Your data matches 8 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000772
(load all 21 compositions to match this statistic)
Mapping
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
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] => [1,2] => [2,1] => ([(0,1)],2)
 => 1
[1,2,3] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,3,2] => [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[2,1,3] => [2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,3,4,2] => [1,4,3,2] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[1,4,2,3] => [1,4,3,2] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[2,3,1,4] => [3,2,1,4] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[3,1,2,4] => [3,2,1,4] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[1,2,3,4,5] => [1,2,3,4,5] => [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
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,2,4,5,3] => [1,2,5,4,3] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,2,5,3,4] => [1,2,5,4,3] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,2,5,4,3] => [1,2,5,4,3] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,3,4,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,3,4,5,2] => [1,5,3,4,2] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,3,5,2,4] => [1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 1
[1,3,5,4,2] => [1,5,4,3,2] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[1,4,2,3,5] => [1,4,3,2,5] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,2,5,3] => [1,5,3,4,2] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,3,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,3,5,2] => [1,5,3,4,2] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,5,2,3] => [1,5,4,3,2] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[1,4,5,3,2] => [1,5,4,3,2] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[1,5,2,3,4] => [1,5,3,4,2] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,5,2,4,3] => [1,5,3,4,2] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,5,3,2,4] => [1,5,4,3,2] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[1,5,3,4,2] => [1,5,4,3,2] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[1,5,4,2,3] => [1,5,4,3,2] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[1,5,4,3,2] => [1,5,4,3,2] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[2,1,3,4,5] => [2,1,3,4,5] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[2,1,3,5,4] => [2,1,3,5,4] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2
[2,1,4,3,5] => [2,1,4,3,5] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2
[2,1,4,5,3] => [2,1,5,4,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[2,1,5,3,4] => [2,1,5,4,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[2,1,5,4,3] => [2,1,5,4,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[2,3,1,4,5] => [3,2,1,4,5] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,3,1,5,4] => [3,2,1,5,4] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[2,3,4,1,5] => [4,2,3,1,5] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,3,5,1,4] => [4,2,5,1,3] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 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: St000456
(load all 4 compositions to match this statistic)
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 22%●distinct values known / distinct values provided: 17%
Values
[1] => ([],1)
 => ([],1)
 => ? = 1
[1,2] => ([],2)
 => ([],2)
 => ? = 1
[1,2,3] => ([],3)
 => ([],3)
 => ? = 2
[1,3,2] => ([(1,2)],3)
 => ([],2)
 => ? = 1
[2,1,3] => ([(1,2)],3)
 => ([],2)
 => ? = 1
[1,2,3,4] => ([],4)
 => ([],4)
 => ? = 3
[1,2,4,3] => ([(2,3)],4)
 => ([],3)
 => ? = 1
[1,3,2,4] => ([(2,3)],4)
 => ([],3)
 => ? = 1
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 2
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ? = 2
[2,1,3,4] => ([(2,3)],4)
 => ([],3)
 => ? = 1
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([],2)
 => ? = 2
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 2
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 2
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ? = 2
[1,2,3,4,5] => ([],5)
 => ([],5)
 => ? = 4
[1,2,3,5,4] => ([(3,4)],5)
 => ([],4)
 => ? = 1
[1,2,4,3,5] => ([(3,4)],5)
 => ([],4)
 => ? = 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ? = 2
[1,3,2,4,5] => ([(3,4)],5)
 => ([],4)
 => ? = 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => ? = 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? = 2
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ? = 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ? = 2
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ? = 2
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 2
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => ? = 3
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 3
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? = 2
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 2
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 3
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 3
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 3
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 3
[2,1,3,4,5] => ([(3,4)],5)
 => ([],4)
 => ? = 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => ? = 2
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([],3)
 => ? = 2
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 2
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 2
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 2
[2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 2
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 2
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? = 2
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ? = 1
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => 1
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => 1
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => 1
[2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 1
[2,3,6,1,5,4] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
[2,3,6,4,1,5] => ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
[2,3,6,5,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,4,1,6,5,3] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => 1
[2,4,3,6,1,5] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
[2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[2,5,1,4,6,3] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => 1
[2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[2,5,1,6,4,3] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[2,5,3,1,6,4] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => 1
[2,5,4,1,6,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => 1
[2,6,1,3,5,4] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
[2,6,1,4,3,5] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
[2,6,1,4,5,3] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[2,6,1,5,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[2,6,1,5,4,3] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[2,6,3,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
[2,6,3,1,5,4] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[3,1,5,6,2,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[3,1,6,2,5,4] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => 1
[3,1,6,4,2,5] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => 1
[3,1,6,5,2,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[3,2,4,6,1,5] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
[3,2,5,1,6,4] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => 1
[3,2,5,6,1,4] => ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 1
[3,2,6,1,4,5] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
[3,2,6,1,5,4] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[3,2,6,4,1,5] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[3,2,6,5,1,4] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 1
