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Your data matches 72 different statistics following compositions of up to 3 maps.
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Matching statistic: St000439
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> 2
[[],[]]
=> [1,0,1,0]
=> 2
[[[]]]
=> [1,1,0,0]
=> 3
[[],[],[]]
=> [1,0,1,0,1,0]
=> 2
[[],[[]]]
=> [1,0,1,1,0,0]
=> 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> 3
[[[],[]]]
=> [1,1,0,1,0,0]
=> 3
[[[[]]]]
=> [1,1,1,0,0,0]
=> 4
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> 2
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> 2
[[],[[],[]]]
=> [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]
=> 3
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> 3
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> 4
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> 3
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> 3
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> 4
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> 4
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> 5
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[[],[[[]],[]]]
=> [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]
=> 3
[[[]],[],[[]]]
=> [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]
=> 3
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5
Description
The position of the first down step of a Dyck path.
Matching statistic: St000382
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1] => 1 = 2 - 1
[[],[]]
=> [1,0,1,0]
=> [1,1] => 1 = 2 - 1
[[[]]]
=> [1,1,0,0]
=> [2] => 2 = 3 - 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1 = 2 - 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,2] => 1 = 2 - 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [2,1] => 2 = 3 - 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [2,1] => 2 = 3 - 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [3] => 3 = 4 - 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1 = 2 - 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1 = 2 - 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1 = 2 - 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => 1 = 2 - 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1 = 2 - 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2 = 3 - 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2 = 3 - 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => 2 = 3 - 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 3 = 4 - 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => 2 = 3 - 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => 2 = 3 - 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => 3 = 4 - 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => 3 = 4 - 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4 = 5 - 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 1 = 2 - 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 1 = 2 - 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 1 = 2 - 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => 1 = 2 - 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 1 = 2 - 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 1 = 2 - 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 1 = 2 - 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => 1 = 2 - 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 1 = 2 - 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => 1 = 2 - 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 1 = 2 - 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 1 = 2 - 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 1 = 2 - 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 1 = 2 - 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 2 = 3 - 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 2 = 3 - 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 2 = 3 - 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => 2 = 3 - 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 2 = 3 - 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => 2 = 3 - 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 3 = 4 - 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => 2 = 3 - 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 3 = 4 - 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => 2 = 3 - 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 2 = 3 - 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => 3 = 4 - 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => 3 = 4 - 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 4 = 5 - 1
[[],[[],[[],[[[],[]],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,2,2,3,1,1] => ? = 2 - 1
[[],[[],[[[],[]],[[],[]]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [1,2,3,1,2,1] => ? = 2 - 1
[[],[[],[[[],[[],[]]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> [1,2,3,2,1,1] => ? = 2 - 1
[[],[[],[[[[],[]],[]],[]]]]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> [1,2,4,1,1,1] => ? = 2 - 1
[[],[[[],[]],[[],[[],[]]]]]
=> [1,0,1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,3,1,2,2,1] => ? = 2 - 1
[[],[[[],[]],[[[],[]],[]]]]
=> [1,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,3,1,3,1,1] => ? = 2 - 1
[[],[[[],[[],[]]],[[],[]]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0]
=> [1,3,2,1,2,1] => ? = 2 - 1
[[],[[[],[[],[[],[]]]],[]]]
=> [1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> [1,3,2,2,1,1] => ? = 2 - 1
[[],[[[],[[[],[]],[]]],[]]]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> [1,3,3,1,1,1] => ? = 2 - 1
[[],[[[[],[]],[[],[]]],[]]]
=> [1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0]
=> [1,4,1,2,1,1] => ? = 2 - 1
[[],[[[[],[[],[]]],[]],[]]]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> [1,4,2,1,1,1] => ? = 2 - 1
[[[],[]],[[],[[],[[],[]]]]]
=> [1,1,0,1,0,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,2,2,2,1] => ? = 3 - 1
[[[],[]],[[[],[]],[[],[]]]]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [2,1,3,1,2,1] => ? = 3 - 1
[[[],[]],[[[],[[],[]]],[]]]
=> [1,1,0,1,0,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [2,1,3,2,1,1] => ? = 3 - 1
[[[],[]],[[[[],[]],[]],[]]]
=> [1,1,0,1,0,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [2,1,4,1,1,1] => ? = 3 - 1
[[[],[[],[]]],[[],[[],[]]]]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,1,1,0,1,0,0,0]
=> [2,2,1,2,2,1] => ? = 3 - 1
[[[],[[],[]]],[[[],[]],[]]]
=> [1,1,0,1,1,0,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,2,1,3,1,1] => ? = 3 - 1
[[[[],[]],[]],[[],[[],[]]]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [3,1,1,2,2,1] => ? = 4 - 1
[[[],[[[],[]],[]]],[[],[]]]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,1,0,1,0,0]
=> [2,3,1,1,2,1] => ? = 3 - 1
[[[[],[]],[[],[]]],[[],[]]]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,1,2,1] => ? = 4 - 1
[[[[],[[],[]]],[]],[[],[]]]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,1,0,1,0,0]
=> [3,2,1,1,2,1] => ? = 4 - 1
[[[[[],[]],[]],[]],[[],[]]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [4,1,1,1,2,1] => ? = 5 - 1
[[[],[[],[[[],[]],[]]]],[]]
=> [1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [2,2,3,1,1,1] => ? = 3 - 1
[[[],[[[],[]],[[],[]]]],[]]
=> [1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0,1,0]
=> [2,3,1,2,1,1] => ? = 3 - 1
[[[],[[[],[[],[]]],[]]],[]]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [2,3,2,1,1,1] => ? = 3 - 1
[[[],[[[[],[]],[]],[]]],[]]
=> [1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,1,0]
=> [2,4,1,1,1,1] => ? = 3 - 1
[[[[],[]],[[],[[],[]]]],[]]
=> [1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [3,1,2,2,1,1] => ? = 4 - 1
[[[[[],[]],[]],[[],[]]],[]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,0,1,0]
=> [4,1,1,2,1,1] => ? = 5 - 1
[[[[[],[]],[[],[]]],[]],[]]
=> [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0,1,0]
=> [4,1,2,1,1,1] => ? = 5 - 1
[[],[[],[[],[[],[[[],[]],[]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,2,2,2,3,1,1] => ? = 2 - 1
[[],[[],[[],[[[],[]],[[],[]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0,0]
=> [1,2,2,3,1,2,1] => ? = 2 - 1
[[],[[],[[],[[[],[[],[]]],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,0]
=> [1,2,2,3,2,1,1] => ? = 2 - 1
[[],[[],[[],[[[[],[]],[]],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,0]
=> [1,2,2,4,1,1,1] => ? = 2 - 1
[[],[[],[[[],[]],[[],[[],[]]]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,2,3,1,2,2,1] => ? = 2 - 1
[[],[[],[[[],[]],[[[],[]],[]]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,2,3,1,3,1,1] => ? = 2 - 1
[[],[[],[[[],[[],[]]],[[],[]]]]]
=> [1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0,0]
=> [1,2,3,2,1,2,1] => ? = 2 - 1
[[],[[],[[[[],[]],[]],[[],[]]]]]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,0,0]
=> [1,2,4,1,1,2,1] => ? = 2 - 1
[[],[[],[[[],[[],[[],[]]]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0,0]
=> [1,2,3,2,2,1,1] => ? = 2 - 1
[[],[[],[[[],[[[],[]],[]]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0,0]
=> [1,2,3,3,1,1,1] => ? = 2 - 1
[[],[[],[[[[],[]],[[],[]]],[]]]]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0,0]
=> [1,2,4,1,2,1,1] => ? = 2 - 1
[[],[[],[[[[],[[],[]]],[]],[]]]]
=> [1,0,1,1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0,0]
=> [1,2,4,2,1,1,1] => ? = 2 - 1
[[],[[],[[[[[],[]],[]],[]],[]]]]
=> [1,0,1,1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,0]
=> [1,2,5,1,1,1,1] => ? = 2 - 1
[[],[[[],[]],[[],[[],[[],[]]]]]]
=> [1,0,1,1,1,0,1,0,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,3,1,2,2,2,1] => ? = 2 - 1
[[],[[[],[]],[[],[[[],[]],[]]]]]
=> [1,0,1,1,1,0,1,0,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,3,1,2,3,1,1] => ? = 2 - 1
[[],[[[],[]],[[[],[]],[[],[]]]]]
=> [1,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [1,3,1,3,1,2,1] => ? = 2 - 1
[[],[[[],[]],[[[],[[],[]]],[]]]]
=> [1,0,1,1,1,0,1,0,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> [1,3,1,3,2,1,1] => ? = 2 - 1
[[],[[[],[]],[[[[],[]],[]],[]]]]
=> [1,0,1,1,1,0,1,0,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> [1,3,1,4,1,1,1] => ? = 2 - 1
[[],[[[],[[],[]]],[[],[[],[]]]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,3,2,1,2,2,1] => ? = 2 - 1
[[],[[[],[[],[]]],[[[],[]],[]]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,3,2,1,3,1,1] => ? = 2 - 1
[[],[[[[],[]],[]],[[],[[],[]]]]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,1,1,2,2,1] => ? = 2 - 1
Description
The first part of an integer composition.
