Your data matches 116 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001575
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
St001575: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 0
[1,2] => ([],2)
 => ([],2)
 => 0
[2,1] => ([(0,1)],2)
 => ([],1)
 => 0
[1,2,3] => ([],3)
 => ([],3)
 => 0
[1,3,2] => ([(1,2)],3)
 => ([],2)
 => 0
[2,1,3] => ([(1,2)],3)
 => ([],2)
 => 0
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 0
[1,2,3,4] => ([],4)
 => ([],4)
 => 0
[1,2,4,3] => ([(2,3)],4)
 => ([],3)
 => 0
[1,3,2,4] => ([(2,3)],4)
 => ([],3)
 => 0
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
[2,1,3,4] => ([(2,3)],4)
 => ([],3)
 => 0
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([],2)
 => 0
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => 0
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[1,2,3,4,5] => ([],5)
 => ([],5)
 => 0
[1,2,3,5,4] => ([(3,4)],5)
 => ([],4)
 => 0
[1,2,4,3,5] => ([(3,4)],5)
 => ([],4)
 => 0
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[1,3,2,4,5] => ([(3,4)],5)
 => ([],4)
 => 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => 0
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0
[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)
 => 0
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 0
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 0
Description
The minimal number of edges to add or remove to make a graph edge transitive. A graph is edge transitive, if for any two edges, there is an automorphism that maps one edge to the other.
Matching statistic: St001577
(load all 4 compositions to match this statistic)
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
St001577: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 0
[1,2] => ([],2)
 => ([],2)
 => 0
[2,1] => ([(0,1)],2)
 => ([],1)
 => 0
[1,2,3] => ([],3)
 => ([],3)
 => 0
[1,3,2] => ([(1,2)],3)
 => ([],2)
 => 0
[2,1,3] => ([(1,2)],3)
 => ([],2)
 => 0
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 0
[1,2,3,4] => ([],4)
 => ([],4)
 => 0
[1,2,4,3] => ([(2,3)],4)
 => ([],3)
 => 0
[1,3,2,4] => ([(2,3)],4)
 => ([],3)
 => 0
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
[2,1,3,4] => ([(2,3)],4)
 => ([],3)
 => 0
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([],2)
 => 0
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => 0
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[1,2,3,4,5] => ([],5)
 => ([],5)
 => 0
[1,2,3,5,4] => ([(3,4)],5)
 => ([],4)
 => 0
[1,2,4,3,5] => ([(3,4)],5)
 => ([],4)
 => 0
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[1,3,2,4,5] => ([(3,4)],5)
 => ([],4)
 => 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => 0
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0
[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)
 => 0
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 0
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 0
Description
The minimal number of edges to add or remove to make a graph a cograph. A cograph is a graph that can be obtained from the one vertex graph by complementation and disjoint union.
Matching statistic: St001324
(load all 2 compositions to match this statistic)
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
Mp00157: Graphs —connected complement⟶ Graphs
St001324: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => ([],1)
 => 0
[1,2] => ([],2)
 => ([],2)
 => ([],2)
 => 0
[2,1] => ([(0,1)],2)
 => ([],1)
 => ([],1)
 => 0
[1,2,3] => ([],3)
 => ([],3)
 => ([],3)
 => 0
[1,3,2] => ([(1,2)],3)
 => ([],2)
 => ([],2)
 => 0
[2,1,3] => ([(1,2)],3)
 => ([],2)
 => ([],2)
 => 0
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => ([],1)
 => 0
[1,2,3,4] => ([],4)
 => ([],4)
 => ([],4)
 => 0
[1,2,4,3] => ([(2,3)],4)
 => ([],3)
 => ([],3)
 => 0
[1,3,2,4] => ([(2,3)],4)
 => ([],3)
 => ([],3)
 => 0
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ([],2)
 => 0
[2,1,3,4] => ([(2,3)],4)
 => ([],3)
 => ([],3)
 => 0
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([],2)
 => ([],2)
 => 0
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ([],2)
 => 0
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => ([],1)
 => 0
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => 0
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => 0
[1,2,3,4,5] => ([],5)
 => ([],5)
 => ([],5)
 => 0
[1,2,3,5,4] => ([(3,4)],5)
 => ([],4)
 => ([],4)
 => 0
[1,2,4,3,5] => ([(3,4)],5)
 => ([],4)
 => ([],4)
