Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St000772
(load all 8 compositions to match this statistic)
Mapping
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => => [1] => ([],1)
 => 1
[[1],[2]]
 => 1 => [1,1] => ([(0,1)],2)
 => 1
[[1,2],[3]]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[[1],[2],[3]]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[[1,2,3],[4]]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,3],[2,4]]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[1,3],[2],[4]]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[1,2],[3],[4]]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[1],[2],[3],[4]]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,2,3,4],[5]]
 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[1,3,4],[2,5]]
 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[1,2,4],[3,5]]
 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[1,3,4],[2],[5]]
 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[1,2,4],[3],[5]]
 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[1,2,3],[4],[5]]
 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[1,4],[2,5],[3]]
 => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[1,3],[2,4],[5]]
 => 1011 => [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]]
 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[1,4],[2],[3],[5]]
 => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[1,3],[2],[4],[5]]
 => 1011 => [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]]
 => 0111 => [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],[2],[3],[4],[5]]
 => 1111 => [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
[[1,2,3,4,5],[6]]
 => 00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 4
[[1,3,4,5],[2,6]]
 => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,2,4,5],[3,6]]
 => 01001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[[1,2,3,5],[4,6]]
 => 00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[1,3,4,5],[2],[6]]
 => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,2,4,5],[3],[6]]
 => 01001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[[1,2,3,5],[4],[6]]
 => 00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[1,2,3,4],[5],[6]]
 => 00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,3,5],[2,4,6]]
 => 10101 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[1,2,5],[3,4,6]]
 => 01001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[[1,4,5],[2,6],[3]]
 => 11001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[[1,3,5],[2,6],[4]]
 => 10101 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[1,2,5],[3,6],[4]]
 => 01101 => [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
[[1,3,5],[2,4],[6]]
 => 10101 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[1,2,5],[3,4],[6]]
 => 01001 => [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]]
 => 10011 => [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
[[1,2,4],[3,5],[6]]
 => 01011 => [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
[[1,2,3],[4,5],[6]]
 => 00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[1,4,5],[2],[3],[6]]
 => 11001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[[1,3,5],[2],[4],[6]]
 => 10101 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[1,2,5],[3],[4],[6]]
 => 01101 => [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
[[1,3,4],[2],[5],[6]]
 => 10011 => [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
[[1,2,4],[3],[5],[6]]
 => 01011 => [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
