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Your data matches 11 different statistics following compositions of up to 3 maps.
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Matching statistic: St000846
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> 0
[1,0,1,0]
=> [2,1] => ([],2)
=> 0
[1,1,0,0]
=> [1,2] => ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => ([],3)
=> 0
[1,0,1,1,0,0]
=> [2,3,1] => ([(1,2)],3)
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => ([(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => ([(0,2),(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]
=> [4,3,2,1] => ([],4)
=> 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => ([(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => ([(2,3)],4)
=> 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(2,3)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],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]
=> [5,4,3,2,1] => ([],5)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => ([(3,4)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => ([(3,4)],5)
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => ([(3,4)],5)
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => ([(3,4)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
Description
The maximal number of elements covering an element of a poset.
Matching statistic: St001933
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 86% ●values known / values provided: 91%●distinct values known / distinct values provided: 86%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 86% ●values known / values provided: 91%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> [1,0]
=> []
=> []
=> ? = 0
[1,0,1,0]
=> [1,1,0,0]
=> []
=> []
=> ? = 0
[1,1,0,0]
=> [1,0,1,0]
=> [1]
=> [1]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> [2]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> [1,1]
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> [2,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [3]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [2]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2,2]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [3,2]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [2,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [3,1,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [3,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [2,1,1]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2,2,1]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [3,2,1]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [4]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [3]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [3,3]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [4,3]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [2]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [3,2]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [2,2]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [2,2,2]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [4,2,2]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [4,2]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,2,2]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,3,2]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [4,3,2]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [3,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [2,2,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [4,2,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [2,1,1]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [4,1,1,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [4,1,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [3,1,1,1]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [3,3,1,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [4,3,1,1]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [4,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [3,2,1]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [3,1,1]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [2,1,1,1]
=> 3
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [3,2,1,1]
=> 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
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2]
=> [5,5,4,2,2]
=> ? = 2
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> [6,5,4,2,2]
=> ? = 2
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2]
=> [5,5,3,3,2]
=> ? = 2
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2,1]
=> [6,5,3,2,2]
=> ? = 2
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2]
=> [5,5,4,3,2]
=> ? = 2
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2,1]
=> [6,5,4,1,1,1]
=> ? = 3
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2]
=> [5,5,3,3,1,1]
=> ? = 2
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> [6,5,3,3,1,1]
=> ? = 2
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1,1]
=> [6,4,4,3,1,1]
=> ? = 2
[1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1,1]
=> [6,4,3,3,1,1]
=> ? = 2
[1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1]
=> [5,4,4,3,1,1]
=> ? = 2
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2]
=> [5,5,4,3,1,1]
=> ? = 2
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2,1]
=> [6,5,4,3,1,1]
=> ? = 2
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> [6,5,3,2,1,1]
=> ? = 2
[1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,2,1]
=> [6,5,2,2,2,1]
=> ? = 3
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1,1]
=> [6,4,4,2,2,1]
=> ? = 2
[1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1]
=> [5,4,4,2,2,1]
=> ? = 2
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2]
=> [5,5,4,2,2,1]
=> ? = 2
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> [6,5,4,2,2,1]
=> ? = 2
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,1]
=> [6,4,3,2,2,1]
=> ? = 2
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,1,1]
=> [6,3,3,3,2,1]
=> ? = 3
[1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1,1]
=> [6,4,4,2,1,1]
=> ? = 2
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2]
=> [5,5,3,3,2,1]
=> ? = 2
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> [6,5,3,3,2,1]
=> ? = 2
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1]
=> [5,4,3,3,2,1]
=> ? = 2
[1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> [5,5,4,3,1]
=> ? = 2
[1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> [5,5,4,2,1,1]
=> ? = 2
[1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2]
=> [5,5,3,2,2,1]
=> ? = 2
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> [4,4,4,3,2,1]
=> ? = 3
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,1]
=> [6,4,4,3,2,1]
=> ? = 2
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> [6,5,4,2,1,1]
=> ? = 2
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2,1]
=> [6,5,3,2,2,1]
=> ? = 2
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,1]
=> [6,4,3,3,2,1]
=> ? = 2
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1]
=> [5,4,4,3,2,1]
=> ? = 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> [5,5,4,3,2,1]
=> ? = 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [4,4,4,3,2,1]
=> [6,5,4,3]
=> ? = 1
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> ? = 1
[1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [8,6,6,4,4,2,2]
=> [7,7,5,5,3,3,1,1]
=> ? = 2
[1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [8,6,4,4,2,2]
=> ?
