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Your data matches 48 different statistics following compositions of up to 3 maps.
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Matching statistic: St000340
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(load all 12 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000340: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> 0
[[],[]]
=> [1,0,1,0]
=> 0
[[[]]]
=> [1,1,0,0]
=> 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> 0
[[],[[]]]
=> [1,0,1,1,0,0]
=> 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> 0
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> 3
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> 2
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[[],[[],[]],[]]
=> [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]
=> 1
[[],[[],[[]]]]
=> [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]
=> 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 4
[[[]],[[],[]]]
=> [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]
=> 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
Description
The number of non-final maximal constant sub-paths of length greater than one.
This is the total number of occurrences of the patterns $110$ and $001$.
Matching statistic: St000925
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 88%
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 88%
Values
[[]]
=> [1,0]
=> [1,0]
=> {{1}}
=> ? = 0 + 1
[[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> {{1,2}}
=> 1 = 0 + 1
[[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> {{1},{2}}
=> 2 = 1 + 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 1 = 0 + 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 2 = 1 + 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3 = 2 + 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 2 = 1 + 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2 = 1 + 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1 = 0 + 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 2 = 1 + 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 3 = 2 + 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 2 = 1 + 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2 = 1 + 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 3 = 2 + 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 4 = 3 + 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 3 = 2 + 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 3 = 2 + 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2 = 1 + 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 3 = 2 + 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3 = 2 + 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2 = 1 + 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2 = 1 + 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1 = 0 + 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 2 = 1 + 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> {{1,2,5},{3},{4}}
=> 3 = 2 + 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> 2 = 1 + 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> 2 = 1 + 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> 3 = 2 + 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> 4 = 3 + 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> 3 = 2 + 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> {{1,5},{2,4},{3}}
=> 3 = 2 + 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 2 = 1 + 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> 3 = 2 + 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> {{1,5},{2,3},{4}}
=> 3 = 2 + 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> {{1,4,5},{2,3}}
=> 2 = 1 + 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> {{1,5},{2,3,4}}
=> 2 = 1 + 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 3 = 2 + 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 4 = 3 + 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 5 = 4 + 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 4 = 3 + 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 4 = 3 + 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 3 = 2 + 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> 3 = 2 + 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 4 = 3 + 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> 4 = 3 + 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 3 = 2 + 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 4 = 3 + 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 4 = 3 + 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 3 = 2 + 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 3 = 2 + 1
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 2 = 1 + 1
[[],[],[],[],[],[[]],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> {{1,2,3,4,5,8},{6},{7}}
=> ? = 2 + 1
[[],[],[],[],[[]],[],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5},{6,7}}
=> ? = 2 + 1
[[],[],[],[],[[]],[[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5},{6},{7}}
=> ? = 3 + 1
[[],[],[],[],[[],[]],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> {{1,2,3,4,6,8},{5},{7}}
=> ? = 2 + 1
[[],[],[],[],[[[]]],[]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,7},{6}}
=> ? = 2 + 1
[[],[],[],[],[[],[[]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> {{1,2,3,4,7,8},{5},{6}}
=> ? = 2 + 1
[[],[],[],[],[[[]],[]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6},{7}}
=> ? = 2 + 1
[[],[],[],[],[[[[]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6,7}}
=> ? = 1 + 1
[[],[],[],[[]],[],[],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4},{5,6,7}}
=> ? = 2 + 1
[[],[],[],[[]],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4},{5,7},{6}}
=> ? = 3 + 1
[[],[],[],[[]],[[]],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> {{1,2,3,8},{4},{5},{6},{7}}
=> ? = 4 + 1
[[],[],[],[[]],[[],[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4},{5},{6,7}}
=> ? = 3 + 1
[[],[],[],[[]],[[[]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,1,0,0,0,0]
=> {{1,2,3,8},{4},{5,6},{7}}
=> ? = 3 + 1
[[],[],[],[[],[]],[],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> {{1,2,3,5,8},{4},{6,7}}
=> ? = 2 + 1
[[],[],[],[[[]]],[],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,6,7},{5}}
=> ? = 2 + 1
[[],[],[],[[],[]],[[]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,5,8},{4},{6},{7}}
=> ? = 3 + 1
[[],[],[],[[[]]],[[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,7},{5},{6}}
=> ? = 3 + 1
[[],[],[],[[],[],[]],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> {{1,2,3,5,6,8},{4},{7}}
=> ? = 2 + 1
[[],[],[],[[],[[]]],[]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> {{1,2,3,7,8},{4},{5},{6}}
=> ? = 3 + 1
[[],[],[],[[[]],[]],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5},{6},{7}}
=> ? = 3 + 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]
=> {{1,2,3,7,8},{4,5},{6}}
=> ? = 2 + 1
[[],[],[],[[[[]]]],[]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5,7},{6}}
=> ? = 2 + 1
[[],[],[],[[],[],[[]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> {{1,2,3,5,7,8},{4},{6}}
=> ? = 2 + 1
[[],[],[],[[],[[]],[]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> {{1,2,3,6,8},{4},{5},{7}}
=> ? = 3 + 1
[[],[],[],[[],[[],[]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> {{1,2,3,6,7,8},{4},{5}}
