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Your data matches 29 different statistics following compositions of up to 3 maps.
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Matching statistic: St000312
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
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%
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]
=> [1,0]
=> [[]]
=> ([(0,1)],2)
=> 2
[1,0,1,0]
=> [1,1,0,0]
=> [[[]]]
=> ([(0,2),(1,2)],3)
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> [[],[]]
=> ([(0,2),(1,2)],3)
=> 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[[[]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [[[],[]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [[[],[[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [[[],[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[[[[]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[[[[],[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[[[],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[[[[]],[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[[],[[]],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[[],[[],[]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[[],[[[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[[],[],[[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[[[],[]],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[[[]],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[[[]]],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[],[[[],[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(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: St001036
(load all 153 compositions to match this statistic)
(load all 153 compositions to match this statistic)
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 86%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> [1,0]
=> []
=> []
=> ? = 2 - 2
[1,0,1,0]
=> [1,1,0,0]
=> []
=> []
=> ? = 2 - 2
[1,1,0,0]
=> [1,0,1,0]
=> [1]
=> [1,0,1,0]
=> 0 = 2 - 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? = 2 - 2
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 2 - 2
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 3 - 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1 = 3 - 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 2 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 5 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 4 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 3 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 4 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 3 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 2 - 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> []
=> []
=> ? = 2 - 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 4 - 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4]
=> ?
=> ? = 4 - 2
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
=> [5,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 4 - 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,4,4,3]
=> ?
=> ? = 5 - 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,4,3]
=> ?
=> ? = 5 - 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0,1,0]
=> [7,3,3,3]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0,1,0]
=> ? = 5 - 2
[1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,3,3,3,2]
=> ?
=> ? = 5 - 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,3,3,2]
=> ?
=> ? = 5 - 2
[1,0,1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [6,3,3,3,3,2]
=> ?
=> ? = 5 - 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,3,2]
=> ?
=> ? = 4 - 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,4,4,3,2]
=> ?
=> ? = 5 - 2
[1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,4,3,2]
=> ?
=> ? = 5 - 2
[1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,4,4,3,2]
=> ?
=> ? = 4 - 2
[1,0,1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,3,2]
=> ?
=> ? = 5 - 2
[1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [4,4,3,3,3,2]
=> ?
=> ? = 5 - 2
[1,0,1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,4,3,3,3,2]
=> ?
=> ? = 5 - 2
[1,0,1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [5,5,4,3,3,2]
=> ?
=> ? = 5 - 2
[1,0,1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,3,2]
=> ?
=> ? = 5 - 2
[1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,2,2,2]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 5 - 2
[1,0,1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0,1,0]
=> [7,6,2,2,2]
=> ?
=> ? = 5 - 2
[1,0,1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,1,0,0]
=> [6,6,5,2]
=> ?
=> ? = 5 - 2
[1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]
=> [7,5,4,2]
=> ?
=> ? = 5 - 2
[1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0,1,0]
=> [7,3,3,3,2]
=> ?
=> ? = 5 - 2
[1,0,1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,1,0,0]
=> [6,6,5,3,2]
=> ?
=> ? = 5 - 2
[1,0,1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2,2]
=> ?
=> ? = 5 - 2
[1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [6,6,5,1,1,1,1]
=> ?
=> ? = 5 - 2
[1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5,1,1,1,1]
=> ?
=> ? = 4 - 2
[1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,1,1,1]
=> ?
=> ? = 5 - 2
[1,1,0,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,1,1,1]
=> ?
=> ? = 5 - 2
[1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [5,5,4,1,1,1,1]
=> ?
=> ? = 5 - 2
[1,1,0,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [7,5,4,1,1,1,1]
=> ?
=> ? = 5 - 2
[1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [6,6,5,2,2,2,1]
=> ?
=> ? = 5 - 2
[1,1,0,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [5,5,2,2,2,2,1]
=> ?
=> ? = 5 - 2
[1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,3,3,3,2,1]
=> ?
=> ? = 4 - 2
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,3,2,1]
=> ?
=> ? = 4 - 2
[1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,4,3,3,3,2,1]
=> ?
=> ? = 4 - 2
[1,1,0,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,6,5,3,2,2,1]
=> ?
=> ? = 5 - 2
[1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,3,2,2,2,1]
=> ?
=> ? = 5 - 2
[1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [6,4,3,3,2,2,1]
=> ?
=> ? = 5 - 2
[1,1,0,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,2,1,1]
=> ?
=> ? = 5 - 2
[1,1,0,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [6,6,5,2,1,1,1]
=> ?
=> ? = 5 - 2
[1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [6,5,2,2,1,1,1]
=> ?
=> ? = 5 - 2
Description
The number of inner corners of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St001499
(load all 64 compositions to match this statistic)
(load all 64 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 86%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> []
=> []
=> []
=> ? = 2 - 1
[1,0,1,0]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0]
=> []
=> []
=> []
=> ? = 2 - 1
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> []
=> []
=> []
=> ? = 2 - 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [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]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> []
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 2 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 4 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,3,2,1]
=> ?
=> ?
=> ? = 4 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,3,2,1]
=> ?
=> ?
=> ? = 4 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,3,2,1]
=> ?
=> ?
=> ? = 4 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,3,3,3,2,1]
=> ?
=> ?
=> ? = 4 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,3,3,3,2,1]
=> ?
=> ?
=> ? = 4 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,6,5,3,2,2,1]
=> ?
=> ?
=> ? = 5 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,4,3,2,2,1]
=> ?
=> ?
=> ? = 5 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [6,6,5,2,2,2,1]
=> ?
=> ?
=> ? = 5 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [5,3,3,2,2,2,1]
=> ?
=> ?
=> ? = 5 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [5,5,2,2,2,2,1]
=> ?
=> ?
=> ? = 5 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 4 - 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 5 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [6,5,5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 5 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,2,1,1]
=> ?
=> ?
=> ? = 5 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,1,1]
=> ?
=> ?
=> ? = 4 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [6,6,5,3,2,1,1]
=> ?
=> ?
