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Your data matches 208 different statistics following compositions of up to 3 maps.
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Matching statistic: St000397
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000397: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000397: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[]]
=> 1
[1,0,1,0]
=> [[],[]]
=> 2
[1,1,0,0]
=> [[[]]]
=> 1
[1,0,1,0,1,0]
=> [[],[],[]]
=> 2
[1,0,1,1,0,0]
=> [[],[[]]]
=> 2
[1,1,0,0,1,0]
=> [[[]],[]]
=> 2
[1,1,0,1,0,0]
=> [[[],[]]]
=> 2
[1,1,1,0,0,0]
=> [[[[]]]]
=> 1
[1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 2
[1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 2
[1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 2
[1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> 2
[1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 2
[1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 2
[1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 2
[1,1,0,1,0,0,1,0]
=> [[[],[]],[]]
=> 2
[1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> 2
[1,1,0,1,1,0,0,0]
=> [[[],[[]]]]
=> 2
[1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 2
[1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 2
[1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> 2
[1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [[],[],[[]],[]]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [[],[],[[],[]]]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [[],[[]],[],[]]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [[],[[],[]],[]]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [[],[[],[],[]]]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [[],[[],[[]]]]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [[],[[[]],[]]]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [[],[[[],[]]]]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [[[]],[],[],[]]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [[[]],[],[[]]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [[[]],[[],[]]]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [[[]],[[[]]]]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [[[],[]],[],[]]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [[[],[]],[[]]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[[],[],[]],[]]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[]]]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [[[],[],[[]]]]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [[[],[[]]],[]]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [[[],[[]],[]]]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [[[],[[],[]]]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [[[],[[[]]]]]
=> 2
Description
The Strahler number of a rooted tree.
Matching statistic: St000535
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000535: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St000535: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> 0 = 1 - 1
[1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0]
=> [1,2] => ([],2)
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,3,2] => ([(1,2)],3)
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => ([(2,3)],4)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => ([(2,3)],4)
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
Description
The rank-width of a graph.
Matching statistic: St001494
(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
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001494: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001494: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> ([],1)
=> 1
[1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,0]
=> [1,2] => ([],2)
=> ([],1)
=> 1
[1,0,1,0,1,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[1,0,1,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ([],1)
=> 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ([],1)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
Description
The Alon-Tarsi number of a graph.
Let $G$ be a graph with vertices $\{1,\dots,n\}$ and edge set $E$. Let $P_G=\prod_{i < j, (i,j)\in E} x_i-x_j$ be its graph polynomial. Then the Alon-Tarsi number is the smallest number $k$ such that $P_G$ contains a monomial with exponents strictly less than $k$.
Matching statistic: St000183
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000183: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000183: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> []
=> 0 = 1 - 1
[1,0,1,0]
=> [1,2] => [2]
=> []
=> 0 = 1 - 1
[1,1,0,0]
=> [2,1] => [1,1]
=> [1]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [3]
=> []
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> [1]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,1]
=> [1]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1]
=> [1]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1]
=> [1]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,1]
=> [1]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,1]
=> [1]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2]
=> [2]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1]
=> [1]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2]
=> [2]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2]
=> [2]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2]
=> [2]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> [2]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
Description
The side length of the Durfee square of an integer partition.
Given a partition $\lambda = (\lambda_1,\ldots,\lambda_n)$, the Durfee square is the largest partition $(s^s)$ whose diagram fits inside the diagram of $\lambda$. In symbols, $s = \max\{ i \mid \lambda_i \geq i \}$.
This is also known as the Frobenius rank.
