- FindStat

Your data matches 9 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000454
(load all 2 compositions to match this statistic)

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => ([],1)
 => ([],1)
 => 0
[1,0,1,0]
 => [1,2] => ([],2)
 => ([],2)
 => 0
[1,1,0,0]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,0,1,0,1,0]
 => [1,2,3] => ([],3)
 => ([],3)
 => 0
[1,0,1,1,0,0]
 => [1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 1
[1,1,0,0,1,0]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => 1
[1,1,1,0,0,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => ([],4)
 => ([],4)
 => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => ([],5)
 => ([],5)
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ([(3,4)],5)
 => ([(3,4)],5)
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => ([(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
[1,0,1,1,1,1,0,0,0,0]
 => [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
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,1,0,1,0,0,0,1,0]
 => [4,2,3,1,5] => ([(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
[1,1,1,0,1,0,1,0,0,0]
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,1,0,1,1,0,0,0,0]
 => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => ([(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
[1,1,1,1,0,0,1,0,0,0]
 => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,1,1,0,1,0,0,0,0]
 => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,1,1,1,0,0,0,0,0]
 => [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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => ([],6)
 => ([],6)
 => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => ([(4,5)],6)
 => ([(4,5)],6)
 => 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => ([(4,5)],6)
 => ([(4,5)],6)
 => 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => 2
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,5,6] => ([(4,5)],6)
 => ([(4,5)],6)
 => 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
 => ([(2,5),(3,4)],6)
 => 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => 2
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,6,4,5,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,6,5,4,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,3,2,4,5,6] => ([(4,5)],6)
 => ([(4,5)],6)
 => 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
 => ([(2,5),(3,4)],6)
 => 1

Description

The largest eigenvalue of a graph if it is integral. If a graph is

$d$ -regular, then its largest eigenvalue equals

$d$ . One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.

Matching statistic: St001270

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
St001270: Graphs ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 86%

Values

[1,0]
 => [1] => ([],1)
 => ([],1)
 => 0
[1,0,1,0]
 => [1,2] => ([],2)
 => ([],2)
 => 0
[1,1,0,0]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,0,1,0,1,0]
 => [1,2,3] => ([],3)
 => ([],3)
 => 0
[1,0,1,1,0,0]
 => [1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 1
[1,1,0,0,1,0]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => 1
[1,1,1,0,0,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => ([],4)
 => ([],4)
 => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => ([],5)
 => ([],5)
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ([(3,4)],5)
 => ([(3,4)],5)
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => ([(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
[1,0,1,1,1,1,0,0,0,0]
 => [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
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,1,0,1,0,0,0,1,0]
 => [4,2,3,1,5] => ([(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
[1,1,1,0,1,0,1,0,0,0]
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,1,0,1,1,0,0,0,0]
 => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => ([(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
[1,1,1,1,0,0,1,0,0,0]
 => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,1,1,0,1,0,0,0,0]
 => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,1,1,1,0,0,0,0,0]
 => [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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => ([],6)
 => ([],6)
 => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => ([(4,5)],6)
 => ([(4,5)],6)
 => 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => ([(4,5)],6)
 => ([(4,5)],6)
 => 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => 2
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,5,6] => ([(4,5)],6)
 => ([(4,5)],6)
 => 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
 => ([(2,5),(3,4)],6)
 => 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => 2
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,6,4,5,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,6,5,4,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,3,2,4,5,6] => ([(4,5)],6)
 => ([(4,5)],6)
 => 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
 => ([(2,5),(3,4)],6)
 => 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,7,3,5,6,4,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 5
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,7,3,6,5,4,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 5
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,7,4,3,6,5,2] => ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 5
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,7,4,5,3,6,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 5
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,7,4,5,6,3,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 5
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,7,4,6,5,3,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 5
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,7,5,4,3,6,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 5
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,7,5,4,6,3,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 5
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,7,6,4,5,3,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 5
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 5
[1,1,1,0,1,1,0,1,0,0,0,0,1,0]
 => [6,2,4,5,3,1,7] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 5
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [7,2,4,6,5,3,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[1,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => [6,2,5,4,3,1,7] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 5
[1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => [7,2,5,4,6,3,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [7,2,6,4,5,3,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [7,2,6,5,4,3,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [6,3,2,5,4,1,7] => ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 5
[1,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => [7,3,2,5,6,4,1] => ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [7,3,2,6,5,4,1] => ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[1,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => [6,3,4,2,5,1,7] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 5
[1,1,1,1,0,1,0,0,1,1,0,0,0,0]
 => [7,3,4,2,6,5,1] => ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [6,3,4,5,2,1,7] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 5
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [7,3,4,6,5,2,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [6,3,5,4,2,1,7] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 5
[1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [7,3,5,4,2,6,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [7,3,5,4,6,2,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [7,3,6,4,5,2,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [7,3,6,5,4,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [6,4,3,2,5,1,7] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 5
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => [7,4,3,2,6,5,1] => ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [6,4,3,5,2,1,7] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 5
[1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [7,4,3,5,2,6,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [7,4,3,5,6,2,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [7,4,3,6,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [6,5,3,4,2,1,7] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 5
[1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [7,5,3,4,2,6,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [7,5,3,4,6,2,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [7,6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [7,6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [6,5,4,3,2,1,7] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 5
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [7,5,4,3,2,6,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [7,5,4,3,6,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [7,6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [7,6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6

Description

The bandwidth of a graph. The bandwidth of a graph is the smallest number

$k$ such that the vertices of the graph can be ordered as

$v_1,\dots,v_n$ with

$k \cdot d(v_i,v_j) \geq |i-j|$ . We adopt the convention that the singleton graph has bandwidth

$0$ , consistent with the bandwith of the complete graph on

$n$ vertices having bandwidth

$n-1$ , but in contrast to any path graph on more than one vertex having bandwidth

$1$ . The bandwidth of a disconnected graph is the maximum of the bandwidths of the connected components.