=> ? = 5 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,1,1]
=> ?
=> ?
=> ? = 4 - 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,4,3,2,1,1]
=> ?
=> ?
=> ? = 5 - 1
[1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,2,1,1]
=> ?
=> ?
=> ? = 4 - 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [6,5,3,3,2,1,1]
=> ?
=> ?
=> ? = 5 - 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,3,3,2,1,1]
=> ?
=> ?
=> ? = 5 - 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [6,4,3,3,2,1,1]
=> ?
=> ?
=> ? = 5 - 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [7,3,3,3,2,1,1]
=> ?
=> ?
=> ? = 5 - 1
[1,0,1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [5,3,3,3,2,1,1]
=> ?
=> ?
=> ? = 5 - 1
[1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,2,1,1]
=> ?
=> ?
=> ? = 4 - 1
[1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 4 - 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,1,1,1]
=> ?
=> ?
=> ? = 5 - 1
[1,0,1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [6,5,5,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 5 - 1
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,1,1,1]
=> ?
=> ?
=> ? = 5 - 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [7,5,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 5 - 1
[1,0,1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [5,5,3,2,1,1,1]
=> ?
=> ?
=> ? = 5 - 1
[1,0,1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [5,4,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,1,0,0,0]
=> ? = 5 - 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [6,3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 5 - 1
[1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [5,3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,1,0,0,0,0]
=> ? = 5 - 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [7,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,1,0,0,0]
=> ? = 5 - 1
[1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [4,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,1,0,0,0]
=> ? = 5 - 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5,1,1,1,1]
=> ?
=> ?
=> ? = 4 - 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [6,6,5,1,1,1,1]
=> ?
=> ?
=> ? = 5 - 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [7,5,4,1,1,1,1]
=> ?
=> ?
=> ? = 5 - 1
[1,0,1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [5,4,4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,1,0,0]
=> ? = 5 - 1
[1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [4,4,4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 5 - 1
[1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [5,4,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 4 - 1
Description
The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra.
We use the bijection in the code by Christian Stump to have a bijection to Dyck paths.
Matching statistic: St000292
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 71%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> []
=> => => ? = 2 - 2
[1,0,1,0]
=> [1]
=> 10 => 00 => 0 = 2 - 2
[1,1,0,0]
=> []
=> => => ? = 2 - 2
[1,0,1,0,1,0]
=> [2,1]
=> 1010 => 0000 => 0 = 2 - 2
[1,0,1,1,0,0]
=> [1,1]
=> 110 => 001 => 1 = 3 - 2
[1,1,0,0,1,0]
=> [2]
=> 100 => 010 => 1 = 3 - 2
[1,1,0,1,0,0]
=> [1]
=> 10 => 00 => 0 = 2 - 2
[1,1,1,0,0,0]
=> []
=> => => ? = 2 - 2
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101010 => 000000 => 0 = 2 - 2
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 11010 => 00001 => 1 = 3 - 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 100110 => 001010 => 2 = 4 - 2
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 10110 => 00100 => 1 = 3 - 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 0011 => 1 = 3 - 2
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 10100 => 01000 => 1 = 3 - 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 1100 => 0101 => 2 = 4 - 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => 00010 => 1 = 3 - 2
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => 0000 => 0 = 2 - 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 001 => 1 = 3 - 2
[1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 0110 => 1 = 3 - 2
[1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 010 => 1 = 3 - 2
[1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 00 => 0 = 2 - 2
[1,1,1,1,0,0,0,0]
=> []
=> => => ? = 2 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 10101010 => 00000000 => 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1101010 => 0000001 => 1 = 3 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 10011010 => 00001010 => 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1011010 => 0000100 => 1 = 3 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 111010 => 000011 => 1 = 3 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 10100110 => 00101000 => 2 = 4 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1100110 => 0010101 => 3 = 5 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 10010110 => 00100010 => 2 = 4 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1010110 => 0010000 => 1 = 3 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 110110 => 001001 => 2 = 4 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 10001110 => 00110110 => 2 = 4 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1001110 => 0011010 => 2 = 4 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => 001100 => 1 = 3 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 00111 => 1 = 3 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1010100 => 0100000 => 1 = 3 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 110100 => 010001 => 2 = 4 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1001100 => 0101010 => 3 = 5 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 101100 => 010100 => 2 = 4 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 11100 => 01011 => 2 = 4 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1010010 => 0001000 => 1 = 3 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 110010 => 000101 => 2 = 4 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1001010 => 0000010 => 1 = 3 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 101010 => 000000 => 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 11010 => 00001 => 1 = 3 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1000110 => 0010110 => 2 = 4 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 100110 => 001010 => 2 = 4 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 10110 => 00100 => 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1110 => 0011 => 1 = 3 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 101000 => 011000 => 1 = 3 - 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 11000 => 01101 => 2 = 4 - 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 100100 => 010010 => 2 = 4 - 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 10100 => 01000 => 1 = 3 - 2
[1,1,1,1,1,0,0,0,0,0]
=> []
=> => => ? = 2 - 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> 1010001110 => 0011011000 => ? = 4 - 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> 1001001110 => 0011010010 => ? = 5 - 2
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> 1000101110 => 0011000110 => ? = 4 - 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> 1000011110 => 0011101110 => ? = 4 - 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> => => ? = 2 - 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> 11001101010 => ? => ? = 5 - 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> 10101101010 => ? => ? = 3 - 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,1,1]
=> 11010100110 => ? => ? = 5 - 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> 11001100110 => ? => ? = 7 - 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,1,1]
=> 10101100110 => ? => ? = 5 - 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,1,1]
=> 11001010110 => ? => ? = 5 - 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,1]
=> 10101010110 => ? => ? = 3 - 2
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [5,3,2,2,1,1]
