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Your data matches 48 different statistics following compositions of up to 3 maps.
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Matching statistic: St000974
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000974: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000974: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[]]
=> 1
[1,0,1,0]
=> [[],[]]
=> 0
[1,1,0,0]
=> [[[]]]
=> 2
[1,0,1,0,1,0]
=> [[],[],[]]
=> 0
[1,0,1,1,0,0]
=> [[],[[]]]
=> 0
[1,1,0,0,1,0]
=> [[[]],[]]
=> 0
[1,1,0,1,0,0]
=> [[[],[]]]
=> 1
[1,1,1,0,0,0]
=> [[[[]]]]
=> 3
[1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 0
[1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 0
[1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 0
[1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> 0
[1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 0
[1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 0
[1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 0
[1,1,0,1,0,0,1,0]
=> [[[],[]],[]]
=> 0
[1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> 1
[1,1,0,1,1,0,0,0]
=> [[[],[[]]]]
=> 1
[1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 0
[1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 1
[1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> 2
[1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [[],[],[[]],[]]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [[],[],[[],[]]]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [[],[[]],[],[]]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [[],[[],[]],[]]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [[],[[],[],[]]]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [[],[[],[[]]]]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [[],[[[]],[]]]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [[],[[[],[]]]]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [[[]],[],[],[]]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [[[]],[],[[]]]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [[[]],[[],[]]]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [[[]],[[[]]]]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [[[],[]],[],[]]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [[[],[]],[[]]]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [[[],[],[]],[]]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[]]]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [[[],[],[[]]]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [[[],[[]]],[]]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [[[],[[]],[]]]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[[],[[],[]]]]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [[[],[[[]]]]]
=> 1
Description
The length of the trunk of an ordered tree.
This is the length of the path from the root to the first vertex which has not exactly one child.
Matching statistic: St000445
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 1
[1,0,1,0]
=> [2,1] => [2,1] => [1,1,0,0]
=> 0
[1,1,0,0]
=> [1,2] => [1,2] => [1,0,1,0]
=> 2
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> 0
[1,1,0,0,1,0]
=> [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> 0
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
Description
The number of rises of length 1 of a Dyck path.
Matching statistic: St000475
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> [1]
=> 1
[1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> [2]
=> 0
[1,1,0,0]
=> [1,2] => ([],2)
=> [1,1]
=> 2
[1,0,1,0,1,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
[1,0,1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> 0
[1,1,0,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> 0
[1,1,0,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> [2,1]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> [1,1,1]
=> 3
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 0
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 0
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> [1,1,1,1]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 1
Description
The number of parts equal to 1 in a partition.
Matching statistic: St001691
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001691: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001691: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 1
[1,0,1,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 0
[1,1,0,0]
=> [1,2] => [1,2] => ([],2)
=> 2
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 0
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 0
[1,1,0,0,1,0]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 0
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => ([],3)
=> 3
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
Description
The number of kings in a graph.
A vertex of a graph is a king, if all its neighbours have smaller degree. In particular, an isolated vertex is a king.
Matching statistic: St001107
(load all 54 compositions to match this statistic)
(load all 54 compositions to match this statistic)
St001107: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> ? = 1
[1,0,1,0]
=> 0
[1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> 0
[1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> 0
[1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> 0
Description
The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path.
In other words, this is the lowest height of a valley of a Dyck path, or its semilength in case of the unique path without valleys.
Matching statistic: St000674
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> ? = 1
[1,0,1,0]
=> [2,1] => [1,1,0,0]
=> 0
[1,1,0,0]
=> [1,2] => [1,0,1,0]
=> 2
[1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> 0
[1,1,0,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
Description
The number of hills of a Dyck path.
A hill is a peak with up step starting and down step ending at height zero.