=> ? = 2 + 1
[[],[],[],[[],[[[]]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
=> {{1,2,3,7,8},{4},{5,6}}
=> ? = 2 + 1
[[],[],[],[[[]],[],[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5},{6,7}}
=> ? = 2 + 1
[[],[],[],[[[]],[[]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> {{1,2,3,8},{4,6},{5},{7}}
=> ? = 3 + 1
[[],[],[],[[[],[]],[]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
=> {{1,2,3,6,8},{4,5},{7}}
=> ? = 2 + 1
[[],[],[],[[[[]]],[]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,7},{5,6}}
=> ? = 2 + 1
[[],[],[],[[[],[[]]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> {{1,2,3,7,8},{4,6},{5}}
=> ? = 2 + 1
[[],[],[],[[[[]],[]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5,6},{7}}
=> ? = 2 + 1
[[],[],[],[[[[],[]]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 1 + 1
[[],[],[],[[[[[]]]]]]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5,6,7}}
=> ? = 1 + 1
[[],[],[[]],[],[],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> {{1,2,8},{3},{4,5,6,7}}
=> ? = 2 + 1
[[],[],[[]],[],[],[[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0]
=> {{1,2,8},{3},{4,5,7},{6}}
=> ? = 3 + 1
[[],[],[[]],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
=> {{1,2,8},{3},{4,7},{5},{6}}
=> ? = 4 + 1
[[],[],[[]],[],[[],[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,1,0,0,0]
=> {{1,2,8},{3},{4,6,7},{5}}
=> ? = 3 + 1
[[],[],[[]],[],[[[]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,1,0,0,0]
=> {{1,2,8},{3},{4,7},{5,6}}
=> ? = 3 + 1
[[],[],[[]],[[]],[],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,1,0,0,0]
=> {{1,2,8},{3},{4},{5,6},{7}}
=> ? = 4 + 1
[[],[],[[]],[[]],[[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> {{1,2,8},{3},{4},{5},{6},{7}}
=> ? = 5 + 1
[[],[],[[]],[[],[]],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,1,0,0,0]
=> {{1,2,8},{3},{4},{5,7},{6}}
=> ? = 4 + 1
[[],[],[[]],[[[]]],[]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,1,0,0,0]
=> {{1,2,8},{3},{4,6},{5},{7}}
=> ? = 4 + 1
[[],[],[[]],[[],[],[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,1,0,0,0]
=> {{1,2,8},{3},{4},{5,6,7}}
=> ? = 3 + 1
[[],[],[[]],[[],[[]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,1,0,0,0]
=> {{1,2,8},{3},{4},{5},{6,7}}
=> ? = 4 + 1
[[],[],[[]],[[[]],[]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,1,0,0,0]
=> {{1,2,8},{3},{4,5},{6},{7}}
=> ? = 4 + 1
[[],[],[[]],[[[],[]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> {{1,2,8},{3},{4,5},{6,7}}
=> ? = 3 + 1
[[],[],[[]],[[[[]]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,1,0,0,0]
=> {{1,2,8},{3},{4,5,6},{7}}
=> ? = 3 + 1
[[],[],[[],[]],[],[],[]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> {{1,2,4,8},{3},{5,6,7}}
=> ? = 2 + 1
Description
The number of topologically connected components of a set partition.
For example, the set partition $\{\{1,5\},\{2,3\},\{4,6\}\}$ has the two connected components $\{1,4,5,6\}$ and $\{2,3\}$.
The number of set partitions with only one block is [[oeis:A099947]].
Matching statistic: St000053
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 88%
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 88%
Values
[[]]
=> [1,0]
=> [1,0]
=> 0
[[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[[],[],[],[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[],[],[],[],[],[],[[]]]
=> [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
[[],[],[],[],[],[[]],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 2
[[],[],[],[],[],[[],[]]]
=> [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
[[],[],[],[],[],[[[]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 1
[[],[],[],[],[[]],[],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> ? = 2
[[],[],[],[],[[]],[[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 3
[[],[],[],[],[[],[]],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> ? = 2
[[],[],[],[],[[[]]],[]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> ? = 2
[[],[],[],[],[[],[],[]]]
=> [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
[[],[],[],[],[[],[[]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> ? = 2
[[],[],[],[],[[[]],[]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> ? = 2
[[],[],[],[],[[[],[]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? = 1
[[],[],[],[],[[[[]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? = 1
[[],[],[],[[]],[],[],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 2
[[],[],[],[[]],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 3
[[],[],[],[[]],[[]],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 4
[[],[],[],[[]],[[],[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,1,0,0,0,0]
=> ? = 3
[[],[],[],[[]],[[[]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 3
[[],[],[],[[],[]],[],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 2
[[],[],[],[[[]]],[],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> ? = 2
[[],[],[],[[],[]],[[]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 3
[[],[],[],[[[]]],[[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> ? = 3
[[],[],[],[[],[],[]],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 2
[[],[],[],[[],[[]]],[]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 3
[[],[],[],[[[]],[]],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> ? = 3
[[],[],[],[[[],[]]],[]]
=> [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]
=> ? = 2
[[],[],[],[[[[]]]],[]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> ? = 2
[[],[],[],[[],[],[],[]]]
=> [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
[[],[],[],[[],[],[[]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 2
[[],[],[],[[],[[]],[]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 3
[[],[],[],[[],[[],[]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 2
[[],[],[],[[],[[[]]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
=> ? = 2
[[],[],[],[[[]],[],[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
=> ? = 2
[[],[],[],[[[]],[[]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> ? = 3
[[],[],[],[[[],[]],[]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
=> ? = 2
[[],[],[],[[[[]]],[]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> ? = 2
[[],[],[],[[[],[],[]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[],[],[],[[[],[[]]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> ? = 2
[[],[],[],[[[[]],[]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> ? = 2
[[],[],[],[[[[],[]]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> ? = 1
[[],[],[],[[[[[]]]]]]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 1
[[],[],[[]],[],[],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 2
[[],[],[[]],[],[],[[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 3
[[],[],[[]],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 4
[[],[],[[]],[],[[],[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 3
[[],[],[[]],[],[[[]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,1,0,0,0]
=> ? = 3
[[],[],[[]],[[]],[],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,1,0,0,0]
=> ? = 4
[[],[],[[]],[[]],[[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 5
[[],[],[[]],[[],[]],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,1,0,0,0]
=> ? = 4
Description
The number of valleys of the Dyck path.