Matching statistic: St001645
(load all 5 compositions to match this statistic)

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => ([],1)
 => ([],1)
 => 1 = 0 + 1
[1,0,1,0]
 => [1,2] => ([],2)
 => ([],2)
 => ? = 0 + 1
[1,1,0,0]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,2,3] => ([],3)
 => ([],3)
 => ? = 0 + 1
[1,0,1,1,0,0]
 => [1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 1 + 1
[1,1,0,0,1,0]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 1 + 1
[1,1,1,0,0,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => ([],4)
 => ([],4)
 => ? = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => ? = 1 + 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => ? = 1 + 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => ? = 1 + 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ? = 1 + 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => ([],5)
 => ([],5)
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ([(3,4)],5)
 => ([(3,4)],5)
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => ([(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 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [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 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [4,2,3,1,5] => ([(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 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => ([(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 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => ([],6)
 => ([],6)
 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => ([(4,5)],6)
 => ([(4,5)],6)
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => ([(4,5)],6)
 => ([(4,5)],6)
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,5,6] => ([(4,5)],6)
 => ([(4,5)],6)
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
 => ([(2,5),(3,4)],6)
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,6,4,5,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,6,5,4,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,3,2,4,5,6] => ([(4,5)],6)
 => ([(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
 => ([(2,5),(3,4)],6)
 => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
 => ([(2,5),(3,4)],6)
 => ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,6,5,4] => ([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,4,3,2,5,6] => ([(3,4),(3,5),(4,5)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,4,3,2,6,5] => ([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,5,3,4,2,6] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,6,3,4,5,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,6,3,5,4,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,5,4,3,2,6] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [6,2,4,5,3,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [6,2,5,4,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [6,3,2,5,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [6,3,4,2,5,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [6,4,3,2,5,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [7,2,4,6,5,3,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => [7,2,5,4,6,3,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [7,2,6,4,5,3,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [7,2,6,5,4,3,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => [7,3,2,5,6,4,1] => ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [7,3,2,6,5,4,1] => ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,1,1,1,0,1,0,0,1,1,0,0,0,0]
 => [7,3,4,2,6,5,1] => ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [7,3,4,6,5,2,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [7,3,5,4,2,6,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [7,3,5,4,6,2,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [7,3,6,4,5,2,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [7,3,6,5,4,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => [7,4,3,2,6,5,1] => ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [7,4,3,5,2,6,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [7,4,3,5,6,2,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [7,4,3,6,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [7,5,3,4,2,6,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [7,5,3,4,6,2,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [7,6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [7,6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [7,5,4,3,2,6,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [7,5,4,3,6,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [7,6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [7,6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1

Description

The pebbling number of a connected graph.

Matching statistic: St001330

Mapping

Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,0]
 => [1] => ([],1)
 => 1 = 0 + 1
[1,0,1,0]
 => [1,1,0,0]
 => [1,2] => ([],2)
 => 1 = 0 + 1
[1,1,0,0]
 => [1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => ([],3)
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [2,1,3] => ([(1,2)],3)
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([],4)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [2,1,3,4] => ([(2,3)],4)
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 + 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([],5)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => ([(3,4)],5)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [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)
 => 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => ([],6)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [2,3,4,1,5,6] => ([(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [3,4,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [3,4,5,2,1,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [6,5,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,1,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 2 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [3,4,2,1,5,6] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [4,5,3,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [5,4,1,2,3,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [6,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [6,5,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [4,5,3,2,1,6] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [6,5,2,3,1,4] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [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)
 => ? = 4 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [2,1,3,4,5,6] => ([(4,5)],6)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,1,6] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [4,3,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [3,2,4,1,5,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [4,3,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [4,3,5,2,1,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [6,5,2,1,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [5,4,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [5,3,4,6,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [4,5,1,2,3,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 2 + 1
[1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
[1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [4,3,2,5,6,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [4,3,2,5,1,6] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [6,3,2,1,4,5] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [5,4,3,6,2,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [4,3,2,1,5,6] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 3 + 1
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,1,6] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [6,5,4,3,2,1,7] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 6 = 5 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1

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: St001118
(load all 2 compositions to match this statistic)

Mapping

Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001118: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 29%

Values

[1,0]
 => ([],1)
 => ([],1)
 => ? = 0 - 1
[1,0,1,0]
 => ([(0,1)],2)
 => ([],2)
 => ? = 0 - 1
[1,1,0,0]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1 - 1
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 0 - 1
[1,0,1,1,0,0]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1 - 1
[1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1 - 1
[1,1,1,0,0,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 0 - 1
[1,0,1,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1 - 1
[1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1 - 1
[1,0,1,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1 - 1
[1,1,0,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1 - 1
[1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 0 - 1
[1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => 1 = 2 - 1
[1,1,1,0,1,0,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => 2 = 3 - 1
[1,1,1,0,1,0,1,0,0,0]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 3 - 1
[1,1,1,0,1,1,0,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? = 4 - 1
[1,1,1,1,0,0,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => 2 = 3 - 1
[1,1,1,1,0,0,1,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? = 4 - 1
[1,1,1,1,0,1,0,0,0,0]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 4 - 1
[1,1,1,1,1,0,0,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
 => ? = 4 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 0 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => ([(5,6)],7)
 => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(5,6)],7)
 => 1 = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(4,7),(5,6),(6,7)],8)
 => ? = 3 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(4,7),(5,6),(6,7)],8)
 => ? = 3 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 1 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => ([(5,6)],7)
 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 2 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(5,6)],7)
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(5,6)],7)
 => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(4,7),(5,6),(6,7)],8)
 => ? = 3 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 3 - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ? = 4 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(4,7),(5,6),(6,7)],8)
 => ? = 3 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ? = 4 - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 4 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
 => ? = 4 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 1 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 1 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => ([(5,6)],7)
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 1 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 1 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(5,6)],7)
 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(4,7),(5,6),(6,7)],8)
 => ? = 3 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(4,7),(5,6),(6,7)],8)
 => ? = 3 - 1
[1,1,0,1,0,0,1,1,1,0,0,0]
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => ([(5,6)],7)
 => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0,1,0]
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => ([(5,6)],7)
 => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,1,0,0]
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => ([(5,6)],7)
 => 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => ([(5,6)],7)
 => 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,1,0,0]
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => ([(5,6)],7)
 => 1 = 2 - 1

Description

The acyclic chromatic index of a graph. An acyclic edge coloring of a graph is a proper colouring of the edges of a graph such that the union of the edges colored with any two given colours is a forest. The smallest number of colours such that such a colouring exists is the acyclic chromatic index.