Matching statistic: St000383
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 88%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 88%
Values
[[]]
=> [1,0]
=> [1] => [1] => 1 = 2 - 1
[[],[]]
=> [1,0,1,0]
=> [1,1] => [1,1] => 1 = 2 - 1
[[[]]]
=> [1,1,0,0]
=> [2] => [2] => 2 = 3 - 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,1,1] => 1 = 2 - 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,2] => [2,1] => 1 = 2 - 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [2,1] => [1,2] => 2 = 3 - 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [2,1] => [1,2] => 2 = 3 - 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [3] => [3] => 3 = 4 - 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1] => 1 = 2 - 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1] => 1 = 2 - 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,2,1] => 1 = 2 - 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,2,1] => 1 = 2 - 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [3,1] => 1 = 2 - 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,2] => 2 = 3 - 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [2,2] => 2 = 3 - 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,2] => 2 = 3 - 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [1,3] => 3 = 4 - 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,2] => 2 = 3 - 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => [2,2] => 2 = 3 - 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => [1,3] => 3 = 4 - 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => [1,3] => 3 = 4 - 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4] => [4] => 4 = 5 - 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1] => 1 = 2 - 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1] => 1 = 2 - 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,2,1,1] => 1 = 2 - 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,2,1,1] => 1 = 2 - 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1] => 1 = 2 - 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2,1] => 1 = 2 - 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1] => 1 = 2 - 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,1,2,1] => 1 = 2 - 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,3,1] => 1 = 2 - 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,1,2,1] => 1 = 2 - 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1] => 1 = 2 - 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,3,1] => 1 = 2 - 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,3,1] => 1 = 2 - 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1] => 1 = 2 - 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => 2 = 3 - 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,1,2] => 2 = 3 - 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,2,2] => 2 = 3 - 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,2,2] => 2 = 3 - 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2] => 2 = 3 - 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => 2 = 3 - 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,1,3] => 3 = 4 - 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [2,1,2] => 2 = 3 - 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [2,3] => 3 = 4 - 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => 2 = 3 - 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,2,2] => 2 = 3 - 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [1,1,3] => 3 = 4 - 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => [1,1,3] => 3 = 4 - 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,4] => 4 = 5 - 1
[[],[],[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1] => ? = 2 - 1
[[],[],[],[],[],[[[]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,3] => [3,1,1,1,1,1] => ? = 2 - 1
[[],[],[],[],[[[[]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,4] => [4,1,1,1,1] => ? = 2 - 1
[[],[],[],[[]],[[[]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,2,3] => [3,2,1,1,1] => ? = 2 - 1
[[],[],[],[[[[]]]],[]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,4,1] => [1,4,1,1,1] => ? = 2 - 1
[[],[],[],[[],[[[]]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,2,3] => [3,2,1,1,1] => ? = 2 - 1
[[],[],[],[[[[]]],[]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,4,1] => [1,4,1,1,1] => ? = 2 - 1
[[],[],[],[[[[]],[]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,4,1] => [1,4,1,1,1] => ? = 2 - 1
[[],[],[],[[[[],[]]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,4,1] => [1,4,1,1,1] => ? = 2 - 1
[[],[],[[]],[],[[[]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,2,1,3] => [3,1,2,1,1] => ? = 2 - 1
[[],[],[[]],[[[[]]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,2,4] => [4,2,1,1] => ? = 2 - 1
[[],[],[[],[]],[[[]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,2,1,3] => [3,1,2,1,1] => ? = 2 - 1
[[],[],[[[[]]]],[],[]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,4,1,1] => [1,1,4,1,1] => ? = 2 - 1
[[],[],[[[[]]]],[[]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,4,2] => [2,4,1,1] => ? = 2 - 1
[[],[],[[[[]]],[]],[]]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,4,1,1] => [1,1,4,1,1] => ? = 2 - 1
[[],[],[[[[]],[]]],[]]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,4,1,1] => [1,1,4,1,1] => ? = 2 - 1
[[],[],[[[[],[]]]],[]]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,4,1,1] => [1,1,4,1,1] => ? = 2 - 1
[[],[],[[],[],[[[]]]]]
=> [1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,2,1,3] => [3,1,2,1,1] => ? = 2 - 1
[[],[],[[],[[[[]]]]]]
=> [1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,2,4] => [4,2,1,1] => ? = 2 - 1
[[],[],[[[[]]],[],[]]]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,4,1,1] => [1,1,4,1,1] => ? = 2 - 1
[[],[],[[[[]]],[[]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,4,2] => [2,4,1,1] => ? = 2 - 1
[[],[],[[[[]],[]],[]]]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,4,1,1] => [1,1,4,1,1] => ? = 2 - 1
[[],[],[[[[],[]]],[]]]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,4,1,1] => [1,1,4,1,1] => ? = 2 - 1
[[],[],[[[[]],[],[]]]]
=> [1,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,4,1,1] => [1,1,4,1,1] => ? = 2 - 1
[[],[],[[[[]],[[]]]]]
=> [1,0,1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,4,2] => [2,4,1,1] => ? = 2 - 1
[[],[],[[[[],[]],[]]]]
=> [1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,4,1,1] => [1,1,4,1,1] => ? = 2 - 1
[[],[],[[[[],[],[]]]]]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,4,1,1] => [1,1,4,1,1] => ? = 2 - 1
[[],[],[[[[],[[]]]]]]
=> [1,0,1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,4,2] => [2,4,1,1] => ? = 2 - 1
[[],[[]],[],[],[[[]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,1,1,3] => [3,1,1,2,1] => ? = 2 - 1
[[],[[],[]],[],[[[]]]]
=> [1,0,1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,2,1,1,3] => [3,1,1,2,1] => ? = 2 - 1
[[],[[[[]]]],[],[],[]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,4,1,1,1] => [1,1,1,4,1] => ? = 2 - 1
[[],[[],[],[]],[[[]]]]
=> [1,0,1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [1,2,1,1,3] => [3,1,1,2,1] => ? = 2 - 1
[[],[[[[]]],[]],[],[]]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [1,4,1,1,1] => [1,1,1,4,1] => ? = 2 - 1
[[],[[[[]],[]]],[],[]]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,4,1,1,1] => [1,1,1,4,1] => ? = 2 - 1
[[],[[[[],[]]]],[],[]]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,4,1,1,1] => [1,1,1,4,1] => ? = 2 - 1
[[],[[[[[]]]]],[[]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,5,2] => [2,5,1] => ? = 2 - 1
[[],[[[[]]],[],[]],[]]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [1,4,1,1,1] => [1,1,1,4,1] => ? = 2 - 1
[[],[[[[]],[]],[]],[]]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,4,1,1,1] => [1,1,1,4,1] => ? = 2 - 1
[[],[[[[],[]]],[]],[]]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [1,4,1,1,1] => [1,1,1,4,1] => ? = 2 - 1
[[],[[[[]],[],[]]],[]]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,4,1,1,1] => [1,1,1,4,1] => ? = 2 - 1
[[],[[[[],[]],[]]],[]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,4,1,1,1] => [1,1,1,4,1] => ? = 2 - 1
[[],[[[[],[],[]]]],[]]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,4,1,1,1] => [1,1,1,4,1] => ? = 2 - 1
[[],[[],[],[],[[[]]]]]
=> [1,0,1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,2,1,1,3] => [3,1,1,2,1] => ? = 2 - 1
[[],[[[[]]],[],[],[]]]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,4,1,1,1] => [1,1,1,4,1] => ? = 2 - 1
[[],[[[[]],[]],[],[]]]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [1,4,1,1,1] => [1,1,1,4,1] => ? = 2 - 1
[[],[[[[],[]]],[],[]]]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,4,1,1,1] => [1,1,1,4,1] => ? = 2 - 1
[[],[[[[[]]]],[[]]]]
=> [1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,5,2] => [2,5,1] => ? = 2 - 1
[[],[[[[]],[],[]],[]]]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,4,1,1,1] => [1,1,1,4,1] => ? = 2 - 1
[[],[[[[],[]],[]],[]]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,4,1,1,1] => [1,1,1,4,1] => ? = 2 - 1
[[],[[[[],[],[]]],[]]]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,4,1,1,1] => [1,1,1,4,1] => ? = 2 - 1
Description
The last part of an integer composition.
Matching statistic: St000297
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [.,.]
=> [1] => => ? = 2 - 2
[[],[]]
=> [[.,.],.]
=> [1,2] => 0 => 0 = 2 - 2
[[[]]]
=> [.,[.,.]]
=> [2,1] => 1 => 1 = 3 - 2
[[],[],[]]
=> [[[.,.],.],.]
=> [1,2,3] => 00 => 0 = 2 - 2
[[],[[]]]
=> [[.,.],[.,.]]
=> [1,3,2] => 01 => 0 = 2 - 2
[[[]],[]]
=> [[.,[.,.]],.]
=> [2,1,3] => 10 => 1 = 3 - 2
[[[],[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 10 => 1 = 3 - 2
[[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 11 => 2 = 4 - 2
[[],[],[],[]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 000 => 0 = 2 - 2
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [1,2,4,3] => 001 => 0 = 2 - 2
[[],[[]],[]]
=> [[[.,.],[.,.]],.]
=> [1,3,2,4] => 010 => 0 = 2 - 2
[[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> [1,3,4,2] => 010 => 0 = 2 - 2
[[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 011 => 0 = 2 - 2
[[[]],[],[]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 100 => 1 = 3 - 2
[[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 101 => 1 = 3 - 2
[[[],[]],[]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 100 => 1 = 3 - 2
[[[[]]],[]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 110 => 2 = 4 - 2
[[[],[],[]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 100 => 1 = 3 - 2
[[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 101 => 1 = 3 - 2
[[[[]],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 110 => 2 = 4 - 2
[[[[],[]]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 110 => 2 = 4 - 2
[[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 111 => 3 = 5 - 2
[[],[],[],[],[]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 0000 => 0 = 2 - 2
[[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 0001 => 0 = 2 - 2
[[],[],[[]],[]]
=> [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 0010 => 0 = 2 - 2
[[],[],[[],[]]]
=> [[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => 0010 => 0 = 2 - 2
[[],[],[[[]]]]
=> [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 0011 => 0 = 2 - 2
[[],[[]],[],[]]
=> [[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => 0100 => 0 = 2 - 2
[[],[[]],[[]]]
=> [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 0101 => 0 = 2 - 2
[[],[[],[]],[]]
=> [[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => 0100 => 0 = 2 - 2
[[],[[[]]],[]]
=> [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => 0110 => 0 = 2 - 2
[[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => 0100 => 0 = 2 - 2
[[],[[],[[]]]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 0101 => 0 = 2 - 2
[[],[[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => 0110 => 0 = 2 - 2
[[],[[[],[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => 0110 => 0 = 2 - 2
[[],[[[[]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 0111 => 0 = 2 - 2
[[[]],[],[],[]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 1000 => 1 = 3 - 2
[[[]],[],[[]]]
=> [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 1001 => 1 = 3 - 2
[[[]],[[]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => 1010 => 1 = 3 - 2
[[[]],[[],[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 1010 => 1 = 3 - 2
[[[]],[[[]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 1011 => 1 = 3 - 2
[[[],[]],[],[]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => 1000 => 1 = 3 - 2
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 1100 => 2 = 4 - 2
[[[],[]],[[]]]
=> [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 1001 => 1 = 3 - 2
[[[[]]],[[]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 1101 => 2 = 4 - 2
[[[],[],[]],[]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => 1000 => 1 = 3 - 2
[[[],[[]]],[]]
=> [[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => 1010 => 1 = 3 - 2
[[[[]],[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => 1100 => 2 = 4 - 2
[[[[],[]]],[]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => 1100 => 2 = 4 - 2
[[[[[]]]],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 1110 => 3 = 5 - 2
[[[],[],[],[]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1000 => 1 = 3 - 2
[[],[],[[]],[[[]]],[]]
=> [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
=> [1,2,4,3,7,6,5,8] => ? => ? = 2 - 2
[[],[],[[],[]],[],[],[]]
=> [[[[[[.,.],.],[[.,.],.]],.],.],.]
=> [1,2,4,5,3,6,7,8] => ? => ? = 2 - 2
[[],[],[[],[]],[],[[]]]
=> [[[[[.,.],.],[[.,.],.]],.],[.,.]]
