Your data matches 7 different statistics following compositions of up to 3 maps.
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Matching statistic: St001486
(load all 4 compositions to match this statistic)
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ 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
([],2)
 => [2] => [1] => 1
([],3)
 => [3] => [1] => 1
([(1,2)],3)
 => [1,2] => [1,1] => 2
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [1,1] => 2
([],4)
 => [4] => [1] => 1
([(2,3)],4)
 => [1,3] => [1,1] => 2
([(1,3),(2,3)],4)
 => [1,1,2] => [2,1] => 3
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,1,1] => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [1,1,1] => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [1,1] => 2
([],5)
 => [5] => [1] => 1
([(3,4)],5)
 => [1,4] => [1,1] => 2
([(2,4),(3,4)],5)
 => [1,1,3] => [2,1] => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,1,1] => 2
([(1,4),(2,3)],5)
 => [2,3] => [1,1] => 2
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [3,1] => 3
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [3,1] => 3
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [1,1] => 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [3,1] => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [2,1,1] => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,1] => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [2,1,1] => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => 3
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,1] => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [2,1] => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [1,1] => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => 3
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => [1,1] => 2
([],6)
 => [6] => [1] => 1
([(4,5)],6)
 => [1,5] => [1,1] => 2
([(3,5),(4,5)],6)
 => [1,1,4] => [2,1] => 3
([(2,5),(3,5),(4,5)],6)
 => [1,2,3] => [1,1,1] => 2
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,3,2] => [1,1,1] => 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,4,1] => [1,1,1] => 2
([(2,5),(3,4)],6)
 => [2,4] => [1,1] => 2
([(2,5),(3,4),(4,5)],6)
 => [1,1,1,3] => [3,1] => 3
([(1,2),(3,5),(4,5)],6)
 => [1,1,1,3] => [3,1] => 3
([(3,4),(3,5),(4,5)],6)
 => [2,4] => [1,1] => 2
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => 3
([(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,3] => [3,1] => 3
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,3,1] => [2,1,1] => 3
([(2,4),(2,5),(3,4),(3,5)],6)
 => [1,2,3] => [1,1,1] => 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [1,1,1] => 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => 3
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,2,1] => [3,1,1] => 3
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,1,3,1] => [2,1,1] => 3
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,3,1] => [1,1,1] => 2
Description
The number of corners of the ribbon associated with an integer composition. We associate a ribbon shape to a composition $c=(c_1,\dots,c_n)$ with $c_i$ cells in the $i$-th row from bottom to top, such that the cells in two rows overlap in precisely one cell. This statistic records the total number of corners of the ribbon shape.
Matching statistic: St000453
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ 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
([],2)
 => [2] => [1] => ([],1)
 => 1
([],3)
 => [3] => [1] => ([],1)
 => 1
([(1,2)],3)
 => [1,2] => [1,1] => ([(0,1)],2)
 => 2
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2
([],4)
 => [4] => [1] => ([],1)
 => 1
([(2,3)],4)
 => [1,3] => [1,1] => ([(0,1)],2)
 => 2
([(1,3),(2,3)],4)
 => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 3
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [1,1] => ([(0,1)],2)
 => 2
([],5)
 => [5] => [1] => ([],1)
 => 1
([(3,4)],5)
 => [1,4] => [1,1] => ([(0,1)],2)
 => 2
([(2,4),(3,4)],5)
 => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
([(1,4),(2,3)],5)
 => [2,3] => [1,1] => ([(0,1)],2)
 => 2
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [1,1] => ([(0,1)],2)
 => 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => [1,1] => ([(0,1)],2)
 => 2
([],6)
 => [6] => [1] => ([],1)
 => 1
([(4,5)],6)
 => [1,5] => [1,1] => ([(0,1)],2)
 => 2
([(3,5),(4,5)],6)
 => [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
 => 3
([(2,5),(3,5),(4,5)],6)
 => [1,2,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
([(2,5),(3,4)],6)
 => [2,4] => [1,1] => ([(0,1)],2)
 => 2
([(2,5),(3,4),(4,5)],6)
 => [1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
([(1,2),(3,5),(4,5)],6)
 => [1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
([(3,4),(3,5),(4,5)],6)
 => [2,4] => [1,1] => ([(0,1)],2)
 => 2
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
([(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(2,4),(2,5),(3,4),(3,5)],6)
 => [1,2,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,1,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
Description
The number of distinct Laplacian eigenvalues of a graph.