Matching statistic: St000455
(load all 2 compositions to match this statistic)

Mapping

Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 14%

Values

[1,0]
 => ([],1)
 => ([],1)
 => ? = 0 - 2
[1,0,1,0]
 => ([(0,1)],2)
 => ([],2)
 => ? = 0 - 2
[1,1,0,0]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1 - 2
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 0 - 2
[1,0,1,1,0,0]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1 - 2
[1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1 - 2
[1,1,1,0,0,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 0 = 2 - 2
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 0 - 2
[1,0,1,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1 - 2
[1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1 - 2
[1,0,1,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0 = 2 - 2
[1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1 - 2
[1,1,0,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1 - 2
[1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => 0 = 2 - 2
[1,1,1,0,1,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ? = 3 - 2
[1,1,1,1,0,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ? = 3 - 2
[1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 0 - 2
[1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1 - 2
[1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1 - 2
[1,0,1,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => 0 = 2 - 2
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1 - 2
[1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1 - 2
[1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0 = 2 - 2
[1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ? = 3 - 2
[1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ? = 3 - 2
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1 - 2
[1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1 - 2
[1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1 - 2
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => 0 = 2 - 2
[1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 2 - 2
[1,1,1,0,0,0,1,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => 0 = 2 - 2
[1,1,1,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => 0 = 2 - 2
[1,1,1,0,1,0,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ? = 3 - 2
[1,1,1,0,1,0,1,0,0,0]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 3 - 2
[1,1,1,0,1,1,0,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? = 4 - 2
[1,1,1,1,0,0,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ? = 3 - 2
[1,1,1,1,0,0,1,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? = 4 - 2
[1,1,1,1,0,1,0,0,0,0]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 4 - 2
[1,1,1,1,1,0,0,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
 => ? = 4 - 2
[1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 0 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 1 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 1 - 2
[1,0,1,0,1,0,1,1,1,0,0,0]
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => ([(5,6)],7)
 => 0 = 2 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 1 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 1 - 2
[1,0,1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(5,6)],7)
 => 0 = 2 - 2
[1,0,1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(4,7),(5,6),(6,7)],8)
 => ? = 3 - 2
[1,0,1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(4,7),(5,6),(6,7)],8)
 => ? = 3 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 1 - 2
[1,0,1,1,0,0,1,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 1 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 1 - 2
[1,0,1,1,0,0,1,1,1,0,0,0]
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => ([(5,6)],7)
 => 0 = 2 - 2
[1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 2 - 2
[1,0,1,1,1,0,0,0,1,0,1,0]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(5,6)],7)
 => 0 = 2 - 2
[1,0,1,1,1,0,0,0,1,1,0,0]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(5,6)],7)
 => 0 = 2 - 2
[1,0,1,1,1,0,1,0,0,0,1,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(4,7),(5,6),(6,7)],8)
 => ? = 3 - 2
[1,0,1,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 3 - 2
[1,0,1,1,1,0,1,1,0,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ? = 4 - 2
[1,0,1,1,1,1,0,0,0,0,1,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(4,7),(5,6),(6,7)],8)
 => ? = 3 - 2
[1,0,1,1,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ? = 4 - 2
[1,0,1,1,1,1,0,1,0,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 4 - 2
[1,0,1,1,1,1,1,0,0,0,0,0]
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
 => ? = 4 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 1 - 2
[1,1,0,0,1,0,1,1,1,0,0,0]
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => ([(5,6)],7)
 => 0 = 2 - 2
[1,1,0,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(5,6)],7)
 => 0 = 2 - 2
[1,1,0,1,0,0,1,1,1,0,0,0]
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => ([(5,6)],7)
 => 0 = 2 - 2
[1,1,1,0,0,0,1,0,1,0,1,0]
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => ([(5,6)],7)
 => 0 = 2 - 2
[1,1,1,0,0,0,1,0,1,1,0,0]
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => ([(5,6)],7)
 => 0 = 2 - 2
[1,1,1,0,0,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => ([(5,6)],7)
 => 0 = 2 - 2
[1,1,1,0,0,0,1,1,0,1,0,0]
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => ([(5,6)],7)
 => 0 = 2 - 2

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: St000028

Mapping

Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00116: Perfect matchings —Kasraoui-Zeng⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St000028: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 86%