=> [1,2,4,5,3,6,8,7] => ? => ? = 2 - 2
[[],[],[[[]]],[[]],[]]
=> [[[[[.,.],.],[.,[.,.]]],[.,.]],.]
=> [1,2,5,4,3,7,6,8] => ? => ? = 2 - 2
[[],[],[[[]],[]],[],[]]
=> [[[[[.,.],.],[[.,[.,.]],.]],.],.]
=> [1,2,5,4,6,3,7,8] => ? => ? = 2 - 2
[[],[],[[],[[]]],[[]]]
=> [[[[.,.],.],[[.,.],[.,.]]],[.,.]]
=> [1,2,4,6,5,3,8,7] => ? => ? = 2 - 2
[[],[],[[[]],[]],[[]]]
=> [[[[.,.],.],[[.,[.,.]],.]],[.,.]]
=> [1,2,5,4,6,3,8,7] => ? => ? = 2 - 2
[[],[],[[[],[]]],[[]]]
=> [[[[.,.],.],[.,[[.,.],.]]],[.,.]]
=> [1,2,5,6,4,3,8,7] => ? => ? = 2 - 2
[[],[],[[[[]]]],[[]]]
=> [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
=> [1,2,6,5,4,3,8,7] => ? => ? = 2 - 2
[[],[],[[],[],[[]]],[]]
=> [[[[.,.],.],[[[.,.],.],[.,.]]],.]
=> [1,2,4,5,7,6,3,8] => ? => ? = 2 - 2
[[],[],[[],[[]],[]],[]]
=> [[[[.,.],.],[[[.,.],[.,.]],.]],.]
=> [1,2,4,6,5,7,3,8] => ? => ? = 2 - 2
[[],[],[[],[[[]]]],[]]
=> [[[[.,.],.],[[.,.],[.,[.,.]]]],.]
=> [1,2,4,7,6,5,3,8] => ? => ? = 2 - 2
[[],[],[[[]],[[]]],[]]
=> [[[[.,.],.],[[.,[.,.]],[.,.]]],.]
=> [1,2,5,4,7,6,3,8] => ? => ? = 2 - 2
[[],[],[[[[]]],[]],[]]
=> [[[[.,.],.],[[.,[.,[.,.]]],.]],.]
=> [1,2,6,5,4,7,3,8] => ? => ? = 2 - 2
[[],[],[[[[]],[]]],[]]
=> [[[[.,.],.],[.,[[.,[.,.]],.]]],.]
=> [1,2,6,5,7,4,3,8] => ? => ? = 2 - 2
[[],[],[[],[],[[]],[]]]
=> [[[.,.],.],[[[[.,.],.],[.,.]],.]]
=> [1,2,4,5,7,6,8,3] => ? => ? = 2 - 2
[[],[],[[],[[],[]],[]]]
=> [[[.,.],.],[[[.,.],[[.,.],.]],.]]
=> [1,2,4,6,7,5,8,3] => ? => ? = 2 - 2
[[],[],[[[],[]],[[]]]]
=> [[[.,.],.],[[.,[[.,.],.]],[.,.]]]
=> [1,2,5,6,4,8,7,3] => ? => ? = 2 - 2
[[],[],[[[[]],[]],[]]]
=> [[[.,.],.],[[.,[[.,[.,.]],.]],.]]
=> [1,2,6,5,7,4,8,3] => ? => ? = 2 - 2
[[],[],[[[[],[]]],[]]]
=> [[[.,.],.],[[.,[.,[[.,.],.]]],.]]
=> [1,2,6,7,5,4,8,3] => ? => ? = 2 - 2
[[],[],[[[],[[],[]]]]]
=> [[[.,.],.],[.,[[.,.],[[.,.],.]]]]
=> [1,2,5,7,8,6,4,3] => ? => ? = 2 - 2
[[],[],[[[[],[]],[]]]]
=> [[[.,.],.],[.,[[.,[[.,.],.]],.]]]
=> [1,2,6,7,5,8,4,3] => ? => ? = 2 - 2
[[],[],[[[[],[[]]]]]]
=> [[[.,.],.],[.,[.,[[.,.],[.,.]]]]]
=> [1,2,6,8,7,5,4,3] => ? => ? = 2 - 2
[[],[[]],[],[[],[]],[]]
=> [[[[[.,.],[.,.]],.],[[.,.],.]],.]
=> [1,3,2,4,6,7,5,8] => ? => ? = 2 - 2
[[],[[]],[],[[[],[]]]]
=> [[[[.,.],[.,.]],.],[.,[[.,.],.]]]
=> [1,3,2,4,7,8,6,5] => ? => ? = 2 - 2
[[],[[]],[[]],[],[[]]]
=> [[[[[.,.],[.,.]],[.,.]],.],[.,.]]
=> [1,3,2,5,4,6,8,7] => ? => ? = 2 - 2
[[],[[]],[[[[],[]]]]]
=> [[[.,.],[.,.]],[.,[.,[[.,.],.]]]]
=> [1,3,2,7,8,6,5,4] => ? => ? = 2 - 2
[[],[[],[]],[[]],[],[]]
=> [[[[[.,.],[[.,.],.]],[.,.]],.],.]
=> [1,3,4,2,6,5,7,8] => ? => ? = 2 - 2
[[],[[[]]],[[]],[],[]]
=> [[[[[.,.],[.,[.,.]]],[.,.]],.],.]
=> [1,4,3,2,6,5,7,8] => ? => ? = 2 - 2
[[],[[[]]],[[]],[[]]]
=> [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,4,3,2,6,5,8,7] => ? => ? = 2 - 2
[[],[[],[]],[[[]]],[]]
=> [[[[.,.],[[.,.],.]],[.,[.,.]]],.]
=> [1,3,4,2,7,6,5,8] => ? => ? = 2 - 2
[[],[[[]]],[[[],[]]]]
=> [[[.,.],[.,[.,.]]],[.,[[.,.],.]]]
=> [1,4,3,2,7,8,6,5] => ? => ? = 2 - 2
[[],[[[]],[]],[],[],[]]
=> [[[[[.,.],[[.,[.,.]],.]],.],.],.]
=> [1,4,3,5,2,6,7,8] => ? => ? = 2 - 2
[[],[[[]],[]],[],[[]]]
=> [[[[.,.],[[.,[.,.]],.]],.],[.,.]]
=> [1,4,3,5,2,6,8,7] => ? => ? = 2 - 2
[[],[[[[]]]],[],[[]]]
=> [[[[.,.],[.,[.,[.,.]]]],.],[.,.]]
=> [1,5,4,3,2,6,8,7] => ? => ? = 2 - 2
[[],[[],[[]]],[[[]]]]
=> [[[.,.],[[.,.],[.,.]]],[.,[.,.]]]
=> [1,3,5,4,2,8,7,6] => ? => ? = 2 - 2
[[],[[],[],[[]]],[],[]]
=> [[[[.,.],[[[.,.],.],[.,.]]],.],.]
=> [1,3,4,6,5,2,7,8] => ? => ? = 2 - 2
[[],[[],[[[]]]],[],[]]
=> [[[[.,.],[[.,.],[.,[.,.]]]],.],.]
=> [1,3,6,5,4,2,7,8] => ? => ? = 2 - 2
[[],[[],[],[[]]],[[]]]
=> [[[.,.],[[[.,.],.],[.,.]]],[.,.]]
=> [1,3,4,6,5,2,8,7] => ? => ? = 2 - 2
[[],[[],[[]],[]],[[]]]
=> [[[.,.],[[[.,.],[.,.]],.]],[.,.]]
=> [1,3,5,4,6,2,8,7] => ? => ? = 2 - 2
[[],[[],[[],[]]],[[]]]
=> [[[.,.],[[.,.],[[.,.],.]]],[.,.]]
=> [1,3,5,6,4,2,8,7] => ? => ? = 2 - 2
[[],[[[]],[],[]],[[]]]
=> [[[.,.],[[[.,[.,.]],.],.]],[.,.]]
=> [1,4,3,5,6,2,8,7] => ? => ? = 2 - 2
[[],[[[]],[[]]],[[]]]
=> [[[.,.],[[.,[.,.]],[.,.]]],[.,.]]
=> [1,4,3,6,5,2,8,7] => ? => ? = 2 - 2
[[],[[[],[]],[]],[[]]]
=> [[[.,.],[[.,[[.,.],.]],.]],[.,.]]
=> [1,4,5,3,6,2,8,7] => ? => ? = 2 - 2
[[],[[[[]]],[]],[[]]]
=> [[[.,.],[[.,[.,[.,.]]],.]],[.,.]]
=> [1,5,4,3,6,2,8,7] => ? => ? = 2 - 2
[[],[[[],[],[]]],[[]]]
=> [[[.,.],[.,[[[.,.],.],.]]],[.,.]]
=> [1,4,5,6,3,2,8,7] => ? => ? = 2 - 2
[[],[[[],[[]]]],[[]]]
=> [[[.,.],[.,[[.,.],[.,.]]]],[.,.]]
=> [1,4,6,5,3,2,8,7] => ? => ? = 2 - 2
[[],[[],[],[[[]]]],[]]
=> [[[.,.],[[[.,.],.],[.,[.,.]]]],.]
=> [1,3,4,7,6,5,2,8] => ? => ? = 2 - 2
[[],[[[]],[],[],[]],[]]
=> [[[.,.],[[[[.,[.,.]],.],.],.]],.]
=> [1,4,3,5,6,7,2,8] => ? => ? = 2 - 2
Description
The number of leading ones in a binary word.
Matching statistic: St000745
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [.,.]
=> [1] => [[1]]
=> 1 = 2 - 1
[[],[]]
=> [[.,.],.]
=> [1,2] => [[1,2]]
=> 1 = 2 - 1
[[[]]]
=> [.,[.,.]]
=> [2,1] => [[1],[2]]
=> 2 = 3 - 1
[[],[],[]]
=> [[[.,.],.],.]
=> [1,2,3] => [[1,2,3]]
=> 1 = 2 - 1
[[],[[]]]
=> [[.,.],[.,.]]
=> [1,3,2] => [[1,2],[3]]
=> 1 = 2 - 1
[[[]],[]]
=> [[.,[.,.]],.]
