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Your data matches 56 different statistics following compositions of up to 3 maps.
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Matching statistic: St000328
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000328: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000328: 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]
=> [[],[]]
=> 2
[1,1,0,0]
=> [[[]]]
=> 1
[1,0,1,0,1,0]
=> [[],[],[]]
=> 3
[1,0,1,1,0,0]
=> [[],[[]]]
=> 2
[1,1,0,0,1,0]
=> [[[]],[]]
=> 2
[1,1,0,1,0,0]
=> [[[],[]]]
=> 2
[1,1,1,0,0,0]
=> [[[[]]]]
=> 1
[1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 4
[1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 3
[1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 3
[1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> 2
[1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 2
[1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 3
[1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 2
[1,1,0,1,0,0,1,0]
=> [[[],[]],[]]
=> 2
[1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> 3
[1,1,0,1,1,0,0,0]
=> [[[],[[]]]]
=> 2
[1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 2
[1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 2
[1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> 2
[1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [[],[],[[]],[]]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [[],[],[[],[]]]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [[],[[]],[],[]]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [[],[[],[]],[]]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [[],[[],[],[]]]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [[],[[],[[]]]]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [[],[[[]],[]]]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [[],[[[],[]]]]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [[[]],[],[],[]]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [[[]],[],[[]]]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [[[]],[[],[]]]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [[[]],[[[]]]]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [[[],[]],[],[]]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [[[],[]],[[]]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[[],[],[]],[]]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[]]]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [[[],[],[[]]]]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [[[],[[]]],[]]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [[[],[[]],[]]]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [[[],[[],[]]]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [[[],[[[]]]]]
=> 2
Description
The maximum number of child nodes in a tree.
Matching statistic: St000308
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 2
[1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 3
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => 2
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 3
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 2
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 2
Description
The height of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The statistic is given by the height of this tree.
See also [[St000325]] for the width of this tree.
Matching statistic: St000381
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => 1
[1,0,1,0]
=> [1,1,0,0]
=> [2] => 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,1] => 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => 3
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,1] => 2
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1] => 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => 3
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 3
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => 2
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => 2
Description
The largest part of an integer composition.
Matching statistic: St000392
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00093: Dyck paths —to binary word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 10 => 1
[1,0,1,0]
=> [1,1,0,0]
=> 1100 => 2
[1,1,0,0]
=> [1,0,1,0]
=> 1010 => 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 111000 => 3
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 110100 => 2
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 110010 => 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 101100 => 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 101010 => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 4
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 3
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 11011000 => 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 3
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 2
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 2
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1111100000 => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1111010000 => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1111001000 => 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1110100100 => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1110011000 => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 2
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000444
(load all 31 compositions to match this statistic)
(load all 31 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[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
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[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]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 2
Description
The length of the maximal rise of a Dyck path.
Matching statistic: St000846
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00047: Ordered trees —to poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00047: Ordered trees —to poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[]]
=> ([(0,1)],2)
=> 1
[1,0,1,0]
=> [[],[]]
=> ([(0,2),(1,2)],3)
=> 2
[1,1,0,0]
=> [[[]]]
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0]
=> [[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,0]
=> [[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,0,1,0]
=> [[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,1,0,0]
=> [[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[1,1,1,0,0,0]
=> [[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,0]
=> [[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> 3
[1,1,0,1,1,0,0,0]
=> [[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [[[],[]],[],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[[],[],[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [[[],[],[[]]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [[[],[[]],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [[[],[[],[]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 2
Description
The maximal number of elements covering an element of a poset.
Matching statistic: St001239
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001239: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001239: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 3
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
Description
The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra.
Matching statistic: St001372
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00093: Dyck paths —to binary word⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 10 => 1
[1,0,1,0]
=> [1,1,0,0]
=> 1100 => 2
[1,1,0,0]
=> [1,0,1,0]
=> 1010 => 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 111000 => 3
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 110100 => 2
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 110010 => 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 101100 => 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 101010 => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 4
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 3
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 11011000 => 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 3
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 2
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 2
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1111100000 => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1111010000 => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1111001000 => 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1110100100 => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1110011000 => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 2
Description
The length of a longest cyclic run of ones of a binary word.
Consider the binary word as a cyclic arrangement of ones and zeros. Then this statistic is the length of the longest continuous sequence of ones in this arrangement.
Matching statistic: St001192
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001192: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001192: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
Description
The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$.
Matching statistic: St000013
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1,0]
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> [2] => [1,1,0,0]
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,1] => [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => [1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,1] => [1,1,0,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1] => [1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => [1,0,1,1,0,0]
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
Description
The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
The following 46 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000147The largest part of an integer partition. St000451The length of the longest pattern of the form k 1 2. St000845The maximal number of elements covered by an element in a poset. St001062The maximal size of a block of a set partition. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000141The maximum drop size of a permutation. St000209Maximum difference of elements in cycles. St000662The staircase size of the code of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000956The maximal displacement of a permutation. St001933The largest multiplicity of a part in an integer partition. St000271The chromatic index of a graph. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St000732The number of double deficiencies of a permutation. St000035The number of left outer peaks of a permutation. St000153The number of adjacent cycles of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000778The metric dimension of a graph. St000996The number of exclusive left-to-right maxima of a permutation. St000171The degree of the graph. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000834The number of right outer peaks of a permutation. St000871The number of very big ascents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001118The acyclic chromatic index of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001530The depth of a Dyck path. St001570The minimal number of edges to add to make a graph Hamiltonian. St001742The difference of the maximal and the minimal degree in a graph. St000983The length of the longest alternating subword. St001578The minimal number of edges to add or remove to make a graph a line graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001624The breadth of a lattice. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001877Number of indecomposable injective modules with projective dimension 2.
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