Matching statistic: St000022
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 1
[1,0,1,0]
=> [1,2] => [1,2] => [2,1] => 0
[1,1,0,0]
=> [2,1] => [2,1] => [1,2] => 2
[1,0,1,0,1,0]
=> [1,2,3] => [1,3,2] => [3,1,2] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [3,1,2] => 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,3,1] => 0
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => [2,1,3] => 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [1,2,3] => 3
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,4,3,2] => [4,1,2,3] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,4,3,2] => [4,1,2,3] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,4,3,2] => [4,1,2,3] => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [4,1,2,3] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => [4,1,2,3] => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => [3,1,2,4] => 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => [3,1,2,4] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [2,3,1,4] => 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => [2,1,3,4] => 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => [4,5,1,2,3] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => [4,5,1,2,3] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => [4,5,1,2,3] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [4,5,1,2,3] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => [4,5,1,2,3] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => [4,1,5,2,3] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => [4,1,5,2,3] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => [4,1,2,5,3] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => [4,1,2,3,5] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,5,4,3,1] => [4,1,2,3,5] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,5,4,1,3] => [4,1,2,5,3] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,5,4,3,1] => [4,1,2,3,5] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,5,4,3,1] => [4,1,2,3,5] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => [4,1,2,3,5] => 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 0
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [2,3,1,4,5,7,6] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 0
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [2,3,1,4,6,5,7] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 0
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [2,3,1,4,6,7,5] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 0
[1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,7,6,5] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 0
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,4,6,7] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 0
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,7,6] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 0
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [2,3,1,5,6,4,7] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 0
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,6,7,4] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 0
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [2,3,1,5,7,6,4] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 0
[1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,1,6,5,4,7] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 0
[1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [2,3,1,6,5,7,4] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 0
[1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [2,3,1,6,7,5,4] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 0
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [2,3,1,7,6,5,4] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 0
[1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [3,2,4,1,5,6,7] => [3,2,7,1,6,5,4] => [5,6,1,7,2,3,4] => ? = 0
[1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [3,2,4,1,5,7,6] => [3,2,7,1,6,5,4] => [5,6,1,7,2,3,4] => ? = 0
[1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [3,2,4,1,6,5,7] => [3,2,7,1,6,5,4] => [5,6,1,7,2,3,4] => ? = 0
[1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [3,2,4,1,6,7,5] => [3,2,7,1,6,5,4] => [5,6,1,7,2,3,4] => ? = 0
[1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [3,2,4,1,7,6,5] => [3,2,7,1,6,5,4] => [5,6,1,7,2,3,4] => ? = 0
[1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [3,2,4,5,1,6,7] => [3,2,7,6,1,5,4] => [5,6,1,2,7,3,4] => ? = 0
[1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [3,2,4,5,1,7,6] => [3,2,7,6,1,5,4] => [5,6,1,2,7,3,4] => ? = 0
[1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [3,2,5,4,1,6,7] => [3,2,7,6,1,5,4] => [5,6,1,2,7,3,4] => ? = 0
[1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [3,2,5,4,1,7,6] => [3,2,7,6,1,5,4] => [5,6,1,2,7,3,4] => ? = 0
[1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [3,4,2,1,5,6,7] => [3,7,2,1,6,5,4] => [5,1,6,7,2,3,4] => ? = 0
[1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [3,4,2,1,5,7,6] => [3,7,2,1,6,5,4] => [5,1,6,7,2,3,4] => ? = 0
[1,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> [3,4,2,1,6,5,7] => [3,7,2,1,6,5,4] => [5,1,6,7,2,3,4] => ? = 0
[1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [3,4,2,1,6,7,5] => [3,7,2,1,6,5,4] => [5,1,6,7,2,3,4] => ? = 0
[1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [3,4,2,1,7,6,5] => [3,7,2,1,6,5,4] => [5,1,6,7,2,3,4] => ? = 0