Values

[1,0]
 => [(1,2)]
 => [(1,2)]
 => [2,1] => 1 = 0 + 1
[1,0,1,0]
 => [(1,2),(3,4)]
 => [(1,2),(3,4)]
 => [2,1,4,3] => 1 = 0 + 1
[1,1,0,0]
 => [(1,4),(2,3)]
 => [(1,3),(2,4)]
 => [3,4,1,2] => 2 = 1 + 1
[1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 1 = 0 + 1
[1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => 2 = 1 + 1
[1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => 2 = 1 + 1
[1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => [(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [(1,2),(3,4),(5,7),(6,8)]
 => [2,1,4,3,7,8,5,6] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => [(1,2),(3,5),(4,6),(7,8)]
 => [2,1,5,6,3,4,8,7] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => [(1,2),(3,6),(4,7),(5,8)]
 => [2,1,6,7,8,3,4,5] => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => [(1,3),(2,4),(5,6),(7,8)]
 => [3,4,1,2,6,5,8,7] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [(1,3),(2,4),(5,7),(6,8)]
 => [3,4,1,2,7,8,5,6] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [(1,4),(2,5),(3,6),(7,8)]
 => [4,5,6,1,2,3,8,7] => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [(1,4),(2,6),(3,7),(5,8)]
 => [4,6,7,1,8,2,3,5] => 4 = 3 + 1
[1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => [(1,5),(2,6),(3,7),(4,8)]
 => [5,6,7,8,1,2,3,4] => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => [(1,2),(3,4),(5,6),(7,9),(8,10)]
 => [2,1,4,3,6,5,9,10,7,8] => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,10)]
 => [(1,2),(3,4),(5,7),(6,8),(9,10)]
 => [2,1,4,3,7,8,5,6,10,9] => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8)]
 => [(1,2),(3,4),(5,8),(6,9),(7,10)]
 => [2,1,4,3,8,9,10,5,6,7] => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,10)]
 => [(1,2),(3,5),(4,6),(7,8),(9,10)]
 => [2,1,5,6,3,4,8,7,10,9] => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9)]
 => [(1,2),(3,5),(4,6),(7,9),(8,10)]
 => [2,1,5,6,3,4,9,10,7,8] => ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [(1,2),(3,8),(4,7),(5,6),(9,10)]
 => [(1,2),(3,6),(4,7),(5,8),(9,10)]
 => [2,1,6,7,8,3,4,5,10,9] => ? = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8)]
 => [(1,2),(3,6),(4,8),(5,9),(7,10)]
 => [2,1,6,8,9,3,10,4,5,7] => ? = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => [(1,2),(3,7),(4,8),(5,9),(6,10)]
 => [2,1,7,8,9,10,3,4,5,6] => ? = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10)]
 => [(1,3),(2,4),(5,6),(7,8),(9,10)]
 => [3,4,1,2,6,5,8,7,10,9] => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9)]
 => [(1,3),(2,4),(5,6),(7,9),(8,10)]
 => [3,4,1,2,6,5,9,10,7,8] => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10)]
 => [(1,3),(2,4),(5,7),(6,8),(9,10)]
 => [3,4,1,2,7,8,5,6,10,9] => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8)]
 => [(1,3),(2,4),(5,8),(6,9),(7,10)]
 => [3,4,1,2,8,9,10,5,6,7] => ? = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => [(1,3),(2,5),(4,7),(6,9),(8,10)]
 => [3,5,1,7,2,9,4,10,6,8] => ? = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,10)]
 => [(1,4),(2,5),(3,6),(7,8),(9,10)]
 => [4,5,6,1,2,3,8,7,10,9] => ? = 2 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9)]
 => [(1,4),(2,5),(3,6),(7,9),(8,10)]
 => [4,5,6,1,2,3,9,10,7,8] => ? = 2 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [(1,8),(2,7),(3,4),(5,6),(9,10)]
 => [(1,4),(2,6),(3,7),(5,8),(9,10)]
 => [4,6,7,1,8,2,3,5,10,9] => ? = 3 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [(1,10),(2,9),(3,4),(5,6),(7,8)]
 => [(1,4),(2,6),(3,8),(5,9),(7,10)]
 => [4,6,8,1,9,2,10,3,5,7] => ? = 3 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [(1,10),(2,9),(3,4),(5,8),(6,7)]
 => [(1,4),(2,7),(3,8),(5,9),(6,10)]
 => [4,7,8,1,9,10,2,3,5,6] => ? = 4 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10)]
 => [(1,5),(2,6),(3,7),(4,8),(9,10)]
 => [5,6,7,8,1,2,3,4,10,9] => ? = 3 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [(1,10),(2,9),(3,6),(4,5),(7,8)]
 => [(1,5),(2,6),(3,8),(4,9),(7,10)]
 => [5,6,8,9,1,2,10,3,4,7] => ? = 4 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,5),(6,7)]
 => [(1,5),(2,7),(3,8),(4,9),(6,10)]
 => [5,7,8,9,1,10,2,3,4,6] => ? = 4 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,7),(5,6)]
 => [(1,6),(2,7),(3,8),(4,9),(5,10)]
 => [6,7,8,9,10,1,2,3,4,5] => 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => [2,1,4,3,6,5,8,7,10,9,12,11] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
 => [(1,2),(3,4),(5,6),(7,8),(9,11),(10,12)]
 => [2,1,4,3,6,5,8,7,11,12,9,10] => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
 => [(1,2),(3,4),(5,6),(7,9),(8,10),(11,12)]
 => [2,1,4,3,6,5,9,10,7,8,12,11] => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
 => [(1,2),(3,4),(5,6),(7,10),(8,11),(9,12)]
 => [2,1,4,3,6,5,10,11,12,7,8,9] => ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
 => [(1,2),(3,4),(5,7),(6,8),(9,10),(11,12)]
 => [2,1,4,3,7,8,5,6,10,9,12,11] => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
 => [(1,2),(3,4),(5,7),(6,8),(9,11),(10,12)]
 => [2,1,4,3,7,8,5,6,11,12,9,10] => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)]
 => [(1,2),(3,4),(5,8),(6,9),(7,10),(11,12)]
 => [2,1,4,3,8,9,10,5,6,7,12,11] => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)]
 => [(1,2),(3,4),(5,8),(6,10),(7,11),(9,12)]
 => [2,1,4,3,8,10,11,5,12,6,7,9] => ? = 3 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
 => [(1,2),(3,4),(5,9),(6,10),(7,11),(8,12)]
 => [2,1,4,3,9,10,11,12,5,6,7,8] => ? = 3 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
 => [(1,2),(3,5),(4,6),(7,8),(9,10),(11,12)]
 => [2,1,5,6,3,4,8,7,10,9,12,11] => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
 => [(1,2),(3,5),(4,6),(7,8),(9,11),(10,12)]
 => [2,1,5,6,3,4,8,7,11,12,9,10] => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
 => [(1,2),(3,5),(4,6),(7,9),(8,10),(11,12)]
 => [2,1,5,6,3,4,9,10,7,8,12,11] => ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
 => [(1,2),(3,5),(4,6),(7,10),(8,11),(9,12)]
 => [2,1,5,6,3,4,10,11,12,7,8,9] => ? = 2 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
 => [(1,2),(3,5),(4,7),(6,9),(8,11),(10,12)]
 => [2,1,5,7,3,9,4,11,6,12,8,10] => ? = 2 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [(1,2),(3,8),(4,7),(5,6),(9,10),(11,12)]
 => [(1,2),(3,6),(4,7),(5,8),(9,10),(11,12)]
 => [2,1,6,7,8,3,4,5,10,9,12,11] => ? = 2 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
 => [(1,2),(3,6),(4,7),(5,8),(9,11),(10,12)]
 => [2,1,6,7,8,3,4,5,11,12,9,10] => ? = 2 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8),(11,12)]
 => [(1,2),(3,6),(4,8),(5,9),(7,10),(11,12)]
 => [2,1,6,8,9,3,10,4,5,7,12,11] => ? = 3 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [(1,2),(3,12),(4,11),(5,6),(7,8),(9,10)]
 => [(1,2),(3,6),(4,8),(5,10),(7,11),(9,12)]
 => [2,1,6,8,10,3,11,4,12,5,7,9] => ? = 3 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
 => [(1,2),(3,6),(4,9),(5,10),(7,11),(8,12)]
 => [2,1,6,9,10,3,11,12,4,5,7,8] => ? = 4 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7),(11,12)]
 => [(1,2),(3,7),(4,8),(5,9),(6,10),(11,12)]
 => [2,1,7,8,9,10,3,4,5,6,12,11] => ? = 3 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
 => [(1,2),(3,7),(4,8),(5,10),(6,11),(9,12)]
 => [2,1,7,8,10,11,3,4,12,5,6,9] => ? = 4 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
 => [(1,2),(3,7),(4,9),(5,10),(6,11),(8,12)]
 => [2,1,7,9,10,11,3,12,4,5,6,8] => ? = 4 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
 => [(1,2),(3,8),(4,9),(5,10),(6,11),(7,12)]
 => [2,1,8,9,10,11,12,3,4,5,6,7] => ? = 4 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)]
 => [(1,3),(2,4),(5,6),(7,8),(9,10),(11,12)]
 => [3,4,1,2,6,5,8,7,10,9,12,11] => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)]
 => [(1,3),(2,4),(5,6),(7,8),(9,11),(10,12)]
 => [3,4,1,2,6,5,8,7,11,12,9,10] => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
 => [(1,3),(2,4),(5,6),(7,9),(8,10),(11,12)]
 => [3,4,1,2,6,5,9,10,7,8,12,11] => ? = 1 + 1
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10)]
 => [(1,3),(2,4),(5,6),(7,10),(8,11),(9,12)]
 => [3,4,1,2,6,5,10,11,12,7,8,9] => ? = 2 + 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
 => [(1,3),(2,4),(5,7),(6,8),(9,10),(11,12)]
 => [3,4,1,2,7,8,5,6,10,9,12,11] => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
 => [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12)]
 => [3,4,1,2,7,8,5,6,11,12,9,10] => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
 => [(1,3),(2,4),(5,8),(6,9),(7,10),(11,12)]
 => [3,4,1,2,8,9,10,5,6,7,12,11] => ? = 2 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12)]
 => [7,8,9,10,11,12,1,2,3,4,5,6] => 6 = 5 + 1

