Your data matches 7 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000451
(load all 3 compositions to match this statistic)
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => 1
[1,0,1,0]
 => [1,2] => [1,2] => 1
[1,1,0,0]
 => [2,1] => [2,1] => 2
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 1
[1,0,1,1,0,0]
 => [1,3,2] => [2,3,1] => 2
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => 2
[1,1,0,1,0,0]
 => [2,3,1] => [3,1,2] => 3
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => 2
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,3,4,1] => 2
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,3,1,4] => 2
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,4,1,3] => 3
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [3,4,2,1] => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => 2
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [3,2,4,1] => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,1,2,4] => 3
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,1,2,3] => 4
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [4,2,3,1] => 3
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => 2
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [4,2,1,3] => 3
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [4,1,3,2] => 3
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,3,4,5,1] => 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [2,3,4,1,5] => 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [2,3,5,1,4] => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [3,4,5,2,1] => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,3,1,4,5] => 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [3,4,2,5,1] => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [2,4,1,3,5] => 3
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,5,1,3,4] => 4
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [3,5,2,4,1] => 3
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [3,4,2,1,5] => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [3,5,2,1,4] => 3
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [3,5,1,4,2] => 3
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [4,5,3,2,1] => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [3,2,4,5,1] => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [3,2,4,1,5] => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [3,2,5,1,4] => 3
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [4,3,5,2,1] => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,1,2,4,5] => 3
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [4,2,3,5,1] => 3
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,1,2,3,5] => 4
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => 5
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [5,2,3,4,1] => 4
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [4,2,3,1,5] => 3
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [5,2,3,1,4] => 4
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [5,2,1,4,3] => 3
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [5,3,4,2,1] => 3
Description
The length of the longest pattern of the form k 1 2...(k-1).
Matching statistic: St001330
(load all 5 compositions to match this statistic)
Mapping
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 73%
Values
[1,0]
 => [1] => [1] => ([],1)
 => 1
[1,0,1,0]
 => [1,2] => [1,2] => ([],2)
 => 1
[1,1,0,0]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 2
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => ([],3)
 => 1
[1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => ([(1,2)],3)
 => 2
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => ([(1,2)],3)
 => 2
[1,1,0,1,0,0]
 => [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,1,1,0,0,0]
 => [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 2
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => 2
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => 2
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => 2
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 2
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
 => 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ? = 3
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => 2
[1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 2
[1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => [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
[1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,1,1,0,1,0,0,0,1,0]
 => [3,4,1,2,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,1,1,0,1,0,0,1,0,0]
 => [3,4,1,5,2] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
[1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
[1,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => 2
[1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,1,1,1,0,0,1,0,0,0]
 => [4,1,5,2,3] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ([(4,5)],6)
 => 2
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ([(4,5)],6)
 => 2
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
 => 3
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,4,5] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
 => 2
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ([(4,5)],6)
 => 2
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
 => 2
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,2,4,6,3,5] => [1,2,6,4,3,5] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,5,3,6,4] => [1,2,6,5,3,4] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,5,6,3,4] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,3,4,6,2,5] => [1,6,4,3,2,5] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,3,5,2,4,6] => [1,5,3,2,4,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,3,5,2,6,4] => [1,6,5,3,2,4] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,3,5,6,2,4] => [1,6,3,2,5,4] => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = 3
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,3,6,2,4,5] => [1,6,3,2,4,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3,6] => [1,5,4,2,3,6] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,4,2,5,6,3] => [1,6,5,4,2,3] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,4,2,6,3,5] => [1,6,4,2,3,5] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,4,5,2,3,6] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,4,5,2,6,3] => [1,6,5,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,4,5,6,2,3] => [1,6,2,5,4,3] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,4,6,2,3,5] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,5,2,3,6,4] => [1,6,5,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,5,2,6,3,4] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,5,6,2,3,4] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [2,1,4,6,3,5] => [2,1,6,4,3,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,6,4] => [2,1,6,5,3,4] => ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [2,1,5,6,3,4] => [2,1,6,3,5,4] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,6,1,5] => [6,4,3,2,1,5] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [2,3,5,1,4,6] => [5,3,2,1,4,6] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,5,1,6,4] => [6,5,3,2,1,4] => ([(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)
 => ? = 5
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [2,3,5,6,1,4] => [6,3,2,1,5,4] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,3,6,1,4,5] => [6,3,2,1,4,5] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
[1,1,0,1,1,0,0,0,1,0,1,0]