 => 0
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ([],3)
 => 0
[1,3,2,4,5] => ([(3,4)],5)
 => ([],4)
 => ([],4)
 => 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => ([],3)
 => 0
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ([],3)
 => 0
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => ([],2)
 => 0
Description
The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. A graph is chordal if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,c)$ and $(b,c)$ are edges and $(a,b)$ is not an edge. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001326
(load all 2 compositions to match this statistic)
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
Mp00157: Graphs —connected complement⟶ Graphs
St001326: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => ([],1)
 => 0
[1,2] => ([],2)
 => ([],2)
 => ([],2)
 => 0
[2,1] => ([(0,1)],2)
 => ([],1)
 => ([],1)
 => 0
[1,2,3] => ([],3)
 => ([],3)
 => ([],3)
 => 0
[1,3,2] => ([(1,2)],3)
 => ([],2)
 => ([],2)
 => 0
[2,1,3] => ([(1,2)],3)
 => ([],2)
 => ([],2)
 => 0
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => ([],1)
 => 0
[1,2,3,4] => ([],4)
 => ([],4)
 => ([],4)
 => 0
[1,2,4,3] => ([(2,3)],4)
 => ([],3)
 => ([],3)
 => 0
[1,3,2,4] => ([(2,3)],4)
 => ([],3)
 => ([],3)
 => 0
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ([],2)
 => 0
[2,1,3,4] => ([(2,3)],4)
 => ([],3)
 => ([],3)
 => 0
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([],2)
 => ([],2)
 => 0
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ([],2)
 => 0
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => ([],1)
 => 0
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => 0
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => 0
[1,2,3,4,5] => ([],5)
 => ([],5)
 => ([],5)
 => 0
[1,2,3,5,4] => ([(3,4)],5)
 => ([],4)
 => ([],4)
 => 0
[1,2,4,3,5] => ([(3,4)],5)
 => ([],4)
 => ([],4)
 => 0
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ([],3)
 => 0
[1,3,2,4,5] => ([(3,4)],5)
 => ([],4)
 => ([],4)
 => 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => ([],3)
 => 0
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ([],3)
 => 0
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => ([],2)
 => 0
Description
The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. A graph is an interval graph if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,c)$ is an edge and $(a,b)$ is not an edge. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001335
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
Mp00157: Graphs —connected complement⟶ Graphs
St001335: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => ([],1)
 => 0
[1,2] => ([],2)
 => ([],2)
 => ([],2)
 => 0
[2,1] => ([(0,1)],2)
 => ([],1)
 => ([],1)
 => 0
[1,2,3] => ([],3)
 => ([],3)
 => ([],3)
 => 0
[1,3,2] => ([(1,2)],3)
 => ([],2)
 => ([],2)
 => 0
[2,1,3] => ([(1,2)],3)
 => ([],2)
 => ([],2)
 => 0
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => ([],1)
 => 0
[1,2,3,4] => ([],4)
 => ([],4)
 => ([],4)
 => 0
[1,2,4,3] => ([(2,3)],4)
 => ([],3)
 => ([],3)
 => 0
[1,3,2,4] => ([(2,3)],4)
 => ([],3)
 => ([],3)
 => 0
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ([],2)
 => 0
[2,1,3,4] => ([(2,3)],4)
 => ([],3)
 => ([],3)
 => 0
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([],2)
 => ([],2)
 => 0
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ([],2)
 => 0
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => ([],1)
 => 0
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => 0
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => 0
[1,2,3,4,5] => ([],5)
 => ([],5)
 => ([],5)
 => 0
[1,2,3,5,4] => ([(3,4)],5)
 => ([],4)
 => ([],4)
 => 0
[1,2,4,3,5] => ([(3,4)],5)
 => ([],4)
 => ([],4)
 => 0
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ([],3)
 => 0
[1,3,2,4,5] => ([(3,4)],5)
 => ([],4)
 => ([],4)
 => 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => ([],3)
 => 0
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ([],3)
 => 0
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => ([],2)
 => 0
Description
The cardinality of a minimal cycle-isolating set of a graph. Let $\mathcal F$ be a set of graphs. A set of vertices $S$ is $\mathcal F$-isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of $S$ does not contain any graph in $\mathcal F$. This statistic returns the cardinality of the smallest isolating set when $\mathcal F$ contains all cycles.