[3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[3,4,2,6,1,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[3,5,1,2,6,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[3,5,2,1,6,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[3,6,1,2,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
[3,6,1,2,5,4] => ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 1
[3,6,2,1,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
[3,6,2,1,5,4] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 1
[4,1,3,6,2,5] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => 1
[4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[4,1,6,3,2,5] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
Description
The monochromatic index of a connected graph. This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path. For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St000302
(load all 2 compositions to match this statistic)
Mapping
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000302: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 17%
Values
[1] => [1] => ([],1)
 => 0 = 1 - 1
[1,2] => [1,2] => ([],2)
 => ? = 1 - 1
[1,2,3] => [1,2,3] => ([],3)
 => ? = 2 - 1
[1,3,2] => [1,3,2] => ([(1,2)],3)
 => ? = 1 - 1
[2,1,3] => [2,1,3] => ([(1,2)],3)
 => ? = 1 - 1
[1,2,3,4] => [1,2,3,4] => ([],4)
 => ? = 3 - 1
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => ? = 1 - 1
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ? = 1 - 1
[1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 - 1
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 - 1
[1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 - 1
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => ? = 1 - 1
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? = 2 - 1
[2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 - 1
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 - 1
[3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? = 4 - 1
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ? = 1 - 1
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ? = 1 - 1
[1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[1,2,5,4,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => ? = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ? = 2 - 1
[1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[1,3,4,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[1,3,5,2,4] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 1 - 1
[1,3,5,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[1,4,2,5,3] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[1,4,3,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[1,4,3,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[1,4,5,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
[1,4,5,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
[1,5,2,3,4] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[1,5,2,4,3] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[1,5,3,2,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
[1,5,3,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
[1,5,4,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
[1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
[2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => ? = 1 - 1
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ? = 2 - 1
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ? = 2 - 1
[2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[2,1,5,3,4] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[2,1,5,4,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[2,3,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[2,3,5,1,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[2,4,1,3,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 1 - 1
[2,4,1,5,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[2,4,3,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
[2,5,1,3,4] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[2,5,1,4,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[3,1,5,2,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[3,2,5,1,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[2,3,4,6,1,5] => [5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[2,3,5,1,6,4] => [4,2,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => 0 = 1 - 1
[2,3,5,6,1,4] => [5,2,6,4,1,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[2,3,6,1,4,5] => [4,2,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => 0 = 1 - 1
[2,3,6,1,5,4] => [4,2,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => 0 = 1 - 1
[2,3,6,4,1,5] => [5,2,6,4,1,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[2,3,6,5,1,4] => [5,2,6,4,1,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[2,4,1,5,6,3] => [3,6,1,4,5,2] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[2,4,1,6,3,5] => [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => 0 = 1 - 1
[2,4,1,6,5,3] => [3,6,1,5,4,2] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[2,4,3,6,1,5] => [5,3,2,6,1,4] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[2,4,5,1,6,3] => [4,6,3,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[2,5,1,3,6,4] => [3,6,1,4,5,2] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[2,5,1,4,6,3] => [3,6,1,4,5,2] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[2,5,1,6,3,4] => [3,6,1,5,4,2] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[2,5,1,6,4,3] => [3,6,1,5,4,2] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[2,5,3,1,6,4] => [4,6,3,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[2,5,4,1,6,3] => [4,6,3,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[2,6,1,3,4,5] => [3,6,1,4,5,2] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[2,6,1,3,5,4] => [3,6,1,4,5,2] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[2,6,1,4,3,5] => [3,6,1,5,4,2] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[2,6,1,4,5,3] => [3,6,1,5,4,2] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[2,6,1,5,3,4] => [3,6,1,5,4,2] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[2,6,1,5,4,3] => [3,6,1,5,4,2] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[2,6,3,1,4,5] => [4,6,3,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[2,6,3,1,5,4] => [4,6,3,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[3,1,4,6,2,5] => [5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[3,1,5,2,6,4] => [4,2,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => 0 = 1 - 1
[3,1,5,6,2,4] => [5,2,6,4,1,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[3,1,6,2,4,5] => [4,2,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => 0 = 1 - 1
[3,1,6,2,5,4] => [4,2,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => 0 = 1 - 1
[3,1,6,4,2,5] => [5,2,6,4,1,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[3,1,6,5,2,4] => [5,2,6,4,1,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[3,2,4,6,1,5] => [5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[3,2,5,1,6,4] => [4,2,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => 0 = 1 - 1
[3,2,5,6,1,4] => [5,2,6,4,1,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[3,2,6,1,4,5] => [4,2,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => 0 = 1 - 1
[3,2,6,1,5,4] => [4,2,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => 0 = 1 - 1
[3,2,6,4,1,5] => [5,2,6,4,1,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[3,2,6,5,1,4] => [5,2,6,4,1,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[3,4,1,6,2,5] => [5,3,2,6,1,4] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[3,4,2,6,1,5] => [5,3,2,6,1,4] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[3,5,1,2,6,4] => [4,6,3,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
Description
The determinant of the distance matrix of a connected graph.