 => [2,4,1,3,5,6] => [4,2,1,3,5,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
[1,1,0,1,1,0,0,0,1,1,0,0]
 => [2,4,1,3,6,5] => [4,2,1,3,6,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [2,4,1,5,3,6] => [5,4,2,1,3,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
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: St000454
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 64%
Values
[1,0]
 => [1] => [1] => ([],1)
 => 0 = 1 - 1
[1,0,1,0]
 => [1,2] => [1,2] => ([],2)
 => 0 = 1 - 1
[1,1,0,0]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => ([],3)
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => ([(1,2)],3)
 => 1 = 2 - 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => ([(1,2)],3)
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[1,1,1,0,0,0]
 => [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
 => ? = 2 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? = 2 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 - 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? = 2 - 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 - 1
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 - 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
 => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [1,5,3,4,2] => ([(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] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ? = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
[1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
[1,1,1,0,1,0,0,0,1,0]
 => [3,4,2,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
[1,1,1,0,1,0,0,1,0,0]
 => [3,4,2,5,1] => [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
[1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,2,1] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
[1,1,1,0,1,1,0,0,0,0]
 => [3,5,4,2,1] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
[1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ? = 3 - 1
[1,1,1,1,0,0,1,0,0,0]
 => [4,3,5,2,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
[1,1,1,1,0,1,0,0,0,0]
 => [4,5,3,2,1] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
[1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ([(4,5)],6)
 => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ([(4,5)],6)
 => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,5,4] => [1,2,3,5,6,4] => ([(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] => [1,2,4,3,5,6] => ([(4,5)],6)
 => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,4,5,3,6] => [1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,4,5,6,3] => [1,2,6,5,4,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,2,4,6,5,3] => [1,2,5,6,4,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,5,4,3,6] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
 => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,5,4,6,3] => [1,2,4,6,5,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,5,6,4,3] => [1,2,6,4,5,3] => ([(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] => [1,2,5,4,6,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,3,2,4,5,6] => [1,3,2,4,5,6] => ([(4,5)],6)
 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,3,2,4,6,5] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
 => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4,6] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
 => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,3,2,5,6,4] => [1,3,2,6,5,4] => ([(1,2),(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,6,5,4] => [1,3,2,5,6,4] => ([(1,2),(3,5),(4,5)],6)
 => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,3,4,2,5,6] => [1,4,3,2,5,6] => ([(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,3,4,2,6,5] => [1,4,3,2,6,5] => ([(1,2),(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,3,4,5,2,6] => [1,5,4,3,2,6] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,2] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 5 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,3,4,6,5,2] => [1,5,6,4,3,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,3,5,4,2,6] => [1,4,5,3,2,6] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,3,5,4,6,2] => [1,4,6,5,3,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,3,5,6,4,2] => [1,6,4,5,3,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,3,6,5,4,2] => [1,5,4,6,3,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,4,3,2,5,6] => [1,3,4,2,5,6] => ([(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] => [1,3,4,2,6,5] => ([(1,2),(3,5),(4,5)],6)
 => ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,4,3,5,2,6] => [1,3,5,4,2,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,4,3,5,6,2] => [1,3,6,5,4,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,4,3,6,5,2] => [1,3,5,6,4,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(4,5)],6)
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,1,3,4,6,5] => [2,1,3,4,6,5] => ([(2,5),(3,4)],6)
 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,1,3,5,4,6] => [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
 => 1 = 2 - 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: St000455
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 18%
Values
[1,0]
 => [1] => ([],1)
 => ([(0,1)],2)
 => -1 = 1 - 2
[1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => ([(0,1),(0,2),(1,2)],3)
 => -1 = 1 - 2
[1,1,0,0]
 => [1,2] => ([],2)
 => ([(0,2),(1,2)],3)
 => 0 = 2 - 2
[1,0,1,0,1,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => -1 = 1 - 2
[1,0,1,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 2 - 2
[1,1,0,0,1,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 2 - 2
[1,1,0,1,0,0]
 => [2,1,3] => ([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 - 2
[1,1,1,0,0,0]
 => [1,2,3] => ([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => 0 = 2 - 2
[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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => -1 = 1 - 2
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 2 - 2
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 2 - 2
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 2
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 2 - 2
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 2 - 2
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 0 = 2 - 2
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 2
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 2
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 2
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 2 - 2
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 2
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 2 - 2
[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)
 => ([(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)
 => -1 = 1 - 2
[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)
 => ([(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 = 2 - 2
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(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 = 2 - 2
[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)
 => ([(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 - 2
[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)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 2 - 2
[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)
 => ([(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 = 2 - 2
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 2 - 2