[1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [3,4,2,5,1,6,7] => [3,7,2,6,1,5,4] => [5,1,6,2,7,3,4] => ? = 0
[1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [3,4,2,5,1,7,6] => [3,7,2,6,1,5,4] => [5,1,6,2,7,3,4] => ? = 0
[1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [3,4,5,2,1,6,7] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 0
[1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> [3,4,5,2,1,7,6] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 0
[1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [3,4,5,2,6,1,7] => [3,7,6,2,5,1,4] => [5,1,2,6,3,7,4] => ? = 0
[1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [3,5,4,2,1,6,7] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 0
[1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [3,5,4,2,1,7,6] => [3,7,6,2,1,5,4] => [5,1,2,6,7,3,4] => ? = 0
[1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [3,5,4,2,6,1,7] => [3,7,6,2,5,1,4] => [5,1,2,6,3,7,4] => ? = 0
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7] => [4,3,2,1,7,6,5] => [4,5,6,7,1,2,3] => ? = 0
[1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => [4,3,2,1,7,6,5] => [4,5,6,7,1,2,3] => ? = 0
[1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => [4,3,2,1,7,6,5] => [4,5,6,7,1,2,3] => ? = 0
[1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> [4,3,2,1,6,7,5] => [4,3,2,1,7,6,5] => [4,5,6,7,1,2,3] => ? = 0
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => [4,3,2,1,7,6,5] => [4,5,6,7,1,2,3] => ? = 0
[1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [4,3,2,5,1,6,7] => [4,3,2,7,1,6,5] => [4,5,6,1,7,2,3] => ? = 0
[1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [4,3,2,5,1,7,6] => [4,3,2,7,1,6,5] => [4,5,6,1,7,2,3] => ? = 0
[1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [4,3,2,5,6,1,7] => [4,3,2,7,6,1,5] => [4,5,6,1,2,7,3] => ? = 0
[1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [4,3,2,6,5,1,7] => [4,3,2,7,6,1,5] => [4,5,6,1,2,7,3] => ? = 0
[1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [4,3,5,2,1,6,7] => [4,3,7,2,1,6,5] => [4,5,1,6,7,2,3] => ? = 0
[1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [4,3,5,2,1,7,6] => [4,3,7,2,1,6,5] => [4,5,1,6,7,2,3] => ? = 0
[1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [4,3,5,2,6,1,7] => [4,3,7,2,6,1,5] => [4,5,1,6,2,7,3] => ? = 0
[1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [4,3,5,6,2,1,7] => [4,3,7,6,2,1,5] => [4,5,1,2,6,7,3] => ? = 0
[1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [4,3,5,6,2,7,1] => [4,3,7,6,2,5,1] => [4,5,1,2,6,3,7] => ? = 1
Description
The number of fixed points of a permutation.
Matching statistic: St000297
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [2,1] => 1 => 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 01 => 0
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 11 => 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 001 => 0
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 011 => 0
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 011 => 0
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 101 => 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 111 => 3
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0001 => 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 0011 => 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 0011 => 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => 0101 => 0
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 0111 => 0
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 0101 => 0
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => 0111 => 0
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => 0011 => 0
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => 1001 => 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => 1011 => 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 0111 => 0
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => 1011 => 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => 1101 => 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 1111 => 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 00001 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,5,1] => 00011 => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,4,6,1] => 00011 => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,6,4,1] => 00101 => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,5,4,1] => 00111 => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,3,5,6,1] => 00101 => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,3,6,5,1] => 00111 => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,4,5,3,6,1] => 00011 => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,5,6,3,1] => 01001 => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,4,6,5,3,1] => 01011 => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,4,3,6,1] => 00111 => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,5,4,6,3,1] => 01011 => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,5,6,4,3,1] => 01101 => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => 01111 => 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,1] => 01001 => 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,2,4,6,5,1] => 01011 => 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [3,2,5,4,6,1] => 01011 => 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,5,6,4,1] => 01101 => 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,2,6,5,4,1] => 01111 => 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [3,4,2,5,6,1] => 00101 => 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,4,2,6,5,1] => 00111 => 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,2,6,1] => 00011 => 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,2,1] => 10001 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,6,5,2,1] => 10011 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [3,5,4,2,6,1] => 00111 => 0