=> 10010110110 => ? => ? = 5 - 2
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,5,1,1,1,1]
=> 11000011110 => ? => ? = 5 - 2
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [5,3,1,1,1,1]
=> 10010011110 => ? => ? = 5 - 2
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2]
=> 10011010100 => ? => ? = 5 - 2
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2]
=> 10101001100 => ? => ? = 5 - 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2]
=> 10011001100 => ? => ? = 7 - 2
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2]
=> 10010101100 => ? => ? = 5 - 2
[1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,2]
=> 10000111100 => ? => ? = 5 - 2
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1]
=> 10011001010 => ? => ? = 5 - 2
[1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [6,4,2,1,1]
=> 10010010110 => ? => ? = 5 - 2
[1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,2,1,1]
=> 10000110110 => ? => ? = 5 - 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1]
=> 10101010101010 => 00000000000000 => ? = 2 - 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,5,4,3,2,1]
=> 10011010101010 => ? => ? = 4 - 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,3,2,1]
=> ? => ? => ? = 4 - 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,3,2,1]
=> ? => ? => ? = 4 - 2
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,3,2,1]
=> ? => ? => ? = 4 - 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,3,3,3,2,1]
=> ? => ? => ? = 4 - 2
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,3,3,3,2,1]
=> ? => ? => ? = 4 - 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,6,5,3,2,2,1]
=> ? => ? => ? = 5 - 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,4,3,2,2,1]
=> ? => ? => ? = 5 - 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [6,6,5,2,2,2,1]
=> ? => ? => ? = 5 - 2
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [5,3,3,2,2,2,1]
=> ? => ? => ? = 5 - 2
[1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [5,5,2,2,2,2,1]
=> ? => ? => ? = 5 - 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,1,1]
=> 10101010100110 => ? => ? = 4 - 2
[1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,2,1,1]
=> 1101010010110 => ? => ? = 5 - 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [6,5,5,4,2,1,1]
=> 1011010010110 => ? => ? = 5 - 2
[1,0,1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,2,1,1]
=> ? => ? => ? = 5 - 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,1,1]
=> ? => ? => ? = 4 - 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [6,6,5,3,2,1,1]
=> ? => ? => ? = 5 - 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,1,1]
=> ? => ? => ? = 4 - 2
[1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,4,3,2,1,1]
=> ? => ? => ? = 5 - 2
[1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,2,1,1]
=> 101101010110 => ? => ? = 4 - 2
[1,0,1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [6,5,3,3,2,1,1]
=> ? => ? => ? = 5 - 2
Description
The number of ascents of a binary word.
Matching statistic: St000390
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 71%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> []
=> => => ? = 2 - 2
[1,0,1,0]
=> [1]
=> 10 => 00 => 0 = 2 - 2
[1,1,0,0]
=> []
=> => => ? = 2 - 2
[1,0,1,0,1,0]
=> [2,1]
=> 1010 => 0000 => 0 = 2 - 2
[1,0,1,1,0,0]
=> [1,1]
=> 110 => 001 => 1 = 3 - 2
[1,1,0,0,1,0]
=> [2]
=> 100 => 010 => 1 = 3 - 2
[1,1,0,1,0,0]
=> [1]
=> 10 => 00 => 0 = 2 - 2
[1,1,1,0,0,0]
=> []
=> => => ? = 2 - 2
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101010 => 000000 => 0 = 2 - 2
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 11010 => 00001 => 1 = 3 - 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 100110 => 001010 => 2 = 4 - 2
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 10110 => 00100 => 1 = 3 - 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 0011 => 1 = 3 - 2
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 10100 => 01000 => 1 = 3 - 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 1100 => 0101 => 2 = 4 - 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => 00010 => 1 = 3 - 2
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => 0000 => 0 = 2 - 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 001 => 1 = 3 - 2
[1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 0110 => 1 = 3 - 2
[1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 010 => 1 = 3 - 2
[1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 00 => 0 = 2 - 2
[1,1,1,1,0,0,0,0]
=> []
=> => => ? = 2 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 10101010 => 00000000 => 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1101010 => 0000001 => 1 = 3 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 10011010 => 00001010 => 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1011010 => 0000100 => 1 = 3 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 111010 => 000011 => 1 = 3 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 10100110 => 00101000 => 2 = 4 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1100110 => 0010101 => 3 = 5 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 10010110 => 00100010 => 2 = 4 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1010110 => 0010000 => 1 = 3 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 110110 => 001001 => 2 = 4 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 10001110 => 00110110 => 2 = 4 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1001110 => 0011010 => 2 = 4 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => 001100 => 1 = 3 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 00111 => 1 = 3 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1010100 => 0100000 => 1 = 3 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 110100 => 010001 => 2 = 4 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1001100 => 0101010 => 3 = 5 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 101100 => 010100 => 2 = 4 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 11100 => 01011 => 2 = 4 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1010010 => 0001000 => 1 = 3 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 110010 => 000101 => 2 = 4 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1001010 => 0000010 => 1 = 3 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 101010 => 000000 => 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 11010 => 00001 => 1 = 3 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1000110 => 0010110 => 2 = 4 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 100110 => 001010 => 2 = 4 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 10110 => 00100 => 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1110 => 0011 => 1 = 3 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 101000 => 011000 => 1 = 3 - 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 11000 => 01101 => 2 = 4 - 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 100100 => 010010 => 2 = 4 - 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 10100 => 01000 => 1 = 3 - 2
[1,1,1,1,1,0,0,0,0,0]
=> []
=> => => ? = 2 - 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> 1010001110 => 0011011000 => ? = 4 - 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> 1001001110 => 0011010010 => ? = 5 - 2
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> 1000101110 => 0011000110 => ? = 4 - 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> 1000011110 => 0011101110 => ? = 4 - 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> => => ? = 2 - 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> 11001101010 => ? => ? = 5 - 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> 10101101010 => ? => ? = 3 - 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,1,1]
=> 11010100110 => ? => ? = 5 - 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> 11001100110 => ? => ? = 7 - 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,1,1]
=> 10101100110 => ? => ? = 5 - 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,1,1]
