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Your data matches 14 different statistics following compositions of up to 3 maps.
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Matching statistic: St000661
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000661: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000661: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1] => [2] => [1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,2] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,0,1,0]
=> [2,1] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,2] => [2] => [1,1,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 0
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,0,1,0]
=> 0
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 0
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 0
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 0
Description
The number of rises of length 3 of a Dyck path.
Matching statistic: St001141
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001141: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001141: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1] => [2] => [1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,2] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,0,1,0]
=> [2,1] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,2] => [2] => [1,1,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 0
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,0,1,0]
=> 0
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 0
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 0
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 0
Description
The number of occurrences of hills of size 3 in a Dyck path.
A hill of size three is a subpath beginning at height zero, consisting of three up steps followed by three down steps.
Matching statistic: St000264
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Values
[1,0,1,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> ? = 0 + 4
[1,0,1,0,1,0]
=> [2,3,1] => [2,1,3] => ([(1,2)],3)
=> ? = 1 + 4
[1,0,1,1,0,0]
=> [2,1,3] => [2,3,1] => ([(0,2),(1,2)],3)
=> ? = 0 + 4
[1,1,0,0,1,0]
=> [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? = 0 + 4
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,1,3,4] => ([(2,3)],4)
=> ? = 0 + 4
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 0 + 4
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 4
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 4
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? = 0 + 4
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 4
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 4
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [2,1,3,4,5] => ([(3,4)],5)
=> ? = 0 + 4
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? = 1 + 4
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 0 + 4
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 0 + 4
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 4
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? = 0 + 4
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 4
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? = 0 + 4
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 4
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 0 + 4
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? = 0 + 4
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 0 + 4
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 0 + 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 0 + 4
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? = 1 + 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? = 0 + 4
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 0 + 4
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 0 + 4
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 0 + 4
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 4
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 0 + 4
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 4
[1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 0 + 4
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 0 + 4
[1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 0 + 4
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => [2,1,3,4,5,6] => ([(4,5)],6)
=> ? = 0 + 4
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> ? = 0 + 4
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => [2,1,6,3,4,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 4
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,6,1,5] => [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? = 1 + 4
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => [2,3,4,1,5,6] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1 + 4
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> ? = 0 + 4
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,6] => [2,3,1,5,4,6] => ([(1,2),(3,5),(4,5)],6)
=> ? = 0 + 4
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,5,1,6,4] => [2,1,5,3,4,6] => ([(1,2),(3,5),(4,5)],6)
=> ? = 0 + 4
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,5,6,1,4] => [2,5,1,3,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? = 0 + 4
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4,6] => [2,3,5,1,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? = 0 + 4
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => [2,1,3,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ? = 0 + 4
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,6,4,5] => [2,3,1,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> ? = 0 + 4
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,6,1,4,5] => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 0 + 4
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,5,6] => [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 0 + 4
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,6] => [2,4,5,1,3,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,6,1,3,5] => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,4,1,3,5,6] => [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 0 + 4
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,5,1,3,6,4] => [2,1,5,6,3,4] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 0 + 4
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,5,1,6,3,4] => [2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,5,6,1,3,4] => [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,5,1,3,4,6] => [2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,4,6,2,5] => [3,6,1,2,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,3,5,6,2,4] => [3,5,1,2,4,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [3,1,4,5,2,6] => [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,1,4,6,2,5] => [3,4,1,2,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 0 + 4
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,1,4,2,5,6] => [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 0 + 4
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [3,4,1,5,2,6] => [3,4,1,5,2,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [3,5,1,2,6,4] => [3,1,5,6,2,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5] => [4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 0 + 4
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,2,4,6,3,5] => [4,6,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 0 + 4
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,4,5,2,6,3] => [4,1,5,2,3,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [4,1,5,2,6,3] => [4,1,5,2,6,3] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [4,5,1,2,6,3] => [4,1,5,6,2,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,2,5,3,6,4] => [5,1,6,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 0 + 4
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,5,2,3,6,4] => [5,1,2,6,3,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,3,5,1,4,7,6] => [2,1,5,7,3,4,6] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 4 = 0 + 4
