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Your data matches 19 different statistics following compositions of up to 3 maps.
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Matching statistic: St001257
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
St001257: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1
[1,0,1,0]
=> 2
[1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> 2
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> 1
Description
The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J.
Matching statistic: St000908
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000908: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
St000908: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> 1
[1,0,1,0]
=> [2,1] => ([],2)
=> 2
[1,1,0,0]
=> [1,2] => ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [2,3,1] => ([(1,2)],3)
=> 2
[1,0,1,1,0,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => ([(1,2)],3)
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
Description
The length of the shortest maximal antichain in a poset.
Matching statistic: St000273
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000273: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000273: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 1
[1,0,1,0]
=> [2,1] => [1,2] => ([],2)
=> 2
[1,1,0,0]
=> [1,2] => [2,1] => ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [2,3,1] => [1,3,2] => ([(1,2)],3)
=> 2
[1,0,1,1,0,0]
=> [2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => [2,1,3] => ([(1,2)],3)
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
Description
The domination number of a graph.
The domination number of a graph is given by the minimum size of a dominating set of vertices. A dominating set of vertices is a subset of the vertex set of such that every vertex is either in this subset or adjacent to an element of this subset.
Matching statistic: St001322
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001322: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001322: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 1
[1,0,1,0]
=> [2,1] => [1,2] => ([],2)
=> 2
[1,1,0,0]
=> [1,2] => [2,1] => ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [2,3,1] => [1,3,2] => ([(1,2)],3)
=> 2
[1,0,1,1,0,0]
=> [2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => [2,1,3] => ([(1,2)],3)
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
Description
The size of a minimal independent dominating set in a graph.
Matching statistic: St001339
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001339: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001339: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 1
[1,0,1,0]
=> [2,1] => [1,2] => ([],2)
=> 2
[1,1,0,0]
=> [1,2] => [2,1] => ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [2,3,1] => [1,3,2] => ([(1,2)],3)
=> 2
[1,0,1,1,0,0]
=> [2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => [2,1,3] => ([(1,2)],3)
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
Description
The irredundance number of a graph.
A set $S$ of vertices is irredundant, if there is no vertex in $S$, whose closed neighbourhood is contained in the union of the closed neighbourhoods of the other vertices of $S$.
The irredundance number is the smallest size of a maximal irredundant set.
Matching statistic: St001829
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001829: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001829: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 1
[1,0,1,0]
=> [2,1] => [1,2] => ([],2)
=> 2
[1,1,0,0]
=> [1,2] => [2,1] => ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [2,3,1] => [1,3,2] => ([(1,2)],3)
=> 2
[1,0,1,1,0,0]
=> [2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => [2,1,3] => ([(1,2)],3)
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
Description
The common independence number of a graph.
The common independence number of a graph $G$ is the greatest integer $r$ such that every vertex of $G$ belongs to some independent set $X$ of vertices of cardinality at least $r$.
Matching statistic: St000674
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 0 = 1 - 1
Description
The number of hills of a Dyck path.
A hill is a peak with up step starting and down step ending at height zero.
Matching statistic: St001271
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001271: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001271: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => ([],1)
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => ([],2)
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
Description
The competition number of a graph.
The competition graph of a digraph $D$ is a (simple undirected) graph which has the same vertex set as $D$ and has an edge between $x$ and $y$ if and only if there exists a vertex $v$ in $D$ such that $(x, v)$ and $(y, v)$ are arcs of $D$. For any graph, $G$ together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number $k(G)$ is the smallest number of such isolated vertices.
Matching statistic: St000906
(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
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000906: Posets ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000906: Posets ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ([],1)
=> ? = 1
[1,0,1,0]
=> [2,1] => [1,2] => ([(0,1)],2)
=> 2
[1,1,0,0]
=> [1,2] => [2,1] => ([],2)
=> 1
[1,0,1,0,1,0]
=> [2,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[1,0,1,1,0,0]
=> [2,1,3] => [3,1,2] => ([(1,2)],3)
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [2,3,1] => ([(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => ([],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,3,2,1] => ([],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => ([(1,2),(1,3),(1,4)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [5,4,1,3,2] => ([(2,3),(2,4)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => ([(3,4)],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [5,2,1,4,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> 1
Description
The length of the shortest maximal chain in a poset.
Matching statistic: St001217
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001217: Dyck paths ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001217: Dyck paths ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [2,1] => [2]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0]
=> [1,2] => [1,1]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,6,1,5] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,5,1,6,4] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,6,1,4,5] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,4,1,5,6,3] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,1,3] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,6,3,5] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,5,1,3,6,4] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,5,6,1,3,4] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,6,1,3,4,5] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,4,5,6,2] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,1,4,6,2,5] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,1,6,2] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,4,6,1,2,5] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [3,1,5,2,6,4] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,6,2,4,5] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [4,1,2,5,6,3] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,1,5,6,2,3] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [4,5,1,2,6,3] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,2,6,3,5] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [4,6,1,2,3,5] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,6,4] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [5,1,6,2,3,4] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,1,2,3,4,5] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 - 1
Description
The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1.
The following 9 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000546The number of global descents of a permutation. St000260The radius of a connected graph. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001545The second Elser number of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
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