Matching statistic: St001630
(load all 6 compositions to match this statistic)
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 17%
Values
[1] => ([],1)
 => ([],1)
 => ? = 1
[1,2] => ([],2)
 => ([],1)
 => ? = 1
[1,2,3] => ([],3)
 => ([],1)
 => ? = 2
[1,3,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? = 1
[2,1,3] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? = 1
[1,2,3,4] => ([],4)
 => ([],1)
 => ? = 3
[1,2,4,3] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 1
[1,3,2,4] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 1
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[2,1,3,4] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 1
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,2,3,4,5] => ([],5)
 => ([],1)
 => ? = 4
[1,2,3,5,4] => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 1
[1,2,4,3,5] => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,3,2,4,5] => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 2
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 3
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 3
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 2
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 3
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 3
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
 => ? = 3
[2,1,3,4,5] => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 2
[2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 1
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 1
[3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 2
[3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 2
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 1
[3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 3
[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 3
[4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 2
[4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3
[4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 3
[1,2,3,5,6,4] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,3,6,4,5] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,4,5,3,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,5,3,4,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,3,2,4,6,5] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,2,5,4,6] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,4,2,5,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,4,2,3,5,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,4,3,2,5,6] => ([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[2,1,3,4,6,5] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,3,5,4,6] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,4,3,5,6] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,3,1,4,5,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,1,2,4,5,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,2,3,5,4,7,6] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,3,5,6,4,7] => ([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,3,6,4,5,7] => ([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,3,6,5,4,7] => ([(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,2,4,3,5,7,6] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,4,3,6,5,7] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,4,5,3,6,7] => ([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,5,3,4,6,7] => ([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,5,4,3,6,7] => ([(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,3,2,4,5,7,6] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St001878
(load all 6 compositions to match this statistic)
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 17%
Values
[1] => ([],1)
 => ([],1)
 => ? = 1
[1,2] => ([],2)
 => ([],1)
 => ? = 1
[1,2,3] => ([],3)
 => ([],1)
 => ? = 2
[1,3,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? = 1
[2,1,3] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? = 1
[1,2,3,4] => ([],4)
 => ([],1)
 => ? = 3
[1,2,4,3] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 1
[1,3,2,4] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 1
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[2,1,3,4] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 1
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,2,3,4,5] => ([],5)
 => ([],1)
 => ? = 4
[1,2,3,5,4] => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 1
[1,2,4,3,5] => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,3,2,4,5] => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 2
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 3
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 3
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 2
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 3
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 3
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
 => ? = 3
[2,1,3,4,5] => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 2
[2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 1
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 1
[3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 2
