Your data matches 213 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001036
(load all 26 compositions to match this statistic)
Mapping
St001036: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => 0 = 2 - 2
[1,0,1,0]
 => 0 = 2 - 2
[1,1,0,0]
 => 0 = 2 - 2
[1,0,1,0,1,0]
 => 0 = 2 - 2
[1,0,1,1,0,0]
 => 1 = 3 - 2
[1,1,0,0,1,0]
 => 1 = 3 - 2
[1,1,0,1,0,0]
 => 0 = 2 - 2
[1,1,1,0,0,0]
 => 0 = 2 - 2
[1,0,1,0,1,0,1,0]
 => 0 = 2 - 2
[1,0,1,0,1,1,0,0]
 => 1 = 3 - 2
[1,0,1,1,0,0,1,0]
 => 2 = 4 - 2
[1,0,1,1,0,1,0,0]
 => 1 = 3 - 2
[1,0,1,1,1,0,0,0]
 => 1 = 3 - 2
[1,1,0,0,1,0,1,0]
 => 1 = 3 - 2
[1,1,0,0,1,1,0,0]
 => 2 = 4 - 2
[1,1,0,1,0,0,1,0]
 => 1 = 3 - 2
[1,1,0,1,0,1,0,0]
 => 0 = 2 - 2
[1,1,0,1,1,0,0,0]
 => 1 = 3 - 2
[1,1,1,0,0,0,1,0]
 => 1 = 3 - 2
[1,1,1,0,0,1,0,0]
 => 1 = 3 - 2
[1,1,1,0,1,0,0,0]
 => 0 = 2 - 2
[1,1,1,1,0,0,0,0]
 => 0 = 2 - 2
[1,0,1,0,1,0,1,0,1,0]
 => 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0]
 => 1 = 3 - 2
[1,0,1,0,1,1,0,0,1,0]
 => 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,0]
 => 1 = 3 - 2
[1,0,1,0,1,1,1,0,0,0]
 => 1 = 3 - 2
[1,0,1,1,0,0,1,0,1,0]
 => 2 = 4 - 2
[1,0,1,1,0,0,1,1,0,0]
 => 3 = 5 - 2
[1,0,1,1,0,1,0,0,1,0]
 => 2 = 4 - 2
[1,0,1,1,0,1,0,1,0,0]
 => 1 = 3 - 2
[1,0,1,1,0,1,1,0,0,0]
 => 2 = 4 - 2
[1,0,1,1,1,0,0,0,1,0]
 => 2 = 4 - 2
[1,0,1,1,1,0,0,1,0,0]
 => 2 = 4 - 2
[1,0,1,1,1,0,1,0,0,0]
 => 1 = 3 - 2
[1,0,1,1,1,1,0,0,0,0]
 => 1 = 3 - 2
[1,1,0,0,1,0,1,0,1,0]
 => 1 = 3 - 2
[1,1,0,0,1,0,1,1,0,0]
 => 2 = 4 - 2
[1,1,0,0,1,1,0,0,1,0]
 => 3 = 5 - 2
[1,1,0,0,1,1,0,1,0,0]
 => 2 = 4 - 2
[1,1,0,0,1,1,1,0,0,0]
 => 2 = 4 - 2
[1,1,0,1,0,0,1,0,1,0]
 => 1 = 3 - 2
[1,1,0,1,0,0,1,1,0,0]
 => 2 = 4 - 2
[1,1,0,1,0,1,0,0,1,0]
 => 1 = 3 - 2
[1,1,0,1,0,1,0,1,0,0]
 => 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0]
 => 1 = 3 - 2
[1,1,0,1,1,0,0,0,1,0]
 => 2 = 4 - 2
[1,1,0,1,1,0,0,1,0,0]
 => 2 = 4 - 2
[1,1,0,1,1,0,1,0,0,0]
 => 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0]
 => 1 = 3 - 2
Description
The number of inner corners of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000340
(load all 9 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000340: 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 = 2 - 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 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]
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 2 = 3 - 1
Description
The number of non-final maximal constant sub-paths of length greater than one. This is the total number of occurrences of the patterns $110$ and $001$.