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,5,4,6,2,1] => 10011 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,5,6,4,2,1] => 10101 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,6,5,4,2,1] => 10111 => 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [2,3,5,4,6,8,7,1] => ? => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,4,8,7,1] => ? => ? = 0
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [2,3,6,5,7,4,8,1] => ? => ? = 0
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [2,3,6,5,7,8,4,1] => ? => ? = 0
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [2,3,6,7,5,4,8,1] => ? => ? = 0
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [2,3,6,7,5,8,4,1] => ? => ? = 0
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,6,8,7,5,4,1] => ? => ? = 0
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,3,7,6,8,5,4,1] => ? => ? = 0
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,7,8,6,5,4,1] => ? => ? = 0
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [2,4,3,5,7,8,6,1] => ? => ? = 0
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [2,4,3,7,6,8,5,1] => ? => ? = 0
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [2,4,3,7,8,6,5,1] => ? => ? = 0
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [2,4,5,3,7,6,8,1] => ? => ? = 0
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [2,4,5,7,6,3,8,1] => ? => ? = 0
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [2,4,5,7,6,8,3,1] => ? => ? = 0
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [2,4,5,8,7,6,3,1] => ? => ? = 0
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [2,4,7,6,5,3,8,1] => ? => ? = 0
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [2,4,7,6,8,5,3,1] => ? => ? = 0
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [2,5,4,6,3,8,7,1] => ? => ? = 0
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [2,5,4,6,8,7,3,1] => ? => ? = 0
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [2,5,6,4,3,8,7,1] => ? => ? = 0
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [2,5,6,4,7,3,8,1] => ? => ? = 0
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [2,5,7,6,4,3,8,1] => ? => ? = 0
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [2,5,7,6,4,8,3,1] => ? => ? = 0
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [2,6,5,4,7,3,8,1] => ? => ? = 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]
=> [2,6,5,7,4,8,3,1] => ? => ? = 0
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [2,6,7,5,4,3,8,1] => ? => ? = 0
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,1,0,0,0,0]
=> [3,2,4,5,8,7,6,1] => ? => ? = 0
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> [3,2,4,6,5,7,8,1] => ? => ? = 0
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,1,0,0]
=> [3,2,4,6,7,5,8,1] => ? => ? = 0
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,4,6,7,8,5,1] => ? => ? = 0
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,1,0,0,0,0]
=> [3,2,4,6,8,7,5,1] => ? => ? = 0
[1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,1,0,0,0]
=> [3,2,4,7,6,8,5,1] => ? => ? = 0
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,1,0,0,0,0]
=> [3,2,5,4,8,7,6,1] => ? => ? = 0
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,1,0,0,0]
=> [3,2,5,6,4,8,7,1] => ? => ? = 0
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,1,0,0,0,0]
=> [3,2,5,6,8,7,4,1] => ? => ? = 0
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,1,0,0,0,0,0]
=> [3,2,5,8,7,6,4,1] => ? => ? = 0
[1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,1,0,0]
=> [3,2,6,5,4,7,8,1] => ? => ? = 0
[1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2,6,5,7,8,4,1] => ? => ? = 0
[1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,1,0,0,0]
=> [3,2,6,7,5,8,4,1] => ? => ? = 0
[1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,1,0,0,0,0]
=> [3,2,6,7,8,5,4,1] => ? => ? = 0
[1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,1,0,0,0]
=> [3,2,7,6,5,8,4,1] => ? => ? = 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]
=> [3,2,7,6,8,5,4,1] => ? => ? = 0
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,1,0,0,0]
=> [3,4,2,6,7,8,5,1] => ? => ? = 0
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,1,0,0,0,0]
=> [3,4,2,6,8,7,5,1] => ? => ? = 0
[1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,1,0,0,0]
=> [3,4,2,7,6,8,5,1] => ? => ? = 0
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,1,0,0,0]
=> [3,4,5,2,6,8,7,1] => ? => ? = 0
[1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,1,0,0,0,0]
=> [3,4,5,2,8,7,6,1] => ? => ? = 0
[1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,1,0,0,0]
=> [3,4,6,5,2,8,7,1] => ? => ? = 0
[1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,4,6,5,8,7,2,1] => ? => ? = 1
Description
The number of leading ones in a binary word.