=> [2,1,3] => [[1,3],[2]]
=> 2 = 3 - 1
[[[],[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => [[1,3],[2]]
=> 2 = 3 - 1
[[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => [[1],[2],[3]]
=> 3 = 4 - 1
[[],[],[],[]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [[1,2,3,4]]
=> 1 = 2 - 1
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 1 = 2 - 1
[[],[[]],[]]
=> [[[.,.],[.,.]],.]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 1 = 2 - 1
[[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> [1,3,4,2] => [[1,2,4],[3]]
=> 1 = 2 - 1
[[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [[1,2],[3],[4]]
=> 1 = 2 - 1
[[[]],[],[]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 2 = 3 - 1
[[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 2 = 3 - 1
[[[],[]],[]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [[1,3,4],[2]]
=> 2 = 3 - 1
[[[[]]],[]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> 3 = 4 - 1
[[[],[],[]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [[1,3,4],[2]]
=> 2 = 3 - 1
[[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [[1,3],[2],[4]]
=> 2 = 3 - 1
[[[[]],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [[1,4],[2],[3]]
=> 3 = 4 - 1
[[[[],[]]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [[1,4],[2],[3]]
=> 3 = 4 - 1
[[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 4 = 5 - 1
[[],[],[],[],[]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [[1,2,3,4,5]]
=> 1 = 2 - 1
[[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1 = 2 - 1
[[],[],[[]],[]]
=> [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1 = 2 - 1
[[],[],[[],[]]]
=> [[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [[1,2,3,5],[4]]
=> 1 = 2 - 1
[[],[],[[[]]]]
=> [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 1 = 2 - 1
[[],[[]],[],[]]
=> [[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1 = 2 - 1
[[],[[]],[[]]]
=> [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 1 = 2 - 1
[[],[[],[]],[]]
=> [[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => [[1,2,4,5],[3]]
=> 1 = 2 - 1
[[],[[[]]],[]]
=> [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => [[1,2,5],[3],[4]]
=> 1 = 2 - 1
[[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [[1,2,4,5],[3]]
=> 1 = 2 - 1
[[],[[],[[]]]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [[1,2,4],[3],[5]]
=> 1 = 2 - 1
[[],[[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [[1,2,5],[3],[4]]
=> 1 = 2 - 1
[[],[[[],[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [[1,2,5],[3],[4]]
=> 1 = 2 - 1
[[],[[[[]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 1 = 2 - 1
[[[]],[],[],[]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 2 = 3 - 1
[[[]],[],[[]]]
=> [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 2 = 3 - 1
[[[]],[[]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => [[1,3,5],[2,4]]
=> 2 = 3 - 1
[[[]],[[],[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [[1,3,5],[2,4]]
=> 2 = 3 - 1
[[[]],[[[]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 2 = 3 - 1
[[[],[]],[],[]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [[1,3,4,5],[2]]
=> 2 = 3 - 1
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [[1,4,5],[2],[3]]
=> 3 = 4 - 1
[[[],[]],[[]]]
=> [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [[1,3,4],[2,5]]
=> 2 = 3 - 1
[[[[]]],[[]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [[1,4],[2,5],[3]]
=> 3 = 4 - 1
[[[],[],[]],[]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [[1,3,4,5],[2]]
=> 2 = 3 - 1
[[[],[[]]],[]]
=> [[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [[1,3,5],[2],[4]]
=> 2 = 3 - 1
[[[[]],[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [[1,4,5],[2],[3]]
=> 3 = 4 - 1
[[[[],[]]],[]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [[1,4,5],[2],[3]]
=> 3 = 4 - 1
[[[[[]]]],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [[1,5],[2],[3],[4]]
=> 4 = 5 - 1
[[],[],[[]],[[[]]],[]]
=> [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
=> [1,2,4,3,7,6,5,8] => ?
=> ? = 2 - 1
[[],[],[[]],[[],[[]]]]
=> [[[[.,.],.],[.,.]],[[.,.],[.,.]]]
=> [1,2,4,3,6,8,7,5] => ?
=> ? = 2 - 1
[[],[],[[],[]],[],[],[]]
=> [[[[[[.,.],.],[[.,.],.]],.],.],.]
=> [1,2,4,5,3,6,7,8] => ?
=> ? = 2 - 1
[[],[],[[],[]],[],[[]]]
=> [[[[[.,.],.],[[.,.],.]],.],[.,.]]
=> [1,2,4,5,3,6,8,7] => ?
=> ? = 2 - 1
[[],[],[[[]]],[[]],[]]
=> [[[[[.,.],.],[.,[.,.]]],[.,.]],.]
=> [1,2,5,4,3,7,6,8] => ?
=> ? = 2 - 1
[[],[],[[[]],[]],[],[]]
=> [[[[[.,.],.],[[.,[.,.]],.]],.],.]
=> [1,2,5,4,6,3,7,8] => ?
=> ? = 2 - 1
[[],[],[[[],[]]],[],[]]
=> [[[[[.,.],.],[.,[[.,.],.]]],.],.]
=> [1,2,5,6,4,3,7,8] => ?
=> ? = 2 - 1
[[],[],[[],[[]]],[[]]]
=> [[[[.,.],.],[[.,.],[.,.]]],[.,.]]
=> [1,2,4,6,5,3,8,7] => ?
=> ? = 2 - 1
[[],[],[[[]],[]],[[]]]
=> [[[[.,.],.],[[.,[.,.]],.]],[.,.]]
=> [1,2,5,4,6,3,8,7] => ?
=> ? = 2 - 1
[[],[],[[[],[]]],[[]]]
=> [[[[.,.],.],[.,[[.,.],.]]],[.,.]]
=> [1,2,5,6,4,3,8,7] => ?
=> ? = 2 - 1
[[],[],[[[[]]]],[[]]]
=> [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
=> [1,2,6,5,4,3,8,7] => ?
=> ? = 2 - 1
[[],[],[[],[],[[]]],[]]
=> [[[[.,.],.],[[[.,.],.],[.,.]]],.]
=> [1,2,4,5,7,6,3,8] => ?
=> ? = 2 - 1
[[],[],[[],[[]],[]],[]]
=> [[[[.,.],.],[[[.,.],[.,.]],.]],.]
=> [1,2,4,6,5,7,3,8] => ?
=> ? = 2 - 1
[[],[],[[],[[[]]]],[]]
=> [[[[.,.],.],[[.,.],[.,[.,.]]]],.]
=> [1,2,4,7,6,5,3,8] => ?
=> ? = 2 - 1
[[],[],[[[]],[[]]],[]]
=> [[[[.,.],.],[[.,[.,.]],[.,.]]],.]
=> [1,2,5,4,7,6,3,8] => ?
=> ? = 2 - 1
[[],[],[[[[]]],[]],[]]
=> [[[[.,.],.],[[.,[.,[.,.]]],.]],.]
=> [1,2,6,5,4,7,3,8] => ?
=> ? = 2 - 1
[[],[],[[[[]],[]]],[]]
=> [[[[.,.],.],[.,[[.,[.,.]],.]]],.]
=> [1,2,6,5,7,4,3,8] => ?
=> ? = 2 - 1
[[],[],[[],[],[[]],[]]]
=> [[[.,.],.],[[[[.,.],.],[.,.]],.]]
=> [1,2,4,5,7,6,8,3] => ?
=> ? = 2 - 1
[[],[],[[],[[]],[],[]]]
=> [[[.,.],.],[[[[.,.],[.,.]],.],.]]
=> [1,2,4,6,5,7,8,3] => ?
=> ? = 2 - 1
[[],[],[[],[[]],[[]]]]
=> [[[.,.],.],[[[.,.],[.,.]],[.,.]]]
=> [1,2,4,6,5,8,7,3] => ?
=> ? = 2 - 1
[[],[],[[],[[],[]],[]]]
=> [[[.,.],.],[[[.,.],[[.,.],.]],.]]
=> [1,2,4,6,7,5,8,3] => ?
=> ? = 2 - 1
[[],[],[[],[[[]],[]]]]
=> [[[.,.],.],[[.,.],[[.,[.,.]],.]]]
=> [1,2,4,7,6,8,5,3] => ?
=> ? = 2 - 1
[[],[],[[[]],[[],[]]]]
=> [[[.,.],.],[[.,[.,.]],[[.,.],.]]]
=> [1,2,5,4,7,8,6,3] => ?
=> ? = 2 - 1
[[],[],[[[],[]],[[]]]]
=> [[[.,.],.],[[.,[[.,.],.]],[.,.]]]
=> [1,2,5,6,4,8,7,3] => ?
=> ? = 2 - 1
[[],[],[[[[]]],[[]]]]
=> [[[.,.],.],[[.,[.,[.,.]]],[.,.]]]
=> [1,2,6,5,4,8,7,3] => ?
=> ? = 2 - 1
[[],[],[[[[]],[]],[]]]
=> [[[.,.],.],[[.,[[.,[.,.]],.]],.]]
=> [1,2,6,5,7,4,8,3] => ?
=> ? = 2 - 1
[[],[],[[[[],[]]],[]]]
=> [[[.,.],.],[[.,[.,[[.,.],.]]],.]]
=> [1,2,6,7,5,4,8,3] => ?
=> ? = 2 - 1
[[],[],[[[],[[],[]]]]]
=> [[[.,.],.],[.,[[.,.],[[.,.],.]]]]
=> [1,2,5,7,8,6,4,3] => ?
=> ? = 2 - 1
[[],[],[[[[],[]],[]]]]
=> [[[.,.],.],[.,[[.,[[.,.],.]],.]]]
=> [1,2,6,7,5,8,4,3] => ?
=> ? = 2 - 1
[[],[],[[[[],[[]]]]]]
=> [[[.,.],.],[.,[.,[[.,.],[.,.]]]]]
=> [1,2,6,8,7,5,4,3] => ?
=> ? = 2 - 1
[[],[[]],[],[[],[]],[]]
=> [[[[[.,.],[.,.]],.],[[.,.],.]],.]
=> [1,3,2,4,6,7,5,8] => ?
=> ? = 2 - 1
[[],[[]],[],[[[],[]]]]
=> [[[[.,.],[.,.]],.],[.,[[.,.],.]]]
=> [1,3,2,4,7,8,6,5] => ?
=> ? = 2 - 1
[[],[[]],[[]],[],[[]]]
=> [[[[[.,.],[.,.]],[.,.]],.],[.,.]]
=> [1,3,2,5,4,6,8,7] => ?
=> ? = 2 - 1
[[],[[]],[[[]],[]],[]]
=> [[[[.,.],[.,.]],[[.,[.,.]],.]],.]
=> [1,3,2,6,5,7,4,8] => ?
=> ? = 2 - 1
[[],[[]],[[],[[]],[]]]
=> [[[.,.],[.,.]],[[[.,.],[.,.]],.]]
=> [1,3,2,5,7,6,8,4] => ?
=> ? = 2 - 1
[[],[[]],[[[[],[]]]]]
=> [[[.,.],[.,.]],[.,[.,[[.,.],.]]]]
=> [1,3,2,7,8,6,5,4] => ?
=> ? = 2 - 1
[[],[[],[]],[],[[[]]]]
=> [[[[.,.],[[.,.],.]],.],[.,[.,.]]]
=> [1,3,4,2,5,8,7,6] => ?
=> ? = 2 - 1
[[],[[],[]],[[]],[],[]]
=> [[[[[.,.],[[.,.],.]],[.,.]],.],.]
=> [1,3,4,2,6,5,7,8] => ?