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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 - 2
[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)
 => ([(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 - 2
[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)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 2
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 2 - 2
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 2
[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)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 2
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 2 - 2
[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)
 => ([(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 = 2 - 2
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 2 - 2
[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)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 2 - 2
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 3 - 2
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 0 = 2 - 2
[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)
 => ([(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 - 2
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 3 - 2
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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 - 2
[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)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 2
[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)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 2
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 2
[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)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 2 - 2
[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)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 0 = 2 - 2
[1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 2
[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)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 3 - 2
[1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 2
[1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,1,1,0,1,1,0,0,0,0]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 2
[1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 2
[1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 2
[1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => ([(3,4)],5)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 2
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 2 - 2
[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)
 => ([(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)
 => -1 = 1 - 2
[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)
 => ([(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 = 2 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,2,1] => ([(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)
 => ([(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 = 2 - 2
[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)
 => ([(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)
 => ? = 3 - 2
[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)
 => ([(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 = 2 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [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)
 => ([(0,2),(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 = 2 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(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 = 2 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [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)
 => ([(0,3),(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)
 => ? = 3 - 2
[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)
 => ([(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)
 => ? = 4 - 2
[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)
 => ([(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)
 => ? = 3 - 2
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [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)
 => ([(0,3),(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 = 2 - 2
[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)
 => ([(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)
 => ? = 3 - 2
[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)
 => ([(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)
 => ? = 3 - 2
[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)
 => ([(0,4),(0,5),(0,6),(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 = 2 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,3,1] => ([(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)
 => ([(0,2),(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 = 2 - 2
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [5,6,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(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 = 2 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(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 = 2 - 2
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [5,4,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(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),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 2
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [4,5,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 2 - 2
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,4,1] => ([(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)
 => ([(0,3),(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)
 => ? = 3 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [5,6,3,2,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(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),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [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)
 => ([(0,4),(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)
 => ? = 4 - 2
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,6,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => ? = 5 - 2
[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)
 => ([(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)
 => ? = 4 - 2
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [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)
 => ([(0,4),(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)
 => ? = 3 - 2
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [5,3,4,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => ? = 4 - 2
[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)
 => ([(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)
 => ? = 3 - 2
[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)
 => ([(0,5),(0,6),(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)
 => ? = 3 - 2
[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)
 => ([(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 = 2 - 2
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [5,6,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 2 - 2
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [6,4,2,3,5,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(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)
 => ? = 3 - 2
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(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 = 2 - 2
[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)
 => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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 = 2 - 2