Matching statistic: St000455
(load all 2 compositions to match this statistic)
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => ? = 0
[1,2] => ([],2)
 => ([],2)
 => ? = 0
[2,1] => ([(0,1)],2)
 => ([],1)
 => ? = 0
[1,2,3] => ([],3)
 => ([],3)
 => ? = 0
[1,3,2] => ([(1,2)],3)
 => ([],2)
 => ? = 0
[2,1,3] => ([(1,2)],3)
 => ([],2)
 => ? = 0
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => ? = 0
[1,2,3,4] => ([],4)
 => ([],4)
 => ? = 0
[1,2,4,3] => ([(2,3)],4)
 => ([],3)
 => ? = 0
[1,3,2,4] => ([(2,3)],4)
 => ([],3)
 => ? = 0
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ? = 0
[2,1,3,4] => ([(2,3)],4)
 => ([],3)
 => ? = 0
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([],2)
 => ? = 0
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ? = 0
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => ? = 0
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ? = 0
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ? = 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ? = 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ? = 0
[1,2,3,4,5] => ([],5)
 => ([],5)
 => ? = 0
[1,2,3,5,4] => ([(3,4)],5)
 => ([],4)
 => ? = 0
[1,2,4,3,5] => ([(3,4)],5)
 => ([],4)
 => ? = 0
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ? = 0
[1,3,2,4,5] => ([(3,4)],5)
 => ([],4)
 => ? = 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => ? = 0
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0
[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)
 => 0
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ? = 0
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 0
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => ? = 0
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 0
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 0
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 0
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 0
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 0
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 0
[2,1,3,4,5] => ([(3,4)],5)
 => ([],4)
 => ? = 0
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => ? = 0
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([],3)
 => ? = 0
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 0
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 0
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 0
[2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 0
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0
[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
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => 1
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 0
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 0
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 0
[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
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 0
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 0
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 0
[3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 0
[3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => 1
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 1
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ? = 0
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 0
[3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 0
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
[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
[3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => ? = 0
[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 0
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ? = 0
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? = 0
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? = 0
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? = 0
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ? = 0
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? = 0
[4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 1
[4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 0
Description
The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.
Matching statistic: St001330
(load all 2 compositions to match this statistic)
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 50%
Values
[1] => [1,0]
 => [2,1] => ([(0,1)],2)
 => 2 = 0 + 2
[1,2] => [1,0,1,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 2 = 0 + 2
[2,1] => [1,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 2 = 0 + 2
[1,2,3] => [1,0,1,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[1,3,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 2 = 0 + 2
[2,1,3] => [1,1,0,0,1,0]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 2 = 0 + 2
[2,3,1] => [1,1,0,1,0,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[3,1,2] => [1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[3,2,1] => [1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2 = 0 + 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 0 + 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2 = 0 + 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2 = 0 + 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2 = 0 + 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 0 + 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2 = 0 + 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2 = 0 + 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? = 0 + 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? = 0 + 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 0 + 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 2 = 0 + 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 2 = 0 + 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 0 + 2
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 0 + 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 2 = 0 + 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 0 + 2
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 0 + 2
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 2 = 0 + 2
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 2 = 0 + 2
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 0 + 2
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 0 + 2
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 0 + 2
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 2 = 0 + 2
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
 => [2,5,4,1,6,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 1 + 2
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 2 = 0 + 2
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
 => [2,5,4,1,6,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 1 + 2
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => [2,5,4,1,6,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 0 + 2
[3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 0 + 2
[3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 0 + 2
[3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => [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)
 => ? = 0 + 2
[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [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)
 => ? = 0 + 2
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 0 + 2
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 0 + 2
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 0 + 2
Description
The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000264
(load all 5 compositions to match this statistic)