Matching statistic: St000777
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ 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
([],2)
 => [2] => [1] => ([],1)
 => 1
([],3)
 => [3] => [1] => ([],1)
 => 1
([(1,2)],3)
 => [1,2] => [1,1] => ([(0,1)],2)
 => 2
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2
([],4)
 => [4] => [1] => ([],1)
 => 1
([(2,3)],4)
 => [1,3] => [1,1] => ([(0,1)],2)
 => 2
([(1,3),(2,3)],4)
 => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 3
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [1,1] => ([(0,1)],2)
 => 2
([],5)
 => [5] => [1] => ([],1)
 => 1
([(3,4)],5)
 => [1,4] => [1,1] => ([(0,1)],2)
 => 2
([(2,4),(3,4)],5)
 => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
([(1,4),(2,3)],5)
 => [2,3] => [1,1] => ([(0,1)],2)
 => 2
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [1,1] => ([(0,1)],2)
 => 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => [1,1] => ([(0,1)],2)
 => 2
([],6)
 => [6] => [1] => ([],1)
 => 1
([(4,5)],6)
 => [1,5] => [1,1] => ([(0,1)],2)
 => 2
([(3,5),(4,5)],6)
 => [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
 => 3
([(2,5),(3,5),(4,5)],6)
 => [1,2,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
([(2,5),(3,4)],6)
 => [2,4] => [1,1] => ([(0,1)],2)
 => 2
([(2,5),(3,4),(4,5)],6)
 => [1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
([(1,2),(3,5),(4,5)],6)
 => [1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
([(3,4),(3,5),(4,5)],6)
 => [2,4] => [1,1] => ([(0,1)],2)
 => 2
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
([(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(2,4),(2,5),(3,4),(3,5)],6)
 => [1,2,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,1,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St001035
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001035: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 99%●distinct values known / distinct values provided: 83%
Values
([],1)
 => [1] => [1] => [1,0]
 => ? = 1 - 2
([],2)
 => [2] => [1] => [1,0]
 => ? = 1 - 2
([],3)
 => [3] => [1] => [1,0]
 => ? = 1 - 2
([(1,2)],3)
 => [1,2] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
([],4)
 => [4] => [1] => [1,0]
 => ? = 1 - 2
([(2,3)],4)
 => [1,3] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
([(1,3),(2,3)],4)
 => [1,1,2] => [2,1] => [1,1,0,0,1,0]
 => 1 = 3 - 2
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 2 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 2 - 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
([],5)
 => [5] => [1] => [1,0]
 => ? = 1 - 2
([(3,4)],5)
 => [1,4] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
([(2,4),(3,4)],5)
 => [1,1,3] => [2,1] => [1,1,0,0,1,0]
 => 1 = 3 - 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 2 - 2
([(1,4),(2,3)],5)
 => [2,3] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 3 - 2
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 3 - 2
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 3 - 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 3 - 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 2 - 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 3 - 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => [1,1,0,0,1,0]
 => 1 = 3 - 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 2 - 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [2,1] => [1,1,0,0,1,0]
 => 1 = 3 - 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => [1,1,0,0,1,0]
 => 1 = 3 - 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
([],6)
 => [6] => [1] => [1,0]
 => ? = 1 - 2
([(4,5)],6)
 => [1,5] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
([(3,5),(4,5)],6)
 => [1,1,4] => [2,1] => [1,1,0,0,1,0]
 => 1 = 3 - 2
([(2,5),(3,5),(4,5)],6)
 => [1,2,3] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 2 - 2
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,3,2] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 2 - 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,4,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 2 - 2
([(2,5),(3,4)],6)
 => [2,4] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
([(2,5),(3,4),(4,5)],6)
 => [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 3 - 2
([(1,2),(3,5),(4,5)],6)
 => [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 3 - 2
([(3,4),(3,5),(4,5)],6)
 => [2,4] => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 3 - 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 3 - 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 3 - 2
([(2,4),(2,5),(3,4),(3,5)],6)
 => [1,2,3] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 2 - 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 3 - 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 2 - 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 3 - 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 3 - 2
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 1 = 3 - 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 3 - 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,3,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 2 - 2
([(1,5),(2,4),(3,4),(3,5)],6)
 => [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 3 - 2
([(0,1),(2,5),(3,4),(4,5)],6)
 => [1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0 = 2 - 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 2 - 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 3 - 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 3 - 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0 = 2 - 2
([],7)
 => [7] => [1] => [1,0]
 => ? = 1 - 2
([],8)
 => [8] => [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: St000455
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 79%●distinct values known / distinct values provided: 33%
Values
([],1)
 => [1] => [1] => ([],1)
 => ? = 1 - 3
([],2)
 => [2] => [1] => ([],1)
 => ? = 1 - 3
([],3)
 => [3] => [1] => ([],1)
 => ? = 1 - 3
([(1,2)],3)
 => [1,2] => [1,1] => ([(0,1)],2)
 => -1 = 2 - 3
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [1,1] => ([(0,1)],2)
 => -1 = 2 - 3
([],4)
 => [4] => [1] => ([],1)
 => ? = 1 - 3
([(2,3)],4)
 => [1,3] => [1,1] => ([(0,1)],2)
 => -1 = 2 - 3
([(1,3),(2,3)],4)
 => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 0 = 3 - 3
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 2 - 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 2 - 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [1,1] => ([(0,1)],2)
 => -1 = 2 - 3
([],5)
 => [5] => [1] => ([],1)
 => ? = 1 - 3
([(3,4)],5)
 => [1,4] => [1,1] => ([(0,1)],2)
 => -1 = 2 - 3
([(2,4),(3,4)],5)
 => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
 => 0 = 3 - 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 2 - 3
([(1,4),(2,3)],5)
 => [2,3] => [1,1] => ([(0,1)],2)
 => -1 = 2 - 3
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 3 - 3
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 3 - 3
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [1,1] => ([(0,1)],2)
 => -1 = 2 - 3
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 3 - 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 3 - 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 2 - 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 3 - 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 0 = 3 - 3
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 2 - 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 0 = 3 - 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => -1 = 2 - 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 0 = 3 - 3
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => [1,1] => ([(0,1)],2)