Matching statistic: St000312
Mapping
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000312: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [[]]
 => ([(0,1)],2)
 => 2
[1,0,1,0]
 => [[],[]]
 => ([(0,2),(1,2)],3)
 => 2
[1,1,0,0]
 => [[[]]]
 => ([(0,2),(1,2)],3)
 => 2
[1,0,1,0,1,0]
 => [[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,0,1,1,0,0]
 => [[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,1,0,0,1,0]
 => [[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,1,0,1,0,0]
 => [[[],[]]]
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,1,1,0,0,0]
 => [[[[]]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,0,1,0,1,1,0,0]
 => [[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,0,1,1,0,0,1,0]
 => [[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,0,1,1,0,1,0,0]
 => [[],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,0,0,1,1,0,0]
 => [[[]],[[]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,0,1,0,1,0,0]
 => [[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,1,0,1,1,0,0,0]
 => [[[],[[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,1,0,1,0,0,0]
 => [[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 4
[1,0,1,1,0,1,0,1,0,0]
 => [[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [[],[[],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [[],[[[]],[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [[],[[[],[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [[],[[[[]]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [[[]],[[],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [[[]],[[[]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [[[],[]],[[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [[[],[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => [[[],[],[[]]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [[[],[[]]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [[[],[[]],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,1,0,1,1,0,1,0,0,0]
 => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [[[],[[[]]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
Description
The number of leaves in a graph. That is, the number of vertices of a graph that have degree 1.
Matching statistic: St000636
Mapping
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000636: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [[]]
 => ([(0,1)],2)
 => 2
[1,0,1,0]
 => [[],[]]
 => ([(0,2),(1,2)],3)
 => 2
[1,1,0,0]
 => [[[]]]
 => ([(0,2),(1,2)],3)
 => 2
[1,0,1,0,1,0]
 => [[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,0,1,1,0,0]
 => [[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,1,0,0,1,0]
 => [[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,1,0,1,0,0]
 => [[[],[]]]
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,1,1,0,0,0]
 => [[[[]]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,0,1,0,1,1,0,0]
 => [[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,0,1,1,0,0,1,0]
 => [[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,0,1,1,0,1,0,0]
 => [[],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,0,0,1,1,0,0]
 => [[[]],[[]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,0,1,0,1,0,0]
 => [[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,1,0,1,1,0,0,0]
 => [[[],[[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,1,0,1,0,0,0]
 => [[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 4
[1,0,1,1,0,1,0,1,0,0]
 => [[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [[],[[],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [[],[[[]],[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [[],[[[],[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [[],[[[[]]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [[[]],[[],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [[[]],[[[]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [[[],[]],[[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [[[],[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => [[[],[],[[]]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [[[],[[]]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [[[],[[]],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,1,0,1,1,0,1,0,0,0]
 => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [[[],[[[]]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
Description
The hull number of a graph. The convex hull of a set of vertices $S$ of a graph is the smallest set $h(S)$ such that for any pair $u,v\in h(S)$ all vertices on a shortest path from $u$ to $v$ are also in $h(S)$. The hull number is the size of the smallest set $S$ such that $h(S)$ is the set of all vertices.
Matching statistic: St001068
(load all 10 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001068: 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]
 => [1,0,1,1,0,0]
 => 2
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => 2
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 3
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 3
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 3
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 4
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 4
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 3
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 3
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 5
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 4
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 4
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 4
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => 4
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 3
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 4
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 5
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => 4
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => 4
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => 4
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 2
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => 4
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => 4
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => 3
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
Matching statistic: St001654
Mapping
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St001654: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [[]]
 => ([(0,1)],2)
 => 2
[1,0,1,0]
 => [[],[]]
 => ([(0,2),(1,2)],3)
 => 2
[1,1,0,0]
 => [[[]]]
 => ([(0,2),(1,2)],3)
 => 2
[1,0,1,0,1,0]
 => [[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,0,1,1,0,0]
 => [[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,1,0,0,1,0]
 => [[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,1,0,1,0,0]
 => [[[],[]]]
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,1,1,0,0,0]
 => [[[[]]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,0,1,0,1,1,0,0]
 => [[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,0,1,1,0,0,1,0]
 => [[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,0,1,1,0,1,0,0]
 => [[],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,0,0,1,1,0,0]
 => [[[]],[[]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,0,1,0,1,0,0]
 => [[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,1,0,1,1,0,0,0]
 => [[[],[[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,1,0,1,0,0,0]
 => [[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 4
[1,0,1,1,0,1,0,1,0,0]
 => [[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [[],[[],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [[],[[[]],[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [[],[[[],[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [[],[[[[]]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [[[]],[[],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [[[]],[[[]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [[[],[]],[[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [[[],[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => [[[],[],[[]]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [[[],[[]]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [[[],[[]],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,1,0,1,1,0,1,0,0,0]
 => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [[[],[[[]]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
Description
The monophonic hull number of a graph. The monophonic hull of a set of vertices $M$ of a graph $G$ is the set of vertices that lie on at least one induced path between vertices in $M$. The monophonic hull number is the size of the smallest set $M$ such that the monophonic hull of $M$ is all of $G$. For example, the monophonic hull number of a graph $G$ with $n$ vertices is $n$ if and only if $G$ is a disjoint union of complete graphs.