Description

The number of stack-sorts needed to sort a permutation. A permutation is (West)

$t$ -stack sortable if it is sortable using

$t$ stacks in series. Let

$W_t(n,k)$ be the number of permutations of size

$n$ with

$k$ descents which are

$t$ -stack sortable. Then the polynomials

$W_{n,t}(x) = \sum_{k=0}^n W_t(n,k)x^k$ are symmetric and unimodal. We have

$W_{n,1}(x) = A_n(x)$ , the Eulerian polynomials. One can show that

$W_{n,1}(x)$ and

$W_{n,2}(x)$ are real-rooted. Precisely the permutations that avoid the pattern

$231$ have statistic at most

$1$ , see [3]. These are counted by

$\frac{1}{n+1}\binom{2n}{n}$ ([[OEIS:A000108]]). Precisely the permutations that avoid the pattern

$2341$ and the barred pattern

$3\bar 5241$ have statistic at most

$2$ , see [4]. These are counted by

$\frac{2(3n)!}{(n+1)!(2n+1)!}$ ([[OEIS:A000139]]).

Matching statistic: St000141

Mapping

Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00116: Perfect matchings —Kasraoui-Zeng⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 86%

Values

[1,0]
 => [(1,2)]
 => [(1,2)]
 => [2,1] => 1 = 0 + 1
[1,0,1,0]
 => [(1,2),(3,4)]
 => [(1,2),(3,4)]
 => [2,1,4,3] => 1 = 0 + 1
[1,1,0,0]
 => [(1,4),(2,3)]
 => [(1,3),(2,4)]
 => [3,4,1,2] => 2 = 1 + 1
[1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 1 = 0 + 1
[1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => 2 = 1 + 1
[1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => 2 = 1 + 1
[1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => [(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [(1,2),(3,4),(5,7),(6,8)]
 => [2,1,4,3,7,8,5,6] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => [(1,2),(3,5),(4,6),(7,8)]
 => [2,1,5,6,3,4,8,7] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => [(1,2),(3,6),(4,7),(5,8)]
 => [2,1,6,7,8,3,4,5] => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => [(1,3),(2,4),(5,6),(7,8)]
 => [3,4,1,2,6,5,8,7] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [(1,3),(2,4),(5,7),(6,8)]
 => [3,4,1,2,7,8,5,6] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [(1,4),(2,5),(3,6),(7,8)]
 => [4,5,6,1,2,3,8,7] => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [(1,4),(2,6),(3,7),(5,8)]
 => [4,6,7,1,8,2,3,5] => 4 = 3 + 1
[1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => [(1,5),(2,6),(3,7),(4,8)]
 => [5,6,7,8,1,2,3,4] => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => [(1,2),(3,4),(5,6),(7,9),(8,10)]
 => [2,1,4,3,6,5,9,10,7,8] => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,10)]
 => [(1,2),(3,4),(5,7),(6,8),(9,10)]
 => [2,1,4,3,7,8,5,6,10,9] => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8)]
 => [(1,2),(3,4),(5,8),(6,9),(7,10)]
 => [2,1,4,3,8,9,10,5,6,7] => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,10)]
 => [(1,2),(3,5),(4,6),(7,8),(9,10)]
 => [2,1,5,6,3,4,8,7,10,9] => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9)]
 => [(1,2),(3,5),(4,6),(7,9),(8,10)]
 => [2,1,5,6,3,4,9,10,7,8] => ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [(1,2),(3,8),(4,7),(5,6),(9,10)]
 => [(1,2),(3,6),(4,7),(5,8),(9,10)]
 => [2,1,6,7,8,3,4,5,10,9] => ? = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8)]
 => [(1,2),(3,6),(4,8),(5,9),(7,10)]
 => [2,1,6,8,9,3,10,4,5,7] => ? = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => [(1,2),(3,7),(4,8),(5,9),(6,10)]
 => [2,1,7,8,9,10,3,4,5,6] => ? = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10)]
 => [(1,3),(2,4),(5,6),(7,8),(9,10)]
 => [3,4,1,2,6,5,8,7,10,9] => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9)]
 => [(1,3),(2,4),(5,6),(7,9),(8,10)]
 => [3,4,1,2,6,5,9,10,7,8] => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10)]
 => [(1,3),(2,4),(5,7),(6,8),(9,10)]
 => [3,4,1,2,7,8,5,6,10,9] => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8)]
 => [(1,3),(2,4),(5,8),(6,9),(7,10)]
 => [3,4,1,2,8,9,10,5,6,7] => ? = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => [(1,3),(2,5),(4,7),(6,9),(8,10)]
 => [3,5,1,7,2,9,4,10,6,8] => ? = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,10)]
 => [(1,4),(2,5),(3,6),(7,8),(9,10)]
 => [4,5,6,1,2,3,8,7,10,9] => ? = 2 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9)]
 => [(1,4),(2,5),(3,6),(7,9),(8,10)]
 => [4,5,6,1,2,3,9,10,7,8] => ? = 2 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [(1,8),(2,7),(3,4),(5,6),(9,10)]
 => [(1,4),(2,6),(3,7),(5,8),(9,10)]
 => [4,6,7,1,8,2,3,5,10,9] => ? = 3 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [(1,10),(2,9),(3,4),(5,6),(7,8)]
 => [(1,4),(2,6),(3,8),(5,9),(7,10)]
 => [4,6,8,1,9,2,10,3,5,7] => ? = 3 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [(1,10),(2,9),(3,4),(5,8),(6,7)]
 => [(1,4),(2,7),(3,8),(5,9),(6,10)]
 => [4,7,8,1,9,10,2,3,5,6] => ? = 4 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10)]
 => [(1,5),(2,6),(3,7),(4,8),(9,10)]
 => [5,6,7,8,1,2,3,4,10,9] => ? = 3 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [(1,10),(2,9),(3,6),(4,5),(7,8)]