 => -1 = 2 - 3
([],6)
 => [6] => [1] => ([],1)
 => ? = 1 - 3
([(4,5)],6)
 => [1,5] => [1,1] => ([(0,1)],2)
 => -1 = 2 - 3
([(3,5),(4,5)],6)
 => [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
 => 0 = 3 - 3
([(2,5),(3,5),(4,5)],6)
 => [1,2,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 2 - 3
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 2 - 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 2 - 3
([(2,5),(3,4)],6)
 => [2,4] => [1,1] => ([(0,1)],2)
 => -1 = 2 - 3
([(2,5),(3,4),(4,5)],6)
 => [1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 3 - 3
([(1,2),(3,5),(4,5)],6)
 => [1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 3 - 3
([(3,4),(3,5),(4,5)],6)
 => [2,4] => [1,1] => ([(0,1)],2)
 => -1 = 2 - 3
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
([(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 3 - 3
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 3 - 3
([(2,4),(2,5),(3,4),(3,5)],6)
 => [1,2,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 2 - 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 2 - 3
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 3 - 3
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,1,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 3 - 3
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 2 - 3
([(1,5),(2,4),(3,4),(3,5)],6)
 => [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
([(0,1),(2,5),(3,4),(4,5)],6)
 => [1,2,1,2] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => -1 = 2 - 3
([(1,2),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 2 - 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,2,1,2] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => -1 = 2 - 3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,2,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 4 - 3
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,2,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 4 - 3
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 4 - 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,2,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 4 - 3
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => [1,2,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 4 - 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,2,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 4 - 3
([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,2,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 4 - 3
([],7)
 => [7] => [1] => ([],1)
 => ? = 1 - 3
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 3
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
 => [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
([(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => [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
([(1,4),(1,5),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
 => [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
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [2,1,1,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 4 - 3
([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => [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
([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [1,1,2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 3
([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
 => [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
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => [2,1,1,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 3
([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [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
([(0,6),(1,5),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [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
([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [2,1,1,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 3
([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [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
([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [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
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => [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
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
 => [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
([(1,2),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [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
([(0,1),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [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
([(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [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
([(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => [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
([(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)
 => [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),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [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
([(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [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
([(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [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
([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [2,1,1,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 3
([(0,1),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [3,1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 4 - 3
([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [3,1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 4 - 3
([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [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
([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [2,1,1,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 3
([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [3,1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 4 - 3
([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => [1,1,2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 3
([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => [1,1,2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 3
([(0,7),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [1,3,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 4 - 3
([(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [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)
 => ? = 5 - 3
([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [3,2,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 4 - 3
([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [1,2,2,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 4 - 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: St001488
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001488: Skew partitions ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 83%
Values
([],1)
 => [1] => [1] => [[1],[]]
 => 1
([],2)
 => [2] => [1] => [[1],[]]
 => 1
([],3)
 => [3] => [1] => [[1],[]]
 => 1
([(1,2)],3)
 => [1,2] => [1,1] => [[1,1],[]]
 => 2
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [1,1] => [[1,1],[]]
 => 2
([],4)
 => [4] => [1] => [[1],[]]
 => 1
([(2,3)],4)
 => [1,3] => [1,1] => [[1,1],[]]
 => 2
([(1,3),(2,3)],4)
 => [1,1,2] => [2,1] => [[2,2],[1]]
 => 3
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,1,1] => [[1,1,1],[]]
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [1,1,1] => [[1,1,1],[]]
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [1,1] => [[1,1],[]]
 => 2
([],5)
 => [5] => [1] => [[1],[]]