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
Mp00157: Graphs —connected complement⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Values
[1] => ([],1)
 => ([],1)
 => ([],1)
 => ? = 0 + 2
[1,2] => ([],2)
 => ([],2)
 => ([],2)
 => ? = 0 + 2
[2,1] => ([(0,1)],2)
 => ([],1)
 => ([],1)
 => ? = 0 + 2
[1,2,3] => ([],3)
 => ([],3)
 => ([],3)
 => ? = 0 + 2
[1,3,2] => ([(1,2)],3)
 => ([],2)
 => ([],2)
 => ? = 0 + 2
[2,1,3] => ([(1,2)],3)
 => ([],2)
 => ([],2)
 => ? = 0 + 2
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ? = 0 + 2
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ? = 0 + 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => ([],1)
 => ? = 0 + 2
[1,2,3,4] => ([],4)
 => ([],4)
 => ([],4)
 => ? = 0 + 2
[1,2,4,3] => ([(2,3)],4)
 => ([],3)
 => ([],3)
 => ? = 0 + 2
[1,3,2,4] => ([(2,3)],4)
 => ([],3)
 => ([],3)
 => ? = 0 + 2
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 0 + 2
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 0 + 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ([],2)
 => ? = 0 + 2
[2,1,3,4] => ([(2,3)],4)
 => ([],3)
 => ([],3)
 => ? = 0 + 2
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([],2)
 => ([],2)
 => ? = 0 + 2
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 0 + 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ? = 0 + 2
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ? = 0 + 2
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 0 + 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ([],2)
 => ? = 0 + 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ? = 0 + 2
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => ([],1)
 => ? = 0 + 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ? = 0 + 2
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ? = 0 + 2
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ? = 0 + 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ? = 0 + 2
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ? = 0 + 2
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ? = 0 + 2
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ? = 0 + 2
[1,2,3,4,5] => ([],5)
 => ([],5)
 => ([],5)
 => ? = 0 + 2
[1,2,3,5,4] => ([(3,4)],5)
 => ([],4)
 => ([],4)
 => ? = 0 + 2
[1,2,4,3,5] => ([(3,4)],5)
 => ([],4)
 => ([],4)
 => ? = 0 + 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 0 + 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 0 + 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ([],3)
 => ? = 0 + 2
[1,3,2,4,5] => ([(3,4)],5)
 => ([],4)
 => ([],4)
 => ? = 0 + 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => ([],3)
 => ? = 0 + 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 0 + 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 2
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 0 + 2
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 0 + 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ([],3)
 => ? = 0 + 2
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 0 + 2
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => ([],2)
 => ? = 0 + 2
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ([],2)
 => ? = 0 + 2
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 2
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 0 + 2
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 0 + 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)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[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)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[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)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[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)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[1,3,6,2,5,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[1,3,6,4,2,5] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[1,4,2,6,5,3] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[1,4,3,6,2,5] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[1,5,3,2,6,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[2,1,4,6,3,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[2,1,5,3,6,4] => ([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[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)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[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)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[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)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[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)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[2,4,6,3,1,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[2,5,1,4,3,6] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[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)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[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)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[2,5,3,1,4,6] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[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)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[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)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[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)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[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)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[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)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[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)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[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)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[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)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[2,6,3,4,1,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[2,6,4,1,3,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[2,6,4,3,1,5] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[3,1,4,2,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[3,1,4,6,5,2] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[3,1,5,4,2,6] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[3,1,5,4,6,2] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
Description
The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.
Matching statistic: St001604
Mapping
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 50%
Values
[1] => [1]
 => []
 => ?
 => ? = 0
[1,2] => [1,1]
 => [1]
 => []
 => ? = 0
[2,1] => [2]
 => []
 => ?
 => ? = 0
[1,2,3] => [1,1,1]
 => [1,1]
 => [1]
 => ? = 0
[1,3,2] => [2,1]
 => [1]
 => []
 => ? = 0
[2,1,3] => [2,1]
 => [1]
 => []
 => ? = 0
[2,3,1] => [3]
 => []
 => ?
 => ? = 0
[3,1,2] => [3]
 => []
 => ?
 => ? = 0
[3,2,1] => [2,1]
 => [1]
 => []
 => ? = 0
[1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => ? = 0
[1,2,4,3] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[1,3,2,4] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[1,3,4,2] => [3,1]
 => [1]
 => []
 => ? = 0
[1,4,2,3] => [3,1]
 => [1]
 => []
 => ? = 0
[1,4,3,2] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[2,1,3,4] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[2,1,4,3] => [2,2]
 => [2]
 => []
 => ? = 0
[2,3,1,4] => [3,1]
 => [1]
 => []
 => ? = 0
[2,3,4,1] => [4]
 => []
 => ?
 => ? = 0
[2,4,1,3] => [4]
 => []
 => ?