=> 11001010110 => ? => ? = 5 - 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,1]
=> 10101010110 => ? => ? = 3 - 2
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [5,3,2,2,1,1]
=> 10010110110 => ? => ? = 5 - 2
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,5,1,1,1,1]
=> 11000011110 => ? => ? = 5 - 2
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [5,3,1,1,1,1]
=> 10010011110 => ? => ? = 5 - 2
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2]
=> 10011010100 => ? => ? = 5 - 2
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2]
=> 10101001100 => ? => ? = 5 - 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2]
=> 10011001100 => ? => ? = 7 - 2
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2]
=> 10010101100 => ? => ? = 5 - 2
[1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,2]
=> 10000111100 => ? => ? = 5 - 2
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1]
=> 10011001010 => ? => ? = 5 - 2
[1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [6,4,2,1,1]
=> 10010010110 => ? => ? = 5 - 2
[1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,2,1,1]
=> 10000110110 => ? => ? = 5 - 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1]
=> 10101010101010 => 00000000000000 => ? = 2 - 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,5,4,3,2,1]
=> 10011010101010 => ? => ? = 4 - 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,3,2,1]
=> ? => ? => ? = 4 - 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,3,2,1]
=> ? => ? => ? = 4 - 2
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,3,2,1]
=> ? => ? => ? = 4 - 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,3,3,3,2,1]
=> ? => ? => ? = 4 - 2
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,3,3,3,2,1]
=> ? => ? => ? = 4 - 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,6,5,3,2,2,1]
=> ? => ? => ? = 5 - 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,4,3,2,2,1]
=> ? => ? => ? = 5 - 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [6,6,5,2,2,2,1]
=> ? => ? => ? = 5 - 2
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [5,3,3,2,2,2,1]
=> ? => ? => ? = 5 - 2
[1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [5,5,2,2,2,2,1]
=> ? => ? => ? = 5 - 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,1,1]
=> 10101010100110 => ? => ? = 4 - 2
[1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,2,1,1]
=> 1101010010110 => ? => ? = 5 - 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [6,5,5,4,2,1,1]
=> 1011010010110 => ? => ? = 5 - 2
[1,0,1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,2,1,1]
=> ? => ? => ? = 5 - 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,1,1]
=> ? => ? => ? = 4 - 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [6,6,5,3,2,1,1]
=> ? => ? => ? = 5 - 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,1,1]
=> ? => ? => ? = 4 - 2
[1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,4,3,2,1,1]
=> ? => ? => ? = 5 - 2
[1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,2,1,1]
=> 101101010110 => ? => ? = 4 - 2
[1,0,1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [6,5,3,3,2,1,1]
=> ? => ? => ? = 5 - 2
Description
The number of runs of ones in a binary word.
Matching statistic: St000249
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000249: Set partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 86%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000249: Set partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> [1,0]
=> [1] => {{1}}
=> ? = 2
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => {{1,2}}
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => {{1},{2}}
=> 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => {{1,3},{2}}
=> 2
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> 3
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => {{1},{2},{3}}
=> 3
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> 2
[1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => {{1,2},{3}}
=> 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => {{1,4},{2,3}}
=> 2
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => {{1,4},{2},{3}}
=> 3
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 4
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => {{1,2,4},{3}}
=> 3
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => {{1,3,4},{2}}
=> 3
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 3
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 4
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> 3
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => {{1},{2,4},{3}}
=> 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 3
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 3
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> 3
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => {{1,3},{2},{4}}
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => {{1,5},{2,3,4}}
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => {{1,5},{2},{3,4}}
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => {{1,2,4,5},{3}}
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> 5
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => {{1,2,5},{3},{4}}
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => {{1,2,5},{3,4}}
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => {{1,2,3,5},{4}}
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => {{1,4,5},{2},{3}}
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => {{1,3,4,5},{2}}
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => {{1,3,5},{2},{4}}
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => {{1,4,5},{2,3}}
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => {{1},{2,4,5},{3}}
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => {{1},{2,5},{3,4}}
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => {{1,2,4},{3},{5}}
=> 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8,7,6,5,4,3,2,1] => {{1,8},{2,7},{3,6},{4,5}}
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [8,7,4,5,6,3,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? = 4
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> [8,3,6,5,4,7,2,1] => {{1,8},{2,3,6,7},{4,5}}
=> ? = 4
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> [8,3,7,5,6,4,2,1] => {{1,8},{2,3,7},{4,5,6}}
=> ? = 4
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> [8,3,4,7,6,5,2,1] => {{1,8},{2,3,4,7},{5,6}}
=> ? = 4
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [8,6,4,5,3,7,2,1] => ?
=> ? = 4
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [8,5,4,3,6,7,2,1] => {{1,8},{2,5,6,7},{3,4}}
=> ? = 4
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> [8,2,6,4,5,7,3,1] => {{1,8},{2},{3,6,7},{4},{5}}
=> ? = 5
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [8,2,4,7,5,6,3,1] => {{1,8},{2},{3,4,7},{5},{6}}
=> ? = 5
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> [8,3,6,4,5,2,7,1] => {{1,8},{2,3,6},{4},{5},{7}}
=> ? = 5
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [8,3,4,2,6,7,5,1] => ?
=> ? = 5
[1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> [8,5,3,4,6,2,7,1] => ?
=> ? = 5
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [2,7,6,5,4,3,8,1] => {{1,2,7,8},{3,6},{4,5}}
=> ? = 4
[1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [2,8,6,4,5,3,7,1] => ?
=> ? = 5
[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [2,8,4,6,5,3,7,1] => {{1,2,8},{3,4,6},{5},{7}}
=> ? = 5
[1,0,1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [2,8,5,4,6,3,7,1] => ?
=> ? = 5
[1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [2,8,6,5,4,7,3,1] => ?
=> ? = 4
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [2,8,6,4,5,7,3,1] => ?
=> ? = 5
[1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,8,7,5,6,4,3,1] => {{1,2,8},{3,7},{4,5,6}}
=> ? = 4
[1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,8,4,7,5,6,3,1] => ?
=> ? = 5
[1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,8,4,7,6,5,3,1] => {{1,2,8},{3,4,7},{5,6}}
=> ? = 4
[1,0,1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [2,8,3,6,5,7,4,1] => ?