=> ? = 2 - 1
[[],[[[]]],[[]],[],[]]
=> [[[[[.,.],[.,[.,.]]],[.,.]],.],.]
=> [1,4,3,2,6,5,7,8] => ?
=> ? = 2 - 1
[[],[[[]]],[[]],[[]]]
=> [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,4,3,2,6,5,8,7] => ?
=> ? = 2 - 1
[[],[[],[]],[[[]]],[]]
=> [[[[.,.],[[.,.],.]],[.,[.,.]]],.]
=> [1,3,4,2,7,6,5,8] => ?
=> ? = 2 - 1
[[],[[[]]],[[[],[]]]]
=> [[[.,.],[.,[.,.]]],[.,[[.,.],.]]]
=> [1,4,3,2,7,8,6,5] => ?
=> ? = 2 - 1
[[],[[],[[]]],[],[],[]]
=> [[[[[.,.],[[.,.],[.,.]]],.],.],.]
=> [1,3,5,4,2,6,7,8] => ?
=> ? = 2 - 1
[[],[[[]],[]],[],[],[]]
=> [[[[[.,.],[[.,[.,.]],.]],.],.],.]
=> [1,4,3,5,2,6,7,8] => ?
=> ? = 2 - 1
[[],[[],[],[]],[],[[]]]
=> [[[[.,.],[[[.,.],.],.]],.],[.,.]]
=> [1,3,4,5,2,6,8,7] => ?
=> ? = 2 - 1
[[],[[[]],[]],[],[[]]]
=> [[[[.,.],[[.,[.,.]],.]],.],[.,.]]
=> [1,4,3,5,2,6,8,7] => ?
=> ? = 2 - 1
[[],[[[[]]]],[],[[]]]
=> [[[[.,.],[.,[.,[.,.]]]],.],[.,.]]
=> [1,5,4,3,2,6,8,7] => ?
=> ? = 2 - 1
[[],[[[]],[]],[[]],[]]
=> [[[[.,.],[[.,[.,.]],.]],[.,.]],.]
=> [1,4,3,5,2,7,6,8] => ?
=> ? = 2 - 1
[[],[[],[[]]],[[[]]]]
=> [[[.,.],[[.,.],[.,.]]],[.,[.,.]]]
=> [1,3,5,4,2,8,7,6] => ?
=> ? = 2 - 1
[[],[[[]],[]],[[],[]]]
=> [[[.,.],[[.,[.,.]],.]],[[.,.],.]]
=> [1,4,3,5,2,7,8,6] => ?
=> ? = 2 - 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000011
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00048: Ordered trees —left-right symmetry⟶ Ordered trees
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [[]]
=> [1,0]
=> [1,0]
=> 1 = 2 - 1
[[],[]]
=> [[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 2 - 1
[[[]]]
=> [[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 3 - 1
[[],[],[]]
=> [[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[[],[[]]]
=> [[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[[[]],[]]
=> [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[[[],[]]]
=> [[[],[]]]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[[[[]]]]
=> [[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
[[],[],[],[]]
=> [[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[],[],[[]]]
=> [[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[[],[[]],[]]
=> [[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[[],[[],[]]]
=> [[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[[],[[[]]]]
=> [[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[[[]],[],[]]
=> [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[[]],[[]]]
=> [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[[],[]],[]]
=> [[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[[[]]],[]]
=> [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[[[],[],[]]]
=> [[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[[],[[]]]]
=> [[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[[[]],[]]]
=> [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[[[[],[]]]]
=> [[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[[[[[]]]]]
=> [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[[],[],[],[],[]]
=> [[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[[],[],[],[[]]]
=> [[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[],[],[[]],[]]
=> [[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[[],[],[[],[]]]
=> [[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[[],[],[[[]]]]
=> [[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 2 - 1
[[],[[]],[],[]]
=> [[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[[],[[]],[[]]]
=> [[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[[],[[],[]],[]]
=> [[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[[],[[[]]],[]]
=> [[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[[],[[],[],[]]]
=> [[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[[],[[],[[]]]]
=> [[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
[[],[[[]],[]]]
=> [[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[[],[[[],[]]]]
=> [[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 2 - 1
[[],[[[[]]]]]
=> [[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[[[]],[],[],[]]
=> [[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[[[]],[],[[]]]
=> [[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[[]],[[]],[]]
=> [[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[[[]],[[],[]]]
=> [[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[[]],[[[]]]]
=> [[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[[[],[]],[],[]]
=> [[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[[[[]]],[],[]]
=> [[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[[],[]],[[]]]
=> [[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[[[]]],[[]]]
=> [[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[[[],[],[]],[]]
=> [[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[[[],[[]]],[]]
=> [[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[[[[]],[]],[]]
=> [[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[[[[],[]]],[]]
=> [[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[[[[[]]]],[]]
=> [[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
[[],[],[],[],[[[],[]]]]
=> [[[[],[]]],[],[],[],[]]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[[]],[[],[]]]
=> [[[],[]],[[]],[],[],[]]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[[]],[[[]]]]
=> [[[[]]],[[]],[],[],[]]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[[],[]],[[]]]
=> [[[]],[[],[]],[],[],[]]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[[[]]],[[]]]
=> [[[]],[[[]]],[],[],[]]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[[],[[],[]]]]
=> [[[[],[]],[]],[],[],[]]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[[[],[]],[]]]
=> [[[],[[],[]]],[],[],[]]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[[[],[],[]]]]
=> [[[[],[],[]]],[],[],[]]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[[[],[[]]]]]
=> [[[[[]],[]]],[],[],[]]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[[[[]],[]]]]
=> [[[[],[[]]]],[],[],[]]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[[[[],[]]]]]
=> [[[[[],[]]]],[],[],[]]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[[]],[],[[[]]]]
=> [[[[]]],[],[[]],[],[]]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[[]],[[]],[[]]]
=> [[[]],[[]],[[]],[],[]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[[]],[[],[[]]]]
=> [[[[]],[]],[[]],[],[]]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[[]],[[[]],[]]]
=> [[[],[[]]],[[]],[],[]]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[[]],[[[],[]]]]
=> [[[[],[]]],[[]],[],[]]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[[]],[[[[]]]]]
=> [[[[[]]]],[[]],[],[]]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[[],[]],[],[[]]]
=> [[[]],[],[[],[]],[],[]]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[[],[]],[[],[]]]
=> [[[],[]],[[],[]],[],[]]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[[],[]],[[[]]]]
=> [[[[]]],[[],[]],[],[]]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[[[]]],[[],[]]]
=> [[[],[]],[[[]]],[],[]]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[[[]]],[[[]]]]
=> [[[[]]],[[[]]],[],[]]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[[],[[]]],[[]]]
=> [[[]],[[[]],[]],[],[]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[[[]],[]],[[]]]
=> [[[]],[[],[[]]],[],[]]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[[[],[]]],[[]]]
=> [[[]],[[[],[]]],[],[]]
=> [1,1,0,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[[[[]]]],[[]]]
=> [[[]],[[[[]]]],[],[]]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[[],[],[[],[]]]]
=> [[[[],[]],[],[]],[],[]]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,1,0,0,0]
=> ? = 2 - 1
[[],[],[[],[[],[]],[]]]
=> [[[],[[],[]],[]],[],[]]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> ? = 2 - 1
[[],[],[[],[[],[],[]]]]
=> [[[[],[],[]],[]],[],[]]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,1,0,0,0]
=> ? = 2 - 1
[[],[],[[],[[],[[]]]]]
=> [[[[[]],[]],[]],[],[]]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,1,0,0,0]
=> ? = 2 - 1
[[],[],[[],[[[]],[]]]]
=> [[[[],[[]]],[]],[],[]]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,1,0,0,0]
=> ? = 2 - 1
[[],[],[[],[[[],[]]]]]
=> [[[[[],[]]],[]],[],[]]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,1,0,0,0]
=> ? = 2 - 1
[[],[],[[[]],[[],[]]]]
=> [[[[],[]],[[]]],[],[]]
=> [1,1,1,0,1,0,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 2 - 1
[[],[],[[[],[]],[],[]]]
=> [[[],[],[[],[]]],[],[]]
=> [1,1,0,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 2 - 1
[[],[],[[[],[]],[[]]]]
=> [[[[]],[[],[]]],[],[]]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 2 - 1
[[],[],[[[],[],[]],[]]]
=> [[[],[[],[],[]]],[],[]]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> ? = 2 - 1
[[],[],[[[],[[]]],[]]]
=> [[[],[[[]],[]]],[],[]]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 2 - 1
[[],[],[[[[]],[]],[]]]
=> [[[],[[],[[]]]],[],[]]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 2 - 1
[[],[],[[[[],[]]],[]]]
=> [[[],[[[],[]]]],[],[]]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> ? = 2 - 1
[[],[],[[[],[],[[]]]]]
=> [[[[[]],[],[]]],[],[]]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,1,0,0,0]
=> ? = 2 - 1
[[],[],[[[],[[]],[]]]]
=> [[[[],[[]],[]]],[],[]]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 2 - 1
[[],[],[[[],[[],[]]]]]
=> [[[[[],[]],[]]],[],[]]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 2 - 1
[[],[],[[[],[[[]]]]]]
=> [[[[[[]]],[]]],[],[]]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> ? = 2 - 1
[[],[],[[[[]],[],[]]]]
=> [[[[],[],[[]]]],[],[]]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 2 - 1
[[],[],[[[[]],[[]]]]]
=> [[[[[]],[[]]]],[],[]]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 2 - 1
[[],[],[[[[],[]],[]]]]
=> [[[[],[[],[]]]],[],[]]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,1,0,0,0]
=> ? = 2 - 1
[[],[],[[[[[]]],[]]]]
=> [[[[],[[[]]]]],[],[]]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 2 - 1
[[],[],[[[[],[],[]]]]]
=> [[[[[],[],[]]]],[],[]]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,1,0,0,0]
=> ? = 2 - 1
[[],[],[[[[],[[]]]]]]
=> [[[[[[]],[]]]],[],[]]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,1,0,0,0]
=> ? = 2 - 1
[[],[],[[[[[]],[]]]]]
=> [[[[[],[[]]]]],[],[]]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 2 - 1
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000288
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [.,.]
=> [1] => => ? = 2 - 2
[[],[]]
=> [.,[.,.]]
=> [2,1] => 0 => 0 = 2 - 2
[[[]]]
=> [[.,.],.]
=> [1,2] => 1 => 1 = 3 - 2
[[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 00 => 0 = 2 - 2
[[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 00 => 0 = 2 - 2
[[[]],[]]
=> [[.,[.,.]],.]
=> [2,1,3] => 01 => 1 = 3 - 2
[[[],[]]]
=> [[.,.],[.,.]]