=> ? = 2
Description
The largest multiplicity of a part in an integer partition.
Matching statistic: St000845
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000845: Posets ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000845: Posets ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 0
[1,0,1,0]
=> [1,2] => [2,1] => ([],2)
=> 0
[1,1,0,0]
=> [2,1] => [1,2] => ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => ([],3)
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => ([(1,2)],3)
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [3,1,2] => ([(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => ([],4)
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => ([(3,4)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [5,4,1,3,2] => ([(2,3),(2,4)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => ([(1,2),(1,3),(1,4)],5)
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [5,1,3,4,2] => ([(1,3),(1,4),(4,2)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,3,2,6,7,5,4] => [4,5,7,6,2,3,1] => ([(1,5),(2,6),(6,3),(6,4)],7)
=> ? = 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,3,4,2,6,7,5] => [5,7,6,2,4,3,1] => ([(1,5),(1,6),(2,3),(2,4)],7)
=> ? = 2
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,3,4,2,7,6,5] => [5,6,7,2,4,3,1] => ([(1,6),(2,4),(2,5),(6,3)],7)
=> ? = 2
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,6,7,5,4,2] => [2,4,5,7,6,3,1] => ([(1,4),(1,5),(5,6),(6,2),(6,3)],7)
=> ? = 2
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,7,6,5,4,2] => [2,4,5,6,7,3,1] => ([(1,3),(1,6),(4,5),(5,2),(6,4)],7)
=> ? = 2
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,4,3,2,6,7,5] => [5,7,6,2,3,4,1] => ([(1,6),(2,4),(2,5),(6,3)],7)
=> ? = 2
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,4,5,3,2,7,6] => [6,7,2,3,5,4,1] => ([(1,5),(2,6),(6,3),(6,4)],7)
=> ? = 2
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,4,5,7,6,3,2] => [2,3,6,7,5,4,1] => ([(1,6),(5,4),(6,2),(6,3),(6,5)],7)
=> ? = 3
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,4,6,5,7,3,2] => [2,3,7,5,6,4,1] => ([(1,6),(5,4),(6,2),(6,3),(6,5)],7)
=> ? = 3
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,7,5,3,2] => [2,3,5,7,6,4,1] => ([(1,6),(5,3),(5,4),(6,2),(6,5)],7)
=> ? = 2
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,5,4,6,7,3,2] => [2,3,7,6,4,5,1] => ([(1,6),(5,4),(6,2),(6,3),(6,5)],7)
=> ? = 3
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,5,6,4,3,7,2] => [2,7,3,4,6,5,1] => ([(1,4),(1,5),(5,6),(6,2),(6,3)],7)
=> ? = 2
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,5,6,4,7,3,2] => [2,3,7,4,6,5,1] => ([(1,6),(5,3),(5,4),(6,2),(6,5)],7)
=> ? = 2
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,5,4,3,7,2] => [2,7,3,4,5,6,1] => ([(1,3),(1,6),(4,5),(5,2),(6,4)],7)
=> ? = 2
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => [2,3,4,5,7,6,1] => ([(1,5),(4,6),(5,4),(6,2),(6,3)],7)
=> ? = 2
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [2,1,3,6,7,5,4] => [4,5,7,6,3,1,2] => ([(1,5),(2,6),(6,3),(6,4)],7)
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,3,6,7,5] => [5,7,6,3,4,1,2] => ([(0,6),(1,5),(2,3),(2,4)],7)
=> ? = 2
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [2,1,4,5,3,7,6] => [6,7,3,5,4,1,2] => ([(0,6),(1,5),(2,3),(2,4)],7)
=> ? = 2
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [2,1,4,5,7,6,3] => [3,6,7,5,4,1,2] => ([(0,5),(1,3),(1,4),(1,6),(6,2)],7)
=> ? = 3
[1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [2,1,4,6,5,7,3] => [3,7,5,6,4,1,2] => ([(0,5),(1,3),(1,4),(1,6),(6,2)],7)
=> ? = 3
[1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [2,1,4,6,7,5,3] => [3,5,7,6,4,1,2] => ([(0,5),(1,4),(1,6),(6,2),(6,3)],7)
=> ? = 2
[1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [2,1,5,4,6,7,3] => [3,7,6,4,5,1,2] => ([(0,5),(1,3),(1,4),(1,6),(6,2)],7)
=> ? = 3
[1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,5,6,4,3,7] => [7,3,4,6,5,1,2] => ([(1,5),(2,6),(6,3),(6,4)],7)
=> ? = 2
[1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,1,5,6,4,7,3] => [3,7,4,6,5,1,2] => ([(0,5),(1,4),(1,6),(6,2),(6,3)],7)
=> ? = 2
[1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [2,1,5,6,7,4,3] => [3,4,7,6,5,1,2] => ([(0,5),(1,6),(6,2),(6,3),(6,4)],7)
=> ? = 3
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,6,7,5,4,3] => [3,4,5,7,6,1,2] => ([(0,4),(1,5),(5,6),(6,2),(6,3)],7)
=> ? = 2
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [2,3,1,4,6,7,5] => [5,7,6,4,1,3,2] => ([(1,5),(1,6),(2,3),(2,4)],7)
=> ? = 2
[1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,7,6,5] => [5,6,7,4,1,3,2] => ([(1,6),(2,4),(2,5),(6,3)],7)
=> ? = 2
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,7,6] => [6,7,4,5,1,3,2] => ([(0,6),(1,5),(2,3),(2,4)],7)
=> ? = 2
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [2,3,1,5,6,4,7] => [7,4,6,5,1,3,2] => ([(1,5),(1,6),(2,3),(2,4)],7)
=> ? = 2
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,6,7,4] => [4,7,6,5,1,3,2] => ([(0,5),(0,6),(1,2),(1,3),(1,4)],7)
=> ? = 3