=> ? = 5
[1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [2,8,3,5,7,6,4,1] => {{1,2,8},{3},{4,5,7},{6}}
=> ? = 5
[1,0,1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [2,8,3,7,5,6,4,1] => {{1,2,8},{3},{4,7},{5},{6}}
=> ? = 5
[1,0,1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [2,8,4,5,3,7,6,1] => ?
=> ? = 5
[1,0,1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [2,8,4,3,6,7,5,1] => {{1,2,8},{3,4},{5,6,7}}
=> ? = 5
[1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,8,7,6,5,4,1] => {{1,2,3,8},{4,7},{5,6}}
=> ? = 4
[1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [2,5,4,3,8,7,6,1] => {{1,2,5,8},{3,4},{6,7}}
=> ? = 4
[1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,1,0,0]
=> [7,2,6,4,5,3,8,1] => {{1,7,8},{2},{3,6},{4},{5}}
=> ? = 5
[1,0,1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,1,0,0]
=> [7,2,4,6,5,3,8,1] => {{1,7,8},{2},{3,4,6},{5}}
=> ? = 5
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [7,2,5,4,6,3,8,1] => ?
=> ? = 5
[1,0,1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,1,0,0,0]
=> [3,2,8,5,6,4,7,1] => {{1,3,8},{2},{4,5,6},{7}}
=> ? = 5
[1,0,1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> [3,2,8,4,6,7,5,1] => ?
=> ? = 5
[1,0,1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,1,0,0,0]
=> [4,2,3,8,6,5,7,1] => ?
=> ? = 5
[1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,1,0,0,0,0]
=> [4,2,3,8,5,7,6,1] => {{1,4,8},{2},{3},{5},{6,7}}
=> ? = 5
[1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,1,0,0,0,0]
=> [4,2,3,8,6,7,5,1] => ?
=> ? = 5
[1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,1,0,0,0]
=> [6,2,4,5,3,8,7,1] => ?
=> ? = 5
[1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,1,0,0,0]
=> [5,2,4,3,8,6,7,1] => ?
=> ? = 5
[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,0]
=> [7,3,6,5,4,2,8,1] => {{1,7,8},{2,3,6},{4,5}}
=> ? = 4
[1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> [7,3,6,4,5,2,8,1] => {{1,7,8},{2,3,6},{4},{5}}
=> ? = 5
[1,0,1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,1,0,0]
=> [7,3,2,5,6,4,8,1] => {{1,7,8},{2,3},{4,5,6}}
=> ? = 5
[1,0,1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,1,0,0]
=> [7,3,4,2,6,5,8,1] => {{1,7,8},{2,3,4},{5,6}}
=> ? = 5
[1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> [7,3,5,4,2,6,8,1] => ?
=> ? = 5
[1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> [4,3,2,7,6,5,8,1] => {{1,4,7,8},{2,3},{5,6}}
=> ? = 4
[1,0,1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,1,0,0,0]
=> [5,3,2,4,8,6,7,1] => ?
=> ? = 5
[1,0,1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> [4,3,2,5,8,7,6,1] => {{1,4,5,8},{2,3},{6,7}}
=> ? = 4
[1,0,1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,1,0,0,0,0]
=> [4,3,2,8,6,7,5,1] => {{1,4,8},{2,3},{5,6,7}}
=> ? = 4
[1,0,1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,1,0,0,0]
=> [6,3,4,2,5,8,7,1] => ?
=> ? = 5
[1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,1,0,0,0]
=> [5,3,4,2,8,6,7,1] => {{1,5,8},{2,3,4},{6},{7}}
=> ? = 5
Description
The number of singletons ([[St000247]]) plus the number of antisingletons ([[St000248]]) of a set partition.
Matching statistic: St000340
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 86%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> [1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 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]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 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]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 3 = 4 - 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]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> 3 = 4 - 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]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 3 = 4 - 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]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 2 = 3 - 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]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 5 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0]
=> ? = 5 - 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0,1,0]
=> ? = 5 - 1
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> ? = 5 - 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> ? = 5 - 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 5 - 1
[1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 5 - 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 5 - 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 3 - 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 3 - 1
[1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 5 - 1
[1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 5 - 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 3 - 1
[1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> ? = 5 - 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
=> ? = 3 - 1
[1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 5 - 1
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 3 - 1
[1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 5 - 1
[1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 5 - 1
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 3 - 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 3 - 1
[1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? = 3 - 1
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 3 - 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 3 - 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 4 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 4 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 4 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 4 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0,0]
=> ?
=> ? = 4 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 4 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0,0]
=> ?
=> ? = 5 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,1,1,0,0,0,1,0,0]
=> ?
=> ? = 5 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,1,0,0,1,0,0]
=> ?
=> ? = 5 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0,0]
=> ?
=> ? = 5 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0,0]
=> ?
=> ? = 5 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> ?
=> ? = 5 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 5 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ?
=> ? = 5 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> ?
=> ? = 5 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> ?
=> ? = 5 - 1
[1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ?
=> ? = 4 - 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 5 - 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> ?
=> ? = 5 - 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> ?
