Your data matches 1 statistic following compositions of up to 3 maps.
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Matching statistic: St000771
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => ([],1)
 => 1
[1,1,0,0]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[1,1,0,1,0,0]
 => [2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,1,0,1,0,0,1,0,0]
 => [4,2,3,5,1] => [5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,1,0,1,0,1,0,0,0]
 => [5,2,3,4,1] => [4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,1,0,1,1,0,0,0,0]
 => [5,2,4,3,1] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,1,1,0,0,1,0,0,0]
 => [5,3,2,4,1] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,1,1,0,1,0,0,0,0]
 => [5,3,4,2,1] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 1
[1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,1] => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 4
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,6,5,1] => [6,5,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,5,4,6,1] => [6,1,2,3,5,4] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [2,3,6,4,5,1] => [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,3,6,5,4,1] => [6,5,4,1,2,3] => ([(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)
 => 3
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,4,3,5,6,1] => [6,1,2,4,3,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,4,3,6,5,1] => [6,5,1,2,4,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [2,5,3,4,6,1] => [6,1,2,4,5,3] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [2,6,3,4,5,1] => [5,1,2,4,6,3] => ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,6,3,5,4,1] => [5,4,1,2,6,3] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [2,5,4,3,6,1] => [6,1,2,5,4,3] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [2,6,4,3,5,1] => [5,1,2,6,4,3] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,6,4,5,3,1] => [5,3,1,2,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,6,5,4,3,1] => [6,5,4,3,1,2] => ([(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)
 => 4
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,2,4,5,6,1] => [6,1,3,2,4,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,2,4,6,5,1] => [6,5,1,3,2,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [3,2,5,4,6,1] => [6,1,3,2,5,4] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 2
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,2,6,4,5,1] => [5,1,3,2,6,4] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 1
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,2,6,5,4,1] => [6,5,4,1,3,2] => ([(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)
 => 3
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [4,2,3,5,6,1] => [6,1,3,4,2,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [4,2,3,6,5,1] => [6,5,1,3,4,2] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [5,2,3,4,6,1] => [6,1,3,4,5,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [6,2,3,4,5,1] => [5,1,3,4,6,2] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [6,2,3,5,4,1] => [5,4,1,3,6,2] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [5,2,4,3,6,1] => [6,1,4,3,5,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [6,2,4,3,5,1] => [5,1,4,3,6,2] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [6,2,4,5,3,1] => [5,3,1,4,6,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 1
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 $1$.