[[1,2,3],[4],[5],[6]]
 => 00111 => [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
[[1,4],[2,5],[3,6]]
 => 11011 => [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
[[1,3],[2,5],[4,6]]
 => 10101 => [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],[3,5],[4,6]]
 => 01101 => [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
[[1,5],[2,6],[3],[4]]
 => 11101 => [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
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].
Matching statistic: St000383
Mapping
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => => [] => ? => ? = 1
[[1],[2]]
 => 1 => [1] => [1] => 1
[[1,2],[3]]
 => 01 => [1,1] => [1,1] => 1
[[1],[2],[3]]
 => 11 => [2] => [2] => 2
[[1,2,3],[4]]
 => 001 => [2,1] => [1,2] => 2
[[1,3],[2,4]]
 => 101 => [1,1,1] => [1,1,1] => 1
[[1,3],[2],[4]]
 => 101 => [1,1,1] => [1,1,1] => 1
[[1,2],[3],[4]]
 => 011 => [1,2] => [2,1] => 1
[[1],[2],[3],[4]]
 => 111 => [3] => [3] => 3
[[1,2,3,4],[5]]
 => 0001 => [3,1] => [1,3] => 3
[[1,3,4],[2,5]]
 => 1001 => [1,2,1] => [1,1,2] => 2
[[1,2,4],[3,5]]
 => 0101 => [1,1,1,1] => [1,1,1,1] => 1
[[1,3,4],[2],[5]]
 => 1001 => [1,2,1] => [1,1,2] => 2
[[1,2,4],[3],[5]]
 => 0101 => [1,1,1,1] => [1,1,1,1] => 1
[[1,2,3],[4],[5]]
 => 0011 => [2,2] => [2,2] => 2
[[1,4],[2,5],[3]]
 => 1101 => [2,1,1] => [1,2,1] => 1
[[1,3],[2,4],[5]]
 => 1011 => [1,1,2] => [2,1,1] => 1
[[1,2],[3,4],[5]]
 => 0101 => [1,1,1,1] => [1,1,1,1] => 1
[[1,4],[2],[3],[5]]
 => 1101 => [2,1,1] => [1,2,1] => 1
[[1,3],[2],[4],[5]]
 => 1011 => [1,1,2] => [2,1,1] => 1
[[1,2],[3],[4],[5]]
 => 0111 => [1,3] => [3,1] => 1
[[1],[2],[3],[4],[5]]
 => 1111 => [4] => [4] => 4
[[1,2,3,4,5],[6]]
 => 00001 => [4,1] => [1,4] => 4
[[1,3,4,5],[2,6]]
 => 10001 => [1,3,1] => [1,1,3] => 3
[[1,2,4,5],[3,6]]
 => 01001 => [1,1,2,1] => [1,1,1,2] => 2
[[1,2,3,5],[4,6]]
 => 00101 => [2,1,1,1] => [1,2,1,1] => 1
[[1,3,4,5],[2],[6]]
 => 10001 => [1,3,1] => [1,1,3] => 3
[[1,2,4,5],[3],[6]]
 => 01001 => [1,1,2,1] => [1,1,1,2] => 2
[[1,2,3,5],[4],[6]]
 => 00101 => [2,1,1,1] => [1,2,1,1] => 1
[[1,2,3,4],[5],[6]]
 => 00011 => [3,2] => [2,3] => 3
[[1,3,5],[2,4,6]]
 => 10101 => [1,1,1,1,1] => [1,1,1,1,1] => 1
[[1,2,5],[3,4,6]]
 => 01001 => [1,1,2,1] => [1,1,1,2] => 2
[[1,4,5],[2,6],[3]]
 => 11001 => [2,2,1] => [1,2,2] => 2
[[1,3,5],[2,6],[4]]
 => 10101 => [1,1,1,1,1] => [1,1,1,1,1] => 1
[[1,2,5],[3,6],[4]]
 => 01101 => [1,2,1,1] => [1,1,2,1] => 1
[[1,3,5],[2,4],[6]]
 => 10101 => [1,1,1,1,1] => [1,1,1,1,1] => 1
[[1,2,5],[3,4],[6]]
 => 01001 => [1,1,2,1] => [1,1,1,2] => 2
[[1,3,4],[2,5],[6]]
 => 10011 => [1,2,2] => [2,1,2] => 2
[[1,2,4],[3,5],[6]]
 => 01011 => [1,1,1,2] => [2,1,1,1] => 1
[[1,2,3],[4,5],[6]]
 => 00101 => [2,1,1,1] => [1,2,1,1] => 1
[[1,4,5],[2],[3],[6]]
 => 11001 => [2,2,1] => [1,2,2] => 2
[[1,3,5],[2],[4],[6]]
 => 10101 => [1,1,1,1,1] => [1,1,1,1,1] => 1
[[1,2,5],[3],[4],[6]]
 => 01101 => [1,2,1,1] => [1,1,2,1] => 1
[[1,3,4],[2],[5],[6]]
 => 10011 => [1,2,2] => [2,1,2] => 2
[[1,2,4],[3],[5],[6]]
 => 01011 => [1,1,1,2] => [2,1,1,1] => 1
[[1,2,3],[4],[5],[6]]
 => 00111 => [2,3] => [3,2] => 2
[[1,4],[2,5],[3,6]]
 => 11011 => [2,1,2] => [2,2,1] => 1
[[1,3],[2,5],[4,6]]
 => 10101 => [1,1,1,1,1] => [1,1,1,1,1] => 1
[[1,2],[3,5],[4,6]]
 => 01101 => [1,2,1,1] => [1,1,2,1] => 1
[[1,5],[2,6],[3],[4]]
 => 11101 => [3,1,1] => [1,3,1] => 1
[[1,4],[2,5],[3],[6]]
 => 11011 => [2,1,2] => [2,2,1] => 1
[]
 => => [] => ? => ? = 1
Description
The last part of an integer composition.