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,3,5,1,7,4,6] => [2,5,1,7,3,4,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 4 = 0 + 4
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,5,7,1,4,6] => [2,5,7,1,3,4,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 4 = 0 + 4
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,1,4,6,7] => [2,3,5,6,1,4,7] => ([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 4 = 0 + 4
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,3,1,6,4,7,5] => [2,1,6,3,7,4,5] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 4 = 0 + 4
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,7,4,5] => [2,6,1,3,7,4,5] => ([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> 4 = 0 + 4
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,3,6,1,4,7,5] => [2,1,6,7,3,4,5] => ([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 4 = 0 + 4
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,6,1,7,4,5] => [2,6,1,7,3,4,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
=> 4 = 0 + 4
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,6,7,1,4,5] => [2,6,7,1,3,4,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 4 = 0 + 4
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,4,5,3,6,7] => [2,4,5,1,6,7,3] => ([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> 4 = 0 + 4
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,3,5,7] => [2,4,6,1,3,7,5] => ([(0,6),(1,4),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> 4 = 0 + 4
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,1,4,7,3,5,6] => [2,4,5,1,3,7,6] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 4 = 0 + 4
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => [2,4,5,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 4 = 0 + 4
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,3,6,7] => [2,4,5,1,6,3,7] => ([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> 4 = 0 + 4
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,4,5,6,1,3,7] => [2,4,5,1,3,6,7] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 4 = 0 + 4
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,4,5,7,1,3,6] => [2,4,7,1,3,5,6] => ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 4 = 0 + 4
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,4,5,1,3,6,7] => [2,4,5,6,1,3,7] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 4 = 0 + 4
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,4,1,3,6,7,5] => [2,1,4,6,7,3,5] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 4 = 0 + 4
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,6,5,7] => [2,4,1,6,7,3,5] => ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> 4 = 0 + 4
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,1,6,3,5,7] => [2,4,6,1,7,3,5] => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 4 = 0 + 4
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,6,7,1,3,5] => [2,4,6,1,3,5,7] => ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> 4 = 0 + 4
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,4,6,1,3,5,7] => [2,4,6,7,1,3,5] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 4 = 0 + 4
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,4,1,7,3,5,6] => [2,4,5,1,7,3,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 4 = 0 + 4
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,4,7,1,3,5,6] => [2,4,5,7,1,3,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 4 = 0 + 4
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St001545
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001545: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001545: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Values
[1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,0,1,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 2
[1,0,1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,1,0,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 0 + 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 0 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 1 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 0 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 0 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 0 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 0 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 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)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 1 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 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)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 0 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 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)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 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)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ? = 0 + 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ? = 0 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ? = 1 + 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 1 + 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ? = 0 + 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => ([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ? = 0 + 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,5,3,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? = 0 + 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,4,5,1,3,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,5,6,3] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ? = 0 + 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ? = 1 + 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,5,6,3,4] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> ? = 0 + 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,5,3,4,6] => ([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ? = 0 + 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,5,1,3,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ? = 0 + 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,1,5,4,6] => ([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ? = 0 + 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,3,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ? = 0 + 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,3,1,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ? = 0 + 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,6,3,4,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ? = 0 + 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,3,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ? = 0 + 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ? = 0 + 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ? = 0 + 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,4,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ? = 0 + 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,4,1,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ? = 0 + 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,4,1,5,6] => ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0 + 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)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [3,5,1,2,6,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,1,5,6,2,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [3,6,1,2,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,5,1,2,6,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [4,1,5,6,2,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [4,5,1,6,2,3] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
Description
The second Elser number of a connected graph.
For a connected graph $G$ the $k$-th Elser number is
$$
els_k(G) = (-1)^{|V(G)|+1} \sum_N (-1)^{|E(N)|} |V(N)|^k
$$
where the sum is over all nuclei of $G$, that is, the connected subgraphs of $G$ whose vertex set is a vertex cover of $G$.
