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Your data matches 268 different statistics following compositions of up to 3 maps.
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Matching statistic: St001481
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
St001481: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1 = 0 + 1
[1,0,1,0]
=> 1 = 0 + 1
[1,1,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
Description
The minimal height of a peak of a Dyck path.
Matching statistic: St000700
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000700: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000700: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[]]
=> 1 = 0 + 1
[1,0,1,0]
=> [[],[]]
=> 1 = 0 + 1
[1,1,0,0]
=> [[[]]]
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [[],[],[]]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [[],[[]]]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [[[]],[]]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [[[],[]]]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [[[[]]]]
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [[[],[]],[]]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [[[],[[]]]]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [[],[],[[]],[]]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [[],[],[[],[]]]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [[],[[]],[],[]]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [[],[[],[]],[]]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [[],[[],[],[]]]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [[],[[],[[]]]]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [[],[[[]],[]]]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [[],[[[],[]]]]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [[[]],[],[],[]]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [[[]],[],[[]]]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [[[]],[[],[]]]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [[[]],[[[]]]]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [[[],[]],[],[]]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [[[],[]],[[]]]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [[[],[],[]],[]]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[]]]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [[[],[],[[]]]]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [[[],[[]]],[]]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [[[],[[]],[]]]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [[[],[[],[]]]]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [[[],[[[]]]]]
=> 2 = 1 + 1
Description
The protection number of an ordered tree.
This is the minimal distance from the root to a leaf.
Matching statistic: St000908
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000908: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000908: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> ([],1)
=> 1 = 0 + 1
[1,0,1,0]
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,1,0,0]
=> ([],2)
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> ([(1,2)],3)
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> ([],3)
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> ([(0,3),(3,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(3,1)],4)
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> ([],4)
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(4,2),(4,3)],5)
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
Description
The length of the shortest maximal antichain in a poset.
Matching statistic: St000310
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000310: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St000310: Graphs ⟶ 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]
=> [1,2] => ([],2)
=> 0
[1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> 0
[1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> 0
[1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
Description
The minimal degree of a vertex of a graph.
Matching statistic: St001107
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 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]
=> 0
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 0
Description
The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path.
In other words, this is the lowest height of a valley of a Dyck path, or its semilength in case of the unique path without valleys.
Matching statistic: St000392
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1 => 1 = 0 + 1
[1,0,1,0]
=> [1,1] => 11 => 2 = 1 + 1
[1,1,0,0]
=> [2] => 10 => 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,1] => 111 => 3 = 2 + 1
[1,0,1,1,0,0]
=> [1,2] => 110 => 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1] => 101 => 1 = 0 + 1
[1,1,0,1,0,0]
=> [3] => 100 => 1 = 0 + 1
[1,1,1,0,0,0]
=> [3] => 100 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,3] => 1100 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [3,1] => 1001 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [4] => 1000 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [4] => 1000 => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [4] => 1000 => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [4] => 1000 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [4] => 1000 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 11111 => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 11110 => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 11101 => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 11100 => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 11011 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 11010 => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 11001 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 11000 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 11000 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 11001 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 11000 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 11000 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 10110 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 10101 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 10100 => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 10011 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 10010 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 10001 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => 10000 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => 10000 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 10001 => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => 10000 => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => 10000 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => 10000 => 1 = 0 + 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St001316
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001316: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St001316: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> 1 = 0 + 1
[1,0,1,0]
=> [1,2] => ([],2)
=> 1 = 0 + 1
[1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
Description
The domatic number of a graph.
This is the maximal size of a partition of the vertices into dominating sets.