Matching statistic: St001655
Mapping
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St001655: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [[]]
 => ([(0,1)],2)
 => 2
[1,0,1,0]
 => [[],[]]
 => ([(0,2),(1,2)],3)
 => 2
[1,1,0,0]
 => [[[]]]
 => ([(0,2),(1,2)],3)
 => 2
[1,0,1,0,1,0]
 => [[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,0,1,1,0,0]
 => [[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,1,0,0,1,0]
 => [[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,1,0,1,0,0]
 => [[[],[]]]
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,1,1,0,0,0]
 => [[[[]]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,0,1,0,1,1,0,0]
 => [[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,0,1,1,0,0,1,0]
 => [[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,0,1,1,0,1,0,0]
 => [[],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,0,0,1,1,0,0]
 => [[[]],[[]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,0,1,0,1,0,0]
 => [[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,1,0,1,1,0,0,0]
 => [[[],[[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,1,0,1,0,0,0]
 => [[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 4
[1,0,1,1,0,1,0,1,0,0]
 => [[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [[],[[],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [[],[[[]],[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [[],[[[],[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [[],[[[[]]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [[[]],[[],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [[[]],[[[]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [[[],[]],[[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [[[],[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => [[[],[],[[]]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [[[],[[]]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [[[],[[]],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,1,0,1,1,0,1,0,0,0]
 => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [[[],[[[]]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
Description
The general position number of a graph. A set $S$ of vertices in a graph $G$ is a general position set if no three vertices of $S$ lie on a shortest path between any two of them.
Matching statistic: St001656
Mapping
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St001656: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [[]]
 => ([(0,1)],2)
 => 2
[1,0,1,0]
 => [[],[]]
 => ([(0,2),(1,2)],3)
 => 2
[1,1,0,0]
 => [[[]]]
 => ([(0,2),(1,2)],3)
 => 2
[1,0,1,0,1,0]
 => [[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,0,1,1,0,0]
 => [[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,1,0,0,1,0]
 => [[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,1,0,1,0,0]
 => [[[],[]]]
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,1,1,0,0,0]
 => [[[[]]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,0,1,0,1,1,0,0]
 => [[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,0,1,1,0,0,1,0]
 => [[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,0,1,1,0,1,0,0]
 => [[],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,0,0,1,1,0,0]
 => [[[]],[[]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,0,1,0,1,0,0]
 => [[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,1,0,1,1,0,0,0]
 => [[[],[[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,1,0,1,0,0,0]
 => [[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 4
[1,0,1,1,0,1,0,1,0,0]
 => [[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [[],[[],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [[],[[[]],[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [[],[[[],[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [[],[[[[]]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [[[]],[[],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [[[]],[[[]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [[[],[]],[[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [[[],[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => [[[],[],[[]]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [[[],[[]]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [[[],[[]],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,1,0,1,1,0,1,0,0,0]
 => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [[[],[[[]]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
Description
The monophonic position number of a graph. A subset $M$ of the vertex set of a graph is a monophonic position set if no three vertices of $M$ lie on a common induced path. The monophonic position number is the size of a largest monophonic position set.
Matching statistic: St001883
Mapping
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St001883: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [[]]
 => ([(0,1)],2)
 => 2
[1,0,1,0]
 => [[],[]]
 => ([(0,2),(1,2)],3)
 => 2
[1,1,0,0]
 => [[[]]]
 => ([(0,2),(1,2)],3)
 => 2
[1,0,1,0,1,0]
 => [[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,0,1,1,0,0]
 => [[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,1,0,0,1,0]
 => [[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,1,0,1,0,0]
 => [[[],[]]]
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,1,1,0,0,0]
 => [[[[]]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,0,1,0,1,1,0,0]
 => [[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,0,1,1,0,0,1,0]
 => [[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,0,1,1,0,1,0,0]
 => [[],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,0,0,1,1,0,0]
 => [[[]],[[]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,0,1,0,1,0,0]
 => [[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,1,0,1,1,0,0,0]
 => [[[],[[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,1,0,1,0,0,0]
 => [[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 4
[1,0,1,1,0,1,0,1,0,0]
 => [[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [[],[[],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [[],[[[]],[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [[],[[[],[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [[],[[[[]]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [[[]],[[],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [[[]],[[[]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [[[],[]],[[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [[[],[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => [[[],[],[[]]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [[[],[[]]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [[[],[[]],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,1,0,1,1,0,1,0,0,0]
 => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [[[],[[[]]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
Description
The mutual visibility number of a graph. This is the largest cardinality of a subset $P$ of vertices of a graph $G$, such that for each pair of vertices in $P$ there is a shortest path in $G$ which contains no other point in $P$. In particular, the mutual visibility number of the disjoint union of two graphs is the maximum of their mutual visibility numbers.