=> ? = 5 - 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: St000619
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 71%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> [1,0]
=> [1] => [1] => ? = 2 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => [1,2] => 1 = 2 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => [2,1] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => 2 = 3 - 1
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => 2 = 3 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [3,1,2] => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [1,2,4,3] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,4,3,2] => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,3,4,2] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,4,2,3] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,3,1] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,3,4,1] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,4,2,1] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [4,1,3,2] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [4,3,1,2] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [4,1,2,3] => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => [1,2,3,5,4] => 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => [1,2,5,4,3] => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => [1,2,4,5,3] => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => [1,2,5,3,4] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,5,3,4,2] => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [1,3,5,4,2] => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,3,4,5,2] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,4,5,3,2] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => [1,5,2,4,3] => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [1,5,4,2,3] => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [1,4,5,2,3] => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [1,5,2,3,4] => 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [5,2,3,4,1] => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,3,4,2,1] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [2,5,3,4,1] => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [2,3,5,4,1] => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,3,4,5,1] => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [2,4,5,3,1] => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,5,3,2,1] => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,5,4,2,1] => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,4,5,2,1] => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => 2 = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [5,1,3,4,2] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [2,5,6,4,3,7,1] => [1,7,3,4,6,5,2] => ? = 5 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => [1,7,6,5,4,3,2] => ? = 7 - 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,4,7,1] => [1,7,4,5,6,3,2] => ? = 5 - 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [2,5,4,3,6,7,1] => [1,7,6,3,4,5,2] => ? = 5 - 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,4,7,6,5,1] => [1,5,6,7,4,3,2] => ? = 5 - 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [4,5,6,3,2,7,1] => [1,7,2,3,6,5,4] => ? = 5 - 1
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [4,3,2,5,6,7,1] => [1,7,6,5,2,3,4] => ? = 5 - 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [4,5,3,2,6,7,1] => [1,7,6,2,3,5,4] => ? = 5 - 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [4,3,2,7,6,5,1] => [1,5,6,7,2,3,4] => ? = 3 - 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,5,4,3,2,7,1] => [1,7,2,3,4,5,6] => ? = 3 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,5,4,3,2,7] => [7,2,3,4,5,6,1] => ? = 3 - 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,4,5,6,3,2,7] => [7,2,3,6,5,4,1] => ? = 5 - 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => ? = 5 - 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7 - 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,4,3,7] => [7,3,4,6,5,2,1] => ? = 5 - 1
[1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => ? = 5 - 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,4,5,3,2,6,7] => [7,6,2,3,5,4,1] => ? = 5 - 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,4,3,2,5,6,7] => [7,6,5,2,3,4,1] => ? = 5 - 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,6,5,4,7,3,2] => [2,3,7,4,5,6,1] => ? = 3 - 1
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,4,5,6,7,3,2] => [2,3,7,6,5,4,1] => ? = 5 - 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => [2,3,4,5,7,6,1] => ? = 3 - 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => [2,3,5,6,7,4,1] => ? = 3 - 1
[1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,4,3] => [3,4,7,6,5,2,1] => ? = 5 - 1
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => [3,4,5,6,7,2,1] => ? = 3 - 1
[1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => ? = 5 - 1
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,6,7,5,4] => [4,5,7,6,3,2,1] => ? = 5 - 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,4,3,2,7,6,5] => [5,6,7,2,3,4,1] => ? = 3 - 1
[1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [2,1,4,5,6,7,3] => [3,7,6,5,4,1,2] => ? = 5 - 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [2,1,6,5,4,7,3] => [3,7,4,5,6,1,2] => ? = 3 - 1
[1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,6,7,5,4,3] => [3,4,5,7,6,1,2] => ? = 3 - 1
[1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [2,1,4,7,6,5,3] => [3,5,6,7,4,1,2] => ? = 3 - 1
[1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [2,3,4,5,1,7,6] => [6,7,1,5,4,3,2] => ? = 5 - 1
[1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,6,7,4] => [4,7,6,5,1,3,2] => ? = 5 - 1
[1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [2,3,4,1,6,7,5] => [5,7,6,1,4,3,2] => ? = 5 - 1
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [2,3,1,7,6,5,4] => [4,5,6,7,1,3,2] => ? = 3 - 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [2,5,4,3,1,7,6] => [6,7,1,3,4,5,2] => ? = 3 - 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [3,6,5,4,2,1,7] => [7,1,2,4,5,6,3] => ? = 3 - 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,6,2,1,7] => [7,1,2,6,5,4,3] => ? = 5 - 1
[1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7] => [7,6,5,4,1,2,3] => ? = 5 - 1
[1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => [7,4,5,6,1,2,3] => ? = 3 - 1
[1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [3,4,5,2,1,6,7] => [7,6,1,2,5,4,3] => ? = 5 - 1
[1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [3,4,2,1,5,6,7] => [7,6,5,1,2,4,3] => ? = 5 - 1
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,2,1,4,7,6,5] => [5,6,7,4,1,2,3] => ? = 3 - 1
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [3,2,1,6,7,5,4] => [4,5,7,6,1,2,3] => ? = 3 - 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [3,4,2,1,7,6,5] => [5,6,7,1,2,4,3] => ? = 3 - 1
[1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [4,3,2,5,1,7,6] => [6,7,1,5,2,3,4] => ? = 3 - 1
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> [4,3,2,1,6,7,5] => [5,7,6,1,2,3,4] => ? = 3 - 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [4,5,3,2,1,7,6] => [6,7,1,2,3,5,4] => ? = 3 - 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,4,3,2,1,7] => [7,1,2,3,4,6,5] => ? = 3 - 1
Description
The number of cyclic descents of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is given by the number of indices $1 \leq i \leq n$ such that $\pi(i) > \pi(i+1)$ where we set $\pi(n+1) = \pi(1)$.