Matching statistic: St001322
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001322: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St001322: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> 1 = 0 + 1
[1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,1,0,0]
=> [1,2] => ([],2)
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> 3 = 2 + 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)
=> 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)
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> 4 = 3 + 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)
=> 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)
=> 1 = 0 + 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)
=> 1 = 0 + 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)
=> 1 = 0 + 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)
=> 1 = 0 + 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)
=> 1 = 0 + 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)
=> 1 = 0 + 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)
=> 1 = 0 + 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)
=> 1 = 0 + 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)
=> 1 = 0 + 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)
=> 1 = 0 + 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)
=> 1 = 0 + 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)
=> 1 = 0 + 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)
=> 1 = 0 + 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)
=> 1 = 0 + 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)
=> 1 = 0 + 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)
=> 1 = 0 + 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)
=> 2 = 1 + 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)
=> 2 = 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)
=> 1 = 0 + 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)
=> 2 = 1 + 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)
=> 1 = 0 + 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)
=> 2 = 1 + 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)
=> 2 = 1 + 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)
=> 1 = 0 + 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)
=> 2 = 1 + 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)
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
Description
The size of a minimal independent dominating set in a graph.
Matching statistic: St001652
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St001652: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St001652: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1 = 0 + 1
[1,0,1,0]
=> [2,1] => [2,1] => 1 = 0 + 1
[1,1,0,0]
=> [1,2] => [1,2] => 2 = 1 + 1
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [2,3,1] => [1,3,2] => 1 = 0 + 1
[1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,4,3,2] => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,4,3,1] => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,1,4,3] => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,4,3] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,4,2,1] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,2,4,1] => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,3,2,4] => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,3,1,4] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,5,4,3,2] => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [2,5,4,3,1] => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [2,1,5,4,3] => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,2,5,4,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [3,5,4,2,1] => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,1,5,4,2] => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,2,5,4,1] => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [3,2,1,5,4] => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,3,2,5,4] => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,5,4,1] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,3,1,5,4] => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,1,3,5,4] => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,5,4] => 3 = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [4,5,3,2,1] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [4,1,5,3,2] => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [4,2,5,3,1] => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,2,1,5,3] => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,4,2,5,3] => 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,3,5,2,1] => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,3,1,5,2] => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,3,2,5,1] => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,4,3,2,5] => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,4,3,5,1] => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,4,3,1,5] => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [2,1,4,3,5] => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,2,4,3,5] => 2 = 1 + 1
Description
The length of a longest interval of consecutive numbers.
For a permutation $\pi=\pi_1,\dots,\pi_n$, this statistic returns the length of a longest subsequence $\pi_k,\dots,\pi_\ell$ such that $\pi_{i+1} = \pi_i + 1$ for $i\in\{k,\dots,\ell-1\}$.
Matching statistic: St001829
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001829: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St001829: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> 1 = 0 + 1
[1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,1,0,0]
=> [1,2] => ([],2)
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> 3 = 2 + 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)
=> 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)
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> 4 = 3 + 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)
=> 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)
=> 1 = 0 + 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)
=> 1 = 0 + 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)
=> 1 = 0 + 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)
=> 1 = 0 + 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)
=> 1 = 0 + 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)
=> 1 = 0 + 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)
=> 1 = 0 + 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)
=> 1 = 0 + 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)
=> 1 = 0 + 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)
=> 1 = 0 + 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)
=> 1 = 0 + 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)
=> 1 = 0 + 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)
=> 1 = 0 + 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)
=> 1 = 0 + 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)
=> 1 = 0 + 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)
=> 1 = 0 + 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)
=> 2 = 1 + 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)
=> 2 = 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)
=> 1 = 0 + 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)
=> 2 = 1 + 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)
=> 1 = 0 + 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)
=> 2 = 1 + 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)
=> 2 = 1 + 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)
=> 1 = 0 + 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)
=> 2 = 1 + 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)
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
Description
The common independence number of a graph.
The common independence number of a graph $G$ is the greatest integer $r$ such that every vertex of $G$ belongs to some independent set $X$ of vertices of cardinality at least $r$.