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [2,3,1,5,7,6,4] => [4,6,7,5,1,3,2] => ([(0,3),(0,4),(1,5),(1,6),(6,2)],7)
=> ? = 2
[1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,1,6,5,4,7] => [7,4,5,6,1,3,2] => ([(1,6),(2,4),(2,5),(6,3)],7)
=> ? = 2
[1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [2,3,1,6,5,7,4] => [4,7,5,6,1,3,2] => ([(0,3),(0,4),(1,5),(1,6),(6,2)],7)
=> ? = 2
[1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [2,3,1,6,7,5,4] => [4,5,7,6,1,3,2] => ([(0,6),(1,4),(1,5),(6,2),(6,3)],7)
=> ? = 2
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [2,3,1,7,6,5,4] => [4,5,6,7,1,3,2] => ([(0,6),(1,3),(1,4),(5,2),(6,5)],7)
=> ? = 2
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [2,3,4,1,6,7,5] => [5,7,6,1,4,3,2] => ([(0,5),(0,6),(1,2),(1,3),(1,4)],7)
=> ? = 3
[1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,4,1,7,6,5] => [5,6,7,1,4,3,2] => ([(0,6),(1,3),(1,4),(1,5),(6,2)],7)
=> ? = 3
[1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [2,3,5,4,1,7,6] => [6,7,1,4,5,3,2] => ([(0,5),(1,3),(1,4),(1,6),(6,2)],7)
=> ? = 3
[1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [2,4,3,1,6,7,5] => [5,7,6,1,3,4,2] => ([(0,3),(0,4),(1,5),(1,6),(6,2)],7)
=> ? = 2
[1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,4,3,5,1,7,6] => [6,7,1,5,3,4,2] => ([(0,5),(1,3),(1,4),(1,6),(6,2)],7)
=> ? = 3
[1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [2,4,5,3,1,7,6] => [6,7,1,3,5,4,2] => ([(0,5),(1,4),(1,6),(6,2),(6,3)],7)
=> ? = 2
[1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [2,5,6,4,3,1,7] => [7,1,3,4,6,5,2] => ([(1,4),(1,5),(5,6),(6,2),(6,3)],7)
=> ? = 2
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [2,6,5,4,3,1,7] => [7,1,3,4,5,6,2] => ([(1,3),(1,6),(4,5),(5,2),(6,4)],7)
=> ? = 2
[1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [3,2,1,4,6,7,5] => [5,7,6,4,1,2,3] => ([(1,6),(2,4),(2,5),(6,3)],7)
=> ? = 2
[1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [3,2,1,5,6,4,7] => [7,4,6,5,1,2,3] => ([(1,6),(2,4),(2,5),(6,3)],7)
=> ? = 2
[1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [3,2,1,5,6,7,4] => [4,7,6,5,1,2,3] => ([(0,6),(1,3),(1,4),(1,5),(6,2)],7)
=> ? = 3
[1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [3,2,1,6,7,5,4] => [4,5,7,6,1,2,3] => ([(0,6),(1,5),(5,4),(6,2),(6,3)],7)
=> ? = 2
[1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [3,2,4,1,6,7,5] => [5,7,6,1,4,2,3] => ([(0,3),(0,4),(1,5),(1,6),(6,2)],7)
=> ? = 2
[1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [3,2,4,5,1,7,6] => [6,7,1,5,4,2,3] => ([(0,5),(1,3),(1,4),(1,6),(6,2)],7)
=> ? = 3
Description
The maximal number of elements covered by an element in a poset.
Matching statistic: St000392
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 86%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> [1,0]
=> []
=> => ? = 0
[1,0,1,0]
=> [1,1,0,0]
=> []
=> => ? = 0
[1,1,0,0]
=> [1,0,1,0]
=> [1]
=> 10 => 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> => ? = 0
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> 10 => 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2]
=> 100 => 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 110 => 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1010 => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> => ? = 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1100 => 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 10110 => 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 10100 => 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 100110 => 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 11010 => 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101010 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? = 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 10 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 100 => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 110 => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1010 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1000 => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 10010 => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 1100 => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1110 => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 10110 => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 10100 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 100110 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 11010 => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 101010 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 10000 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 100010 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 100100 => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1000110 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1001010 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 11000 => 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 110010 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 11100 => 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 101100 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1001110 => 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 110110 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1010110 => 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 101000 => 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1010010 => 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1001100 => 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 10001110 => 3