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(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 = 2 - 2
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(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 = 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: St001864
Mapping
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001864: Signed permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 36%
Values
[1,0]
 => [1] => [1] => [1] => 0 = 1 - 1
[1,0,1,0]
 => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[1,1,0,0]
 => [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,3,2] => [3,1,2] => [3,1,2] => 1 = 2 - 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,3,1] => [2,3,1] => 2 = 3 - 1
[1,1,1,0,0,0]
 => [3,1,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [3,1,2,4] => [3,1,2,4] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1,4,2] => [3,1,4,2] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [4,2,1,3] => [4,2,1,3] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,3,4,1] => [2,3,4,1] => 3 = 4 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [2,1,4,3] => [2,1,4,3] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [3,4,1,2] => [3,4,1,2] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [1,3,4,2] => [1,3,4,2] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [4,1,2,5,3] => [4,1,2,5,3] => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [3,1,4,5,2] => [3,1,4,5,2] => ? = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [4,2,1,5,3] => [4,2,1,5,3] => ? = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [5,2,3,1,4] => [5,2,3,1,4] => ? = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [4,2,5,1,3] => [4,2,5,1,3] => ? = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 3 - 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 4 - 1
[1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 3 - 1
[1,1,1,0,1,0,0,0,1,0]
 => [3,4,1,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => 2 = 3 - 1
[1,1,1,0,1,0,0,1,0,0]
 => [3,4,1,5,2] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 4 - 1
[1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => [1,3,4,5,2] => [1,3,4,5,2] => 3 = 4 - 1
[1,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2 = 3 - 1
[1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => [4,1,5,2,3] => [4,1,5,2,3] => ? = 3 - 1
[1,1,1,1,0,0,1,0,0,0]
 => [4,1,5,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => 2 = 3 - 1
[1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => [1,2,4,5,3] => [1,2,4,5,3] => 2 = 3 - 1
[1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [6,1,2,3,4,5] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [5,1,2,3,4,6] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => [5,1,2,3,6,4] => [5,1,2,3,6,4] => ? = 3 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,6,2,3,4,5] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,5,6] => [4,1,2,3,5,6] => [4,1,2,3,5,6] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5] => [6,4,1,2,3,5] => [6,4,1,2,3,5] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,4,5,3,6] => [4,1,2,5,3,6] => [4,1,2,5,3,6] => ? = 3 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,4,5,6,3] => [4,1,2,5,6,3] => [4,1,2,5,6,3] => ? = 4 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,2,4,6,3,5] => [4,1,2,3,6,5] => [4,1,2,3,6,5] => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,5,3,4,6] => [1,5,2,3,4,6] => [1,5,2,3,4,6] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,5,3,6,4] => [5,6,1,2,3,4] => [5,6,1,2,3,4] => ? = 3 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,5,6,3,4] => [1,5,2,3,6,4] => [1,5,2,3,6,4] => ? = 3 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,3,2,4,5,6] => [3,1,2,4,5,6] => [3,1,2,4,5,6] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,3,2,4,6,5] => [6,3,1,2,4,5] => [6,3,1,2,4,5] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4,6] => [5,3,1,2,4,6] => [5,3,1,2,4,6] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,3,2,5,6,4] => [5,3,1,2,6,4] => [5,3,1,2,6,4] => ? = 3 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,6,4,5] => [3,6,1,2,4,5] => [3,6,1,2,4,5] => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,3,4,2,5,6] => [3,1,4,2,5,6] => [3,1,4,2,5,6] => ? = 3 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,3,4,2,6,5] => [6,3,1,4,2,5] => [6,3,1,4,2,5] => ? = 3 - 1
Description
The number of excedances of a signed permutation. For a signed permutation $\pi\in\mathfrak H_n$, this is $\lvert\{i\in[n] \mid \pi(i) > i\}\rvert$.
Matching statistic: St001431
(load all 2 compositions to match this statistic)
Mapping
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001431: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 36%
Values
[1,0]
 => [1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2 = 3 - 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3 = 4 - 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => ? = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => ? = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => ? = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => ? = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => ? = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => ? = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => ? = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => ? = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => ? = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => ? = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => ? = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => ? = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => ? = 3 - 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => ? = 4 - 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => ? = 3 - 1
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => ? = 3 - 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => ? = 4 - 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 4 - 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => ? = 3 - 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => ? = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => ? = 3 - 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => ? = 3 - 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => ? = 3 - 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 2 - 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]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 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]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2 - 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]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => ? = 3 - 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]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 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]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => ? = 3 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => ? = 4 - 1
Description
Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. The modified algebra B is obtained from the stable Auslander algebra kQ/I by deleting all relations which contain walks of length at least three (conjectural this step of deletion is not necessary as the stable higher Auslander algebras might be quadratic) and taking as B then the algebra kQ^(op)/J when J is the quadratic perp of the ideal I. See http://www.findstat.org/DyckPaths/NakayamaAlgebras for the definition of Loewy length and Nakayama algebras associated to Dyck paths.