[3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 2
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 1
[3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 3
[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 3
[4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 2
[4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3
[4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 3
[1,2,3,5,6,4] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,3,6,4,5] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,4,5,3,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,5,3,4,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,3,2,4,6,5] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,2,5,4,6] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,4,2,5,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,4,2,3,5,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,4,3,2,5,6] => ([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[2,1,3,4,6,5] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,3,5,4,6] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,4,3,5,6] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,3,1,4,5,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,1,2,4,5,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,2,3,5,4,7,6] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,3,5,6,4,7] => ([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,3,6,4,5,7] => ([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,3,6,5,4,7] => ([(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,2,4,3,5,7,6] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,4,3,6,5,7] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,4,5,3,6,7] => ([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,5,3,4,6,7] => ([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,5,4,3,6,7] => ([(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,3,2,4,5,7,6] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St001632
(load all 3 compositions to match this statistic)
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 17%
Values
[1] => [1] => [1] => ([],1)
 => ? = 1 - 1
[1,2] => [2,1] => [2,1] => ([],2)
 => ? = 1 - 1
[1,2,3] => [3,2,1] => [3,2,1] => ([],3)
 => ? = 2 - 1
[1,3,2] => [2,3,1] => [2,3,1] => ([(1,2)],3)
 => ? = 1 - 1
[2,1,3] => [3,1,2] => [3,1,2] => ([(1,2)],3)
 => ? = 1 - 1
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => ([],4)
 => ? = 3 - 1
[1,2,4,3] => [3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
 => ? = 1 - 1
[1,3,2,4] => [4,2,3,1] => [4,2,3,1] => ([(2,3)],4)
 => ? = 1 - 1
[1,3,4,2] => [2,4,3,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => ? = 2 - 1
[1,4,2,3] => [3,2,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => ? = 2 - 1
[1,4,3,2] => [2,3,4,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => ? = 2 - 1
[2,1,3,4] => [4,3,1,2] => [4,3,1,2] => ([(2,3)],4)
 => ? = 1 - 1
[2,1,4,3] => [3,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ? = 2 - 1
[2,3,1,4] => [4,1,3,2] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => ? = 2 - 1
[3,1,2,4] => [4,2,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
 => ? = 2 - 1
[3,2,1,4] => [4,1,2,3] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => ? = 2 - 1
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
 => ? = 4 - 1
[1,2,3,5,4] => [4,5,3,2,1] => [4,5,3,2,1] => ([(3,4)],5)
 => ? = 1 - 1
[1,2,4,3,5] => [5,3,4,2,1] => [5,3,4,2,1] => ([(3,4)],5)
 => ? = 1 - 1
[1,2,4,5,3] => [3,5,4,2,1] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
 => ? = 2 - 1
[1,2,5,3,4] => [4,3,5,2,1] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
 => ? = 2 - 1
[1,2,5,4,3] => [3,4,5,2,1] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
 => ? = 2 - 1
[1,3,2,4,5] => [5,4,2,3,1] => [5,4,2,3,1] => ([(3,4)],5)
 => ? = 1 - 1
[1,3,2,5,4] => [4,5,2,3,1] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
 => ? = 2 - 1
[1,3,4,2,5] => [5,2,4,3,1] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
 => ? = 2 - 1
[1,3,4,5,2] => [2,5,4,3,1] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => ? = 2 - 1
[1,3,5,2,4] => [4,2,5,3,1] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
 => ? = 1 - 1
[1,3,5,4,2] => [2,4,5,3,1] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => ? = 3 - 1
[1,4,2,3,5] => [5,3,2,4,1] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
 => ? = 2 - 1
[1,4,2,5,3] => [3,5,2,4,1] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
 => ? = 2 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
 => ? = 2 - 1
[1,4,3,5,2] => [2,5,3,4,1] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => ? = 2 - 1
[1,4,5,2,3] => [3,2,5,4,1] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 3 - 1
[1,4,5,3,2] => [2,3,5,4,1] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => ? = 3 - 1
[1,5,2,3,4] => [4,3,2,5,1] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => ? = 2 - 1
[1,5,2,4,3] => [3,4,2,5,1] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
 => ? = 2 - 1
[1,5,3,2,4] => [4,2,3,5,1] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
