Your data matches 4 different statistics following compositions of up to 3 maps.
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Matching statistic: St001407
Mapping
St001407: Semistandard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => 1
[[2,2]]
 => 1
[[1],[2]]
 => 2
[[1,3]]
 => 1
[[2,3]]
 => 0
[[3,3]]
 => 1
[[1],[3]]
 => 1
[[2],[3]]
 => 1
[[1,1,2]]
 => 2
[[1,2,2]]
 => 2
[[2,2,2]]
 => 2
[[1,1],[2]]
 => 3
[[1,2],[2]]
 => 2
[[1,4]]
 => 1
[[2,4]]
 => 0
[[3,4]]
 => 0
[[4,4]]
 => 1
[[1],[4]]
 => 1
[[2],[4]]
 => 0
[[3],[4]]
 => 1
[[1,1,3]]
 => 2
[[1,2,3]]
 => 1
[[1,3,3]]
 => 2
[[2,2,3]]
 => 1
[[2,3,3]]
 => 1
[[3,3,3]]
 => 2
[[1,1],[3]]
 => 2
[[1,2],[3]]
 => 1
[[1,3],[2]]
 => 2
[[1,3],[3]]
 => 1
[[2,2],[3]]
 => 2
[[2,3],[3]]
 => 1
[[1],[2],[3]]
 => 3
[[1,1,1,2]]
 => 3
[[1,1,2,2]]
 => 3
[[1,2,2,2]]
 => 3
[[2,2,2,2]]
 => 3
[[1,1,1],[2]]
 => 4
[[1,1,2],[2]]
 => 3
[[1,2,2],[2]]
 => 3
[[1,1],[2,2]]
 => 4
[[1,5]]
 => 1
[[2,5]]
 => 0
[[3,5]]
 => 0
[[4,5]]
 => 0
[[5,5]]
 => 1
[[1],[5]]
 => 1
[[2],[5]]
 => 0
[[3],[5]]
 => 0
[[4],[5]]
 => 1
Description
The number of minimal entries in a semistandard tableau. An entry is minimal if replacing it with a smaller entry does not yield a semistandard tableau.
Matching statistic: St000771
Mapping
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 43%
Values
[[1,2]]
 => [[1,2]]
 => [1,2] => ([],2)
 => ? ∊ {1,1,2}
[[2,2]]
 => [[2,2]]
 => [1,2] => ([],2)
 => ? ∊ {1,1,2}
[[1],[2]]
 => [[1,2]]
 => [1,2] => ([],2)
 => ? ∊ {1,1,2}
[[1,3]]
 => [[1,3]]
 => [1,2] => ([],2)
 => ? ∊ {0,1,1,1,1}
[[2,3]]
 => [[2,3]]
 => [1,2] => ([],2)
 => ? ∊ {0,1,1,1,1}
[[3,3]]
 => [[3,3]]
 => [1,2] => ([],2)
 => ? ∊ {0,1,1,1,1}
[[1],[3]]
 => [[1,3]]
 => [1,2] => ([],2)
 => ? ∊ {0,1,1,1,1}
[[2],[3]]
 => [[2,3]]
 => [1,2] => ([],2)
 => ? ∊ {0,1,1,1,1}
[[1,1,2]]
 => [[1,1,2]]
 => [1,2,3] => ([],3)
 => ? ∊ {2,2,2,2,3}
[[1,2,2]]
 => [[1,2,2]]
 => [1,2,3] => ([],3)
 => ? ∊ {2,2,2,2,3}
[[2,2,2]]
 => [[2,2,2]]
 => [1,2,3] => ([],3)
 => ? ∊ {2,2,2,2,3}
[[1,1],[2]]
 => [[1,1,2]]
 => [1,2,3] => ([],3)
 => ? ∊ {2,2,2,2,3}
[[1,2],[2]]
 => [[1,2,2]]
 => [1,2,3] => ([],3)
 => ? ∊ {2,2,2,2,3}
[[1,4]]
 => [[1,4]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,1,1,1,1}
[[2,4]]
 => [[2,4]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,1,1,1,1}
[[3,4]]
 => [[3,4]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,1,1,1,1}
[[4,4]]
 => [[4,4]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,1,1,1,1}
[[1],[4]]
 => [[1,4]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,1,1,1,1}
[[2],[4]]
 => [[2,4]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,1,1,1,1}
[[3],[4]]
 => [[3,4]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,1,1,1,1}
[[1,1,3]]
 => [[1,1,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1,2,2,2,2,2,2,3}
[[1,2,3]]
 => [[1,2,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1,2,2,2,2,2,2,3}
[[1,3,3]]
 => [[1,3,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1,2,2,2,2,2,2,3}
[[2,2,3]]
 => [[2,2,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1,2,2,2,2,2,2,3}
[[2,3,3]]
 => [[2,3,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1,2,2,2,2,2,2,3}
[[3,3,3]]
 => [[3,3,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1,2,2,2,2,2,2,3}
[[1,1],[3]]
 => [[1,1,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1,2,2,2,2,2,2,3}
[[1,2],[3]]
 => [[1,2,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1,2,2,2,2,2,2,3}
[[1,3],[2]]
 => [[1,2],[3]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,3],[3]]
 => [[1,3,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1,2,2,2,2,2,2,3}
[[2,2],[3]]
 => [[2,2,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1,2,2,2,2,2,2,3}
[[2,3],[3]]
 => [[2,3,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1,2,2,2,2,2,2,3}
[[1],[2],[3]]
 => [[1,2],[3]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,1,1,2]]
 => [[1,1,1,2]]
 => [1,2,3,4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,4,4}
[[1,1,2,2]]
 => [[1,1,2,2]]
 => [1,2,3,4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,4,4}
[[1,2,2,2]]
 => [[1,2,2,2]]
 => [1,2,3,4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,4,4}
[[2,2,2,2]]
 => [[2,2,2,2]]
 => [1,2,3,4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,4,4}