Matching statistic: St000253
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000253: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000253: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => {{1}}
=> ? = 1 - 1
[1,0,1,0]
=> [1,2] => [2,1] => {{1,2}}
=> 1 = 2 - 1
[1,1,0,0]
=> [2,1] => [1,2] => {{1},{2}}
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [2,3,1] => {{1,2,3}}
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1,3] => {{1,2},{3}}
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [3,2,1] => {{1,3},{2}}
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => {{1},{2},{3}}
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [1,3,2] => {{1},{2,3}}
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [2,3,4,1] => {{1,2,3,4}}
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,3,1,4] => {{1,2,3},{4}}
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,4,3,1] => {{1,2,4},{3}}
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,3,4] => {{1,2},{3},{4}}
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,4,3] => {{1,2},{3,4}}
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,2,4,1] => {{1,3,4},{2}}
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,2,1,4] => {{1,3},{2},{4}}
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [4,2,3,1] => {{1,4},{2},{3}}
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [4,3,2,1] => {{1,4},{2,3}}
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [1,4,2,3] => {{1},{2,3,4}}
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [2,3,4,5,1] => {{1,2,3,4,5}}
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,3,4,1,5] => {{1,2,3,4},{5}}
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,3,5,4,1] => {{1,2,3,5},{4}}
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,4,3,5,1] => {{1,2,4,5},{3}}
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,4,3,1,5] => {{1,2,4},{3},{5}}
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,5,3,4,1] => {{1,2,5},{3},{4}}
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [2,5,4,3,1] => {{1,2,5},{3,4}}
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [2,1,5,3,4] => {{1,2},{3,4,5}}
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [3,2,4,5,1] => {{1,3,4,5},{2}}
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2,4,1,5] => {{1,3,4},{2},{5}}
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2,5,4,1] => {{1,3,5},{2},{4}}
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [4,2,3,5,1] => {{1,4,5},{2},{3}}
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,2,3,1,5] => {{1,4},{2},{3},{5}}
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [5,2,4,3,1] => {{1,5},{2},{3,4}}
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [1,2,5,3,4] => {{1},{2},{3,4,5}}
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [4,3,2,5,1] => {{1,4,5},{2,3}}
=> 1 = 2 - 1
Description
The crossing number of a set partition.
This is the maximal number of chords in the standard representation of a set partition, that mutually cross.
Matching statistic: St001031
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001031: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001031: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> [1,0]
=> ? = 1 - 1
[1,0,1,0]
=> [1,2] => [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [2,1] => [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [3]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
Description
The height of the bicoloured Motzkin path associated with the Dyck path.
Matching statistic: St001333
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001333: Graphs ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001333: Graphs ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 0 = 1 - 1
[1,0,1,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0]
=> [1,2] => [1,2] => ([],2)
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [7,6,2,3,4,5,1] => [6,5,4,3,2,7,1] => ([(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)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3} - 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [6,7,2,3,4,5,1] => [7,1,6,5,4,3,2] => ([(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)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3} - 1
[1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [7,6,2,3,4,1,5] => [7,5,4,3,2,6,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3} - 1
[1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [6,7,2,3,4,1,5] => [6,1,7,5,4,3,2] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3} - 1
[1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [6,7,2,3,1,4,5] => [7,5,1,6,4,3,2] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3} - 1
[1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [7,5,2,3,1,4,6] => [7,6,4,3,2,5,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3} - 1
[1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [7,4,5,1,2,3,6] => [7,6,3,5,2,4,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3} - 1
[1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [7,4,3,1,2,5,6] => [3,7,6,5,2,4,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3} - 1
[1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [7,4,2,1,3,5,6] => [7,6,5,3,2,4,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3} - 1
[1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [7,5,1,2,3,4,6] => [7,6,4,2,5,3,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3} - 1
[1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => [7,6,5,4,2,3,1] => ([(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)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3} - 1
[1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => [2,7,6,5,4,3,1] => ([(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)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3} - 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3} - 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,1,2,3,4,5,7] => [6,5,4,3,2,1,7] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3} - 1
Description
The cardinality of a minimal edge-isolating set of a graph.
Let $\mathcal F$ be a set of graphs. A set of vertices $S$ is $\mathcal F$-isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of $S$ does not contain any graph in $\mathcal F$.
This statistic returns the cardinality of the smallest isolating set when $\mathcal F$ contains only the graph with one edge.
Matching statistic: St000920
(load all 31 compositions to match this statistic)
(load all 31 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
Description
The logarithmic height of a Dyck path.
