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Your data matches 141 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 = 2 - 1
[1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 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 - 1
[1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 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 = 2 - 1
[1,0,1,0]
=> [[],[]]
=> 1 = 2 - 1
[1,1,0,0]
=> [[[]]]
=> 2 = 3 - 1
[1,0,1,0,1,0]
=> [[],[],[]]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [[],[[]]]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [[[]],[]]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [[[],[]]]
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> [[[[]]]]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [[[],[]],[]]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [[[],[[]]]]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> 1 = 2 - 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 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [[],[],[[],[]]]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [[],[[]],[],[]]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [[],[[],[]],[]]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [[],[[],[],[]]]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [[],[[],[[]]]]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [[],[[[]],[]]]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [[],[[[],[]]]]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [[[]],[],[],[]]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [[[]],[],[[]]]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [[[]],[[],[]]]
=> 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [[[]],[[[]]]]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [[[],[]],[],[]]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [[[],[]],[[]]]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [[[],[],[]],[]]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[]]]
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [[[],[],[[]]]]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [[[],[[]]],[]]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [[[],[[]],[]]]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [[[],[[],[]]]]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [[[],[[[]]]]]
=> 2 = 3 - 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 = 2 - 1
[1,0,1,0]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0]
=> ([],2)
=> 2 = 3 - 1
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> ([(1,2)],3)
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> ([],3)
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> ([(0,3),(3,1),(3,2)],4)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(3,1)],4)
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> ([],4)
=> 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 2 - 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 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 1 = 2 - 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 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(4,2),(4,3)],5)
=> 1 = 2 - 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 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 2 = 3 - 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 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(2,4)],5)
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> ([(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
Description
The length of the shortest maximal antichain in a poset.
Matching statistic: St001038
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001038: 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]
=> 2
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[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]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4
[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]
=> 5
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 2
[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]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 3
Description
The minimal height of a column in the parallelogram polyomino associated with the Dyck path.
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 = 2 - 1
[1,0,1,0]
=> [1,1] => 11 => 2 = 3 - 1
[1,1,0,0]
=> [2] => 10 => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,1] => 111 => 3 = 4 - 1
[1,0,1,1,0,0]
=> [1,2] => 110 => 2 = 3 - 1
[1,1,0,0,1,0]
=> [2,1] => 101 => 1 = 2 - 1
[1,1,0,1,0,0]
=> [3] => 100 => 1 = 2 - 1
[1,1,1,0,0,0]
=> [3] => 100 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [1,3] => 1100 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [3,1] => 1001 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [4] => 1000 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [4] => 1000 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [4] => 1000 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [4] => 1000 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4] => 1000 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 11111 => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 11110 => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 11101 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 11100 => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 11011 => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 11010 => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 11001 => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 11000 => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 11000 => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 11001 => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 11000 => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 11000 => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 10110 => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 10101 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 10100 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 10011 => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 10010 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 10001 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => 10000 => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => 10000 => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 10001 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => 10000 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => 10000 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => 10000 => 1 = 2 - 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 = 2 - 1
[1,0,1,0]
=> [1,2] => ([],2)
=> 1 = 2 - 1
[1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> 2 = 3 - 1
[1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 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 = 4 - 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 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 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 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 2 = 3 - 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 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 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 = 3 - 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 = 3 - 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 = 2 - 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 = 3 - 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 = 3 - 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 = 3 - 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 = 2 - 1
[1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0]
=> [1,2] => ([],2)
=> 2 = 3 - 1
[1,0,1,0,1,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> 3 = 4 - 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 = 2 - 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 = 2 - 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 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 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 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> 4 = 5 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 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 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 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 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 3 - 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 = 3 - 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 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 3 - 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 = 2 - 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 = 3 - 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 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 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 = 2 - 1
[1,0,1,0]
=> [2,1] => [2,1] => 1 = 2 - 1
[1,1,0,0]
=> [1,2] => [1,2] => 2 = 3 - 1
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [2,3,1] => [1,3,2] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => 2 = 3 - 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,4,3,2] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,4,3,1] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,1,4,3] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,4,3] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,4,2,1] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,2,4,1] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,3,2,4] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,3,1,4] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [2,5,4,3,1] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [2,1,5,4,3] => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,2,5,4,3] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [3,5,4,2,1] => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,1,5,4,2] => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,2,5,4,1] => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [3,2,1,5,4] => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,3,2,5,4] => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,5,4,1] => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,3,1,5,4] => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,1,3,5,4] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,5,4] => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [4,5,3,2,1] => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [4,1,5,3,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [4,2,5,3,1] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,2,1,5,3] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,4,2,5,3] => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,3,5,2,1] => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,3,1,5,2] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,3,2,5,1] => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,4,3,2,5] => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,4,3,5,1] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,4,3,1,5] => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [2,1,4,3,5] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,2,4,3,5] => 2 = 3 - 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 = 2 - 1
[1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0]
=> [1,2] => ([],2)
=> 2 = 3 - 1
[1,0,1,0,1,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> 3 = 4 - 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 = 2 - 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 = 2 - 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 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 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 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> 4 = 5 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 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 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 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 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 3 - 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 = 3 - 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 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 3 - 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 = 2 - 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 = 3 - 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 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
Description
The common independence number of a graph.