Matching statistic: St000291
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 71%
Mp00096: Binary words —Foata bijection⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> 10 => 10 => 1 = 2 - 1
[1,0,1,0]
=> 1010 => 1100 => 1 = 2 - 1
[1,1,0,0]
=> 1100 => 0110 => 1 = 2 - 1
[1,0,1,0,1,0]
=> 101010 => 111000 => 1 = 2 - 1
[1,0,1,1,0,0]
=> 101100 => 011010 => 2 = 3 - 1
[1,1,0,0,1,0]
=> 110010 => 101100 => 2 = 3 - 1
[1,1,0,1,0,0]
=> 110100 => 011100 => 1 = 2 - 1
[1,1,1,0,0,0]
=> 111000 => 001110 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> 10101010 => 11110000 => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> 10101100 => 01110010 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> 10110010 => 10110100 => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> 10110100 => 01110100 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> 10111000 => 00110110 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> 11001010 => 11011000 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> 11001100 => 01011010 => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> 11010010 => 10111000 => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> 11010100 => 01111000 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> 11011000 => 00111010 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> 11100010 => 10011100 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> 11100100 => 01011100 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> 11101000 => 00111100 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> 11110000 => 00011110 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 1111100000 => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 0111100010 => 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => 1011100100 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => 0111100100 => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 0011100110 => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => 1101101000 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 0101101010 => 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => 1011101000 => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 0111101000 => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 0011101010 => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 1001101100 => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 0101101100 => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 0011101100 => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 0001101110 => 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 1110110000 => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 0110110010 => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 1010110100 => 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 0110110100 => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 0010110110 => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 1101110000 => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => 0101110010 => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => 1011110000 => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 0111110000 => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => 0011110010 => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 1001110100 => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => 0101110100 => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 0011110100 => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => 0001110110 => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 10101010101100 => 01111110000010 => ? = 3 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 10101011001100 => 01011110001010 => ? = 5 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 10101011010100 => 01111110001000 => ? = 3 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 10101011110000 => 00011110001110 => ? = 3 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> 10110010101100 => 01110110100010 => ? = 5 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 10110011001100 => 01010110101010 => ? = 7 - 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> 10110011010100 => 01110110101000 => ? = 5 - 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> 10110011110000 => 00010110101110 => ? = 5 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> 10110101001100 => 01011110100010 => ? = 5 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 10110101010100 => 01111110100000 => ? = 3 - 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> 10110110100100 => 01011110101000 => ? = 5 - 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> 10110110110000 => 00011110101010 => ? = 5 - 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> 10111100001100 => 01000110111010 => ? = 5 - 1
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> 10111100100100 => 01010110111000 => ? = 5 - 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> 10111100110000 => 00010110111010 => ? = 5 - 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> 10111101010000 => 00011110111000 => ? = 3 - 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 10111111000000 => 00000110111110 => ? = 3 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> 11001010101010 => 11111011000000 => ? = 3 - 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> 11001010110010 => 10111011000100 => ? = 5 - 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> 11001100101010 => 11101011010000 => ? = 5 - 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> 11001100110010 => 10101011010100 => ? = 7 - 1
[1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> 11001101101000 => ? => ? = 5 - 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> 11001111000010 => 10001011011100 => ? = 5 - 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> 11001111001000 => 00101011011100 => ? = 5 - 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> 11010100101010 => 11101111000000 => ? = 3 - 1
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> 11010100110010 => 10101111000100 => ? = 5 - 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> 11010101101000 => 00111111000100 => ? = 3 - 1
[1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> 11011010010010 => 10101111010000 => ? = 5 - 1
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> 11011010101000 => 00111111010000 => ? = 3 - 1
[1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> 11011011000010 => 10001111010100 => ? = 5 - 1
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> 11011011001000 => 00101111010100 => ? = 5 - 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> 11011110100000 => 00001111011100 => ? = 3 - 1
[1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> 11101001001100 => 01010111100010 => ? = 5 - 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> 11101001010100 => 01110111100000 => ? = 3 - 1
[1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> 11101010100100 => 01011111100000 => ? = 3 - 1
[1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> 11101010110000 => 00011111100010 => ? = 3 - 1
[1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> 11101100001100 => 01000111101010 => ? = 5 - 1
[1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> 11101100100100 => 01010111101000 => ? = 5 - 1
[1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> 11101100110000 => 00010111101010 => ? = 5 - 1
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> 11101101010000 => 00011111101000 => ? = 3 - 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> 11101111000000 => 00000111101110 => ? = 3 - 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> 11110000101010 => 11100011110000 => ? = 3 - 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> 11110000110010 => 10100011110100 => ? = 5 - 1
[1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> 11110010010010 => 10101011110000 => ? = 5 - 1
[1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> 11110010101000 => 00111011110000 => ? = 3 - 1
[1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> 11110011000010 => 10001011110100 => ? = 5 - 1
[1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> 11110011001000 => 00101011110100 => ? = 5 - 1
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> 11110101000010 => 10001111110000 => ? = 3 - 1
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> 11110101001000 => 00101111110000 => ? = 3 - 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> 11110110100000 => 00001111110100 => ? = 3 - 1
Description
The number of descents of a binary word.