=> [1,3,2] => 10 => 1 = 3 - 2
[[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 11 => 2 = 4 - 2
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 000 => 0 = 2 - 2
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 000 => 0 = 2 - 2
[[],[[]],[]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 000 => 0 = 2 - 2
[[],[[],[]]]
=> [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 000 => 0 = 2 - 2
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 000 => 0 = 2 - 2
[[[]],[],[]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 001 => 1 = 3 - 2
[[[]],[[]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 001 => 1 = 3 - 2
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 010 => 1 = 3 - 2
[[[[]]],[]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 011 => 2 = 4 - 2
[[[],[],[]]]
=> [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 100 => 1 = 3 - 2
[[[],[[]]]]
=> [[.,.],[[.,.],.]]
=> [1,3,4,2] => 100 => 1 = 3 - 2
[[[[]],[]]]
=> [[[.,.],.],[.,.]]
=> [1,2,4,3] => 110 => 2 = 4 - 2
[[[[],[]]]]
=> [[[.,.],[.,.]],.]
=> [1,3,2,4] => 101 => 2 = 4 - 2
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 111 => 3 = 5 - 2
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 0000 => 0 = 2 - 2
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 0000 => 0 = 2 - 2
[[],[],[[]],[]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 0000 => 0 = 2 - 2
[[],[],[[],[]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 0000 => 0 = 2 - 2
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 0000 => 0 = 2 - 2
[[],[[]],[],[]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 0000 => 0 = 2 - 2
[[],[[]],[[]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 0000 => 0 = 2 - 2
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 0000 => 0 = 2 - 2
[[],[[[]]],[]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 0000 => 0 = 2 - 2
[[],[[],[],[]]]
=> [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 0000 => 0 = 2 - 2
[[],[[],[[]]]]
=> [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 0000 => 0 = 2 - 2
[[],[[[]],[]]]
=> [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 0000 => 0 = 2 - 2
[[],[[[],[]]]]
=> [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 0000 => 0 = 2 - 2
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 0000 => 0 = 2 - 2
[[[]],[],[],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 0001 => 1 = 3 - 2
[[[]],[],[[]]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => 0001 => 1 = 3 - 2
[[[]],[[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => 0001 => 1 = 3 - 2
[[[]],[[],[]]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 1000 => 1 = 3 - 2
[[[]],[[[]]]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => 0001 => 1 = 3 - 2
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 0100 => 1 = 3 - 2
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 0011 => 2 = 4 - 2
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 0100 => 1 = 3 - 2
[[[[]]],[[]]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => 0011 => 2 = 4 - 2
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 0010 => 1 = 3 - 2
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 0010 => 1 = 3 - 2
[[[[]],[]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 0110 => 2 = 4 - 2
[[[[],[]]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => 0101 => 2 = 4 - 2
[[[[[]]]],[]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 0111 => 3 = 5 - 2
[[[],[],[],[]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 1000 => 1 = 3 - 2
[[],[],[],[],[[[]],[]]]
=> [.,[.,[.,[.,[[[.,.],.],[.,.]]]]]]
=> [5,6,8,7,4,3,2,1] => ? => ? = 2 - 2
[[],[],[],[[],[]],[],[]]
=> [.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
=> [5,4,8,7,6,3,2,1] => ? => ? = 2 - 2
[[],[],[],[[],[]],[[]]]
=> [.,[.,[.,[[.,[.,.]],[[.,.],.]]]]]
=> [5,4,7,8,6,3,2,1] => ? => ? = 2 - 2
[[],[],[],[[],[],[]],[]]
=> [.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]
=> [6,5,4,8,7,3,2,1] => ? => ? = 2 - 2
[[],[],[],[[],[[]]],[]]
=> [.,[.,[.,[[.,[[.,.],.]],[.,.]]]]]
=> [5,6,4,8,7,3,2,1] => ? => ? = 2 - 2
[[],[],[],[[[]],[]],[]]
=> [.,[.,[.,[[[.,[.,.]],.],[.,.]]]]]
=> [5,4,6,8,7,3,2,1] => ? => ? = 2 - 2
[[],[],[[]],[],[[]],[]]
=> [.,[.,[[.,[.,[[.,[.,.]],.]]],.]]]
=> [6,5,7,4,3,8,2,1] => ? => ? = 2 - 2
[[],[],[[]],[[]],[[]]]
=> [.,[.,[[.,[[.,[[.,.],.]],.]],.]]]
=> [5,6,4,7,3,8,2,1] => ? => ? = 2 - 2
[[],[],[[]],[[],[]],[]]
=> [.,[.,[[.,.],[[.,[.,.]],[.,.]]]]]
=> [3,6,5,8,7,4,2,1] => ? => ? = 2 - 2
[[],[],[[]],[[[],[]]]]
=> [.,[.,[[.,.],[[[.,.],[.,.]],.]]]]
=> [3,5,7,6,8,4,2,1] => ? => ? = 2 - 2
[[],[],[[],[]],[],[[]]]
=> [.,[.,[[.,[.,.]],[.,[[.,.],.]]]]]
=> [4,3,7,8,6,5,2,1] => ? => ? = 2 - 2
[[],[],[[[]]],[],[[]]]
=> [.,[.,[[[.,[.,[[.,.],.]]],.],.]]]
=> [5,6,4,3,7,8,2,1] => ? => ? = 2 - 2
[[],[],[[],[]],[[]],[]]
=> [.,[.,[[.,[.,.]],[[.,[.,.]],.]]]]
=> [4,3,7,6,8,5,2,1] => ? => ? = 2 - 2
[[],[],[[[]]],[[]],[]]
=> [.,[.,[[[.,[[.,[.,.]],.]],.],.]]]
=> [5,4,6,3,7,8,2,1] => ? => ? = 2 - 2
[[],[],[[],[]],[[],[]]]
=> [.,[.,[[.,[.,.]],[[.,.],[.,.]]]]]
=> [4,3,6,8,7,5,2,1] => ? => ? = 2 - 2
[[],[],[[[]]],[[],[]]]
=> [.,[.,[[[.,.],.],[[.,.],[.,.]]]]]
=> [3,4,6,8,7,5,2,1] => ? => ? = 2 - 2
[[],[],[[[]]],[[[]]]]
=> [.,[.,[[[.,[[[.,.],.],.]],.],.]]]
=> [4,5,6,3,7,8,2,1] => ? => ? = 2 - 2
[[],[],[[],[[]]],[],[]]
=> [.,[.,[[.,[[.,.],.]],[.,[.,.]]]]]
=> [4,5,3,8,7,6,2,1] => ? => ? = 2 - 2
[[],[],[[[]],[]],[],[]]
=> [.,[.,[[[.,[.,.]],.],[.,[.,.]]]]]
=> [4,3,5,8,7,6,2,1] => ? => ? = 2 - 2
[[],[],[[[],[]]],[[]]]
=> [.,[.,[[[.,[.,.]],[[.,.],.]],.]]]
=> [4,3,6,7,5,8,2,1] => ? => ? = 2 - 2
[[],[],[[[[]]]],[[]]]
=> [.,[.,[[[[.,[[.,.],.]],.],.],.]]]
=> [4,5,3,6,7,8,2,1] => ? => ? = 2 - 2
[[],[],[[],[[]],[]],[]]
=> [.,[.,[[.,[[.,[.,.]],.]],[.,.]]]]
=> [5,4,6,3,8,7,2,1] => ? => ? = 2 - 2
[[],[],[[[],[],[]]],[]]
=> [.,[.,[[[.,[.,[.,.]]],[.,.]],.]]]
=> [5,4,3,7,6,8,2,1] => ? => ? = 2 - 2
[[],[],[[[[],[]]]],[]]
=> [.,[.,[[[[.,[.,.]],[.,.]],.],.]]]
=> [4,3,6,5,7,8,2,1] => ? => ? = 2 - 2
[[],[],[[],[],[[]],[]]]
=> [.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> [3,7,6,8,5,4,2,1] => ? => ? = 2 - 2
[[],[],[[],[],[[],[]]]]
=> [.,[.,[[.,[.,[[.,.],[.,.]]]],.]]]
=> [5,7,6,4,3,8,2,1] => ? => ? = 2 - 2
[[],[],[[],[],[[[]]]]]
=> [.,[.,[[.,.],[.,[[[.,.],.],.]]]]]
=> [3,6,7,8,5,4,2,1] => ? => ? = 2 - 2
[[],[],[[],[[]],[],[]]]
=> [.,[.,[[.,.],[[.,[.,[.,.]]],.]]]]
=> [3,7,6,5,8,4,2,1] => ? => ? = 2 - 2
[[],[],[[],[[]],[[]]]]
=> [.,[.,[[.,.],[[.,[[.,.],.]],.]]]]
=> [3,6,7,5,8,4,2,1] => ? => ? = 2 - 2
[[],[],[[],[[],[]],[]]]
=> [.,[.,[[.,[[.,.],[.,.]]],[.,.]]]]
=> [4,6,5,3,8,7,2,1] => ? => ? = 2 - 2
[[],[],[[],[[[]]],[]]]
=> [.,[.,[[.,.],[[[.,[.,.]],.],.]]]]
=> [3,6,5,7,8,4,2,1] => ? => ? = 2 - 2
[[],[],[[],[[],[],[]]]]
=> [.,[.,[[.,[[.,.],[.,[.,.]]]],.]]]
=> [4,7,6,5,3,8,2,1] => ? => ? = 2 - 2
[[],[],[[],[[],[[]]]]]
=> [.,[.,[[.,[[.,.],[[.,.],.]]],.]]]
=> [4,6,7,5,3,8,2,1] => ? => ? = 2 - 2
[[],[],[[],[[[]],[]]]]
=> [.,[.,[[.,[[[.,.],.],[.,.]]],.]]]
=> [4,5,7,6,3,8,2,1] => ? => ? = 2 - 2
[[],[],[[],[[[],[]]]]]
=> [.,[.,[[.,[[[.,.],[.,.]],.]],.]]]
=> [4,6,5,7,3,8,2,1] => ? => ? = 2 - 2
[[],[],[[[]],[],[[]]]]
=> [.,[.,[[[.,.],.],[.,[[.,.],.]]]]]
=> [3,4,7,8,6,5,2,1] => ? => ? = 2 - 2
[[],[],[[[]],[[]],[]]]
=> [.,[.,[[[.,.],.],[[.,[.,.]],.]]]]
=> [3,4,7,6,8,5,2,1] => ? => ? = 2 - 2
[[],[],[[[]],[[],[]]]]
=> [.,[.,[[[.,.],[[.,.],[.,.]]],.]]]
=> [3,5,7,6,4,8,2,1] => ? => ? = 2 - 2
[[],[],[[[],[]],[],[]]]
=> [.,[.,[[[.,.],[.,.]],[.,[.,.]]]]]
=> [3,5,4,8,7,6,2,1] => ? => ? = 2 - 2
[[],[],[[[],[]],[[]]]]
=> [.,[.,[[[.,.],[.,.]],[[.,.],.]]]]