 => ? = 3 - 1
[1,5,3,4,2] => [2,4,3,5,1] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => ? = 3 - 1
[1,5,4,2,3] => [3,2,4,5,1] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 3 - 1
[1,5,4,3,2] => [2,3,4,5,1] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => ? = 3 - 1
[2,1,3,4,5] => [5,4,3,1,2] => [5,4,3,1,2] => ([(3,4)],5)
 => ? = 1 - 1
[2,1,3,5,4] => [4,5,3,1,2] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => ? = 2 - 1
[2,1,4,3,5] => [5,3,4,1,2] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => ? = 2 - 1
[2,1,4,5,3] => [3,5,4,1,2] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
 => ? = 2 - 1
[2,1,5,3,4] => [4,3,5,1,2] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
 => ? = 2 - 1
[2,1,5,4,3] => [3,4,5,1,2] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
 => ? = 2 - 1
[2,3,1,4,5] => [5,4,1,3,2] => [5,4,1,3,2] => ([(2,3),(2,4)],5)
 => ? = 2 - 1
[2,3,1,5,4] => [4,5,1,3,2] => [4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
 => ? = 2 - 1
[2,3,4,1,5] => [5,1,4,3,2] => [5,1,4,3,2] => ([(1,2),(1,3),(1,4)],5)
 => ? = 2 - 1
[2,3,5,1,4] => [4,1,5,3,2] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => 0 = 1 - 1
[2,4,1,3,5] => [5,3,1,4,2] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
 => ? = 1 - 1
[2,4,1,5,3] => [3,5,1,4,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 1 - 1
[2,5,1,3,4] => [4,3,1,5,2] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
 => 0 = 1 - 1
[2,5,1,4,3] => [3,4,1,5,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 1 - 1
[3,1,5,2,4] => [4,2,5,1,3] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0 = 1 - 1
[3,2,5,1,4] => [4,1,5,2,3] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => 0 = 1 - 1
[2,3,4,6,1,5] => [5,1,6,4,3,2] => [5,1,6,4,3,2] => ([(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => 0 = 1 - 1
[2,3,5,1,6,4] => [4,6,1,5,3,2] => [4,6,1,5,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6)
 => 0 = 1 - 1
[2,3,5,6,1,4] => [4,1,6,5,3,2] => [4,1,6,5,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => 0 = 1 - 1
[2,3,6,1,4,5] => [5,4,1,6,3,2] => [5,4,1,6,3,2] => ([(0,5),(1,5),(2,3),(2,4),(2,5)],6)
 => 0 = 1 - 1
[2,3,6,1,5,4] => [4,5,1,6,3,2] => [4,6,1,5,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6)
 => 0 = 1 - 1
[2,3,6,4,1,5] => [5,1,4,6,3,2] => [5,1,6,4,3,2] => ([(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => 0 = 1 - 1
[2,3,6,5,1,4] => [4,1,5,6,3,2] => [4,1,6,5,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => 0 = 1 - 1
[2,4,1,5,6,3] => [3,6,5,1,4,2] => [3,6,5,1,4,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6)
 => 0 = 1 - 1
[2,4,1,6,3,5] => [5,3,6,1,4,2] => [5,3,6,1,4,2] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
 => 0 = 1 - 1
[2,4,1,6,5,3] => [3,5,6,1,4,2] => [3,6,5,1,4,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6)
 => 0 = 1 - 1
[2,4,3,6,1,5] => [5,1,6,3,4,2] => [5,1,6,4,3,2] => ([(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => 0 = 1 - 1
[2,4,5,1,6,3] => [3,6,1,5,4,2] => [3,6,1,5,4,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5)],6)
 => 0 = 1 - 1
[2,5,1,3,6,4] => [4,6,3,1,5,2] => [4,6,3,1,5,2] => ([(0,5),(1,4),(1,5),(2,3),(2,5)],6)
 => 0 = 1 - 1
[2,5,1,4,6,3] => [3,6,4,1,5,2] => [3,6,5,1,4,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6)
 => 0 = 1 - 1
[2,5,1,6,3,4] => [4,3,6,1,5,2] => [4,3,6,1,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6)
 => 0 = 1 - 1
[2,5,1,6,4,3] => [3,4,6,1,5,2] => [3,6,5,1,4,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6)
 => 0 = 1 - 1
[2,5,3,1,6,4] => [4,6,1,3,5,2] => [4,6,1,5,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6)
 => 0 = 1 - 1
[2,5,4,1,6,3] => [3,6,1,4,5,2] => [3,6,1,5,4,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5)],6)
 => 0 = 1 - 1
[2,6,1,3,4,5] => [5,4,3,1,6,2] => [5,4,3,1,6,2] => ([(0,5),(1,5),(2,5),(3,4),(3,5)],6)
 => 0 = 1 - 1
[2,6,1,3,5,4] => [4,5,3,1,6,2] => [4,6,3,1,5,2] => ([(0,5),(1,4),(1,5),(2,3),(2,5)],6)
 => 0 = 1 - 1
[2,6,1,4,3,5] => [5,3,4,1,6,2] => [5,3,6,1,4,2] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
 => 0 = 1 - 1
[2,6,1,4,5,3] => [3,5,4,1,6,2] => [3,6,5,1,4,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6)
 => 0 = 1 - 1
[2,6,1,5,3,4] => [4,3,5,1,6,2] => [4,3,6,1,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6)
 => 0 = 1 - 1
[2,6,1,5,4,3] => [3,4,5,1,6,2] => [3,6,5,1,4,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6)
 => 0 = 1 - 1
[2,6,3,1,4,5] => [5,4,1,3,6,2] => [5,4,1,6,3,2] => ([(0,5),(1,5),(2,3),(2,4),(2,5)],6)
 => 0 = 1 - 1
[2,6,3,1,5,4] => [4,5,1,3,6,2] => [4,6,1,5,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6)
 => 0 = 1 - 1
[3,1,4,6,2,5] => [5,2,6,4,1,3] => [5,2,6,4,1,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5)],6)
 => 0 = 1 - 1
[3,1,5,2,6,4] => [4,6,2,5,1,3] => [4,6,2,5,1,3] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
 => 0 = 1 - 1
[3,1,5,6,2,4] => [4,2,6,5,1,3] => [4,2,6,5,1,3] => ([(0,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => 0 = 1 - 1