Matching statistic: St000980
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000980: Dyck paths ⟶ ℤResult quality: 12% ●values known / values provided: 68%●distinct values known / distinct values provided: 12%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000980: Dyck paths ⟶ ℤResult quality: 12% ●values known / values provided: 68%●distinct values known / distinct values provided: 12%
Values
[1,0]
=> [1] => [1] => [1,0]
=> ? = 1
[1,0,1,0]
=> [1,1] => [2] => [1,1,0,0]
=> 0
[1,1,0,0]
=> [2] => [1] => [1,0]
=> ? = 2
[1,0,1,0,1,0]
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,2] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,0,1,0]
=> [2,1] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,1,0,0]
=> [3] => [1] => [1,0]
=> ? = 1
[1,1,1,0,0,0]
=> [3] => [1] => [1,0]
=> ? = 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,2] => [2] => [1,1,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [4] => [1] => [1,0]
=> ? = 1
[1,1,0,1,1,0,0,0]
=> [4] => [1] => [1,0]
=> ? = 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 0
[1,1,1,0,0,1,0,0]
=> [4] => [1] => [1,0]
=> ? = 1
[1,1,1,0,1,0,0,0]
=> [4] => [1] => [1,0]
=> ? = 2
[1,1,1,1,0,0,0,0]
=> [4] => [1] => [1,0]
=> ? = 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [1] => [1,0]
=> ? = 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [1] => [1,0]
=> ? = 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [1] => [1,0]
=> ? = 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [1] => [1,0]
=> ? = 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [1] => [1,0]
=> ? = 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 0
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,0,1,0]
=> 0
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 0
[1,1,1,0,0,1,0,1,0,0]
=> [5] => [1] => [1,0]
=> ? = 1
[1,1,1,0,0,1,1,0,0,0]
=> [5] => [1] => [1,0]
=> ? = 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 0
[1,1,1,0,1,0,0,1,0,0]
=> [5] => [1] => [1,0]
=> ? = 1
[1,1,1,0,1,0,1,0,0,0]
=> [5] => [1] => [1,0]
=> ? = 2
[1,1,1,0,1,1,0,0,0,0]
=> [5] => [1] => [1,0]
=> ? = 2
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 0
[1,1,1,1,0,0,0,1,0,0]
=> [5] => [1] => [1,0]
=> ? = 1
[1,1,1,1,0,0,1,0,0,0]
=> [5] => [1] => [1,0]
=> ? = 2
[1,1,1,1,0,1,0,0,0,0]
=> [5] => [1] => [1,0]
=> ? = 3
[1,1,1,1,1,0,0,0,0,0]
=> [5] => [1] => [1,0]
=> ? = 5
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 2
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 2
Description
The number of boxes weakly below the path and above the diagonal that lie below at least two peaks.
For example, the path $111011010000$ has three peaks in positions $03, 15, 26$. The boxes below $03$ are $01,02,\textbf{12}$, the boxes below $15$ are $\textbf{12},13,14,\textbf{23},\textbf{24},\textbf{34}$, and the boxes below $26$ are $\textbf{23},\textbf{24},25,\textbf{34},35,45$.
We thus obtain the four boxes in positions $12,23,24,34$ that are below at least two peaks.