 => [(1,5),(2,6),(3,8),(4,9),(7,10)]
 => [5,6,8,9,1,2,10,3,4,7] => ? = 4 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,5),(6,7)]
 => [(1,5),(2,7),(3,8),(4,9),(6,10)]
 => [5,7,8,9,1,10,2,3,4,6] => ? = 4 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,7),(5,6)]
 => [(1,6),(2,7),(3,8),(4,9),(5,10)]
 => [6,7,8,9,10,1,2,3,4,5] => 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => [2,1,4,3,6,5,8,7,10,9,12,11] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
 => [(1,2),(3,4),(5,6),(7,8),(9,11),(10,12)]
 => [2,1,4,3,6,5,8,7,11,12,9,10] => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
 => [(1,2),(3,4),(5,6),(7,9),(8,10),(11,12)]
 => [2,1,4,3,6,5,9,10,7,8,12,11] => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
 => [(1,2),(3,4),(5,6),(7,10),(8,11),(9,12)]
 => [2,1,4,3,6,5,10,11,12,7,8,9] => ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
 => [(1,2),(3,4),(5,7),(6,8),(9,10),(11,12)]
 => [2,1,4,3,7,8,5,6,10,9,12,11] => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
 => [(1,2),(3,4),(5,7),(6,8),(9,11),(10,12)]
 => [2,1,4,3,7,8,5,6,11,12,9,10] => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)]
 => [(1,2),(3,4),(5,8),(6,9),(7,10),(11,12)]
 => [2,1,4,3,8,9,10,5,6,7,12,11] => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)]
 => [(1,2),(3,4),(5,8),(6,10),(7,11),(9,12)]
 => [2,1,4,3,8,10,11,5,12,6,7,9] => ? = 3 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
 => [(1,2),(3,4),(5,9),(6,10),(7,11),(8,12)]
 => [2,1,4,3,9,10,11,12,5,6,7,8] => ? = 3 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
 => [(1,2),(3,5),(4,6),(7,8),(9,10),(11,12)]
 => [2,1,5,6,3,4,8,7,10,9,12,11] => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
 => [(1,2),(3,5),(4,6),(7,8),(9,11),(10,12)]
 => [2,1,5,6,3,4,8,7,11,12,9,10] => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
 => [(1,2),(3,5),(4,6),(7,9),(8,10),(11,12)]
 => [2,1,5,6,3,4,9,10,7,8,12,11] => ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
 => [(1,2),(3,5),(4,6),(7,10),(8,11),(9,12)]
 => [2,1,5,6,3,4,10,11,12,7,8,9] => ? = 2 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
 => [(1,2),(3,5),(4,7),(6,9),(8,11),(10,12)]
 => [2,1,5,7,3,9,4,11,6,12,8,10] => ? = 2 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [(1,2),(3,8),(4,7),(5,6),(9,10),(11,12)]
 => [(1,2),(3,6),(4,7),(5,8),(9,10),(11,12)]
 => [2,1,6,7,8,3,4,5,10,9,12,11] => ? = 2 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
 => [(1,2),(3,6),(4,7),(5,8),(9,11),(10,12)]
 => [2,1,6,7,8,3,4,5,11,12,9,10] => ? = 2 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8),(11,12)]
 => [(1,2),(3,6),(4,8),(5,9),(7,10),(11,12)]
 => [2,1,6,8,9,3,10,4,5,7,12,11] => ? = 3 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [(1,2),(3,12),(4,11),(5,6),(7,8),(9,10)]
 => [(1,2),(3,6),(4,8),(5,10),(7,11),(9,12)]
 => [2,1,6,8,10,3,11,4,12,5,7,9] => ? = 3 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
 => [(1,2),(3,6),(4,9),(5,10),(7,11),(8,12)]
 => [2,1,6,9,10,3,11,12,4,5,7,8] => ? = 4 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7),(11,12)]
 => [(1,2),(3,7),(4,8),(5,9),(6,10),(11,12)]
 => [2,1,7,8,9,10,3,4,5,6,12,11] => ? = 3 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
 => [(1,2),(3,7),(4,8),(5,10),(6,11),(9,12)]
 => [2,1,7,8,10,11,3,4,12,5,6,9] => ? = 4 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
 => [(1,2),(3,7),(4,9),(5,10),(6,11),(8,12)]
 => [2,1,7,9,10,11,3,12,4,5,6,8] => ? = 4 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
 => [(1,2),(3,8),(4,9),(5,10),(6,11),(7,12)]
 => [2,1,8,9,10,11,12,3,4,5,6,7] => ? = 4 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)]
 => [(1,3),(2,4),(5,6),(7,8),(9,10),(11,12)]
 => [3,4,1,2,6,5,8,7,10,9,12,11] => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)]
 => [(1,3),(2,4),(5,6),(7,8),(9,11),(10,12)]
 => [3,4,1,2,6,5,8,7,11,12,9,10] => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
 => [(1,3),(2,4),(5,6),(7,9),(8,10),(11,12)]
 => [3,4,1,2,6,5,9,10,7,8,12,11] => ? = 1 + 1
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10)]
 => [(1,3),(2,4),(5,6),(7,10),(8,11),(9,12)]
 => [3,4,1,2,6,5,10,11,12,7,8,9] => ? = 2 + 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
 => [(1,3),(2,4),(5,7),(6,8),(9,10),(11,12)]
 => [3,4,1,2,7,8,5,6,10,9,12,11] => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
 => [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12)]
 => [3,4,1,2,7,8,5,6,11,12,9,10] => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
 => [(1,3),(2,4),(5,8),(6,9),(7,10),(11,12)]
 => [3,4,1,2,8,9,10,5,6,7,12,11] => ? = 2 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12)]
 => [7,8,9,10,11,12,1,2,3,4,5,6] => 6 = 5 + 1

Description

The maximum drop size of a permutation. The maximum drop size of a permutation

$\pi$ of

$[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of

$i-\pi(i)$ .