[3,1,6,2,4,5] => [5,4,2,6,1,3] => [5,4,2,6,1,3] => ([(0,5),(1,5),(2,4),(3,4),(3,5)],6)
 => 0 = 1 - 1
[3,1,6,2,5,4] => [4,5,2,6,1,3] => [4,6,2,5,1,3] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
 => 0 = 1 - 1
[3,1,6,4,2,5] => [5,2,4,6,1,3] => [5,2,6,4,1,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5)],6)
 => 0 = 1 - 1
[3,1,6,5,2,4] => [4,2,5,6,1,3] => [4,2,6,5,1,3] => ([(0,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => 0 = 1 - 1
[3,2,4,6,1,5] => [5,1,6,4,2,3] => [5,1,6,4,3,2] => ([(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => 0 = 1 - 1
[3,2,5,1,6,4] => [4,6,1,5,2,3] => [4,6,1,5,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6)
 => 0 = 1 - 1
[3,2,5,6,1,4] => [4,1,6,5,2,3] => [4,1,6,5,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => 0 = 1 - 1
[3,2,6,1,4,5] => [5,4,1,6,2,3] => [5,4,1,6,3,2] => ([(0,5),(1,5),(2,3),(2,4),(2,5)],6)
 => 0 = 1 - 1
[3,2,6,1,5,4] => [4,5,1,6,2,3] => [4,6,1,5,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6)
 => 0 = 1 - 1
[3,2,6,4,1,5] => [5,1,4,6,2,3] => [5,1,6,4,3,2] => ([(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => 0 = 1 - 1
[3,2,6,5,1,4] => [4,1,5,6,2,3] => [4,1,6,5,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => 0 = 1 - 1
[3,4,1,6,2,5] => [5,2,6,1,4,3] => [5,2,6,1,4,3] => ([(0,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => 0 = 1 - 1
[3,4,2,6,1,5] => [5,1,6,2,4,3] => [5,1,6,4,3,2] => ([(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => 0 = 1 - 1
[3,5,1,2,6,4] => [4,6,2,1,5,3] => [4,6,2,1,5,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6)
 => 0 = 1 - 1
[3,5,2,1,6,4] => [4,6,1,2,5,3] => [4,6,1,5,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6)
 => 0 = 1 - 1
Description
The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
Matching statistic: St001060
(load all 11 compositions to match this statistic)
Mapping
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 17%
Values
[1] => [1] => [1] => ([],1)
 => ? = 1 + 1
[1,2] => [1,2] => [1,2] => ([],2)
 => ? = 1 + 1
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
 => ? = 2 + 1
[1,3,2] => [1,3,2] => [1,2,3] => ([],3)
 => ? = 1 + 1
[2,1,3] => [2,1,3] => [1,2,3] => ([],3)
 => ? = 1 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? = 3 + 1
[1,2,4,3] => [1,2,4,3] => [1,2,3,4] => ([],4)
 => ? = 1 + 1
[1,3,2,4] => [1,3,2,4] => [1,2,3,4] => ([],4)
 => ? = 1 + 1
[1,3,4,2] => [1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
 => ? = 2 + 1
[1,4,2,3] => [1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
 => ? = 2 + 1
[1,4,3,2] => [1,3,4,2] => [1,2,3,4] => ([],4)
 => ? = 2 + 1
[2,1,3,4] => [2,1,3,4] => [1,2,3,4] => ([],4)
 => ? = 1 + 1
[2,1,4,3] => [2,1,4,3] => [1,2,3,4] => ([],4)
 => ? = 2 + 1
[2,3,1,4] => [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ? = 2 + 1
[3,1,2,4] => [3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
 => ? = 2 + 1
[3,2,1,4] => [2,3,1,4] => [1,2,3,4] => ([],4)
 => ? = 2 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? = 4 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4,5] => ([],5)
 => ? = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => ([],5)
 => ? = 1 + 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ? = 2 + 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
 => ? = 2 + 1
[1,2,5,4,3] => [1,2,4,5,3] => [1,2,3,4,5] => ([],5)
 => ? = 2 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => ([],5)
 => ? = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,2,3,4,5] => ([],5)
 => ? = 2 + 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ? = 2 + 1
[1,3,4,5,2] => [1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,3,5,2,4] => [1,5,4,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ? = 1 + 1
[1,3,5,4,2] => [1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
 => ? = 3 + 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ? = 2 + 1
[1,4,2,5,3] => [1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,4,3,2,5] => [1,3,4,2,5] => [1,2,3,4,5] => ([],5)
 => ? = 2 + 1
[1,4,3,5,2] => [1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ? = 2 + 1
[1,4,5,2,3] => [1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? = 3 + 1
[1,4,5,3,2] => [1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ? = 3 + 1
[1,5,2,3,4] => [1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ? = 2 + 1
[1,5,2,4,3] => [1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
 => ? = 2 + 1
[1,5,3,2,4] => [1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
 => ? = 3 + 1
[1,5,3,4,2] => [1,3,4,5,2] => [1,2,3,4,5] => ([],5)
 => ? = 3 + 1
[1,5,4,2,3] => [1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ? = 3 + 1
[1,5,4,3,2] => [1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? = 3 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? = 1 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [1,2,3,4,5] => ([],5)
 => ? = 2 + 1
[2,1,4,3,5] => [2,1,4,3,5] => [1,2,3,4,5] => ([],5)
 => ? = 2 + 1
[2,1,4,5,3] => [2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ? = 2 + 1
[2,1,5,3,4] => [2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
 => ? = 2 + 1