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 10010110 => 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
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> => ? = 0
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [6,2,2,1]
=> 1000011010 => ? = 2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,1]
=> 1001000110 => ? = 2
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [6,3,3,1]
=> 1000110010 => ? = 2
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2]
=> 1000011100 => ? = 3
[1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,1,1]
=> 10001001110 => ? = 3
[1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,2,1,1]
=> 10000110110 => ? = 2
[1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,1,1]
=> 10010001110 => ? = 3
[1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [6,3,3,2]
=> 1000110100 => ? = 2
[1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [6,3,3,1,1]
=> 10001100110 => ? = 2
[1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,1]
=> 10000111010 => ? = 3
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [6,3,3,2,1]
=> 10001101010 => ? = 2
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [5,5,1,1,1]
=> 1100001110 => ? = 3
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [5,5,2,1,1]
=> 1100010110 => ? = 2
[1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,1]
=> 1100011010 => ? = 2
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [4,3,1,1,1,1]
=> 1010011110 => ? = 4
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [4,2,2,1,1,1]
=> 1001101110 => ? = 3
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1,1,1]
=> 1010101110 => ? = 3
[1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [5,4,4,1,1]
=> 1011000110 => ? = 2
[1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,1]
=> 1001110010 => ? = 3
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [5,3,1,1,1,1]
=> 10010011110 => ? = 4
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,2,1,1,1]
=> 10001101110 => ? = 3
[1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1,1,1]
=> 10010101110 => ? = 3
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,4,1,1,1,1]
=> 1100011110 => ? = 4
[1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,4,2,1,1,1]
=> 1100101110 => ? = 3
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [4,3,3,1,1,1]
=> 1011001110 => ? = 3
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [4,2,2,2,1,1]
=> 1001110110 => ? = 3
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,1,1]
=> 1010110110 => ? = 2
[1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [5,4,2,2,2]
=> 1010011100 => ? = 3
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [5,4,1,1,1,1]
=> 10100011110 => ? = 4
[1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [5,4,2,1,1,1]
=> 10100101110 => ? = 3
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [5,3,3,1,1,1]
=> 10011001110 => ? = 3
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [5,2,2,2,1,1]
=> 10001110110 => ? = 3
[1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [5,3,2,2,1,1]
=> 10010110110 => ? = 2
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [4,4,2,2,1,1]
=> 1100110110 => ? = 2
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [4,3,3,2,1,1]
=> 1011010110 => ? = 2
[1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [5,4,3,1,1,1]
=> 10101001110 => ? = 3
[1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [5,4,2,2,1,1]
=> 10100110110 => ? = 2
[1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [5,3,3,2,1,1]
=> 10011010110 => ? = 2
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,1,1]
=> 1101010110 => ? = 2
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,1]
=> 10101010110 => ? = 2
[1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,1]
=> 1010000110 => ? = 2
[1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2]
=> 1010001100 => ? = 2
[1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,1,1]
=> 10100001110 => ? = 3
Description
The length of the longest run of ones in a binary word.
Matching statistic: St001372
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 86%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> [1,0]
=> []
=> => ? = 0
[1,0,1,0]
=> [1,1,0,0]
=> []
=> => ? = 0
[1,1,0,0]
=> [1,0,1,0]
=> [1]
=> 10 => 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> => ? = 0
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> 10 => 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2]