=> [3,5,4,7,8,6,2,1] => ? => ? = 2 - 2
[[],[],[[[[]]],[[]]]]
=> [.,[.,[[[[.,.],.],.],[[.,.],.]]]]
=> [3,4,5,7,8,6,2,1] => ? => ? = 2 - 2
[[],[],[[[],[],[]],[]]]
=> [.,[.,[[[.,.],[.,[.,.]]],[.,.]]]]
=> [3,6,5,4,8,7,2,1] => ? => ? = 2 - 2
[[],[],[[[],[[]]],[]]]
=> [.,[.,[[[.,.],[[.,.],.]],[.,.]]]]
=> [3,5,6,4,8,7,2,1] => ? => ? = 2 - 2
[[],[],[[[[]],[]],[]]]
=> [.,[.,[[[[.,.],.],[.,.]],[.,.]]]]
=> [3,4,6,5,8,7,2,1] => ? => ? = 2 - 2
[[],[],[[[[],[]]],[]]]
=> [.,[.,[[[[.,.],[.,.]],.],[.,.]]]]
=> [3,5,4,6,8,7,2,1] => ? => ? = 2 - 2
[[],[],[[[],[],[[]]]]]
=> [.,[.,[[[.,.],[.,[[.,.],.]]],.]]]
=> [3,6,7,5,4,8,2,1] => ? => ? = 2 - 2
[[],[],[[[],[[]],[]]]]
=> [.,[.,[[[.,.],[[.,[.,.]],.]],.]]]
=> [3,6,5,7,4,8,2,1] => ? => ? = 2 - 2
[[],[],[[[],[[],[]]]]]
=> [.,[.,[[[.,[[.,.],[.,.]]],.],.]]]
=> [4,6,5,3,7,8,2,1] => ? => ? = 2 - 2
[[],[],[[[],[[[]]]]]]
=> [.,[.,[[[.,.],[[[.,.],.],.]],.]]]
=> [3,5,6,7,4,8,2,1] => ? => ? = 2 - 2
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000678
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [.,.]
=> [1,0]
=> [1,0]
=> ? = 2 - 1
[[],[]]
=> [.,[.,.]]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1 = 2 - 1
[[[]]]
=> [[.,.],.]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2 = 3 - 1
[[],[],[]]
=> [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[[],[[]]]
=> [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[[[]],[]]
=> [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[[[],[]]]
=> [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[[[[]]]]
=> [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[],[[]],[]]
=> [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[[]],[],[]]
=> [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[[[[]]],[]]
=> [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[],[],[[]],[]]
=> [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[[],[],[[],[]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[],[[]],[],[]]
=> [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[],[[]],[[]]]
=> [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 2 - 1
[[],[[[]]],[]]
=> [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[[],[[],[],[]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[[],[[],[[]]]]
=> [.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[[],[[[]],[]]]
=> [.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[[],[[[],[]]]]
=> [.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[[]],[],[],[]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
[[[]],[],[[]]]
=> [[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [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]
=> 2 = 3 - 1
[[[]],[[],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[[[]],[[[]]]]
=> [[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[[[[]]],[],[]]
=> [[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 4 - 1
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[[[]]],[[]]]
=> [[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[[[[]],[]],[]]
=> [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 4 - 1
[[[[],[]]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 4 - 1
[[[[[]]]],[]]
=> [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[[[],[],[],[]]]
=> [[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 3 - 1
[[],[],[],[],[],[],[],[]]
=> [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[],[],[[]],[]]
=> [.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[],[[]],[],[]]
=> [.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[],[[],[]],[]]
=> [.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[],[[[]]],[]]
=> [.,[.,[.,[.,[[[.,.],.],[.,.]]]]]]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[],[[],[],[]]]
=> [.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[],[[[]],[]]]
=> [.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[[]],[],[],[]]
=> [.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[[]],[[]],[]]
=> [.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[[],[]],[],[]]
=> [.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[[[]]],[],[]]
=> [.,[.,[.,[[[.,.],.],[.,[.,.]]]]]]
=> [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[[],[],[]],[]]
=> [.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[[],[[]]],[]]
=> [.,[.,[.,[[.,[[.,.],.]],[.,.]]]]]
=> [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[[[]],[]],[]]
=> [.,[.,[.,[[[.,.],[.,.]],[.,.]]]]]
=> [1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[[[],[]]],[]]
=> [.,[.,[.,[[[.,[.,.]],.],[.,.]]]]]
=> [1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[[[[]]]],[]]
=> [.,[.,[.,[[[[.,.],.],.],[.,.]]]]]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[[],[],[],[]]]
=> [.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[[],[[]],[]]]
=> [.,[.,[.,[[.,[[.,.],[.,.]]],.]]]]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[[[]],[],[]]]
=> [.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[[[],[]],[]]]
=> [.,[.,[.,[[[.,[.,.]],[.,.]],.]]]]
=> [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[[[[]]],[]]]
=> [.,[.,[.,[[[[.,.],.],[.,.]],.]]]]
=> [1,1,1,1,0,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[[[],[],[]]]]
=> [.,[.,[.,[[[.,[.,[.,.]]],.],.]]]]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[[]],[],[],[],[]]
=> [.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[[]],[],[[]],[]]
=> [.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[[]],[[]],[],[]]
=> [.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[[]],[[],[]],[]]
=> [.,[.,[[.,.],[[.,[.,.]],[.,.]]]]]
=> [1,1,1,0,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[[]],[[[]]],[]]
=> [.,[.,[[.,.],[[[.,.],.],[.,.]]]]]
=> [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[[],[]],[],[],[]]
=> [.,[.,[[.,[.,.]],[.,[.,[.,.]]]]]]
=> [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[[[]]],[],[],[]]
=> [.,[.,[[[.,.],.],[.,[.,[.,.]]]]]]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[[],[]],[[]],[]]
=> [.,[.,[[.,[.,.]],[[.,.],[.,.]]]]]
=> [1,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[[[]]],[[]],[]]
=> [.,[.,[[[.,.],.],[[.,.],[.,.]]]]]
=> [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[[],[],[]],[],[]]
=> [.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[[],[[]]],[],[]]
=> [.,[.,[[.,[[.,.],.]],[.,[.,.]]]]]
=> [1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[[[]],[]],[],[]]
=> [.,[.,[[[.,.],[.,.]],[.,[.,.]]]]]
=> [1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[[[],[]]],[],[]]
=> [.,[.,[[[.,[.,.]],.],[.,[.,.]]]]]
=> [1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[[[[]]]],[],[]]
=> [.,[.,[[[[.,.],.],.],[.,[.,.]]]]]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[[],[],[]],[[]]]
=> [.,[.,[[.,[.,[.,.]]],[[.,.],.]]]]
=> [1,1,1,1,1,0,0,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[[],[[]]],[[]]]
=> [.,[.,[[.,[[.,.],.]],[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[[],[],[],[]],[]]
=> [.,[.,[[.,[.,[.,[.,.]]]],[.,.]]]]
=> [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[[],[],[[]]],[]]
=> [.,[.,[[.,[.,[[.,.],.]]],[.,.]]]]
=> [1,1,1,1,1,0,1,0,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[[],[[]],[]],[]]
=> [.,[.,[[.,[[.,.],[.,.]]],[.,.]]]]
=> [1,1,1,1,0,1,1,0,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[[],[[],[]]],[]]
=> [.,[.,[[.,[[.,[.,.]],.]],[.,.]]]]
=> [1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[[],[[[]]]],[]]
=> [.,[.,[[.,[[[.,.],.],.]],[.,.]]]]
=> [1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[[[]],[],[]],[]]
=> [.,[.,[[[.,.],[.,[.,.]]],[.,.]]]]
=> [1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[[[]],[[]]],[]]
=> [.,[.,[[[.,.],[[.,.],.]],[.,.]]]]
=> [1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[[[],[]],[]],[]]
=> [.,[.,[[[.,[.,.]],[.,.]],[.,.]]]]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[[[[]]],[]],[]]
=> [.,[.,[[[[.,.],.],[.,.]],[.,.]]]]
=> [1,1,1,0,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[[[],[],[]]],[]]
=> [.,[.,[[[.,[.,[.,.]]],.],[.,.]]]]
=> [1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[[[],[[]]]],[]]
=> [.,[.,[[[.,[[.,.],.]],.],[.,.]]]]
=> [1,1,1,1,0,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,1,0,0,0,0]
=> ? = 2 - 1
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000675
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000675: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 88%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000675: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 88%
Values
[[]]
=> [1,0]
=> [1,0]
=> [1,0]
=> ? = 2 - 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]
=> 2 = 3 - 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3 = 4 - 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 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]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 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]
=> 4 = 5 - 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 2 - 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 3 - 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 4 - 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[[],[],[],[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 2 - 1
[[],[],[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 2 - 1
[[],[],[],[],[],[[]],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[[],[],[],[],[],[[],[]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> ? = 2 - 1
[[],[],[],[],[],[[[]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 2 - 1
[[],[],[],[],[[]],[],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? = 2 - 1
[[],[],[],[],[[]],[[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> ? = 2 - 1
[[],[],[],[],[[],[]],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[[],[],[],[],[[[]]],[]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 2 - 1
[[],[],[],[],[[],[],[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 2 - 1
[[],[],[],[],[[],[[]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 2 - 1
[[],[],[],[],[[[]],[]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,1,0,0]
=> ? = 2 - 1
[[],[],[],[],[[[],[]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 2 - 1
[[],[],[],[],[[[[]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[[]],[],[],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> ? = 2 - 1
[[],[],[],[[]],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 2 - 1
[[],[],[],[[]],[[]],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[[],[],[],[[]],[[],[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 2 - 1
[[],[],[],[[]],[[[]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 2 - 1
[[],[],[],[[],[]],[],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> ? = 2 - 1
[[],[],[],[[[]]],[],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> ? = 2 - 1
[[],[],[],[[],[]],[[]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,1,0,0]
=> ? = 2 - 1
[[],[],[],[[[]]],[[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0]
=> ? = 2 - 1
[[],[],[],[[],[],[]],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[[],[],[],[[],[[]]],[]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 2 - 1
[[],[],[],[[[]],[]],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[[],[],[],[[[],[]]],[]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0,1,0]
=> ? = 2 - 1
[[],[],[],[[[[]]]],[]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 2 - 1
[[],[],[],[[],[],[],[]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 2 - 1
[[],[],[],[[],[],[[]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 2 - 1
[[],[],[],[[],[[]],[]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,1,0,1,0,0]
=> ? = 2 - 1
[[],[],[],[[],[[],[]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 2 - 1
[[],[],[],[[],[[[]]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[[[]],[],[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 2 - 1
[[],[],[],[[[]],[[]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 2 - 1
[[],[],[],[[[],[]],[]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,0,1,0,0]
=> ? = 2 - 1
[[],[],[],[[[[]]],[]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,1,0,0]
=> ? = 2 - 1
[[],[],[],[[[],[],[]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 2 - 1
[[],[],[],[[[],[[]]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[[[[]],[]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 2 - 1
[[],[],[],[[[[],[]]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[[[[[]]]]]]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[[]],[],[],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[[],[],[[]],[],[],[[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> ? = 2 - 1
[[],[],[[]],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> ? = 2 - 1
[[],[],[[]],[],[[],[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,1,0,0]
=> ? = 2 - 1
[[],[],[[]],[],[[[]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
=> ? = 2 - 1
[[],[],[[]],[[]],[],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> ? = 2 - 1
[[],[],[[]],[[]],[[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,0]
=> ? = 2 - 1
Description
The number of centered multitunnels of a Dyck path.