[2,1,5,4,3] => [2,1,4,5,3] => [1,2,3,4,5] => ([],5)
 => ? = 2 + 1
[2,3,1,4,5] => [3,1,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => ? = 2 + 1
[2,3,1,5,4] => [3,1,2,5,4] => [1,3,2,4,5] => ([(3,4)],5)
 => ? = 2 + 1
[2,3,4,1,5] => [4,1,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[2,3,5,1,4] => [5,4,1,2,3] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[2,4,1,3,5] => [4,3,1,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ? = 1 + 1
[2,4,1,5,3] => [5,3,1,2,4] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[2,5,1,3,4] => [5,4,3,1,2] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[2,5,1,4,3] => [4,5,3,1,2] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => 2 = 1 + 1
[3,1,5,2,4] => [5,4,2,1,3] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[2,3,4,6,1,5] => [6,5,1,2,3,4] => [1,6,4,2,5,3] => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[2,3,5,1,6,4] => [6,4,1,2,3,5] => [1,6,5,3,2,4] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[2,3,5,6,1,4] => [5,1,2,3,6,4] => [1,5,6,4,3,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[2,3,6,1,4,5] => [6,5,4,1,2,3] => [1,6,3,4,2,5] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[2,3,6,1,5,4] => [5,6,4,1,2,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[2,3,6,4,1,5] => [4,6,5,1,2,3] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => 2 = 1 + 1
[2,3,6,5,1,4] => [6,4,5,1,2,3] => [1,6,3,5,2,4] => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[2,4,1,5,6,3] => [6,3,1,2,4,5] => [1,6,5,4,2,3] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[2,4,1,6,3,5] => [6,5,3,1,2,4] => [1,6,4,2,5,3] => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[2,4,1,6,5,3] => [5,6,3,1,2,4] => [1,5,2,6,4,3] => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[2,4,3,6,1,5] => [3,6,5,1,2,4] => [1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => 2 = 1 + 1
[2,4,5,1,6,3] => [4,1,2,6,3,5] => [1,4,6,5,3,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[2,5,1,3,6,4] => [6,4,3,1,2,5] => [1,6,5,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[2,5,1,4,6,3] => [4,6,3,1,2,5] => [1,4,2,6,5,3] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[2,5,1,6,3,4] => [5,3,1,2,6,4] => [1,5,6,4,2,3] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[2,5,1,6,4,3] => [6,3,1,2,5,4] => [1,6,4,2,3,5] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[2,5,4,1,6,3] => [6,3,4,1,2,5] => [1,6,5,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[2,6,1,3,4,5] => [6,5,4,3,1,2] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[2,6,1,3,5,4] => [5,6,4,3,1,2] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[2,6,1,4,5,3] => [4,5,6,3,1,2] => [1,4,3,6,2,5] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[2,6,1,5,3,4] => [6,4,5,3,1,2] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[2,6,3,1,4,5] => [3,6,5,4,1,2] => [1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => 2 = 1 + 1
[2,6,3,1,5,4] => [3,5,6,4,1,2] => [1,3,6,2,5,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[3,1,4,6,2,5] => [6,5,2,1,3,4] => [1,6,4,2,5,3] => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[3,1,5,2,6,4] => [6,4,2,1,3,5] => [1,6,5,3,2,4] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[3,1,5,6,2,4] => [5,2,1,3,6,4] => [1,5,6,4,3,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[3,1,6,2,4,5] => [6,5,4,2,1,3] => [1,6,3,4,2,5] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[3,1,6,2,5,4] => [5,6,4,2,1,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[3,1,6,4,2,5] => [4,6,5,2,1,3] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => 2 = 1 + 1
[3,1,6,5,2,4] => [6,4,5,2,1,3] => [1,6,3,5,2,4] => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[3,4,1,6,2,5] => [3,1,6,5,2,4] => [1,3,6,4,5,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[3,4,2,6,1,5] => [6,5,1,3,2,4] => [1,6,4,3,2,5] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[3,5,1,2,6,4] => [3,1,6,4,2,5] => [1,3,6,5,2,4] => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[3,5,2,1,6,4] => [6,4,1,3,2,5] => [1,6,5,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[3,6,1,2,4,5] => [3,1,6,5,4,2] => [1,3,6,2,4,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 1 + 1
[3,6,1,2,5,4] => [3,1,5,6,4,2] => [1,3,5,4,6,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[3,6,2,1,4,5] => [6,5,4,1,3,2] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[4,1,2,6,3,5] => [6,5,3,2,1,4] => [1,6,4,2,5,3] => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[4,1,3,6,2,5] => [3,6,5,2,1,4] => [1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => 2 = 1 + 1
[4,1,6,2,3,5] => [4,2,1,6,5,3] => [1,4,6,3,2,5] => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[4,1,6,3,2,5] => [6,5,2,1,4,3] => [1,6,3,2,5,4] => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[4,3,1,6,2,5] => [6,5,2,3,1,4] => [1,6,4,3,2,5] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[4,3,2,6,1,5] => [3,2,6,5,1,4] => [1,3,6,4,5,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
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.