[[1,1,1],[2]]
 => [[1,1,1,2]]
 => [1,2,3,4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,4,4}
[[1,1,2],[2]]
 => [[1,1,2,2]]
 => [1,2,3,4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,4,4}
[[1,2,2],[2]]
 => [[1,2,2,2]]
 => [1,2,3,4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,4,4}
[[1,1],[2,2]]
 => [[1,1,2,2]]
 => [1,2,3,4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,4,4}
[[1,5]]
 => [[1,5]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[[2,5]]
 => [[2,5]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[[3,5]]
 => [[3,5]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[[4,5]]
 => [[4,5]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[[5,5]]
 => [[5,5]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[[1],[5]]
 => [[1,5]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[[2],[5]]
 => [[2,5]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[[3],[5]]
 => [[3,5]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[[4],[5]]
 => [[4,5]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[[1,1,4]]
 => [[1,1,4]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[[1,2,4]]
 => [[1,2,4]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[[1,4],[2]]
 => [[1,2],[4]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,4],[3]]
 => [[1,3],[4]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[2,4],[3]]
 => [[2,3],[4]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1],[2],[4]]
 => [[1,2],[4]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1],[3],[4]]
 => [[1,3],[4]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[2],[3],[4]]
 => [[2,3],[4]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,1,3],[2]]
 => [[1,1,2],[3]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,2,3],[2]]
 => [[1,2,2],[3]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,1],[2],[3]]
 => [[1,1,2],[3]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,2],[2],[3]]
 => [[1,2,2],[3]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,5],[2]]
 => [[1,2],[5]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,5],[3]]
 => [[1,3],[5]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,5],[4]]
 => [[1,4],[5]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[2,5],[3]]
 => [[2,3],[5]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[2,5],[4]]
 => [[2,4],[5]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[3,5],[4]]
 => [[3,4],[5]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1],[2],[5]]
 => [[1,2],[5]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1],[3],[5]]
 => [[1,3],[5]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1],[4],[5]]
 => [[1,4],[5]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[2],[3],[5]]
 => [[2,3],[5]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[2],[4],[5]]
 => [[2,4],[5]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[3],[4],[5]]
 => [[3,4],[5]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,1,4],[2]]
 => [[1,1,2],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,1,4],[3]]
 => [[1,1,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,2,4],[2]]
 => [[1,2,2],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,2,4],[3]]
 => [[1,2,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,3,4],[3]]
 => [[1,3,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[2,2,4],[3]]
 => [[2,2,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[2,3,4],[3]]
 => [[2,3,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,1],[2],[4]]
 => [[1,1,2],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,1],[3],[4]]
 => [[1,1,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,2],[2],[4]]
 => [[1,2,2],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,2],[3],[4]]
 => [[1,2,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[1,3],[3],[4]]
 => [[1,3,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[2,2],[3],[4]]
 => [[2,2,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[2,3],[3],[4]]
 => [[2,3,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[1,1,1,3],[2]]
 => [[1,1,1,2],[3]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[1,1,2,3],[2]]
 => [[1,1,2,2],[3]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[1,2,2,3],[2]]