=> 100 => 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 110 => 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1010 => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> => ? = 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1100 => 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 10110 => 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 10100 => 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 100110 => 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 11010 => 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101010 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? = 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 10 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 100 => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 110 => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1010 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1000 => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 10010 => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 1100 => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1110 => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 10110 => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 10100 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 100110 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 11010 => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 101010 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 10000 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 100010 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 100100 => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1000110 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1001010 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 11000 => 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 110010 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 11100 => 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 101100 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1001110 => 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 110110 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1010110 => 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 101000 => 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1010010 => 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1001100 => 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 10001110 => 3
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 10010110 => 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
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> => ? = 0
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [6,2,2,1]
=> 1000011010 => ? = 2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,1]
=> 1001000110 => ? = 2
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [6,3,3,1]
=> 1000110010 => ? = 2
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2]
=> 1000011100 => ? = 3
[1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,1,1]
=> 10001001110 => ? = 3
[1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,2,1,1]
=> 10000110110 => ? = 2
[1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,1,1]
=> 10010001110 => ? = 3
[1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [6,3,3,2]
=> 1000110100 => ? = 2
[1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [6,3,3,1,1]
=> 10001100110 => ? = 2
[1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,1]
=> 10000111010 => ? = 3
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [6,3,3,2,1]
=> 10001101010 => ? = 2
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [5,5,1,1,1]
=> 1100001110 => ? = 3
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [5,5,2,1,1]
=> 1100010110 => ? = 2
[1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,1]
=> 1100011010 => ? = 2
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [4,3,1,1,1,1]
=> 1010011110 => ? = 4
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [4,2,2,1,1,1]
=> 1001101110 => ? = 3
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1,1,1]
=> 1010101110 => ? = 3
[1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [5,4,4,1,1]
=> 1011000110 => ? = 2
[1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,1]
=> 1001110010 => ? = 3
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [5,3,1,1,1,1]
=> 10010011110 => ? = 4
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,2,1,1,1]
=> 10001101110 => ? = 3
[1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1,1,1]
=> 10010101110 => ? = 3
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,4,1,1,1,1]
=> 1100011110 => ? = 4
[1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,4,2,1,1,1]
=> 1100101110 => ? = 3
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [4,3,3,1,1,1]
=> 1011001110 => ? = 3
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [4,2,2,2,1,1]
=> 1001110110 => ? = 3
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,1,1]
=> 1010110110 => ? = 2
[1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [5,4,2,2,2]
=> 1010011100 => ? = 3
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [5,4,1,1,1,1]
