [1,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 2
[1,3,1,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 3
[1,3,3] => [1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 2
[1,4,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 3
[2,1,1,1,1,1] => [1,5] => ([(4,5)],6)
 => ([(1,2)],3)
 => ? = 2
[2,1,1,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2
[2,1,1,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2
[2,1,2,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[2,1,2,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 3
[2,2,1,1,1] => [2,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[2,2,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
[2,2,3] => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2
[2,3,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 3
[3,1,1,1,1] => [1,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[3,1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2
[3,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 3
[3,2,2] => [1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 2
[3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2
[4,1,1,1] => [1,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 2
[5,1,1] => [1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 2
[7] => [1] => ([],1)
 => ([],1)
 => 1
[1,1,1,1,1,1,1,1] => [8] => ([],8)
 => ?
 => ? = 1
[1,1,1,1,1,1,2] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1)],2)
 => 2
[1,1,1,1,1,3] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2
[1,1,1,1,2,1,1] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2
[1,1,1,1,2,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ? = 2
[1,1,1,1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
[1,1,1,2,1,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2
[1,1,1,2,2,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2
[1,1,1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2
[1,1,1,5] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
[1,1,2,1,1,1,1] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2
[1,1,2,1,1,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
[1,1,2,2,2] => [2,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[1,1,3,1,1,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2
[1,1,4,1,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2
[1,1,6] => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2
[1,2,1,1,1,1,1] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 3
[1,2,1,1,1,2] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[2,2,2,2] => [4] => ([],4)
 => ([],1)
 => 1
[2,2,4] => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2
[3,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2
[4,4] => [2] => ([],2)
 => ([],1)
 => 1
[8] => [1] => ([],1)
 => ([],1)
 => 1
[1,1,1,1,1,1,3] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1)],2)
 => 2
[1,1,1,1,1,4] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2
[1,1,1,1,5] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
[1,1,1,6] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
[1,1,7] => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2
[2,2,2,2,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
[2,2,2,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
[2,2,5] => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2
[3,3,3] => [3] => ([],3)
 => ([],1)
 => 1
[4,4,1] => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2

Description

The pebbling number of a connected graph.