Matching statistic: St000451

Mapping

Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00116: Perfect matchings —Kasraoui-Zeng⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 86%

Values

[1,0]
 => [(1,2)]
 => [(1,2)]
 => [2,1] => 2 = 0 + 2
[1,0,1,0]
 => [(1,2),(3,4)]
 => [(1,2),(3,4)]
 => [2,1,4,3] => 2 = 0 + 2
[1,1,0,0]
 => [(1,4),(2,3)]
 => [(1,3),(2,4)]
 => [3,4,1,2] => 3 = 1 + 2
[1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 2 = 0 + 2
[1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => 3 = 1 + 2
[1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => 3 = 1 + 2
[1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => [(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => 4 = 2 + 2
[1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => 2 = 0 + 2
[1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [(1,2),(3,4),(5,7),(6,8)]
 => [2,1,4,3,7,8,5,6] => 3 = 1 + 2
[1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => [(1,2),(3,5),(4,6),(7,8)]
 => [2,1,5,6,3,4,8,7] => 3 = 1 + 2
[1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => [(1,2),(3,6),(4,7),(5,8)]
 => [2,1,6,7,8,3,4,5] => 4 = 2 + 2
[1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => [(1,3),(2,4),(5,6),(7,8)]
 => [3,4,1,2,6,5,8,7] => 3 = 1 + 2
[1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [(1,3),(2,4),(5,7),(6,8)]
 => [3,4,1,2,7,8,5,6] => 3 = 1 + 2
[1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [(1,4),(2,5),(3,6),(7,8)]
 => [4,5,6,1,2,3,8,7] => 4 = 2 + 2
[1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [(1,4),(2,6),(3,7),(5,8)]
 => [4,6,7,1,8,2,3,5] => 5 = 3 + 2
[1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => [(1,5),(2,6),(3,7),(4,8)]
 => [5,6,7,8,1,2,3,4] => 5 = 3 + 2
[1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => 2 = 0 + 2
[1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => [(1,2),(3,4),(5,6),(7,9),(8,10)]
 => [2,1,4,3,6,5,9,10,7,8] => ? = 1 + 2
[1,0,1,0,1,1,0,0,1,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,10)]
 => [(1,2),(3,4),(5,7),(6,8),(9,10)]
 => [2,1,4,3,7,8,5,6,10,9] => ? = 1 + 2
[1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8)]
 => [(1,2),(3,4),(5,8),(6,9),(7,10)]
 => [2,1,4,3,8,9,10,5,6,7] => ? = 2 + 2
[1,0,1,1,0,0,1,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,10)]
 => [(1,2),(3,5),(4,6),(7,8),(9,10)]
 => [2,1,5,6,3,4,8,7,10,9] => ? = 1 + 2
[1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9)]
 => [(1,2),(3,5),(4,6),(7,9),(8,10)]
 => [2,1,5,6,3,4,9,10,7,8] => ? = 1 + 2
[1,0,1,1,1,0,0,0,1,0]
 => [(1,2),(3,8),(4,7),(5,6),(9,10)]
 => [(1,2),(3,6),(4,7),(5,8),(9,10)]
 => [2,1,6,7,8,3,4,5,10,9] => ? = 2 + 2
[1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8)]
 => [(1,2),(3,6),(4,8),(5,9),(7,10)]
 => [2,1,6,8,9,3,10,4,5,7] => ? = 3 + 2
[1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => [(1,2),(3,7),(4,8),(5,9),(6,10)]
 => [2,1,7,8,9,10,3,4,5,6] => ? = 3 + 2
[1,1,0,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10)]
 => [(1,3),(2,4),(5,6),(7,8),(9,10)]
 => [3,4,1,2,6,5,8,7,10,9] => ? = 1 + 2
[1,1,0,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9)]
 => [(1,3),(2,4),(5,6),(7,9),(8,10)]
 => [3,4,1,2,6,5,9,10,7,8] => ? = 1 + 2
[1,1,0,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10)]
 => [(1,3),(2,4),(5,7),(6,8),(9,10)]
 => [3,4,1,2,7,8,5,6,10,9] => ? = 1 + 2
[1,1,0,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8)]
 => [(1,3),(2,4),(5,8),(6,9),(7,10)]
 => [3,4,1,2,8,9,10,5,6,7] => ? = 2 + 2
[1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => [(1,3),(2,5),(4,7),(6,9),(8,10)]
 => [3,5,1,7,2,9,4,10,6,8] => ? = 2 + 2
[1,1,1,0,0,0,1,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,10)]
 => [(1,4),(2,5),(3,6),(7,8),(9,10)]
 => [4,5,6,1,2,3,8,7,10,9] => ? = 2 + 2
[1,1,1,0,0,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9)]
 => [(1,4),(2,5),(3,6),(7,9),(8,10)]
 => [4,5,6,1,2,3,9,10,7,8] => ? = 2 + 2
[1,1,1,0,1,0,0,0,1,0]
 => [(1,8),(2,7),(3,4),(5,6),(9,10)]
 => [(1,4),(2,6),(3,7),(5,8),(9,10)]
 => [4,6,7,1,8,2,3,5,10,9] => ? = 3 + 2
[1,1,1,0,1,0,1,0,0,0]
 => [(1,10),(2,9),(3,4),(5,6),(7,8)]
 => [(1,4),(2,6),(3,8),(5,9),(7,10)]
 => [4,6,8,1,9,2,10,3,5,7] => ? = 3 + 2
[1,1,1,0,1,1,0,0,0,0]
 => [(1,10),(2,9),(3,4),(5,8),(6,7)]
 => [(1,4),(2,7),(3,8),(5,9),(6,10)]
 => [4,7,8,1,9,10,2,3,5,6] => ? = 4 + 2
[1,1,1,1,0,0,0,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10)]
 => [(1,5),(2,6),(3,7),(4,8),(9,10)]
 => [5,6,7,8,1,2,3,4,10,9] => ? = 3 + 2
[1,1,1,1,0,0,1,0,0,0]
 => [(1,10),(2,9),(3,6),(4,5),(7,8)]
 => [(1,5),(2,6),(3,8),(4,9),(7,10)]
 => [5,6,8,9,1,2,10,3,4,7] => ? = 4 + 2
[1,1,1,1,0,1,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,5),(6,7)]
 => [(1,5),(2,7),(3,8),(4,9),(6,10)]
 => [5,7,8,9,1,10,2,3,4,6] => ? = 4 + 2