This is the number of factorisations $D = A B C$ of a Dyck path, such that $B$ is a Dyck path and $A$ and $B$ have the same length.
Matching statistic: St000054
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 88%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 88%
Values
[[]]
=> [1,0]
=> [1] => [1] => 1 = 2 - 1
[[],[]]
=> [1,0,1,0]
=> [1,2] => [1,2] => 1 = 2 - 1
[[[]]]
=> [1,1,0,0]
=> [2,1] => [2,1] => 2 = 3 - 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,3,2] => 1 = 2 - 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 1 = 2 - 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 2 = 3 - 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => 2 = 3 - 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [3,1,2] => [3,1,2] => 3 = 4 - 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,4,3,2] => 1 = 2 - 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,4,3,2] => 1 = 2 - 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,4,3,2] => 1 = 2 - 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 1 = 2 - 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,4,3,2] => 1 = 2 - 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,4,3] => 2 = 3 - 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 2 = 3 - 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,4,1,3] => 2 = 3 - 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,1,4,2] => 3 = 4 - 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => 2 = 3 - 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,4,1,3] => 2 = 3 - 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [3,1,4,2] => 3 = 4 - 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [3,4,1,2] => 3 = 4 - 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [4,1,3,2] => 4 = 5 - 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => 1 = 2 - 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => 1 = 2 - 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => 1 = 2 - 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => 1 = 2 - 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,5,4,3,2] => 1 = 2 - 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => 1 = 2 - 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => 1 = 2 - 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => 1 = 2 - 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,5,4,3,2] => 1 = 2 - 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => 1 = 2 - 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,4,3,2] => 1 = 2 - 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,5,4,3,2] => 1 = 2 - 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,5,4,3,2] => 1 = 2 - 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,5,4,3,2] => 1 = 2 - 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => 2 = 3 - 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => 2 = 3 - 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => 2 = 3 - 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 2 = 3 - 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,5,4,3] => 2 = 3 - 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => 2 = 3 - 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [3,1,5,4,2] => 3 = 4 - 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => 2 = 3 - 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [3,1,5,4,2] => 3 = 4 - 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => 2 = 3 - 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,5,1,4,3] => 2 = 3 - 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [3,1,5,4,2] => 3 = 4 - 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [3,5,1,4,2] => 3 = 4 - 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [4,1,5,3,2] => 4 = 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] => ? = 3 - 1
[[[[]]],[],[],[],[]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6,7] => [3,1,7,6,5,4,2] => ? = 4 - 1
[[[],[]],[],[],[[]]]
=> [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] => ? = 3 - 1
[[[[]]],[],[],[[]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [3,1,2,4,5,7,6] => [3,1,7,6,5,4,2] => ? = 4 - 1
[[[],[]],[],[[]],[]]
=> [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] => ? = 3 - 1
[[[[]]],[],[[]],[]]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [3,1,2,4,6,5,7] => [3,1,7,6,5,4,2] => ? = 4 - 1
[[[],[]],[],[[],[]]]
=> [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] => ? = 3 - 1
[[[],[]],[],[[[]]]]
=> [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,7,5,6] => [2,7,1,6,5,4,3] => ? = 3 - 1
[[[[]]],[],[[],[]]]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [3,1,2,4,6,7,5] => [3,1,7,6,5,4,2] => ? = 4 - 1
[[[[]]],[],[[[]]]]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,1,2,4,7,5,6] => [3,1,7,6,5,4,2] => ? = 4 - 1
[[[],[]],[[]],[],[]]
=> [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] => ? = 3 - 1
[[[[]]],[[]],[],[]]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,1,2,5,4,6,7] => [3,1,7,6,5,4,2] => ? = 4 - 1
[[[],[]],[[]],[[]]]
=> [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] => ? = 3 - 1
[[[[]]],[[]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,1,2,5,4,7,6] => [3,1,7,6,5,4,2] => ? = 4 - 1
[[[],[]],[[],[]],[]]
=> [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] => ? = 3 - 1
[[[],[]],[[[]]],[]]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,1,6,4,5,7] => [2,7,1,6,5,4,3] => ? = 3 - 1
[[[[]]],[[],[]],[]]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [3,1,2,5,6,4,7] => [3,1,7,6,5,4,2] => ? = 4 - 1
[[[[]]],[[[]]],[]]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,1,2,6,4,5,7] => [3,1,7,6,5,4,2] => ? = 4 - 1
[[[],[]],[[],[],[]]]
=> [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] => ? = 3 - 1
[[[],[]],[[],[[]]]]
=> [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [2,3,1,5,7,4,6] => [2,7,1,6,5,4,3] => ? = 3 - 1
[[[],[]],[[[]],[]]]
=> [1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [2,3,1,6,4,7,5] => [2,7,1,6,5,4,3] => ? = 3 - 1
[[[],[]],[[[],[]]]]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [2,3,1,6,7,4,5] => [2,7,1,6,5,4,3] => ? = 3 - 1
[[[],[]],[[[[]]]]]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [2,3,1,7,4,5,6] => [2,7,1,6,5,4,3] => ? = 3 - 1
[[[[]]],[[],[],[]]]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [3,1,2,5,6,7,4] => [3,1,7,6,5,4,2] => ? = 4 - 1
[[[[]]],[[],[[]]]]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [3,1,2,5,7,4,6] => [3,1,7,6,5,4,2] => ? = 4 - 1
[[[[]]],[[[]],[]]]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [3,1,2,6,4,7,5] => [3,1,7,6,5,4,2] => ? = 4 - 1
[[[[]]],[[[],[]]]]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [3,1,2,6,7,4,5] => [3,1,7,6,5,4,2] => ? = 4 - 1
[[[[]]],[[[[]]]]]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,2,7,4,5,6] => [3,1,7,6,5,4,2] => ? = 4 - 1
[[[],[],[]],[],[],[]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,4,1,5,6,7] => [2,7,6,1,5,4,3] => ? = 3 - 1
[[[],[[]]],[],[],[]]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [2,4,1,3,5,6,7] => [2,7,1,6,5,4,3] => ? = 3 - 1
[[[[]],[]],[],[],[]]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5,6,7] => [3,1,7,6,5,4,2] => ? = 4 - 1
[[[[],[]]],[],[],[]]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [3,4,1,2,5,6,7] => [3,7,1,6,5,4,2] => ? = 4 - 1
[[[[[]]]],[],[],[]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,1,2,3,5,6,7] => [4,1,7,6,5,3,2] => ? = 5 - 1
[[[],[],[]],[],[[]]]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [2,3,4,1,5,7,6] => [2,7,6,1,5,4,3] => ? = 3 - 1
[[[],[[]]],[],[[]]]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [2,4,1,3,5,7,6] => [2,7,1,6,5,4,3] => ? = 3 - 1
[[[[]],[]],[],[[]]]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [3,1,4,2,5,7,6] => [3,1,7,6,5,4,2] => ? = 4 - 1
[[[[],[]]],[],[[]]]
=> [1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5,7,6] => [3,7,1,6,5,4,2] => ? = 4 - 1
[[[[[]]]],[],[[]]]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,1,2,3,5,7,6] => [4,1,7,6,5,3,2] => ? = 5 - 1
[[[],[],[]],[[]],[]]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5,7] => [2,7,6,1,5,4,3] => ? = 3 - 1
[[[],[[]]],[[]],[]]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [2,4,1,3,6,5,7] => [2,7,1,6,5,4,3] => ? = 3 - 1
[[[[]],[]],[[]],[]]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [3,1,4,2,6,5,7] => [3,1,7,6,5,4,2] => ? = 4 - 1
[[[[],[]]],[[]],[]]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> [3,4,1,2,6,5,7] => [3,7,1,6,5,4,2] => ? = 4 - 1
[[[[[]]]],[[]],[]]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => [4,1,7,6,5,3,2] => ? = 5 - 1
[[[],[],[]],[[],[]]]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [2,3,4,1,6,7,5] => [2,7,6,1,5,4,3] => ? = 3 - 1
[[[],[],[]],[[[]]]]
=> [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,4,1,7,5,6] => [2,7,6,1,5,4,3] => ? = 3 - 1
[[[],[[]]],[[],[]]]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [2,4,1,3,6,7,5] => [2,7,1,6,5,4,3] => ? = 3 - 1
[[[],[[]]],[[[]]]]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [2,4,1,3,7,5,6] => [2,7,1,6,5,4,3] => ? = 3 - 1
[[[[]],[]],[[],[]]]
=> [1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [3,1,4,2,6,7,5] => [3,1,7,6,5,4,2] => ? = 4 - 1
[[[[]],[]],[[[]]]]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [3,1,4,2,7,5,6] => [3,1,7,6,5,4,2] => ? = 4 - 1
[[[[],[]]],[[],[]]]
=> [1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [3,4,1,2,6,7,5] => [3,7,1,6,5,4,2] => ? = 4 - 1
Description
The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$
\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
$$
The following 62 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000759The smallest missing part in an integer partition. St000971The smallest closer of a set partition. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000069The number of maximal elements of a poset. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St000234The number of global ascents of a permutation. St000068The number of minimal elements in a poset. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000237The number of small exceedances. St000546The number of global descents of a permutation. St000007The number of saliances of the permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000542The number of left-to-right-minima of a permutation. St000654The first descent of a permutation. St000989The number of final rises of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000990The first ascent of a permutation. St000740The last entry of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000738The first entry in the last row of a standard tableau. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000843The decomposition number of a perfect matching. St000203The number of external nodes of a binary tree. St000734The last entry in the first row of a standard tableau. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000084The number of subtrees. St000056The decomposition (or block) number of a permutation. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St000326The position of the first one in a binary word after appending a 1 at the end. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000287The number of connected components of a graph. St000314The number of left-to-right-maxima of a permutation. St000352The Elizalde-Pak rank of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000051The size of the left subtree of a binary tree. St000133The "bounce" of a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St000061The number of nodes on the left branch of a binary tree. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001889The size of the connectivity set of a signed permutation. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. 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. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001937The size of the center of a parking function.
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