The common independence number of a graph $G$ is the greatest integer $r$ such that every vertex of $G$ belongs to some independent set $X$ of vertices of cardinality at least $r$.
Matching statistic: St000310
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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 = 2 - 2
[1,0,1,0]
=> [1,2] => ([],2)
=> 0 = 2 - 2
[1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> 1 = 3 - 2
[1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> 0 = 2 - 2
[1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> 0 = 2 - 2
[1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> 0 = 2 - 2
[1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 3 - 2
[1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 4 - 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> 0 = 2 - 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1 = 3 - 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 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 = 5 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> 0 = 2 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 0 = 2 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 0 = 2 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[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 = 2 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[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 = 2 - 2
[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 = 2 - 2
[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 = 2 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 0 = 2 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 0 = 2 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 1 = 3 - 2
[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 = 3 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[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 = 3 - 2
[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 = 3 - 2
[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 = 2 - 2
[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 = 3 - 2
[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 = 3 - 2
[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 = 3 - 2
Description
The minimal degree of a vertex of a graph.
The following 131 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. 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. 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. 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. 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. St000654The first descent of a permutation. St000297The number of leading ones in a binary word. St000989The number of final rises 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. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000439The position of the first down step of a Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001180Number of indecomposable injective modules with projective dimension at most 1. 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. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. 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 $eA$. 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. St001060The distinguishing index of a graph. 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 $eA$. St000656The number of cuts of a poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000382The first part of an integer composition. St001130The number of two successive successions in a permutation. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000237The number of small exceedances. St001330The hat guessing number of a graph. St001875The number of simple modules with projective dimension at most 1. St000731The number of double exceedences of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000260The radius of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001530The depth of a Dyck path. St000056The decomposition (or block) number of a permutation. St000261The edge connectivity of a graph. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St000221The number of strong fixed points of a permutation. St000241The number of cyclical small excedances. St001545The second Elser number of a connected graph. St000895The number of ones on the main diagonal of an alternating sign matrix. St000264The girth of a graph, which is not a tree. St001498The normalised height of a Nakayama algebra with magnitude 1. St001571The Cartan determinant of the integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St000474Dyson's crank of a partition. St001432The order dimension of the partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000477The weight of a partition according to Alladi. 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. St000667The greatest common divisor of the parts of the partition. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000870The product of the hook lengths of the diagonal cells in an integer partition. 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. St001279The sum of the parts of an integer partition that are at least two. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000883The number of longest increasing subsequences of a permutation. 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. 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. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001846The number of elements which do not have a complement in the lattice. St000741The Colin de Verdière graph invariant. 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. St000649The number of 3-excedences of a permutation. St000877The depth of the binary word interpreted as a path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. 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. St001820The size of the image of the pop stack sorting operator. St001267The length of the Lyndon factorization of the binary word. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000982The length of the longest constant subword. St000117The number of centered tunnels of a Dyck path. 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. 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. St001413Half the length of the longest even length palindromic prefix of a binary word. 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. 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. St000894The trace of an alternating sign matrix. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. 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. 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. St000655The length of the minimal rise of a Dyck path.
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