Matching statistic: St000781
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 68%●distinct values known / distinct values provided: 12%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 68%●distinct values known / distinct values provided: 12%
Values
[1,0]
=> [1] => [1]
=> []
=> ? = 1 + 1
[1,0,1,0]
=> [1,1] => [1,1]
=> [1]
=> 1 = 0 + 1
[1,1,0,0]
=> [2] => [2]
=> []
=> ? = 2 + 1
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,2] => [2,1]
=> [1]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [2,1] => [2,1]
=> [1]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [3] => [3]
=> []
=> ? = 1 + 1
[1,1,1,0,0,0]
=> [3] => [3]
=> []
=> ? = 3 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,3] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> [2]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [3,1] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [4] => [4]
=> []
=> ? = 1 + 1
[1,1,0,1,1,0,0,0]
=> [4] => [4]
=> []
=> ? = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [4] => [4]
=> []
=> ? = 1 + 1
[1,1,1,0,1,0,0,0]
=> [4] => [4]
=> []
=> ? = 2 + 1
[1,1,1,1,0,0,0,0]
=> [4] => [4]
=> []
=> ? = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> [2,1]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> [2,1]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> [2,1]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [3,2]
=> [2]
=> 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> [2]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> [2]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [5]
=> []
=> ? = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [5]
=> []
=> ? = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [5]
=> []
=> ? = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [5]
=> []
=> ? = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [5]
=> []
=> ? = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> [2]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [5] => [5]
=> []
=> ? = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [5] => [5]
=> []
=> ? = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [5] => [5]
=> []
=> ? = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [5] => [5]
=> []
=> ? = 2 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [5] => [5]
=> []
=> ? = 2 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [5] => [5]
=> []
=> ? = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [5] => [5]
=> []
=> ? = 2 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [5] => [5]
=> []
=> ? = 3 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [5] => [5]
=> []
=> ? = 5 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [3,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [3,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [2,2,1,1]
=> [2,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [3,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [4,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [6] => [6]
=> []
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [6] => [6]
=> []
=> ? = 1 + 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [6] => [6]
=> []
=> ? = 1 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [6] => [6]
=> []
=> ? = 1 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [6] => [6]
=> []
=> ? = 1 + 1
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [6] => [6]
=> []
=> ? = 1 + 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [6] => [6]
=> []
=> ? = 1 + 1
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [6] => [6]
=> []
=> ? = 1 + 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [6] => [6]
=> []
=> ? = 1 + 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6] => [6]
=> []
=> ? = 1 + 1
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [6] => [6]
=> []
=> ? = 1 + 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [6] => [6]
=> []
=> ? = 1 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [6] => [6]
=> []
=> ? = 1 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [6] => [6]
=> []
=> ? = 1 + 1
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [6] => [6]
=> []
=> ? = 1 + 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [6] => [6]
=> []
=> ? = 1 + 1
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [6] => [6]
=> []
=> ? = 1 + 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [6] => [6]
=> []
=> ? = 1 + 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [6] => [6]
=> []
=> ? = 1 + 1
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [6] => [6]
=> []
=> ? = 1 + 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [6] => [6]
=> []
=> ? = 1 + 1
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [6] => [6]
=> []
=> ? = 1 + 1
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [6] => [6]
=> []
=> ? = 2 + 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6] => [6]
=> []
=> ? = 2 + 1
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [6] => [6]
=> []
=> ? = 1 + 1
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [6] => [6]
=> []
=> ? = 2 + 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [6] => [6]
=> []
=> ? = 2 + 1
Description
The number of proper colouring schemes of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1].
This statistic is the number of distinct such integer partitions that occur.
The following 38 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000993The multiplicity of the largest part of an integer partition. St000382The first part of an integer composition. St000873The aix statistic of a permutation. St000895The number of ones on the main diagonal of an alternating sign matrix. St000264The girth of a graph, which is not a tree. St000221The number of strong fixed points of a permutation. St000315The number of isolated vertices of a graph. St000461The rix statistic of a permutation. St000907The number of maximal antichains of minimal length in a poset. St000117The number of centered tunnels of a Dyck path. St000241The number of cyclical small excedances. St000338The number of pixed points of a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001826The maximal number of leaves on a vertex of a graph. St001672The restrained domination number of a graph. St001342The number of vertices in the center of a graph. St001368The number of vertices of maximal degree in a graph. St000234The number of global ascents of a permutation. St001479The number of bridges of a graph. St000546The number of global descents of a permutation. St000910The number of maximal chains of minimal length in a poset. St001498The normalised height of a Nakayama algebra with magnitude 1. St000455The second largest eigenvalue of a graph if it is integral. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001545The second Elser number of a connected graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001570The minimal number of edges to add to make a graph Hamiltonian. St000056The decomposition (or block) number of a permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001948The number of augmented double ascents of a permutation. St000335The difference of lower and upper interactions.
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