The following 258 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000546The number of global descents of a permutation. St000974The length of the trunk of an ordered tree. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000383The last part of an integer composition. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000729The minimal arc length of a set partition. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001662The length of the longest factor of consecutive numbers in a permutation. St001933The largest multiplicity of a part in an integer partition. St000504The cardinality of the first block of a set partition. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000906The length of the shortest maximal chain in a poset. St000990The first ascent of a permutation. St000993The multiplicity of the largest part of an integer partition. St000297The number of leading ones in a binary word. St000989The number of final rises of a permutation. St000654The first descent of a permutation. St000234The number of global ascents of a permutation. St000657The smallest part of an integer composition. St000617The number of global maxima of a Dyck path. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001525The number of symmetric hooks on the diagonal of a partition. St001877Number of indecomposable injective modules with projective dimension 2. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000934The 2-degree of an integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000010The length of the partition. St000053The number of valleys of the Dyck path. St000148The number of odd parts of a partition. St000306The bounce count of a Dyck path. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000369The dinv deficit of a Dyck path. St000475The number of parts equal to 1 in a partition. St000548The number of different non-empty partial sums of an integer partition. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000877The depth of the binary word interpreted as a path. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000992The alternating sum of the parts of an integer partition. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero 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 $eA$. 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 $eA$. 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. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in 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. 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. St001413Half the length of the longest even length palindromic prefix of a binary word. St001484The number of singletons of an integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000376The bounce deficit of a Dyck path. St000379The number of Hamiltonian cycles in a graph. St000442The maximal area to the right of an up step of a Dyck path. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000940The number of characters of the symmetric group whose value on the partition is zero. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001280The number of parts of an integer partition that are at least two. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St000929The constant term of the character polynomial of an integer partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000117The number of centered tunnels of a Dyck path. St000221The number of strong fixed points of a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000941The number of characters of the symmetric group whose value on the partition is even. St000455The second largest eigenvalue of a graph if it is integral. St000137The Grundy value of an integer partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001651The Frankl number of a lattice. St000661The number of rises of length 3 of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000931The number of occurrences of the pattern UUU in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001095The number of non-isomorphic posets with precisely one further covering relation. St001141The number of occurrences of hills of size 3 in a Dyck path. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001130The number of two successive successions in a permutation. St000382The first part of an integer composition. St000936The number of even values of the symmetric group character corresponding to the partition. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000237The number of small exceedances. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001330The hat guessing number of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000731The number of double exceedences of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000260The radius of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000241The number of cyclical small excedances. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000056The decomposition (or block) number of a permutation. St000261The edge connectivity of a graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001530The depth of a Dyck path. St000624The normalized sum of the minimal distances to a greater element. St000650The number of 3-rises of a permutation. St000732The number of double deficiencies of a permutation. St000779The tier of a permutation. St000872The number of very big descents of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000944The 3-degree of an integer partition. St001176The size of a partition minus its first part. St001248Sum of the even parts of a partition. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001279The sum of the parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001541The Gini index of an integer partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000895The number of ones on the main diagonal of an alternating sign matrix. St000478Another weight of a partition according to Alladi. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001383The BG-rank of an integer partition. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001571The Cartan determinant of the integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000928The sum of the coefficients of the character polynomial of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001561The value of the elementary symmetric function evaluated at 1. St000883The number of longest increasing subsequences of a permutation. St001568The smallest positive integer that does not appear twice in the partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001964The interval resolution global dimension of a poset. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001846The number of elements which do not have a complement in the lattice. St000741The Colin de Verdière graph invariant. St000649The number of 3-excedences of a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St001889The size of the connectivity set of a signed permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000708The product of the parts of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001128The exponens consonantiae of a partition. St001820The size of the image of the pop stack sorting operator. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001267The length of the Lyndon factorization of the binary word. St000894The trace of an alternating sign matrix. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001811The Castelnuovo-Mumford regularity of a permutation. St001948The number of augmented double ascents of a permutation. St000338The number of pixed points of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000911The number of maximal antichains of maximal size in a poset. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000982The length of the longest constant subword. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000655The length of the minimal rise of a Dyck path.
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