=> 10100011110 => ? = 4
[1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [5,4,2,1,1,1]
=> 10100101110 => ? = 3
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [5,3,3,1,1,1]
=> 10011001110 => ? = 3
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [5,2,2,2,1,1]
=> 10001110110 => ? = 3
[1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [5,3,2,2,1,1]
=> 10010110110 => ? = 2
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [4,4,2,2,1,1]
=> 1100110110 => ? = 2
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [4,3,3,2,1,1]
=> 1011010110 => ? = 2
[1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [5,4,3,1,1,1]
=> 10101001110 => ? = 3
[1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [5,4,2,2,1,1]
=> 10100110110 => ? = 2
[1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [5,3,3,2,1,1]
=> 10011010110 => ? = 2
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,1,1]
=> 1101010110 => ? = 2
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,1]
=> 10101010110 => ? = 2
[1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,1]
=> 1010000110 => ? = 2
[1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2]
=> 1010001100 => ? = 2
[1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,1,1]
=> 10100001110 => ? = 3
Description
The length of a longest cyclic run of ones of a binary word.
Consider the binary word as a cyclic arrangement of ones and zeros. Then this statistic is the length of the longest continuous sequence of ones in this arrangement.
Matching statistic: St001330
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,0,1,0]
=> [1,2] => [1,2] => ([],2)
=> 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => ([],3)
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,1,1,0,0,0]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [1,2,3,4,6,5] => ([(4,5)],6)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [1,2,3,5,4,6] => ([(4,5)],6)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,4,5] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => [1,2,4,3,5,6] => ([(4,5)],6)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,3,5] => [1,2,6,4,3,5] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,3,6,4] => [1,2,6,5,3,4] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,5,6,3,4] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,4,6,2,5] => [1,6,4,3,2,5] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,5,2,4,6] => [1,5,3,2,4,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,5,2,6,4] => [1,6,5,3,2,4] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,3,5,6,2,4] => [1,6,3,2,5,4] => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,6,2,4,5] => [1,6,3,2,4,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3,6] => [1,5,4,2,3,6] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,6,3] => [1,6,5,4,2,3] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,6,3,5] => [1,6,4,2,3,5] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,5,2,3,6] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,4,5,2,6,3] => [1,6,5,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,4,5,6,2,3] => [1,6,2,5,4,3] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,2,3,5] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,6,4] => [1,6,5,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,6,3,4] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,2,3,4] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,3,5] => [2,1,6,4,3,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,6,4] => [2,1,6,5,3,4] => ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,1,5,6,3,4] => [2,1,6,3,5,4] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,1,5] => [6,4,3,2,1,5] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,3,5,1,4,6] => [5,3,2,1,4,6] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,1,6,4] => [6,5,3,2,1,4] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,6,1,4] => [6,3,2,1,5,4] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,1,4,5] => [6,3,2,1,4,5] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,4,1,3,5,6] => [4,2,1,3,5,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,6,5] => [4,2,1,3,6,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,4,1,5,3,6] => [5,4,2,1,3,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000455
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 29%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 29%
Values
[1,0]
=> [1] => ([],1)
=> ([(0,1)],2)
=> -1 = 0 - 1
[1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 0 - 1
[1,1,0,0]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 0 - 1
[1,0,1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> -1 = 0 - 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> -1 = 0 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> -1 = 0 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,2,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,6,3,2,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,5,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,6,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [4,5,3,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,5,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [5,3,4,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [6,4,2,3,5,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St001864