Matching statistic: St000053
(load all 10 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000053: 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]
 => 1 = 2 - 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 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]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => 2 = 3 - 1
Description
The number of valleys of the Dyck path.
The following 203 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000291The number of descents of a binary word. St000325The width of the tree associated to a permutation. St000390The number of runs of ones in a binary word. St000470The number of runs in a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000021The number of descents of a permutation. St000204The number of internal nodes of a binary tree. St000245The number of ascents of a permutation. St000647The number of big descents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000871The number of very big ascents of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000105The number of blocks in the set partition. St000445The number of rises of length 1 of a Dyck path. St000925The number of topologically connected components of a set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000010The length of the partition. St000024The number of double up and double down steps of a Dyck path. St000054The first entry of the permutation. St000062The length of the longest increasing subsequence of the permutation. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000288The number of ones in a binary word. St000314The number of left-to-right-maxima of a permutation. St000482The (zero)-forcing number of a graph. St000822The Hadwiger number of the graph. St001029The size of the core of a graph. St001116The game chromatic number of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St000155The number of exceedances (also excedences) of a permutation. St000157The number of descents of a standard tableau. St000214The number of adjacencies of a permutation. St000272The treewidth of a graph. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000362The size of a minimal vertex cover of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000536The pathwidth of a graph. St000648The number of 2-excedences of a permutation. St000662The staircase size of the code of a permutation. St000703The number of deficiencies of a permutation. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000619The number of cyclic descents of a permutation. St000646The number of big ascents of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St000015The number of peaks of a Dyck path. St000213The number of weak exceedances (also weak excedences) of a permutation. St000308The height of the tree associated to a permutation. St000354The number of recoils of a permutation. St000528The height of a poset. St000542The number of left-to-right-minima of a permutation. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000829The Ulam distance of a permutation to the identity permutation. St000912The number of maximal antichains in a poset. St000991The number of right-to-left minima of a permutation. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000019The cardinality of the support of a permutation. St000030The sum of the descent differences of a permutations. St000080The rank of the poset. St000120The number of left tunnels of a Dyck path. St000238The number of indices that are not small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000691The number of changes of a binary word. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001489The maximum of the number of descents and the number of inverse descents. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001812The biclique partition number of a graph. St000702The number of weak deficiencies of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000795The mad of a permutation. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000957The number of Bruhat lower covers of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001668The number of points of the poset minus the width of the poset. St000292The number of ascents of a binary word. St001315The dissociation number of a graph. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000742The number of big ascents of a permutation after prepending zero. St000451The length of the longest pattern of the form k 1 2. St001427The number of descents of a signed permutation. St000035The number of left outer peaks of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000392The length of the longest run of ones in a binary word. St000628The balance of a binary word. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the 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. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000306The bounce count of a Dyck path. St001864The number of excedances of a signed permutation. St000167The number of leaves of an ordered tree. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000264The girth of a graph, which is not a tree. St000159The number of distinct parts of the integer partition. St001330The hat guessing number of a graph. St000028The number of stack-sorts needed to sort a permutation. St001083The number of boxed occurrences of 132 in a permutation. St000259The diameter of a connected graph. St001394The genus of a permutation. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000164The number of short pairs. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000454The largest eigenvalue of a graph if it is integral. St001488The number of corners of a skew partition. St001902The number of potential covers of a poset. St000052The number of valleys of a Dyck path not on the x-axis. St000455The second largest eigenvalue of a graph if it is integral. St001875The number of simple modules with projective dimension at most 1. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000260The radius of a connected graph. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000443The number of long tunnels of a Dyck path. St000665The number of rafts of a permutation. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001368The number of vertices of maximal degree in a graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000872The number of very big descents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001964The interval resolution global dimension of a poset. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. 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 $eA$. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000714The number of semistandard Young tableau of given shape, with entries at most 2. 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 $eA$. St001469The holeyness of a permutation. St000741The Colin de Verdière graph invariant. St001487The number of inner corners of a skew partition. St001896The number of right descents of a signed permutations. St001867The number of alignments of type EN of a signed permutation. St000779The tier of a permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001060The distinguishing index of a graph. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000834The number of right outer peaks of a permutation. St000731The number of double exceedences of a permutation. St000381The largest part of an integer composition. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001637The number of (upper) dissectors of a poset. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000317The cycle descent number of a permutation. St000767The number of runs in an integer composition. St000862The number of parts of the shifted shape of a permutation. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one 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. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001470The cyclic holeyness of a permutation. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001811The Castelnuovo-Mumford regularity of a permutation. St001823The Stasinski-Voll length of a signed permutation. St001946The number of descents in a parking function. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition.