Your data matches 21 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000700
(load all 5 compositions to match this statistic)
Mapping
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%
Values
[1,0]
 => [[]]
 => 1
[1,0,1,0]
 => [[],[]]
 => 1
[1,1,0,0]
 => [[[]]]
 => 2
[1,0,1,0,1,0]
 => [[],[],[]]
 => 1
[1,0,1,1,0,0]
 => [[],[[]]]
 => 1
[1,1,0,0,1,0]
 => [[[]],[]]
 => 1
[1,1,0,1,0,0]
 => [[[],[]]]
 => 2
[1,1,1,0,0,0]
 => [[[[]]]]
 => 3
[1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => 1
[1,0,1,0,1,1,0,0]
 => [[],[],[[]]]
 => 1
[1,0,1,1,0,0,1,0]
 => [[],[[]],[]]
 => 1
[1,0,1,1,0,1,0,0]
 => [[],[[],[]]]
 => 1
[1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => 1
[1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => 1
[1,1,0,0,1,1,0,0]
 => [[[]],[[]]]
 => 2
[1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => 1
[1,1,0,1,0,1,0,0]
 => [[[],[],[]]]
 => 2
[1,1,0,1,1,0,0,0]
 => [[[],[[]]]]
 => 2
[1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => 1
[1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => 2
[1,1,1,0,1,0,0,0]
 => [[[[],[]]]]
 => 3
[1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[[]]]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [[],[],[[]],[]]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [[],[],[[],[]]]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [[],[],[[[]]]]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [[],[[]],[],[]]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [[],[[]],[[]]]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [[],[[],[]],[]]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [[],[[],[],[]]]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [[],[[],[[]]]]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [[],[[[]]],[]]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [[],[[[]],[]]]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [[],[[[],[]]]]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [[],[[[[]]]]]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [[[]],[],[],[]]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [[[]],[],[[]]]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [[[]],[[]],[]]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [[[]],[[],[]]]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [[[]],[[[]]]]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [[[],[]],[],[]]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [[[],[]],[[]]]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [[[],[],[]],[]]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [[[],[],[],[]]]
 => 2
[1,1,0,1,0,1,1,0,0,0]
 => [[[],[],[[]]]]
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [[[],[[]]],[]]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [[[],[[]],[]]]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [[[],[[],[]]]]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [[[],[[[]]]]]
 => 2
Description
The protection number of an ordered tree. This is the minimal distance from the root to a leaf.
Matching statistic: St001316
(load all 3 compositions to match this statistic)
Mapping
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%
Values
[1,0]
 => [1] => ([],1)
 => 1
[1,0,1,0]
 => [1,2] => ([],2)
 => 1
[1,1,0,0]
 => [2,1] => ([(0,1)],2)
 => 2
[1,0,1,0,1,0]
 => [1,2,3] => ([],3)
 => 1
[1,0,1,1,0,0]
 => [1,3,2] => ([(1,2)],3)
 => 1
[1,1,0,0,1,0]
 => [2,1,3] => ([(1,2)],3)
 => 1
[1,1,0,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 2
[1,1,1,0,0,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => ([],4)
 => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => ([(2,3)],4)
 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => ([(2,3)],4)
 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => ([(2,3)],4)
 => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[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
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => ([],5)
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ([(3,4)],5)
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => ([(3,4)],5)
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => ([(3,4)],5)
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 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
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 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
[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
[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
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => ([(3,4)],5)
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => 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
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 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
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
Description
The domatic number of a graph. This is the maximal size of a partition of the vertices into dominating sets.