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001864: Signed permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 57%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001864: Signed permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 57%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => [3,1,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => [2,3,1] => 2
[1,1,1,0,0,0]
=> [3,1,2] => [1,3,2] => [1,3,2] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => [3,1,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1,4,2] => [3,1,4,2] => 2
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [4,2,1,3] => [4,2,1,3] => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,4,1] => [2,3,4,1] => 3
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [3,4,1,2] => [3,4,1,2] => 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [1,3,4,2] => [1,3,4,2] => 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,1,2,5,3] => [4,1,2,5,3] => ? = 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [3,1,4,5,2] => [3,1,4,5,2] => ? = 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [4,2,1,5,3] => [4,2,1,5,3] => ? = 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [5,2,3,1,4] => [5,2,3,1,4] => ? = 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => [4,2,5,1,3] => ? = 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 3
[1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 2
[1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => 2
[1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 2
[1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => [1,3,4,5,2] => [1,3,4,5,2] => 3
[1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => [4,1,5,2,3] => [4,1,5,2,3] => ? = 2
[1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => [1,2,4,5,3] => [1,2,4,5,3] => 2
[1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [6,1,2,3,4,5] => [6,1,2,3,4,5] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [5,1,2,3,4,6] => [5,1,2,3,4,6] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [5,1,2,3,6,4] => [5,1,2,3,6,4] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,6,2,3,4,5] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => [4,1,2,3,5,6] => [4,1,2,3,5,6] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => [6,4,1,2,3,5] => [6,4,1,2,3,5] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => [4,1,2,5,3,6] => [4,1,2,5,3,6] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => [4,1,2,5,6,3] => [4,1,2,5,6,3] => ? = 3
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,3,5] => [4,1,2,3,6,5] => [4,1,2,3,6,5] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,3,4,6] => [1,5,2,3,4,6] => [1,5,2,3,4,6] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,3,6,4] => [5,6,1,2,3,4] => [5,6,1,2,3,4] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,5,6,3,4] => [1,5,2,3,6,4] => [1,5,2,3,6,4] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => [3,1,2,4,5,6] => [3,1,2,4,5,6] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => [6,3,1,2,4,5] => [6,3,1,2,4,5] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => [5,3,1,2,4,6] => [5,3,1,2,4,6] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,3,2,5,6,4] => [5,3,1,2,6,4] => [5,3,1,2,6,4] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,4,5] => [3,6,1,2,4,5] => [3,6,1,2,4,5] => ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,4,2,5,6] => [3,1,4,2,5,6] => [3,1,4,2,5,6] => ? = 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,4,2,6,5] => [6,3,1,4,2,5] => [6,3,1,4,2,5] => ? = 2
Description
The number of excedances of a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, this is $\lvert\{i\in[n] \mid \pi(i) > i\}\rvert$.
Matching statistic: St001431
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001431: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 57%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001431: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 57%
Values
[1,0]
=> [1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> ? = 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> ? = 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> ? = 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 3
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 2
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 2
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> ? = 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> ? = 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> ? = 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 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]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 3
Description
Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.
