Your data matches 1 statistic following compositions of up to 3 maps.
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Mp00207: Standard tableaux horizontal strip sizesInteger compositions
Mp00172: Integer compositions rotate back to frontInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St000772: Graphs ⟶ 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] => [1,1] => ([(0,1)],2)
=> 1
[[1,3],[2]]
=> [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[[1],[2],[3]]
=> [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[[1,3,4],[2]]
=> [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,4],[2],[3]]
=> [1,1,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,2],[3],[4]]
=> [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1],[2],[3],[4]]
=> [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[1,3,4,5],[2]]
=> [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,4,5],[2],[3]]
=> [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,2,5],[3],[4]]
=> [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,2,3],[4],[5]]
=> [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,2],[3,5],[4]]
=> [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,3],[2,4],[5]]
=> [1,2,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[1,3,4,5,6],[2]]
=> [1,5] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[1,4,5,6],[2],[3]]
=> [1,1,4] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2,5,6],[3],[4]]
=> [2,1,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2,3,6],[4],[5]]
=> [3,1,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,3,4],[5],[6]]
=> [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2,6],[3,5],[4]]
=> [2,1,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,3,6],[2,4],[5]]
=> [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)
=> 1
[[1,2,3],[4,6],[5]]
=> [3,1,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,4],[2,5],[6]]
=> [1,3,1,1] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,4],[3,5],[6]]
=> [2,2,1,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,6],[2],[4],[5]]
=> [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)
=> 1
[[1,2,6],[3],[4],[5]]
=> [2,1,1,2] => [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)
=> 1
[[1,3,4],[2],[5],[6]]
=> [1,3,1,1] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,4],[3],[5],[6]]
=> [2,2,1,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => [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)
=> 2
[[1,3],[2,4],[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)
=> 1
[[1,3],[2,6],[4],[5]]
=> [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)
=> 1
[[1,2],[3,6],[4],[5]]
=> [2,1,1,2] => [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)
=> 1
[[1,4],[2,5],[3],[6]]
=> [1,1,2,1,1] => [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)
=> 1
[[1,3],[2,4],[5],[6]]
=> [1,2,1,1,1] => [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)
=> 1
[[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => [2,1,1,1,1] => ([(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)
=> 1
[[1,4],[2],[3],[5],[6]]
=> [1,1,2,1,1] => [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)
=> 1
[[1,3],[2],[4],[5],[6]]
=> [1,2,1,1,1] => [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)
=> 1
[[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[[1,3,4,5,6,7],[2]]
=> [1,6] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
[[1,4,5,6,7],[2],[3]]
=> [1,1,5] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[[1,2,5,6,7],[3],[4]]
=> [2,1,4] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,2,3,6,7],[4],[5]]
=> [3,1,3] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[[1,2,3,4,7],[5],[6]]
=> [4,1,2] => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
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].