Matching statistic: St001829
(load all 2 compositions to match this statistic)
Mapping
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%
Values
[1,0]
 => [1] => ([],1)
 => 1
[1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => 1
[1,1,0,0]
 => [1,2] => ([],2)
 => 2
[1,0,1,0,1,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,0,1,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[1,1,0,0,1,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[1,1,0,1,0,0]
 => [2,1,3] => ([(1,2)],3)
 => 2
[1,1,1,0,0,0]
 => [1,2,3] => ([],3)
 => 3
[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
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 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
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 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
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 2
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 2
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => ([(2,3)],4)
 => 3
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([],4)
 => 4
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[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
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[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
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[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
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => 2
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: St000729
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000729: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [2,1] => [2,1] => {{1,2}}
 => 1
[1,0,1,0]
 => [3,1,2] => [2,3,1] => {{1,2,3}}
 => 1
[1,1,0,0]
 => [2,3,1] => [3,1,2] => {{1,3},{2}}
 => 2
[1,0,1,0,1,0]
 => [4,1,2,3] => [2,3,4,1] => {{1,2,3,4}}
 => 1
[1,0,1,1,0,0]
 => [3,1,4,2] => [2,4,1,3] => {{1,2,4},{3}}
 => 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [3,1,4,2] => {{1,3,4},{2}}
 => 1
[1,1,0,1,0,0]
 => [4,3,1,2] => [3,4,2,1] => {{1,3},{2,4}}
 => 2
[1,1,1,0,0,0]
 => [2,3,4,1] => [4,1,2,3] => {{1,4},{2},{3}}
 => 3
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [2,3,4,5,1] => {{1,2,3,4,5}}
 => 1
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [2,3,5,1,4] => {{1,2,3,5},{4}}
 => 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [2,4,1,5,3] => {{1,2,4,5},{3}}
 => 1
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [2,4,5,3,1] => {{1,2,4},{3,5}}
 => 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [2,5,1,3,4] => {{1,2,5},{3},{4}}
 => 1
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [3,1,4,5,2] => {{1,3,4,5},{2}}
 => 1
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [3,1,5,2,4] => {{1,3,5},{2},{4}}
 => 2
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [3,4,2,5,1] => {{1,3},{2,4,5}}
 => 1
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [3,4,5,2,1] => {{1,3,5},{2,4}}
 => 2
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [3,5,2,1,4] => {{1,3},{2,5},{4}}
 => 2
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [4,1,2,5,3] => {{1,4,5},{2},{3}}
 => 1
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [4,1,5,3,2] => {{1,4},{2},{3,5}}
 => 2
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [4,5,2,3,1] => {{1,4},{2,5},{3}}
 => 3
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => {{1,5},{2},{3},{4}}
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [2,3,4,5,6,1] => {{1,2,3,4,5,6}}
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [2,3,4,6,1,5] => {{1,2,3,4,6},{5}}
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [2,3,5,1,6,4] => {{1,2,3,5,6},{4}}
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [2,3,5,6,4,1] => {{1,2,3,5},{4,6}}
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [2,3,6,1,4,5] => {{1,2,3,6},{4},{5}}
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [2,4,1,5,6,3] => {{1,2,4,5,6},{3}}
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [2,4,1,6,3,5] => {{1,2,4,6},{3},{5}}
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [2,4,5,3,6,1] => {{1,2,4},{3,5,6}}
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [2,4,5,6,3,1] => {{1,2,4,6},{3,5}}
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [2,4,6,3,1,5] => {{1,2,4},{3,6},{5}}
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [2,5,1,3,6,4] => {{1,2,5,6},{3},{4}}
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [2,5,1,6,4,3] => {{1,2,5},{3},{4,6}}
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [2,5,6,3,4,1] => {{1,2,5},{3,6},{4}}
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [2,6,1,3,4,5] => {{1,2,6},{3},{4},{5}}
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [3,1,4,5,6,2] => {{1,3,4,5,6},{2}}
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [3,1,4,6,2,5] => {{1,3,4,6},{2},{5}}
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [3,1,5,2,6,4] => {{1,3,5,6},{2},{4}}
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [3,1,5,6,4,2] => {{1,3,5},{2},{4,6}}
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [3,1,6,2,4,5] => {{1,3,6},{2},{4},{5}}
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [3,4,2,5,6,1] => {{1,3},{2,4,5,6}}
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [3,4,2,6,1,5] => {{1,3},{2,4,6},{5}}
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [3,4,5,2,6,1] => {{1,3,5,6},{2,4}}
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [3,4,5,6,1,2] => {{1,3,5},{2,4,6}}
 => 2
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [3,4,6,2,1,5] => {{1,3,6},{2,4},{5}}
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [3,5,2,1,6,4] => {{1,3},{2,5,6},{4}}
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [3,5,2,6,4,1] => {{1,3},{2,5},{4,6}}
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [3,5,6,2,4,1] => {{1,3,6},{2,5},{4}}
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [3,6,2,1,4,5] => {{1,3},{2,6},{4},{5}}
 => 2
Description
The minimal arc length of a set partition. The arcs of a set partition are those $i < j$ that are consecutive elements in the blocks. If there are no arcs, the minimal arc length is the size of the ground set (as the minimum of the empty set in the universe of arcs of length less than the size of the ground set).