[1,1,1,1,1,0,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,7),(5,6)]
 => [(1,6),(2,7),(3,8),(4,9),(5,10)]
 => [6,7,8,9,10,1,2,3,4,5] => 6 = 4 + 2
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => [2,1,4,3,6,5,8,7,10,9,12,11] => 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
 => [(1,2),(3,4),(5,6),(7,8),(9,11),(10,12)]
 => [2,1,4,3,6,5,8,7,11,12,9,10] => ? = 1 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
 => [(1,2),(3,4),(5,6),(7,9),(8,10),(11,12)]
 => [2,1,4,3,6,5,9,10,7,8,12,11] => ? = 1 + 2
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
 => [(1,2),(3,4),(5,6),(7,10),(8,11),(9,12)]
 => [2,1,4,3,6,5,10,11,12,7,8,9] => ? = 2 + 2
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
 => [(1,2),(3,4),(5,7),(6,8),(9,10),(11,12)]
 => [2,1,4,3,7,8,5,6,10,9,12,11] => ? = 1 + 2
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
 => [(1,2),(3,4),(5,7),(6,8),(9,11),(10,12)]
 => [2,1,4,3,7,8,5,6,11,12,9,10] => ? = 1 + 2
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)]
 => [(1,2),(3,4),(5,8),(6,9),(7,10),(11,12)]
 => [2,1,4,3,8,9,10,5,6,7,12,11] => ? = 2 + 2
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)]
 => [(1,2),(3,4),(5,8),(6,10),(7,11),(9,12)]
 => [2,1,4,3,8,10,11,5,12,6,7,9] => ? = 3 + 2
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
 => [(1,2),(3,4),(5,9),(6,10),(7,11),(8,12)]
 => [2,1,4,3,9,10,11,12,5,6,7,8] => ? = 3 + 2
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
 => [(1,2),(3,5),(4,6),(7,8),(9,10),(11,12)]
 => [2,1,5,6,3,4,8,7,10,9,12,11] => ? = 1 + 2
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
 => [(1,2),(3,5),(4,6),(7,8),(9,11),(10,12)]
 => [2,1,5,6,3,4,8,7,11,12,9,10] => ? = 1 + 2
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
 => [(1,2),(3,5),(4,6),(7,9),(8,10),(11,12)]
 => [2,1,5,6,3,4,9,10,7,8,12,11] => ? = 1 + 2
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
 => [(1,2),(3,5),(4,6),(7,10),(8,11),(9,12)]
 => [2,1,5,6,3,4,10,11,12,7,8,9] => ? = 2 + 2
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
 => [(1,2),(3,5),(4,7),(6,9),(8,11),(10,12)]
 => [2,1,5,7,3,9,4,11,6,12,8,10] => ? = 2 + 2
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [(1,2),(3,8),(4,7),(5,6),(9,10),(11,12)]
 => [(1,2),(3,6),(4,7),(5,8),(9,10),(11,12)]
 => [2,1,6,7,8,3,4,5,10,9,12,11] => ? = 2 + 2
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
 => [(1,2),(3,6),(4,7),(5,8),(9,11),(10,12)]
 => [2,1,6,7,8,3,4,5,11,12,9,10] => ? = 2 + 2
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8),(11,12)]
 => [(1,2),(3,6),(4,8),(5,9),(7,10),(11,12)]
 => [2,1,6,8,9,3,10,4,5,7,12,11] => ? = 3 + 2
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [(1,2),(3,12),(4,11),(5,6),(7,8),(9,10)]
 => [(1,2),(3,6),(4,8),(5,10),(7,11),(9,12)]
 => [2,1,6,8,10,3,11,4,12,5,7,9] => ? = 3 + 2
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
 => [(1,2),(3,6),(4,9),(5,10),(7,11),(8,12)]
 => [2,1,6,9,10,3,11,12,4,5,7,8] => ? = 4 + 2
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7),(11,12)]
 => [(1,2),(3,7),(4,8),(5,9),(6,10),(11,12)]
 => [2,1,7,8,9,10,3,4,5,6,12,11] => ? = 3 + 2
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
 => [(1,2),(3,7),(4,8),(5,10),(6,11),(9,12)]
 => [2,1,7,8,10,11,3,4,12,5,6,9] => ? = 4 + 2
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
 => [(1,2),(3,7),(4,9),(5,10),(6,11),(8,12)]
 => [2,1,7,9,10,11,3,12,4,5,6,8] => ? = 4 + 2
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
 => [(1,2),(3,8),(4,9),(5,10),(6,11),(7,12)]
 => [2,1,8,9,10,11,12,3,4,5,6,7] => ? = 4 + 2
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)]
 => [(1,3),(2,4),(5,6),(7,8),(9,10),(11,12)]
 => [3,4,1,2,6,5,8,7,10,9,12,11] => ? = 1 + 2
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)]
 => [(1,3),(2,4),(5,6),(7,8),(9,11),(10,12)]
 => [3,4,1,2,6,5,8,7,11,12,9,10] => ? = 1 + 2
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
 => [(1,3),(2,4),(5,6),(7,9),(8,10),(11,12)]
 => [3,4,1,2,6,5,9,10,7,8,12,11] => ? = 1 + 2
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10)]
 => [(1,3),(2,4),(5,6),(7,10),(8,11),(9,12)]
 => [3,4,1,2,6,5,10,11,12,7,8,9] => ? = 2 + 2
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
 => [(1,3),(2,4),(5,7),(6,8),(9,10),(11,12)]
 => [3,4,1,2,7,8,5,6,10,9,12,11] => ? = 1 + 2
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
 => [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12)]
 => [3,4,1,2,7,8,5,6,11,12,9,10] => ? = 1 + 2
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
 => [(1,3),(2,4),(5,8),(6,9),(7,10),(11,12)]
 => [3,4,1,2,8,9,10,5,6,7,12,11] => ? = 2 + 2
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12)]
 => [7,8,9,10,11,12,1,2,3,4,5,6] => 7 = 5 + 2

Description

The length of the longest pattern of the form k 1 2...(k-1).