It is clear that this number is even. It was shown in [1] that it is non-negative.
Matching statistic: St001330
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 67%
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 67%
Values
[1,0,1,0]
=> [1,1,0,0]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 3 = 1 + 2
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 0 + 2
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 0 + 2
[1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [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)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 0 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [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)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 1 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [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)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [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)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 0 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [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)
=> ([(0,3),(0,7),(1,2),(1,7),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [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)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [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)
=> ([(0,3),(0,7),(1,2),(1,7),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 1 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 0 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 0 + 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 0 + 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 0 + 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,5),(0,9),(1,4),(1,9),(2,6),(2,8),(3,7),(3,8),(4,6),(4,10),(5,7),(5,10),(6,11),(7,11),(8,11),(9,10),(10,11)],12)
=> ? = 0 + 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [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)
=> ([(0,8),(1,5),(1,7),(2,4),(2,6),(3,6),(3,7),(4,8),(4,9),(5,8),(5,9),(6,9),(7,9)],10)
=> ? = 0 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,5),(0,9),(1,4),(1,9),(2,6),(2,8),(3,7),(3,8),(4,6),(4,10),(5,7),(5,10),(6,11),(7,11),(8,11),(9,10),(10,11)],12)
=> ? = 1 + 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [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)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> ? = 1 + 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [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)
=> ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ? = 1 + 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ([(0,1),(0,9),(1,8),(2,5),(2,9),(3,5),(3,6),(4,6),(4,7),(5,10),(6,10),(7,8),(7,10),(8,9),(9,10)],11)
=> ? = 0 + 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [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)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> ? = 0 + 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,3),(0,7),(1,2),(1,6),(2,8),(3,9),(4,5),(4,8),(4,9),(5,6),(5,7),(6,8),(7,9)],10)
=> ? = 0 + 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [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)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> ? = 0 + 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ? = 0 + 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ([(0,1),(0,9),(1,8),(2,5),(2,9),(3,5),(3,6),(4,6),(4,7),(5,10),(6,10),(7,8),(7,10),(8,9),(9,10)],11)
=> ? = 0 + 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [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)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> ? = 0 + 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [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)
=> ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [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)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ? = 0 + 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ([(0,1),(0,9),(1,8),(2,5),(2,9),(3,5),(3,6),(4,6),(4,7),(5,10),(6,10),(7,8),(7,10),(8,9),(9,10)],11)
=> ? = 1 + 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ([(0,3),(0,9),(1,2),(1,6),(2,8),(3,5),(4,5),(4,7),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 0 + 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [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)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ? = 0 + 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ([(0,3),(0,9),(1,2),(1,6),(2,8),(3,5),(4,5),(4,7),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 0 + 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ? = 0 + 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,3),(0,7),(1,2),(1,6),(2,8),(3,9),(4,5),(4,8),(4,9),(5,6),(5,7),(6,8),(7,9)],10)
=> ? = 0 + 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ([(0,3),(0,9),(1,2),(1,8),(2,6),(3,7),(4,7),(4,8),(5,6),(5,9),(6,8),(7,9),(8,9)],10)
=> ? = 0 + 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ? = 0 + 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ([(0,3),(0,9),(1,2),(1,8),(2,6),(3,7),(4,7),(4,8),(5,6),(5,9),(6,8),(7,9),(8,9)],10)
=> ? = 0 + 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 0 + 2
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 0 + 2
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 0 + 2
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 0 + 2
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 0 + 2
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 0 + 2
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 0 + 2
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 0 + 2
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 0 + 2
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 0 + 2
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000842
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000842: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000842: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Values
[1,0,1,0]
=> [1]
=> [1,0]
=> [2,1] => 2 = 0 + 2
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 1 + 2
[1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 0 + 2
[1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 0 + 2
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 2 = 0 + 2
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 2 = 0 + 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 2 = 0 + 2
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 2 = 0 + 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 2 = 0 + 2