Matching statistic: St000310
(load all 4 compositions to match this statistic)
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000310: 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
[1,0,1,0]
 => [1,2] => ([],2)
 => ([],1)
 => 0 = 1 - 1
[1,1,0,0]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,2,3] => ([],3)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => 0 = 1 - 1
[1,1,0,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 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)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 0 = 1 - 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),(1,3),(2,3)],4)
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 0 = 1 - 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)],3)
 => 0 = 1 - 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)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 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),(1,3),(2,3)],4)
 => 0 = 1 - 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)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 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),(1,3),(2,3)],4)
 => 0 = 1 - 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),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => 1 = 2 - 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)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => 1 = 2 - 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)],3)
 => 0 = 1 - 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)
 => ([(0,1)],2)
 => 1 = 2 - 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)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 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)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 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)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 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)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 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)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
Description
The minimal degree of a vertex of a graph.
Matching statistic: St001107
(load all 3 compositions to match this statistic)
Mapping
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,0,0]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => 1 = 2 - 1
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: St001038
(load all 5 compositions to match this statistic)
Mapping
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]
 => [1,0]
 => ? = 1
[1,0,1,0]
 => [1,1,0,0]
 => 1
[1,1,0,0]
 => [1,0,1,0]
 => 2
[1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 2
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
Description
The minimal height of a column in the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000906
(load all 7 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000906: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => ([],1)
 => ? = 1
[1,0,1,0]
 => [2,1] => ([],2)
 => 1
[1,1,0,0]
 => [1,2] => ([(0,1)],2)
 => 2
[1,0,1,0,1,0]
 => [3,2,1] => ([],3)
 => 1
[1,0,1,1,0,0]
 => [2,3,1] => ([(1,2)],3)
 => 1
[1,1,0,0,1,0]
 => [3,1,2] => ([(1,2)],3)
 => 1
[1,1,0,1,0,0]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => 2
[1,1,1,0,0,0]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([],4)
 => 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => ([(2,3)],4)
 => 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(2,3)],4)
 => 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(1,3),(2,3)],4)
 => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(2,3)],4)
 => 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => 2
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(1,3),(2,3)],4)
 => 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 2
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => 3
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => ([],5)
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => ([(3,4)],5)
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => ([(3,4)],5)
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => ([(2,4),(3,4)],5)
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => ([(3,4)],5)
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => ([(1,4),(2,3)],5)
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => ([(2,4),(3,4)],5)
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => ([(3,4)],5)
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => ([(2,4),(3,4)],5)
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => 1
Description
The length of the shortest maximal chain in a poset.