This is the floor of the binary logarithm of the usual height increased by one:
$$
\lfloor\log_2(1+height(D))\rfloor
$$
Matching statistic: St001330
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 90%●distinct values known / distinct values provided: 67%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 90%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => ([],1)
=> ([],1)
=> 1
[1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,0]
=> [1,2] => ([],2)
=> ([],1)
=> 1
[1,0,1,0,1,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[1,0,1,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ([],1)
=> 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ([],1)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,3}
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? ∊ {2,2,3}
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [4,1,5,2,6,3] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,3}
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,5,7,1,4,6] => ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,7] => ([(1,6),(2,5),(3,4)],7)
=> ([(1,6),(2,5),(3,4)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,6,3,7,5] => ([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,3,5,7] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,5,7,6] => ([(1,6),(2,5),(3,4)],7)
=> ([(1,6),(2,5),(3,4)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,4,1,5,3,7,6] => ([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,4,5,7,1,3,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,6,5,7] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,1,6,3,5,7] => ([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,4,6,1,3,7,5] => ([(0,6),(1,4),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,4,6,1,7,3,5] => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,6,7,1,3,5] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,4,6,1,3,5,7] => ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,4,1,3,5,7,6] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,4,7,1,3,5,6] => ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ([(1,6),(2,5),(3,4)],7)
=> ([(1,6),(2,5),(3,4)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,1,3,5,7,4,6] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,7] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,1,5,3,7,4,6] => ([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [2,5,1,3,6,4,7] => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [2,5,1,6,3,7,4] => ([(0,6),(1,4),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,5,7,1,3,4,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,1,3,6,4,7,5] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,7,6] => ([(1,6),(2,5),(3,4)],7)
=> ([(1,6),(2,5),(3,4)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,3,2,5,7,4,6] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,5,2,4,7,6] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,5,2,7,4,6] => ([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,3,5,7,2,4,6] => ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,3,2,6,4,7,5] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,3,6,2,4,7,5] => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [3,1,4,2,6,5,7] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [3,1,4,6,2,5,7] => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,4,2,5,7,6] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [3,4,6,1,7,2,5] => ([(0,1),(0,6),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4,7] => ([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [3,1,5,2,4,7,6] => ([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [3,1,5,7,2,4,6] => ([(0,6),(1,4),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [3,5,1,6,2,7,4] => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [3,5,1,6,7,2,4] => ([(0,1),(0,6),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [3,5,1,6,2,4,7] => ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [3,5,6,1,7,2,4] => ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [3,5,1,7,2,4,6] => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [3,5,7,1,2,4,6] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [3,6,1,7,2,4,5] => ([(0,1),(0,6),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,4,2,5,3,7,6] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,4,2,5,7,3,6] => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,4,2,6,3,7,5] => ([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
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: St001568
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 90%●distinct values known / distinct values provided: 33%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 90%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> [1] => [1]
=> []
=> ? = 1
[1,0,1,0]
=> [2,1] => [2]
=> []
=> ? ∊ {1,2}
[1,1,0,0]
=> [1,2] => [1,1]
=> [1]
=> ? ∊ {1,2}
[1,0,1,0,1,0]
=> [3,2,1] => [3]
=> []
=> ? ∊ {1,2,2,2}
[1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> [1]
=> ? ∊ {1,2,2,2}
[1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> [1]
=> ? ∊ {1,2,2,2}
[1,1,0,1,0,0]
=> [2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,2,2,2}
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 2
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2}
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2}
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2}
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> [1,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2}
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> [1,1]
=> 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2}
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2}
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,1,1]
=> [1,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> [1,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,1,1]
=> [1,1]
=> 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2}
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2}
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [4,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2}
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2}
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,1,1]
=> [1,1]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2}
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,1,1]
=> [1,1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [4,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2}
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2}
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,1,1]
=> [1,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [3,1,1]
=> [1,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,1,1]
=> [1,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> [1,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [4,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2}
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,1,1]
=> [1,1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,1,1]
=> [1,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [3,1,1]
=> [1,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2}
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,1]
=> [1,1]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2}
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2}
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,1,1]
=> [1,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,1,1]
=> [1,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,1,1]
=> [1,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,1,1]
=> [1,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [3,1,1]
=> [1,1]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,1,1]
=> [1,1]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [3,1,1]
=> [1,1]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [3,1,1]
=> [1,1]
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [3,1,1]
=> [1,1]
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [3,1,1]
=> [1,1]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [6]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [5,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [5,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => [5,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [4,1,1]
=> [1,1]
=> 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [5,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [4,1,1]
=> [1,1]
=> 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => [5,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,2,1] => [5,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,2,1] => [4,1,1]
=> [1,1]
=> 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [4,1,1]
=> [1,1]
=> 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,2,1] => [4,1,1]
=> [1,1]
=> 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => [4,1,1]
=> [1,1]
=> 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [3,1,1,1]
=> [1,1,1]
=> 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => [5,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => [4,1,1]
=> [1,1]
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [4,1,1]
=> [1,1]
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => [4,1,1]
=> [1,1]
=> 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => [3,1,1,1]
=> [1,1,1]
=> 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,4,1] => [5,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,5,1] => [5,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,6,1] => [5,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => [5,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,3] => [5,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,4] => [5,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,5] => [5,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,6] => [5,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1] => [7]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => [6,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => [6,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => [6,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,2,1] => [6,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,3,2,1] => [6,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,2,1] => [6,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,4,2,1] => [6,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,5,2,1] => [6,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
Description
The smallest positive integer that does not appear twice in the partition.