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 2 = 0 + 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 0 + 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 2 = 0 + 2
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [8,1,4,5,2,7,3,6] => ? = 0 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,3,4,1,6,2,5] => ? = 1 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,8,7,3,6] => ? = 0 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ? = 0 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => 2 = 0 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [7,1,4,8,2,3,5,6] => ? = 0 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => ? = 0 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,1,2,8,7,3,5,6] => ? = 0 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => ? = 0 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => 2 = 0 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [8,1,2,3,7,4,5,6] => ? = 0 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 0 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 2 = 0 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 2 = 0 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 1 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => 2 = 0 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ? = 0 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 2 = 0 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 2 = 0 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => ? = 0 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => 2 = 0 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => ? = 0 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 2 = 0 + 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 2 = 0 + 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 2 = 0 + 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 2 = 0 + 2
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [10,1,4,6,2,7,9,3,5,8] => ? = 0 + 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [9,3,5,1,6,8,2,4,7] => ? = 0 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [4,1,2,10,6,7,9,3,5,8] => ? = 1 + 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [3,1,9,5,6,8,2,4,7] => ? = 1 + 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [2,8,4,5,7,1,3,6] => ? = 1 + 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [7,1,10,6,2,3,9,4,5,8] => ? = 0 + 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [6,9,5,1,2,8,3,4,7] => ? = 0 + 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [10,1,2,7,6,3,9,4,5,8] => ? = 0 + 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [9,1,6,5,2,8,3,4,7] => ? = 0 + 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [8,5,4,1,7,2,3,6] => ? = 0 + 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [10,1,2,3,6,7,9,4,5,8] => ? = 0 + 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [9,1,2,5,6,8,3,4,7] => ? = 0 + 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [8,1,4,5,7,2,3,6] => ? = 0 + 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => ? = 0 + 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [6,1,4,5,2,10,9,3,7,8] => ? = 1 + 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [5,3,4,1,9,8,2,6,7] => ? = 0 + 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [4,1,2,5,6,10,9,3,7,8] => ? = 0 + 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [3,1,4,5,9,8,2,6,7] => ? = 0 + 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [2,3,4,8,7,1,5,6] => ? = 0 + 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [10,1,6,5,2,3,9,4,7,8] => ? = 0 + 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [9,5,4,1,2,8,3,6,7] => ? = 0 + 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [10,1,2,5,6,3,9,4,7,8] => ? = 0 + 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [9,1,4,5,2,8,3,6,7] => ? = 0 + 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [8,3,4,1,7,2,5,6] => ? = 0 + 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [5,1,2,3,6,10,9,4,7,8] => ? = 0 + 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,1,2,5,9,8,3,6,7] => ? = 0 + 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [3,1,4,8,7,2,5,6] => ? = 0 + 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => ? = 0 + 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [10,1,9,5,2,3,4,6,7,8] => ? = 0 + 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [9,8,4,1,2,3,5,6,7] => ? = 0 + 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [10,1,2,5,9,3,4,6,7,8] => ? = 0 + 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [9,1,4,8,2,3,5,6,7] => ? = 0 + 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [8,3,7,1,2,4,5,6] => ? = 0 + 2
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [5,1,2,3,9,10,4,6,7,8] => ? = 0 + 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,1,2,8,9,3,5,6,7] => ? = 0 + 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => 2 = 0 + 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 2 = 0 + 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => 2 = 0 + 2
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 2 = 0 + 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => 2 = 0 + 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => 2 = 0 + 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => 2 = 0 + 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 2 = 0 + 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => 2 = 0 + 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => 2 = 0 + 2
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 2 = 0 + 2
[1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [8,1,2,7,6,3,4,5] => 2 = 0 + 2
[1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => 2 = 0 + 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => 2 = 0 + 2
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,1,2,3,4,5,6] => 2 = 0 + 2
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => 2 = 0 + 2
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,5,5]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [8,7,4,5,6,1,2,3] => 2 = 0 + 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [6,7,8,1,2,3,4,5] => 2 = 0 + 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [8,1,2,3,4,5,6,9,7] => 2 = 0 + 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => 2 = 0 + 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => 2 = 0 + 2
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,9,1,2,3,4,5,6,7] => 2 = 0 + 2
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => 2 = 0 + 2
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [9,1,2,3,4,5,6,7,10,8] => 2 = 0 + 2
Description
The breadth of a permutation.