 => 1
([(3,4)],5)
 => [1,4] => [1,1] => [[1,1],[]]
 => 2
([(2,4),(3,4)],5)
 => [1,1,3] => [2,1] => [[2,2],[1]]
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,1,1] => [[1,1,1],[]]
 => 2
([(1,4),(2,3)],5)
 => [2,3] => [1,1] => [[1,1],[]]
 => 2
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [3,1] => [[3,3],[2]]
 => 3
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [3,1] => [[3,3],[2]]
 => 3
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [1,1] => [[1,1],[]]
 => 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [3,1] => [[3,3],[2]]
 => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
 => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,1] => [[1,1,1],[]]
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => [[2,2],[1]]
 => 3
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,1] => [[1,1,1],[]]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [2,1] => [[2,2],[1]]
 => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [1,1] => [[1,1],[]]
 => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => [[2,2],[1]]
 => 3
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => [1,1] => [[1,1],[]]
 => 2
([],6)
 => [6] => [1] => [[1],[]]
 => 1
([(4,5)],6)
 => [1,5] => [1,1] => [[1,1],[]]
 => 2
([(3,5),(4,5)],6)
 => [1,1,4] => [2,1] => [[2,2],[1]]
 => 3
([(2,5),(3,5),(4,5)],6)
 => [1,2,3] => [1,1,1] => [[1,1,1],[]]
 => 2
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,3,2] => [1,1,1] => [[1,1,1],[]]
 => 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,4,1] => [1,1,1] => [[1,1,1],[]]
 => 2
([(2,5),(3,4)],6)
 => [2,4] => [1,1] => [[1,1],[]]
 => 2
([(2,5),(3,4),(4,5)],6)
 => [1,1,1,3] => [3,1] => [[3,3],[2]]
 => 3
([(1,2),(3,5),(4,5)],6)
 => [1,1,1,3] => [3,1] => [[3,3],[2]]
 => 3
([(3,4),(3,5),(4,5)],6)
 => [2,4] => [1,1] => [[1,1],[]]
 => 2
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => [[4,4],[3]]
 => 3
([(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,3] => [3,1] => [[3,3],[2]]
 => 3
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
 => 3
([(2,4),(2,5),(3,4),(3,5)],6)
 => [1,2,3] => [1,1,1] => [[1,1,1],[]]
 => 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => [[4,4],[3]]
 => 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [1,1,1] => [[1,1,1],[]]
 => 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => [[4,4],[3]]
 => 3
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => [[4,4],[3]]
 => 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
 => 3
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
 => 3
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,3,1] => [1,1,1] => [[1,1,1],[]]
 => 2
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
 => ? = 3
([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
 => ? = 3
([(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
 => ? = 3
([(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,6),(2,3),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,2),(1,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,6),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,2),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,5),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 3
([(0,1),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => ? = 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: St001330
(load all 2 compositions to match this statistic)
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Values
([],1)
 => [1] => [1] => ([],1)
 => 1
([],2)
 => [2] => [2] => ([],2)
 => 1
([],3)
 => [3] => [3] => ([],3)
 => 1
([(1,2)],3)
 => [1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 2
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [1,2] => ([(1,2)],3)
 => 2
([],4)
 => [4] => [4] => ([],4)
 => 1
([(2,3)],4)
 => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
([(1,3),(2,3)],4)
 => [1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [1,3] => ([(2,3)],4)
 => 2
([],5)
 => [5] => [5] => ([],5)
 => 1
([(3,4)],5)
 => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
([(2,4),(3,4)],5)
 => [1,1,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
([(1,4),(2,3)],5)
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [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),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [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
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => 2
([],6)
 => [6] => [6] => ([],6)
 => 1
([(4,5)],6)
 => [1,5] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2
([(3,5),(4,5)],6)
 => [1,1,4] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
([(2,5),(3,5),(4,5)],6)
 => [1,2,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [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
([(2,5),(3,4)],6)
 => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
([(2,5),(3,4),(4,5)],6)
 => [1,1,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
([(1,2),(3,5),(4,5)],6)
 => [1,1,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
([(3,4),(3,5),(4,5)],6)
 => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
([(1,5),(2,5),(3,4),(4,5)],6)
 => [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
([(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [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
([(2,4),(2,5),(3,4),(3,5)],6)
 => [1,2,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [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
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [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
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [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
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,2,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [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
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
([(1,5),(2,4),(3,4),(3,5)],6)
 => [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
([(0,1),(2,5),(3,4),(4,5)],6)
 => [1,2,1,2] => [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)
 => ? = 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [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
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [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
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,2,1,2] => [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)
 => ? = 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [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
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [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
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [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
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [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
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,1,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,2,1,2] => [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)
 => ? = 2
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,1,2] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(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),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [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
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => 2
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1] => [1,5] => ([(4,5)],6)
 => 2
([(5,6)],7)
 => [1,6] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 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.