The following 198 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000897The number of different multiplicities of parts of an integer partition. St001393The induced matching number of a graph. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000396The register function (or Horton-Strahler number) of a binary tree. St000260The radius of a connected graph. St000031The number of cycles in the cycle decomposition of a permutation. St000451The length of the longest pattern of the form k 1 2. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000035The number of left outer peaks of a permutation. St000647The number of big descents of a permutation. St000884The number of isolated descents of a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St000402Half the size of the symmetry class of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001335The cardinality of a minimal cycle-isolating set of a graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000679The pruning number of an ordered tree. St000630The length of the shortest palindromic decomposition of a binary word. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001471The magnitude of a Dyck path. St000741The Colin de Verdière graph invariant. St000662The staircase size of the code of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St000742The number of big ascents of a permutation after prepending zero. St001060The distinguishing index of a graph. St000454The largest eigenvalue of a graph if it is integral. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000862The number of parts of the shifted shape of a permutation. St000842The breadth of a permutation. St000007The number of saliances of the permutation. St000028The number of stack-sorts needed to sort a permutation. St000141The maximum drop size of a permutation. St001625The Möbius invariant of a lattice. St000097The order of the largest clique of the graph. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000834The number of right outer peaks of a permutation. St001261The Castelnuovo-Mumford regularity of a graph. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001029The size of the core of a graph. St001093The detour number of a graph. St001111The weak 2-dynamic chromatic number of a graph. St000058The order of a permutation. St000733The row containing the largest entry of a standard tableau. St001737The number of descents of type 2 in a permutation. St000298The order dimension or Dushnik-Miller dimension of a poset. St000098The chromatic number of a graph. St001810The number of fixed points of a permutation smaller than its largest moved point. St001665The number of pure excedances of a permutation. St000392The length of the longest run of ones in a binary word. St000527The width of the poset. St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000209Maximum difference of elements in cycles. St000956The maximal displacement of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001545The second Elser number of a connected graph. St000259The diameter of a connected graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000640The rank of the largest boolean interval in a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000526The number of posets with combinatorially isomorphic order polytopes. St000455The second largest eigenvalue of a graph if it is integral. St001644The dimension of a graph. St001116The game chromatic number of a graph. St000845The maximal number of elements covered by an element in a poset. St000245The number of ascents of a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000486The number of cycles of length at least 3 of a permutation. St001046The maximal number of arcs nesting a given arc of a perfect matching. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000352The Elizalde-Pak rank of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000646The number of big ascents of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001729The number of visible descents of a permutation. St001928The number of non-overlapping descents in a permutation. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000991The number of right-to-left minima of a permutation. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001530The depth of a Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St000062The length of the longest increasing subsequence of the permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000638The number of up-down runs of a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000700The protection number of an ordered tree. St000783The side length of the largest staircase partition fitting into a partition. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001372The length of a longest cyclic run of ones of a binary word. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001589The nesting number of a perfect matching. St001734The lettericity of a graph. St001962The proper pathwidth of a graph. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000154The sum of the descent bottoms of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000172The Grundy number of a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000353The number of inner valleys of a permutation. St000472The sum of the ascent bottoms of a permutation. St000671The maximin edge-connectivity for choosing a subgraph. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000779The tier of a permutation. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001112The 3-weak dynamic number of a graph. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001353The number of prime nodes in the modular decomposition of a graph. St001469The holeyness of a permutation. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001638The book thickness of a graph. St001963The tree-depth of a graph. St001110The 3-dynamic chromatic number of a graph. St001555The order of a signed permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000306The bounce count of a Dyck path. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000381The largest part of an integer composition. St000764The number of strong records in an integer composition. St001597The Frobenius rank of a skew partition. St001394The genus of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001870The number of positive entries followed by a negative entry in a signed permutation. St000876The number of factors in the Catalan decomposition of a binary word. St000628The balance of a binary word. St000891The number of distinct diagonal sums of a permutation matrix. St001624The breadth of a lattice. St000893The number of distinct diagonal sums of an alternating sign matrix. St000015The number of peaks of a Dyck path. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000758The length of the longest staircase fitting into an integer composition. St000846The maximal number of elements covering an element of a poset. St000903The number of different parts of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St000942The number of critical left to right maxima of the parking functions. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St000632The jump number of the poset. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001569The maximal modular displacement of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001960The number of descents of a permutation minus one if its first entry is not one. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation.
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