Matching statistic: St000925
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 71%
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> {{1},{2}}
=> 2
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 3
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 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]
=> {{1},{2,3,4,5}}
=> 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]
=> {{1},{2,4,5},{3}}
=> 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]
=> {{1},{2},{3,5},{4}}
=> 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]
=> {{1},{2},{3,4,5}}
=> 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]
=> {{1},{2,3},{4,5}}
=> 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]
=> {{1,2},{3,4},{5}}
=> 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]
=> {{1,3},{2},{4},{5}}
=> 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]
=> {{1,2},{3,5},{4}}
=> 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]
=> {{1,2},{3,4,5}}
=> 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]
=> {{1,3},{2},{4,5}}
=> 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]
=> {{1,4},{2,3},{5}}
=> 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]
=> {{1,2,3},{4},{5}}
=> 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]
=> {{1,2,3},{4,5}}
=> 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]
=> {{1,2,3,4},{5}}
=> 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]
=> {{1},{2,3,4,5,6}}
=> 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]
=> {{1},{2,3,5,6},{4}}
=> 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]
=> {{1},{2,4,6},{3},{5}}
=> 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]
=> {{1},{2,4,5,6},{3}}
=> 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]
=> {{1},{2,5,6},{3,4}}
=> 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]
=> {{1},{2},{3,6},{4,5}}
=> 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]
=> {{1},{2},{3},{4,6},{5}}
=> 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]
=> {{1},{2},{3,4,6},{5}}
=> 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]
=> {{1},{2},{3,4,5,6}}
=> 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]
=> {{1},{2},{3},{4,5,6}}
=> 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]
=> {{1},{2,3},{4},{5,6}}
=> 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]
=> {{1},{2,3},{4,6},{5}}
=> 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]
=> {{1},{2,3},{4,5,6}}
=> 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]
=> {{1},{2,3,4},{5,6}}
=> 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]
=> {{1,2},{3,4,5},{6}}
=> 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]
=> {{1,2},{3},{4,5},{6}}
=> 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]
=> {{1,3},{2},{4},{5},{6}}
=> 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]
=> {{1,3},{2},{4,5},{6}}
=> 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]
=> {{1,3,4},{2},{5},{6}}
=> 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]
=> {{1,2},{3,6},{4,5}}
=> 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]
=> {{1,2},{3},{4,6},{5}}
=> 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]
=> {{1,2},{3,4,6},{5}}
=> 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]
=> {{1,2},{3,4,5,6}}
=> 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]
=> {{1,2},{3},{4,5,6}}
=> 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]
=> {{1,3},{2},{4},{5,6}}
=> 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]
=> {{1,3},{2},{4,6},{5}}
=> 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]
=> {{1,3},{2},{4,5,6}}
=> 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]
=> {{1,3,4},{2},{5,6}}
=> 3
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> {{1},{2,3,6,8},{4},{5},{7}}
=> ? = 5
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> {{1},{2,3,7,8},{4,5,6}}
=> ? = 3
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> {{1},{2},{3,8},{4,6,7},{5}}
=> ? = 5
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4},{5,8},{6},{7}}
=> ? = 7
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> {{1},{2},{3},{4,8},{5,6,7}}
=> ? = 5
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> {{1},{2},{3,4,5},{6,8},{7}}
=> ? = 5
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> {{1},{2},{3,4,6,8},{5},{7}}
=> ? = 5
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> {{1},{2},{3},{4,5,6,8},{7}}
=> ? = 5
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> {{1},{2},{3},{4},{5,6,7,8}}
=> ? = 5
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> {{1},{2,6},{3,5},{4},{7,8}}
=> ? = 5
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> {{1},{2,3,4},{5,8},{6},{7}}
=> ? = 5
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> {{1},{2,3,5},{4},{6,8},{7}}
=> ? = 5
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4},{5,6,7,8}}
=> ? = 3
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> {{1},{2,3,4,5,6},{7,8}}
=> ? = 3
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2},{3,4,5,6,7},{8}}
=> ? = 3
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> {{1,2},{3,5,7},{4},{6},{8}}
=> ? = 5
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> {{1,3},{2},{4,5,6},{7},{8}}
=> ? = 5
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> {{1,4},{2},{3},{5},{6},{7},{8}}
=> ? = 7
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> {{1,3},{2},{4,5,7},{6},{8}}
=> ? = 5
[1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> {{1,4},{2},{3},{5,6,7},{8}}
=> ? = 5
[1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> {{1,3,6},{2},{4,5},{7},{8}}
=> ? = 5
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> {{1,3,4,5},{2},{6},{7},{8}}
=> ? = 5
[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]
=> [1,1,0,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> {{1,2},{3,4,8},{5,6,7}}
=> ? = 3
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> {{1,2},{3,5,8},{4},{6},{7}}
=> ? = 5
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> {{1,2},{3,4,5,6,8},{7}}
=> ? = 3
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> {{1,2},{3,5,6,7,8},{4}}
=> ? = 3
[1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> {{1,3},{2},{4,5,8},{6},{7}}
=> ? = 5
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,3},{2},{4,5,6,7,8}}
=> ? = 3
[1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> {{1,4},{2},{3},{5},{6,7,8}}
=> ? = 5
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> {{1,4},{2},{3},{5,6,8},{7}}
=> ? = 5
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> {{1,3,4,5},{2},{6,7,8}}
=> ? = 3
[1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,1,0,0]
=> {{1,2,3},{4},{5,8},{6},{7}}
=> ? = 5
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,1,0,0]
=> {{1,2,3},{4,8},{5,6,7}}
=> ? = 3
[1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3},{4,5,6,8},{7}}
=> ? = 3
[1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> {{1,2,3},{4},{5,6,7,8}}
=> ? = 3
[1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,1,0,0]
=> {{1,6},{2,5},{3},{4},{7,8}}
=> ? = 5
[1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,1,0,1,0,0]
=> {{1,2,4},{3},{5,8},{6},{7}}
=> ? = 5
[1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,0,1,0,0]
=> {{1,2,5},{3},{4},{6,8},{7}}
=> ? = 5
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,4},{3},{5,6,7,8}}
=> ? = 3
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,1,0,0]
=> {{1,2,4,5,6},{3},{7,8}}
=> ? = 3
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,5,6,7},{3,4},{8}}
=> ? = 3
[1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> {{1,2,7},{3,5},{4},{6},{8}}
=> ? = 5
[1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 5
[1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 3
[1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> {{1,2,6},{3,5},{4},{7},{8}}
=> ? = 5
[1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,5},{4},{6},{7},{8}}
=> ? = 5
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> {{1,2,3,4},{5},{6,7,8}}
=> ? = 3
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
=> {{1,2,3,4},{5,6,8},{7}}
=> ? = 3
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0]
=> {{1,2,3,5},{4},{6,7,8}}
=> ? = 3
[1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> {{1,6},{2,3,4,5},{7,8}}
=> ? = 3
Description
The number of topologically connected components of a set partition.
For example, the set partition $\{\{1,5\},\{2,3\},\{4,6\}\}$ has the two connected components $\{1,4,5,6\}$ and $\{2,3\}$.
The number of set partitions with only one block is [[oeis:A099947]].
The following 19 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000871The number of very big ascents of a permutation. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000053The number of valleys of the Dyck path. St001083The number of boxed occurrences of 132 in a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St001488The number of corners of a skew partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001902The number of potential covers of a poset. St000665The number of rafts of a permutation. St000015The number of peaks of a Dyck path. St000083The number of left oriented leafs of a binary tree except the first one. St000354The number of recoils 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. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001960The number of descents of a permutation minus one if its first entry is not one. St000035The number of left outer peaks of a permutation. St000075The orbit size of a standard tableau under promotion. St001637The number of (upper) dissectors of a poset.
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