 => [[1,2,2,2],[3]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[1,1,3],[2,2]]
 => [[1,1,2,2],[3]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[1,1,1],[2],[3]]
 => [[1,1,1,2],[3]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[1,1,2],[2],[3]]
 => [[1,1,2,2],[3]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[1,2,2],[2],[3]]
 => [[1,2,2,2],[3]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[1,1],[2,2],[3]]
 => [[1,1,2,2],[3]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
Description
The largest multiplicity of a 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 $2$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St000772
Mapping
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 43%
Values
[[1,2]]
 => [[1,2]]
 => [1,2] => ([],2)
 => ? ∊ {1,1,2}
[[2,2]]
 => [[2,2]]
 => [1,2] => ([],2)
 => ? ∊ {1,1,2}
[[1],[2]]
 => [[1,2]]
 => [1,2] => ([],2)
 => ? ∊ {1,1,2}
[[1,3]]
 => [[1,3]]
 => [1,2] => ([],2)
 => ? ∊ {0,1,1,1,1}
[[2,3]]
 => [[2,3]]
 => [1,2] => ([],2)
 => ? ∊ {0,1,1,1,1}
[[3,3]]
 => [[3,3]]
 => [1,2] => ([],2)
 => ? ∊ {0,1,1,1,1}
[[1],[3]]
 => [[1,3]]
 => [1,2] => ([],2)
 => ? ∊ {0,1,1,1,1}
[[2],[3]]
 => [[2,3]]
 => [1,2] => ([],2)
 => ? ∊ {0,1,1,1,1}
[[1,1,2]]
 => [[1,1,2]]
 => [1,2,3] => ([],3)
 => ? ∊ {2,2,2,2,3}
[[1,2,2]]
 => [[1,2,2]]
 => [1,2,3] => ([],3)
 => ? ∊ {2,2,2,2,3}
[[2,2,2]]
 => [[2,2,2]]
 => [1,2,3] => ([],3)
 => ? ∊ {2,2,2,2,3}
[[1,1],[2]]
 => [[1,1,2]]
 => [1,2,3] => ([],3)
 => ? ∊ {2,2,2,2,3}
[[1,2],[2]]
 => [[1,2,2]]
 => [1,2,3] => ([],3)
 => ? ∊ {2,2,2,2,3}
[[1,4]]
 => [[1,4]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,1,1,1,1}
[[2,4]]
 => [[2,4]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,1,1,1,1}
[[3,4]]
 => [[3,4]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,1,1,1,1}
[[4,4]]
 => [[4,4]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,1,1,1,1}
[[1],[4]]
 => [[1,4]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,1,1,1,1}
[[2],[4]]
 => [[2,4]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,1,1,1,1}
[[3],[4]]
 => [[3,4]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,1,1,1,1}
[[1,1,3]]
 => [[1,1,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1,2,2,2,2,2,2,3}
[[1,2,3]]
 => [[1,2,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1,2,2,2,2,2,2,3}
[[1,3,3]]
 => [[1,3,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1,2,2,2,2,2,2,3}
[[2,2,3]]
 => [[2,2,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1,2,2,2,2,2,2,3}
[[2,3,3]]
 => [[2,3,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1,2,2,2,2,2,2,3}
[[3,3,3]]
 => [[3,3,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1,2,2,2,2,2,2,3}
[[1,1],[3]]
 => [[1,1,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1,2,2,2,2,2,2,3}
[[1,2],[3]]
 => [[1,2,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1,2,2,2,2,2,2,3}
[[1,3],[2]]
 => [[1,2],[3]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,3],[3]]
 => [[1,3,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1,2,2,2,2,2,2,3}
[[2,2],[3]]
 => [[2,2,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1,2,2,2,2,2,2,3}
[[2,3],[3]]
 => [[2,3,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1,2,2,2,2,2,2,3}
[[1],[2],[3]]
 => [[1,2],[3]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,1,1,2]]
 => [[1,1,1,2]]
 => [1,2,3,4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,4,4}
[[1,1,2,2]]
 => [[1,1,2,2]]
 => [1,2,3,4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,4,4}
[[1,2,2,2]]
 => [[1,2,2,2]]
 => [1,2,3,4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,4,4}
[[2,2,2,2]]
 => [[2,2,2,2]]
 => [1,2,3,4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,4,4}
[[1,1,1],[2]]
 => [[1,1,1,2]]
 => [1,2,3,4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,4,4}
[[1,1,2],[2]]
 => [[1,1,2,2]]
 => [1,2,3,4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,4,4}
[[1,2,2],[2]]
 => [[1,2,2,2]]
 => [1,2,3,4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,4,4}
[[1,1],[2,2]]
 => [[1,1,2,2]]
 => [1,2,3,4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,4,4}
[[1,5]]
 => [[1,5]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[[2,5]]
 => [[2,5]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[[3,5]]
 => [[3,5]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[[4,5]]
 => [[4,5]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[[5,5]]
 => [[5,5]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[[1],[5]]
 => [[1,5]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[[2],[5]]
 => [[2,5]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[[3],[5]]
 => [[3,5]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[[4],[5]]
 => [[4,5]]
 => [1,2] => ([],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[[1,1,4]]
 => [[1,1,4]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[[1,2,4]]
 => [[1,2,4]]
 => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[[1,4],[2]]
 => [[1,2],[4]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,4],[3]]
 => [[1,3],[4]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[2,4],[3]]
 => [[2,3],[4]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1],[2],[4]]
 => [[1,2],[4]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1],[3],[4]]
 => [[1,3],[4]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[2],[3],[4]]
 => [[2,3],[4]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,1,3],[2]]
 => [[1,1,2],[3]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,2,3],[2]]
 => [[1,2,2],[3]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,1],[2],[3]]
 => [[1,1,2],[3]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,2],[2],[3]]
 => [[1,2,2],[3]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,5],[2]]
 => [[1,2],[5]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,5],[3]]
 => [[1,3],[5]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,5],[4]]
 => [[1,4],[5]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[2,5],[3]]
 => [[2,3],[5]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[2,5],[4]]
 => [[2,4],[5]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[3,5],[4]]
 => [[3,4],[5]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1],[2],[5]]
 => [[1,2],[5]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1],[3],[5]]
 => [[1,3],[5]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1],[4],[5]]
 => [[1,4],[5]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[2],[3],[5]]
 => [[2,3],[5]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[2],[4],[5]]
 => [[2,4],[5]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[3],[4],[5]]
 => [[3,4],[5]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[1,1,4],[2]]
 => [[1,1,2],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,1,4],[3]]
 => [[1,1,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,2,4],[2]]
 => [[1,2,2],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,2,4],[3]]
 => [[1,2,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,3,4],[3]]
 => [[1,3,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[2,2,4],[3]]
 => [[2,2,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[2,3,4],[3]]
 => [[2,3,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,1],[2],[4]]
 => [[1,1,2],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,1],[3],[4]]
 => [[1,1,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,2],[2],[4]]
 => [[1,2,2],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,2],[3],[4]]
 => [[1,2,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[1,3],[3],[4]]
 => [[1,3,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[2,2],[3],[4]]
 => [[2,2,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[2,3],[3],[4]]
 => [[2,3,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[1,1,1,3],[2]]
 => [[1,1,1,2],[3]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[1,1,2,3],[2]]
 => [[1,1,2,2],[3]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[1,2,2,3],[2]]
 => [[1,2,2,2],[3]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[1,1,3],[2,2]]
 => [[1,1,2,2],[3]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[1,1,1],[2],[3]]
 => [[1,1,1,2],[3]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[1,1,2],[2],[3]]
 => [[1,1,2,2],[3]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[1,2,2],[2],[3]]
 => [[1,2,2,2],[3]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[1,1],[2,2],[3]]
 => [[1,1,2,2],[3]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 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].
