Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St000090
(load all 2 compositions to match this statistic)
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
St000090: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => [1] => 0
([],2)
 => [2] => [2] => 0
([(0,1)],2)
 => [1,1] => [1,1] => 0
([],3)
 => [3] => [3] => 0
([(1,2)],3)
 => [1,2] => [2,1] => -1
([(0,2),(1,2)],3)
 => [1,1,1] => [1,1,1] => 0
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [1,2] => 1
([],4)
 => [4] => [4] => 0
([(2,3)],4)
 => [1,3] => [3,1] => -2
([(1,3),(2,3)],4)
 => [1,1,2] => [2,1,1] => -1
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,1,2] => 1
([(0,3),(1,2)],4)
 => [2,2] => [2,2] => 0
([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [1,1,1,1] => 0
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [2,2] => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,1,1,1] => 0
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [1,1,2] => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2,1] => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [1,3] => 2
([],5)
 => [5] => [5] => 0
([(3,4)],5)
 => [1,4] => [4,1] => -3
([(2,4),(3,4)],5)
 => [1,1,3] => [3,1,1] => -2
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [2,1,2] => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,1,3] => 2
([(1,4),(2,3)],5)
 => [2,3] => [3,2] => -1
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [2,1,1,1] => -1
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [2,1,1,1] => -1
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [3,2] => -1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1] => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [2,1,1,1] => -1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [1,1,1,2] => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => [2,1,2] => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1] => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [2,2,1] => -1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1] => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1] => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [1,1,1,2] => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1] => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [2,2,1] => -1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1] => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => [1,1,2,1] => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,2,2] => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1] => 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1] => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1] => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [2,3] => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => [1,1,2,1] => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,2,1,1] => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1] => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => 1
Description
The variation of a composition.
Matching statistic: St000772
(load all 2 compositions to match this statistic)
Mapping
St000772: Graphs ⟶ ℤResult quality: 55% ●values known / values provided: 80%●distinct values known / distinct values provided: 55%
Values
([],1)
 => 1 = 0 + 1
([],2)
 => ? = 0 + 1
([(0,1)],2)
 => 1 = 0 + 1
([],3)
 => ? = 0 + 1
([(1,2)],3)
 => ? = -1 + 1
([(0,2),(1,2)],3)
 => 1 = 0 + 1
([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
([],4)
 => ? = 0 + 1
([(2,3)],4)
 => ? = -2 + 1
([(1,3),(2,3)],4)
 => ? = -1 + 1
([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,3),(1,2)],4)
 => ? = 0 + 1
([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
([(1,2),(1,3),(2,3)],4)
 => ? = 0 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
([],5)
 => ? = 0 + 1
([(3,4)],5)
 => ? = -3 + 1
([(2,4),(3,4)],5)
 => ? = -2 + 1
([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(1,4),(2,3)],5)
 => ? = -1 + 1
([(1,4),(2,3),(3,4)],5)
 => ? = -1 + 1
([(0,1),(2,4),(3,4)],5)
 => ? = -1 + 1
([(2,3),(2,4),(3,4)],5)
 => ? = -1 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = -1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = -1 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = -1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2 = 1 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 0 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
([],6)
 => ? = 0 + 1
([(4,5)],6)
 => ? = -4 + 1
([(3,5),(4,5)],6)
 => ? = -3 + 1
([(2,5),(3,5),(4,5)],6)
 => ? = -1 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 1 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 4 = 3 + 1
([(2,5),(3,4)],6)
 => ? = -2 + 1
([(2,5),(3,4),(4,5)],6)
 => ? = -2 + 1
([(1,2),(3,5),(4,5)],6)
 => ? = -2 + 1
([(3,4),(3,5),(4,5)],6)
 => ? = -2 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => ? = -1 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => ? = 0 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = -2 + 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1 = 0 + 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => ? = -1 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => ? = 0 + 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = -1 + 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1 = 0 + 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = -2 + 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = -1 + 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1 = 0 + 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = -1 + 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 0 + 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => 1 = 0 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 1 = 0 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 3 = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
([(0,5),(1,4),(2,3)],6)
 => ? = 0 + 1
([(1,5),(2,4),(3,4),(3,5)],6)
 => ? = -1 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => ? = -1 + 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => ? = -2 + 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1 = 0 + 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = -1 + 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = -1 + 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = -1 + 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ? = 0 + 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => 1 = 0 + 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian $$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$ Its eigenvalues are $0,4,4,6$, so the statistic is $1$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$. The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].