Matching statistic: St001545
Mapping
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001545: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 17%
Values
[1] => [1] => ([],1)
 => ([],1)
 => ? = 1 + 23
[1,2] => [1,2] => ([],2)
 => ([],1)
 => ? = 1 + 23
[1,2,3] => [1,2,3] => ([],3)
 => ([],1)
 => ? = 2 + 23
[1,3,2] => [1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 1 + 23
[2,1,3] => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 1 + 23
[1,2,3,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => ? = 3 + 23
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 1 + 23
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 1 + 23
[1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 23
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 23
[1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 23
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 1 + 23
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ? = 2 + 23
[2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 23
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 23
[3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 23
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => ? = 4 + 23
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 1 + 23
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 1 + 23
[1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 23
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 23
[1,2,5,4,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 23
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 1 + 23
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? = 2 + 23
[1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 23
[1,3,4,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 23
[1,3,5,2,4] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => ? = 1 + 23
[1,3,5,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 23
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 23
[1,4,2,5,3] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 23
[1,4,3,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 23
[1,4,3,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 23
[1,4,5,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 23
[1,4,5,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 23
[1,5,2,3,4] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 23
[1,5,2,4,3] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 23
[1,5,3,2,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 23
[1,5,3,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 23
[1,5,4,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 23
[1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 23
[2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 1 + 23
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? = 2 + 23
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? = 2 + 23
[2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 23
[2,1,5,3,4] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 23
[2,1,5,4,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 23
[2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 23
[2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 23
[2,3,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 23
[2,3,5,1,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[2,4,1,3,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => ? = 1 + 23
[2,4,1,5,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[2,5,1,3,4] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[2,5,1,4,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[3,1,5,2,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[3,2,5,1,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[2,3,4,6,1,5] => [5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[2,4,1,5,6,3] => [3,6,1,4,5,2] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[2,5,1,3,6,4] => [3,6,1,4,5,2] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[2,5,1,4,6,3] => [3,6,1,4,5,2] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[2,6,1,3,4,5] => [3,6,1,4,5,2] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[2,6,1,3,5,4] => [3,6,1,4,5,2] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[3,1,4,6,2,5] => [5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[3,2,4,6,1,5] => [5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[4,1,2,6,3,5] => [5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[4,1,3,6,2,5] => [5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[2,3,4,5,7,1,6] => [6,2,3,4,7,1,5] => ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[2,3,5,7,1,4,6] => [5,2,6,7,1,3,4] => ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[2,4,1,5,6,7,3] => [3,7,1,4,5,6,2] => ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[2,4,5,7,1,3,6] => [5,6,3,7,1,2,4] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[2,4,6,1,3,7,5] => [4,5,7,1,2,6,3] => ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[2,4,6,1,7,3,5] => [4,6,7,1,5,2,3] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[2,4,7,1,3,5,6] => [4,5,7,1,2,6,3] => ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[2,4,7,1,3,6,5] => [4,5,7,1,2,6,3] => ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[2,4,7,1,5,3,6] => [4,6,7,1,5,2,3] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[2,4,7,1,6,3,5] => [4,6,7,1,5,2,3] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[2,5,1,3,6,7,4] => [3,7,1,4,5,6,2] => ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[2,5,1,4,6,7,3] => [3,7,1,4,5,6,2] => ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[2,5,3,7,1,4,6] => [5,6,3,7,1,2,4] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[2,5,4,7,1,3,6] => [5,6,3,7,1,2,4] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[2,5,7,1,3,4,6] => [4,6,7,1,5,2,3] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[2,5,7,1,4,3,6] => [4,6,7,1,5,2,3] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[2,6,1,3,4,7,5] => [3,7,1,4,5,6,2] => ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[2,6,1,3,5,7,4] => [3,7,1,4,5,6,2] => ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[2,7,1,3,4,5,6] => [3,7,1,4,5,6,2] => ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[2,7,1,3,4,6,5] => [3,7,1,4,5,6,2] => ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[3,1,4,5,7,2,6] => [6,2,3,4,7,1,5] => ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[3,1,5,7,2,4,6] => [5,2,6,7,1,3,4] => ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[3,2,4,5,7,1,6] => [6,2,3,4,7,1,5] => ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[3,2,5,7,1,4,6] => [5,2,6,7,1,3,4] => ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[3,5,1,7,2,4,6] => [5,6,3,7,1,2,4] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[3,5,2,7,1,4,6] => [5,6,3,7,1,2,4] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[4,1,2,5,7,3,6] => [6,2,3,4,7,1,5] => ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[4,1,3,5,7,2,6] => [6,2,3,4,7,1,5] => ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[5,1,2,3,7,4,6] => [6,2,3,4,7,1,5] => ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
[5,1,2,4,7,3,6] => [6,2,3,4,7,1,5] => ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 24 = 1 + 23
Description
The second Elser number of a connected graph. For a connected graph $G$ the $k$-th Elser number is $$ els_k(G) = (-1)^{|V(G)|+1} \sum_N (-1)^{|E(N)|} |V(N)|^k $$ where the sum is over all nuclei of $G$, that is, the connected subgraphs of $G$ whose vertex set is a vertex cover of $G$. It is clear that this number is even. It was shown in [1] that it is non-negative.