According to [1, Def.1.6], this is the minimal Manhattan distance between two ones in the permutation matrix of $\pi$: $$\min\{|i-j|+|\pi(i)-\pi(j)|: i\neq j\}.$$
According to [1, Def.1.3], a permutation $\pi$ is $k$-prolific, if the set of permutations obtained from $\pi$ by deleting any $k$ elements and standardising has maximal cardinality, i.e., $\binom{n}{k}$.
By [1, Thm.2.22], a permutation is $k$-prolific if and only if its breath is at least $k+2$.
By [1, Cor.4.3], the smallest permutations that are $k$-prolific have size $\lceil k^2+2k+1\rceil$, and by [1, Thm.4.4], there are $k$-prolific permutations of any size larger than this.
According to [2] the proportion of $k$-prolific permutations in the set of all permutations is asymptotically equal to $\exp(-k^2-k)$.
Matching statistic: St000741
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000741: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000741: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [1,2] => ([],2)
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => ([(1,2)],3)
=> ? = 0 + 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> ? = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ? = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ? = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 0 + 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> ? = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> ? = 0 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [3,4,1,5,2,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,4,5,1,2,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,1,4,2,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [3,1,4,5,2,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,1,4,2,5,6] => ([(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [3,4,1,2,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,1,2,5,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 0 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4,6] => ([(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? = 0 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,4,2,6,5] => ([(1,2),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6] => ([(2,5),(3,5),(4,5)],6)
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => ([(4,5)],6)
=> ? = 0 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,3,6,2,4,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? = 0 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [3,1,5,6,2,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4,6] => ([(1,2),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [3,1,2,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [3,1,2,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [3,1,2,4,6,5] => ([(1,2),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,6,1,2,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 0 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [3,5,1,2,6,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 0 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,1,2,4,5,6] => ([(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,4,1,5,3,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 0 + 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,6,3,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,5,3,6] => ([(1,2),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,6] => ([(1,2),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,5,3,6,4] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,1,3,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> 1 = 0 + 1
Description
The Colin de Verdière graph invariant.
Matching statistic: St001060
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 67%
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 67%
Values
[1,0,1,0]
=> [1,1,0,0]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 0 + 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 3 = 1 + 2
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [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)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 0 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [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)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 1 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [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)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [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)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 0 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [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)
=> ([(0,3),(0,7),(1,2),(1,7),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [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)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [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)
=> ([(0,3),(0,7),(1,2),(1,7),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 0 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 1 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 0 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 0 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 0 + 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 0 + 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 0 + 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,5),(0,9),(1,4),(1,9),(2,6),(2,8),(3,7),(3,8),(4,6),(4,10),(5,7),(5,10),(6,11),(7,11),(8,11),(9,10),(10,11)],12)
=> ? = 0 + 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [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)
=> ([(0,8),(1,5),(1,7),(2,4),(2,6),(3,6),(3,7),(4,8),(4,9),(5,8),(5,9),(6,9),(7,9)],10)
=> ? = 0 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,5),(0,9),(1,4),(1,9),(2,6),(2,8),(3,7),(3,8),(4,6),(4,10),(5,7),(5,10),(6,11),(7,11),(8,11),(9,10),(10,11)],12)
=> ? = 1 + 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [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)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> ? = 1 + 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [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)
=> ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ? = 1 + 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ([(0,1),(0,9),(1,8),(2,5),(2,9),(3,5),(3,6),(4,6),(4,7),(5,10),(6,10),(7,8),(7,10),(8,9),(9,10)],11)
=> ? = 0 + 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [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)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> ? = 0 + 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,3),(0,7),(1,2),(1,6),(2,8),(3,9),(4,5),(4,8),(4,9),(5,6),(5,7),(6,8),(7,9)],10)
=> ? = 0 + 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [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)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> ? = 0 + 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ? = 0 + 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ([(0,1),(0,9),(1,8),(2,5),(2,9),(3,5),(3,6),(4,6),(4,7),(5,10),(6,10),(7,8),(7,10),(8,9),(9,10)],11)
=> ? = 0 + 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [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)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> ? = 0 + 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [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)
=> ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [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)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ? = 0 + 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ([(0,1),(0,9),(1,8),(2,5),(2,9),(3,5),(3,6),(4,6),(4,7),(5,10),(6,10),(7,8),(7,10),(8,9),(9,10)],11)
=> ? = 1 + 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ([(0,3),(0,9),(1,2),(1,6),(2,8),(3,5),(4,5),(4,7),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 0 + 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [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)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ? = 0 + 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ([(0,3),(0,9),(1,2),(1,6),(2,8),(3,5),(4,5),(4,7),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 0 + 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ? = 0 + 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,3),(0,7),(1,2),(1,6),(2,8),(3,9),(4,5),(4,8),(4,9),(5,6),(5,7),(6,8),(7,9)],10)