Matching statistic: St001905
(load all 2 compositions to match this statistic)
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001905: Parking functions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 36%
Values
[1,0]
 => [1] => [1] => [1] => 0 = 1 - 1
[1,0,1,0]
 => [2,1] => [1,2] => [1,2] => 0 = 1 - 1
[1,1,0,0]
 => [1,2] => [2,1] => [2,1] => 1 = 2 - 1
[1,0,1,0,1,0]
 => [2,3,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,0,1,1,0,0]
 => [2,1,3] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,1,0,0,1,0]
 => [1,3,2] => [3,2,1] => [3,2,1] => 1 = 2 - 1
[1,1,0,1,0,0]
 => [3,1,2] => [3,1,2] => [3,1,2] => 2 = 3 - 1
[1,1,1,0,0,0]
 => [1,2,3] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [1,4,2,3] => [1,4,2,3] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [4,2,3,1] => [4,2,3,1] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [3,2,4,1] => [3,2,4,1] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [4,1,3,2] => [4,1,3,2] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [4,1,2,3] => [4,1,2,3] => 3 = 4 - 1
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [3,1,4,2] => [3,1,4,2] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [2,4,3,1] => [2,4,3,1] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [3,4,2,1] => [3,4,2,1] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [3,4,1,2] => [3,4,1,2] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [1,2,5,4,3] => [1,2,5,4,3] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [1,4,3,5,2] => [1,4,3,5,2] => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [1,5,2,4,3] => [1,5,2,4,3] => ? = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [1,3,5,4,2] => [1,3,5,4,2] => ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [1,4,5,3,2] => [1,4,5,3,2] => ? = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [1,4,5,2,3] => [1,4,5,2,3] => ? = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [1,3,4,5,2] => [1,3,4,5,2] => ? = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [4,2,3,5,1] => [4,2,3,5,1] => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [4,2,5,3,1] => [4,2,5,3,1] => ? = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [5,1,3,4,2] => [5,1,3,4,2] => ? = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [4,1,3,5,2] => [4,1,3,5,2] => ? = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [4,1,2,5,3] => [4,1,2,5,3] => ? = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [3,1,5,4,2] => [3,1,5,4,2] => ? = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [4,1,5,3,2] => [4,1,5,3,2] => ? = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [4,1,5,2,3] => [4,1,5,2,3] => ? = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [3,1,4,5,2] => [3,1,4,5,2] => ? = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => [2,5,3,4,1] => [2,5,3,4,1] => ? = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3] => [3,5,2,4,1] => [3,5,2,4,1] => ? = 3 - 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,4,5,2,3] => [4,5,2,3,1] => [4,5,2,3,1] => ? = 4 - 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,4,2,3,5] => [3,4,2,5,1] => [3,4,2,5,1] => ? = 3 - 1
[1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => [3,5,1,4,2] => [3,5,1,4,2] => ? = 3 - 1
[1,1,1,0,1,0,0,1,0,0]
 => [4,1,5,2,3] => [4,5,1,3,2] => [4,5,1,3,2] => ? = 4 - 1
[1,1,1,0,1,0,1,0,0,0]
 => [4,5,1,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 4 - 1
[1,1,1,0,1,1,0,0,0,0]
 => [4,1,2,3,5] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 3 - 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,2,5,3,4] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 3 - 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,5,2,3,4] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 3 - 1
[1,1,1,1,0,1,0,0,0,0]
 => [5,1,2,3,4] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 3 - 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,5,1,6] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [2,3,4,1,6,5] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [2,3,4,6,1,5] => [1,2,3,6,4,5] => [1,2,3,6,4,5] => ? = 3 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [2,3,4,1,5,6] => [1,2,3,5,6,4] => [1,2,3,5,6,4] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [2,3,1,5,6,4] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [2,3,1,5,4,6] => [1,2,5,4,6,3] => [1,2,5,4,6,3] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [2,3,5,1,6,4] => [1,2,6,3,5,4] => [1,2,6,3,5,4] => ? = 3 - 1
Description
The number of preferred parking spots in a parking function less than the index of the car. Let $(a_1,\dots,a_n)$ be a parking function. Then this statistic returns the number of indices $1\leq i\leq n$ such that $a_i < i$.