Matching statistic: St001060
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 14%
Values
[[1,2]]
 => [1,2] => [2] => ([],2)
 => ? ∊ {1,1,2}
[[2,2]]
 => [1,2] => [2] => ([],2)
 => ? ∊ {1,1,2}
[[1],[2]]
 => [2,1] => [2] => ([],2)
 => ? ∊ {1,1,2}
[[1,3]]
 => [1,2] => [2] => ([],2)
 => ? ∊ {0,1,1,1,1}
[[2,3]]
 => [1,2] => [2] => ([],2)
 => ? ∊ {0,1,1,1,1}
[[3,3]]
 => [1,2] => [2] => ([],2)
 => ? ∊ {0,1,1,1,1}
[[1],[3]]
 => [2,1] => [2] => ([],2)
 => ? ∊ {0,1,1,1,1}
[[2],[3]]
 => [2,1] => [2] => ([],2)
 => ? ∊ {0,1,1,1,1}
[[1,1,2]]
 => [1,2,3] => [3] => ([],3)
 => ? ∊ {2,2,2,2,3}
[[1,2,2]]
 => [1,2,3] => [3] => ([],3)
 => ? ∊ {2,2,2,2,3}
[[2,2,2]]
 => [1,2,3] => [3] => ([],3)
 => ? ∊ {2,2,2,2,3}
[[1,1],[2]]
 => [3,1,2] => [3] => ([],3)
 => ? ∊ {2,2,2,2,3}
[[1,2],[2]]
 => [2,1,3] => [3] => ([],3)
 => ? ∊ {2,2,2,2,3}
[[1,4]]
 => [1,2] => [2] => ([],2)
 => ? ∊ {0,0,0,1,1,1,1}
[[2,4]]
 => [1,2] => [2] => ([],2)
 => ? ∊ {0,0,0,1,1,1,1}
[[3,4]]
 => [1,2] => [2] => ([],2)
 => ? ∊ {0,0,0,1,1,1,1}
[[4,4]]
 => [1,2] => [2] => ([],2)
 => ? ∊ {0,0,0,1,1,1,1}
[[1],[4]]
 => [2,1] => [2] => ([],2)
 => ? ∊ {0,0,0,1,1,1,1}
[[2],[4]]
 => [2,1] => [2] => ([],2)
 => ? ∊ {0,0,0,1,1,1,1}
[[3],[4]]
 => [2,1] => [2] => ([],2)
 => ? ∊ {0,0,0,1,1,1,1}
[[1,1,3]]
 => [1,2,3] => [3] => ([],3)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3}
[[1,2,3]]
 => [1,2,3] => [3] => ([],3)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3}
[[1,3,3]]
 => [1,2,3] => [3] => ([],3)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3}
[[2,2,3]]
 => [1,2,3] => [3] => ([],3)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3}
[[2,3,3]]
 => [1,2,3] => [3] => ([],3)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3}
[[3,3,3]]
 => [1,2,3] => [3] => ([],3)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3}
[[1,1],[3]]
 => [3,1,2] => [3] => ([],3)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3}
[[1,2],[3]]
 => [3,1,2] => [3] => ([],3)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3}
[[1,3],[2]]
 => [2,1,3] => [3] => ([],3)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3}
[[1,3],[3]]
 => [2,1,3] => [3] => ([],3)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3}
[[2,2],[3]]
 => [3,1,2] => [3] => ([],3)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3}
[[2,3],[3]]
 => [2,1,3] => [3] => ([],3)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3}
[[1],[2],[3]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2
[[1,1,1,2]]
 => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,4,4}
[[1,1,2,2]]
 => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,4,4}
[[1,2,2,2]]
 => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,4,4}
[[2,2,2,2]]
 => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,4,4}
[[1,1,1],[2]]
 => [4,1,2,3] => [4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,4,4}
[[1,1,2],[2]]
 => [3,1,2,4] => [4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,4,4}
[[1,2,2],[2]]
 => [2,1,3,4] => [4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,4,4}
[[1,1],[2,2]]
 => [3,4,1,2] => [4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,4,4}
[[1,5]]
 => [1,2] => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[[2,5]]
 => [1,2] => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[[3,5]]
 => [1,2] => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[[4,5]]
 => [1,2] => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[[5,5]]
 => [1,2] => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[[1],[5]]
 => [2,1] => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[[2],[5]]
 => [2,1] => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[[3],[5]]
 => [2,1] => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[[4],[5]]
 => [2,1] => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[[1,1,4]]
 => [1,2,3] => [3] => ([],3)
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[[1],[2],[4]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2
[[1],[3],[4]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2
[[2],[3],[4]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2
[[1,2],[2],[3]]
 => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[1,3],[2],[3]]
 => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[1],[2],[5]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2
[[1],[3],[5]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2
[[1],[4],[5]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2
[[2],[3],[5]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2
[[2],[4],[5]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2
[[3],[4],[5]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2
[[1,2],[2],[4]]
 => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[1,3],[2],[4]]
 => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[1,4],[2],[3]]
 => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[1,4],[2],[4]]
 => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[1,3],[3],[4]]
 => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[1,4],[3],[4]]
 => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[2,3],[3],[4]]
 => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[2,4],[3],[4]]
 => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
Description
The distinguishing index of a graph. This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism. If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.