=> ? = 0 + 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ([(0,3),(0,9),(1,2),(1,8),(2,6),(3,7),(4,7),(4,8),(5,6),(5,9),(6,8),(7,9),(8,9)],10)
=> ? = 0 + 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ? = 0 + 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ([(0,3),(0,9),(1,2),(1,8),(2,6),(3,7),(4,7),(4,8),(5,6),(5,9),(6,8),(7,9),(8,9)],10)
=> ? = 0 + 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ([(0,6),(1,2),(1,3),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 0 + 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ([(0,3),(0,9),(1,2),(1,6),(2,8),(3,5),(4,5),(4,7),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 0 + 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ? = 0 + 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [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)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ? = 0 + 2
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ([(0,3),(0,9),(1,2),(1,6),(2,8),(3,5),(4,5),(4,7),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 0 + 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ? = 0 + 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ([(0,1),(0,9),(1,8),(2,5),(2,9),(3,5),(3,6),(4,6),(4,7),(5,10),(6,10),(7,8),(7,10),(8,9),(9,10)],11)
=> ? = 0 + 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ? = 0 + 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ? = 0 + 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ([(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)
=> ([(0,8),(1,5),(1,7),(2,4),(2,6),(3,6),(3,7),(4,8),(4,9),(5,8),(5,9),(6,9),(7,9)],10)
=> ? = 0 + 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [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)
=> ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> ? = 0 + 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St001198
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 67%
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 67%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? = 0 + 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 2
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 0 + 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 0 + 2
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 0 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [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
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [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
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [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
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [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 = 0 + 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [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
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [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 = 0 + 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [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 = 0 + 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> 2 = 0 + 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 2 = 0 + 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [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 = 0 + 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [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 = 0 + 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [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 = 0 + 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [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 = 0 + 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [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 = 0 + 2
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [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 = 0 + 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [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 = 0 + 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [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 = 0 + 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [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 = 0 + 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [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 = 0 + 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 2 = 0 + 2
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [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 = 0 + 2
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [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 = 0 + 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [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 = 0 + 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 2 = 0 + 2
Description
The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001206
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 67%
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 67%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? = 0 + 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 2
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 0 + 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 0 + 2
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 0 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [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
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [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
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [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
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [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 = 0 + 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [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
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [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 = 0 + 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [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 = 0 + 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> 2 = 0 + 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 2 = 0 + 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [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 = 0 + 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [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 = 0 + 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [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 = 0 + 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [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 = 0 + 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [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 = 0 + 2
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [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 = 0 + 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [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 = 0 + 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [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 = 0 + 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [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 = 0 + 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [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 = 0 + 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 2 = 0 + 2
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [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 = 0 + 2
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [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 = 0 + 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [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 = 0 + 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 2 = 0 + 2
Description
The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.
The following 4 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001846The number of elements which do not have a complement in the lattice. St001162The minimum jump of a permutation. St001820The size of the image of the pop stack sorting operator. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
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