The modified algebra B is obtained from the stable Auslander algebra kQ/I by deleting all relations which contain walks of length at least three (conjectural this step of deletion is not necessary as the stable higher Auslander algebras might be quadratic) and taking as B then the algebra kQ^(op)/J when J is the quadratic perp of the ideal I.
See http://www.findstat.org/DyckPaths/NakayamaAlgebras for the definition of Loewy length and Nakayama algebras associated to Dyck paths.
Matching statistic: St001905
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001905: Parking functions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 57%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001905: Parking functions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 57%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [2,1] => [1,2] => [1,2] => 0
[1,1,0,0]
=> [1,2] => [2,1] => [2,1] => 1
[1,0,1,0,1,0]
=> [2,3,1] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [2,1,3] => [1,3,2] => [1,3,2] => 1
[1,1,0,0,1,0]
=> [1,3,2] => [3,2,1] => [3,2,1] => 1
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => [3,1,2] => 2
[1,1,1,0,0,0]
=> [1,2,3] => [2,3,1] => [2,3,1] => 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [1,2,4,3] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [1,4,3,2] => [1,4,3,2] => 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [1,3,4,2] => [1,3,4,2] => 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [4,2,3,1] => [4,2,3,1] => 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [3,2,4,1] => [3,2,4,1] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => [4,1,3,2] => 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,1,2,3] => [4,1,2,3] => 3
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,4,2] => [3,1,4,2] => 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [2,4,3,1] => [2,4,3,1] => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,4,2,1] => [3,4,2,1] => 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [3,4,1,2] => [3,4,1,2] => 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [1,2,5,4,3] => [1,2,5,4,3] => ? = 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => ? = 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [1,4,3,5,2] => [1,4,3,5,2] => ? = 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [1,5,2,4,3] => [1,5,2,4,3] => ? = 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 3
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [1,3,5,4,2] => [1,3,5,4,2] => ? = 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [1,4,5,3,2] => [1,4,5,3,2] => ? = 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [1,4,5,2,3] => [1,4,5,2,3] => ? = 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [1,3,4,5,2] => [1,3,4,5,2] => ? = 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [4,2,3,5,1] => [4,2,3,5,1] => ? = 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [4,2,5,3,1] => [4,2,5,3,1] => ? = 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => [5,1,3,4,2] => ? = 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,1,3,5,2] => [4,1,3,5,2] => ? = 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 3
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 4
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,1,2,5,3] => [4,1,2,5,3] => ? = 3
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,5,4,2] => [3,1,5,4,2] => ? = 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [4,1,5,3,2] => [4,1,5,3,2] => ? = 3
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [4,1,5,2,3] => [4,1,5,2,3] => ? = 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,4,5,2] => [3,1,4,5,2] => ? = 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [2,5,3,4,1] => [2,5,3,4,1] => ? = 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [3,5,2,4,1] => [3,5,2,4,1] => ? = 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [4,5,2,3,1] => [4,5,2,3,1] => ? = 3
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [3,4,2,5,1] => [3,4,2,5,1] => ? = 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [3,5,1,4,2] => [3,5,1,4,2] => ? = 2
[1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [4,5,1,3,2] => [4,5,1,3,2] => ? = 2
[1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 3
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 2
[1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,6,1,5] => [1,2,3,6,4,5] => [1,2,3,6,4,5] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => [1,2,3,5,6,4] => [1,2,3,5,6,4] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,6] => [1,2,5,4,6,3] => [1,2,5,4,6,3] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,5,1,6,4] => [1,2,6,3,5,4] => [1,2,6,3,5,4] => ? = 2
Description
The number of preferred parking spots in a parking function less than the index of the car.
Let $(a_1,\dots,a_n)$ be a parking function. Then this statistic returns the number of indices $1\leq i\leq n$ such that $a_i < i$.
The following 1 statistic also match your data. Click on any of them to see the details.
St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.
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