 => ? = 1
[2,4,3,1] => [3,1]
 => [1]
 => []
 => ? = 0
[3,1,2,4] => [3,1]
 => [1]
 => []
 => ? = 0
[3,1,4,2] => [4]
 => []
 => ?
 => ? = 1
[3,2,1,4] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[3,2,4,1] => [3,1]
 => [1]
 => []
 => ? = 0
[3,4,1,2] => [2,2]
 => [2]
 => []
 => ? = 0
[3,4,2,1] => [4]
 => []
 => ?
 => ? = 0
[4,1,2,3] => [4]
 => []
 => ?
 => ? = 0
[4,1,3,2] => [3,1]
 => [1]
 => []
 => ? = 0
[4,2,1,3] => [3,1]
 => [1]
 => []
 => ? = 0
[4,2,3,1] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 0
[4,3,1,2] => [4]
 => []
 => ?
 => ? = 0
[4,3,2,1] => [2,2]
 => [2]
 => []
 => ? = 0
[1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => ? = 0
[1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => ? = 0
[1,2,4,5,3] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 0
[1,2,5,3,4] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 0
[1,2,5,4,3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => ? = 0
[1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => ? = 0
[1,3,2,5,4] => [2,2,1]
 => [2,1]
 => [1]
 => ? = 0
[1,3,4,2,5] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 0
[1,3,4,5,2] => [4,1]
 => [1]
 => []
 => ? = 0
[1,3,5,2,4] => [4,1]
 => [1]
 => []
 => ? = 1
[1,3,5,4,2] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 0
[1,4,2,3,5] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 0
[1,4,2,5,3] => [4,1]
 => [1]
 => []
 => ? = 1
[1,4,3,2,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => ? = 0
[1,4,3,5,2] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 0
[1,4,5,2,3] => [2,2,1]
 => [2,1]
 => [1]
 => ? = 0
[1,4,5,3,2] => [4,1]
 => [1]
 => []
 => ? = 0
[1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,2,3,4,6,5] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,3,5,4,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,3,6,5,4] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,3,5,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,5,4,3,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,6,4,5,3] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,3,2,4,5,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,4,3,2,5,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,5,3,4,2,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,6,3,4,5,2] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[2,1,3,4,5,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[3,2,1,4,5,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[4,2,3,1,5,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[5,2,3,4,1,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[6,2,3,4,5,1] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,3,4,5,7,6] => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,2,3,4,6,5,7] => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,2,3,4,7,6,5] => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,2,3,5,4,6,7] => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,2,3,5,4,7,6] => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,2,3,5,7,6,4] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,3,6,5,4,7] => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,2,3,6,5,7,4] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,3,6,7,4,5] => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,2,3,7,4,6,5] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,3,7,5,4,6] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,3,7,5,6,4] => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,2,3,7,6,5,4] => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,3,5,6,7] => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,2,4,3,5,7,6] => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,3,6,5,7] => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,3,7,6,5] => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,6,5,3,7] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,7,5,6,3] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,5,4,3,6,7] => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,2,5,4,3,7,6] => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,2,5,4,6,3,7] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,5,4,7,6,3] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,5,6,3,4,7] => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,2,5,7,3,6,4] => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,2,6,3,5,4,7] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,6,4,3,5,7] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,6,4,5,3,7] => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,2,6,4,5,7,3] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,6,4,7,3,5] => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,2,6,5,4,3,7] => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,2,6,7,5,3,4] => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,2,7,3,5,6,4] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001001
Mapping
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 50%
Values
[1] => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 0
[1,2] => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 0
[2,1] => [2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 0
[1,2,3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 0
[1,3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[2,1,3] => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[2,3,1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 0
[3,1,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 0
[3,2,1] => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 0
[1,2,4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 0
[1,3,2,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 0
[1,3,4,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 0
[1,4,2,3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 0
[1,4,3,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 0
[2,1,3,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 0
[2,1,4,3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 0
[2,3,1,4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 0
[2,3,4,1] => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 0
[2,4,1,3] => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 1
[2,4,3,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 0
[3,1,2,4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 0
[3,1,4,2] => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 1
[3,2,1,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 0
[3,2,4,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 0
[3,4,1,2] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 0
[3,4,2,1] => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 0
[4,1,2,3] => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 0
[4,1,3,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 0
[4,2,1,3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 0
[4,2,3,1] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 0
[4,3,1,2] => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 0
[4,3,2,1] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 0
[1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 0
[1,2,3,5,4] => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 0
[1,2,4,3,5] => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 0