Matching statistic: St000974
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000974: Ordered trees ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => [[],[]]
 => 0 = 1 - 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [[],[],[]]
 => 0 = 1 - 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => [[[],[]]]
 => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [[],[[],[]]]
 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => 0 = 1 - 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [[[],[],[]]]
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [[[[],[]]]]
 => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [[],[],[[],[]]]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [[],[[],[]],[]]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[],[[],[],[]]]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[],[[[],[]]]]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[[],[]],[],[]]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [[[],[[]],[]]]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[[],[],[]],[]]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[[],[],[],[]]]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [[[],[[],[]]]]
 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[[[],[]]],[]]
 => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[[[],[]],[]]]
 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [[[[],[],[]]]]
 => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [[[[[],[]]]]]
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[],[]]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [[],[],[],[[],[]]]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [[],[],[[],[]],[]]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [[],[],[[],[],[]]]
 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [[],[],[[[],[]]]]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [[],[[],[]],[],[]]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [[],[[],[[]],[]]]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [[],[[],[],[]],[]]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [[],[[],[],[],[]]]
 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [[],[[],[[],[]]]]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [[],[[[],[]]],[]]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [[],[[[],[]],[]]]
 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [[],[[[],[],[]]]]
 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[],[[[[],[]]]]]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [[[],[]],[],[],[]]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [[[],[]],[[],[]]]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [[[],[[]],[]],[]]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [[[],[[]],[],[]]]
 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [[[],[[[]],[]]]]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [[[],[],[]],[],[]]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [[[],[],[[]],[]]]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [[[],[],[],[]],[]]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [[[],[],[],[],[]]]
 => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [[[],[],[[],[]]]]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [[[],[[],[]]],[]]
 => 0 = 1 - 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 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [[[],[[],[],[]]]]
 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [[[],[[[],[]]]]]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [[[],[],[]],[],[],[],[]]
 => ? = 1 - 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [[[],[],[],[]],[],[],[]]
 => ? = 1 - 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
 => [[[[],[]]],[],[],[],[]]
 => ? = 1 - 1
Description
The length of the trunk of an ordered tree. This is the length of the path from the root to the first vertex which has not exactly one child.
Matching statistic: St000908
(load all 7 compositions to match this statistic)
Mapping
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000908: Posets ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => ([],1)
 => 1
[1,0,1,0]
 => ([(0,1)],2)
 => 1
[1,1,0,0]
 => ([],2)
 => 2
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 1
[1,0,1,1,0,0]
 => ([(0,2),(1,2)],3)
 => 1
[1,1,0,0,1,0]
 => ([(0,1),(0,2)],3)
 => 1
[1,1,0,1,0,0]
 => ([(1,2)],3)
 => 2
[1,1,1,0,0,0]
 => ([],3)
 => 3
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,0,1,0,1,1,0,0]
 => ([(0,3),(1,3),(3,2)],4)
 => 1
[1,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,0,1,1,0,1,0,0]
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[1,0,1,1,1,0,0,0]
 => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,1,0,0,1,0,1,0]
 => ([(0,3),(3,1),(3,2)],4)
 => 1
[1,1,0,0,1,1,0,0]
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(3,1)],4)
 => 1
[1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => 2
[1,1,0,1,1,0,0,0]
 => ([(1,3),(2,3)],4)
 => 2
[1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => 1
[1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => 2
[1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => 3
[1,1,1,1,0,0,0,0]
 => ([],4)
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 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
[1,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 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
[1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,3),(3,4),(4,1),(4,2)],5)
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(4,2),(4,3)],5)
 => 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
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(3,2),(4,1),(4,3)],5)
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => 2
[1,1,0,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(2,4)],5)
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => ([(1,4),(2,4),(3,4)],5)
 => 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7)
 => ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7)
 => ? = 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
 => ? = 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(4,6),(5,4),(6,1),(6,2),(6,3)],7)
 => ? = 1
Description
The length of the shortest maximal antichain in a poset.
The following 11 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001322The size of a minimal independent dominating set in a graph. St001481The minimal height of a peak of a Dyck path. St000234The number of global ascents of a permutation. St000546The number of global descents of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000056The decomposition (or block) number of a permutation. St000261The edge connectivity of a graph. St000221The number of strong fixed points of a permutation. St000264The girth of a graph, which is not a tree. St001413Half the length of the longest even length palindromic prefix of a binary word. St000655The length of the minimal rise of a Dyck path.