[1,2,4,5,3] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
[1,2,5,3,4] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
[1,2,5,4,3] => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 0
[1,3,2,4,5] => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 0
[1,3,2,5,4] => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 0
[1,3,4,2,5] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
[1,3,4,5,2] => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 0
[1,3,5,2,4] => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 1
[1,3,5,4,2] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
[1,4,2,3,5] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
[1,4,2,5,3] => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 1
[1,4,3,2,5] => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 0
[1,4,3,5,2] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
[1,4,5,2,3] => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 0
[1,4,5,3,2] => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 0
[1,5,2,3,4] => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 0
[1,5,2,4,3] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
[1,5,3,2,4] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
[1,5,3,4,2] => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 0
[1,5,4,2,3] => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 0
[1,5,4,3,2] => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 0
[2,1,3,4,5] => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 0
[2,1,3,5,4] => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 0
[2,1,4,3,5] => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 0
[2,1,4,5,3] => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 0
[2,1,5,3,4] => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 0
[2,1,5,4,3] => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 0
[2,3,1,4,5] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
[2,3,1,5,4] => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 0
[2,3,4,1,5] => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 0
[2,3,4,5,1] => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 0
[2,3,5,1,4] => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 1
[2,3,5,4,1] => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 0
[2,4,1,3,5] => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 1
[2,4,3,1,5] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
[2,4,3,5,1] => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 0
[2,4,5,1,3] => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 0
[2,4,5,3,1] => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 0
[2,5,1,3,4] => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 1
[2,5,1,4,3] => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 1
[2,5,3,1,4] => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 1
[2,5,3,4,1] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
[2,5,4,1,3] => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 0
[2,5,4,3,1] => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 0
[3,1,2,4,5] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
[3,1,2,5,4] => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 0
[3,1,4,2,5] => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 1
[3,1,4,5,2] => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 1
[3,1,5,4,2] => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 1
[3,2,1,4,5] => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 0
[3,2,4,5,1] => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 0
[3,2,5,1,4] => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 1
[3,4,2,1,5] => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 0
[3,4,2,5,1] => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 0
[3,4,5,1,2] => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 0
[3,5,2,1,4] => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 0
[3,5,2,4,1] => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 0
[3,5,4,2,1] => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 0
[4,1,2,3,5] => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 0
[4,1,2,5,3] => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 1
[4,1,3,5,2] => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 1
[4,1,5,3,2] => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 0
[4,2,1,5,3] => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 1
[4,2,3,1,5] => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 0
Description
The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.
The following 106 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001964The interval resolution global dimension of a poset. St000068The number of minimal elements in a poset. St001846The number of elements which do not have a complement in the lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001571The Cartan determinant of the integer partition. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000256The number of parts from which one can substract 2 and still get an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001845The number of join irreducibles minus the rank of a lattice. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001877Number of indecomposable injective modules with projective dimension 2. St001875The number of simple modules with projective dimension at most 1. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000914The sum of the values of the Möbius function of a poset. St001490The number of connected components of a skew partition. St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000403The Szeged index minus the Wiener index of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001119The length of a shortest maximal path in a graph. St001271The competition number of a graph. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001323The independence gap of a graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001395The number of strictly unfriendly partitions of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001797The number of overfull subgraphs of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St001316The domatic number of a graph. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000552The number of cut vertices of a graph. St000273The domination number of a graph. St000544The cop number of a graph. St000553The number of blocks of a graph. St000916The packing number of a graph. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001829The common independence number of a graph. St000379The number of Hamiltonian cycles in a graph. St000699The toughness times the least common multiple of 1,. St001281The normalized isoperimetric number of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St000322The skewness of a graph. St000095The number of triangles of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001578The minimal number of edges to add or remove to make a graph a line graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001871The number of triconnected components of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001765The number of connected components of the friends and strangers graph. St001570The minimal number of edges to add to make a graph Hamiltonian.