Matching statistic: St001068
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 88%
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 88%
Values
[[]]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
[[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 2 + 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 2 + 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 1 + 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3 = 2 + 1
[[],[],[],[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 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
[[],[],[],[],[],[[]],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 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
[[],[],[],[],[],[[[]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 1 + 1
[[],[],[],[],[[]],[],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> ? = 2 + 1
[[],[],[],[],[[]],[[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 3 + 1
[[],[],[],[],[[],[]],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> ? = 2 + 1
[[],[],[],[],[[[]]],[]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> ? = 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
[[],[],[],[],[[],[[]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[[],[],[],[],[[[]],[]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[[],[],[],[],[[[],[]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[[],[],[],[],[[[[]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? = 1 + 1
[[],[],[],[[]],[],[],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 2 + 1
[[],[],[],[[]],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 3 + 1
[[],[],[],[[]],[[]],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 4 + 1
[[],[],[],[[]],[[],[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 1
[[],[],[],[[]],[[[]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 3 + 1
[[],[],[],[[],[]],[],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 2 + 1
[[],[],[],[[[]]],[],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> ? = 2 + 1
[[],[],[],[[],[]],[[]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 3 + 1
[[],[],[],[[[]]],[[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> ? = 3 + 1
[[],[],[],[[],[],[]],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 2 + 1
[[],[],[],[[],[[]]],[]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 3 + 1
[[],[],[],[[[]],[]],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 + 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]
=> ? = 2 + 1
[[],[],[],[[[[]]]],[]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> ? = 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
[[],[],[],[[],[],[[]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[[],[],[],[[],[[]],[]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 3 + 1
[[],[],[],[[],[[],[]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[[],[],[],[[],[[[]]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[[],[],[],[[[]],[],[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
=> ? = 2 + 1
[[],[],[],[[[]],[[]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> ? = 3 + 1
[[],[],[],[[[],[]],[]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[[],[],[],[[[[]]],[]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> ? = 2 + 1
[[],[],[],[[[],[],[]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[[],[],[],[[[],[[]]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[[],[],[],[[[[]],[]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> ? = 2 + 1
[[],[],[],[[[[],[]]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> ? = 1 + 1
[[],[],[],[[[[[]]]]]]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 1 + 1
[[],[],[[]],[],[],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 2 + 1
[[],[],[[]],[],[],[[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 3 + 1
[[],[],[[]],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 4 + 1
[[],[],[[]],[],[[],[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 3 + 1
[[],[],[[]],[],[[[]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,1,0,0,0]
=> ? = 3 + 1
[[],[],[[]],[[]],[],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,1,0,0,0]
=> ? = 4 + 1
[[],[],[[]],[[]],[[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 5 + 1
[[],[],[[]],[[],[]],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,1,0,0,0]
=> ? = 4 + 1
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
Matching statistic: St000024
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 88%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 88%
Values
[[]]
=> [1,0]
=> [1,0]
=> [1,0]
=> 0
[[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[[],[],[],[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[[],[],[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[],[],[],[],[],[[]],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[[],[],[],[],[],[[],[]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[],[],[],[],[],[[[]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[],[],[],[],[[]],[],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 2
[[],[],[],[],[[]],[[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[[],[],[],[],[[],[]],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 2
[[],[],[],[],[[[]]],[]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 2
[[],[],[],[],[[],[],[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[],[],[],[],[[],[[]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2
[[],[],[],[],[[[]],[]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2
[[],[],[],[],[[[],[]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[],[],[],[],[[[[]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[],[],[],[[]],[],[],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 2
[[],[],[],[[]],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 3
[[],[],[],[[]],[[]],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 4
[[],[],[],[[]],[[],[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 3
[[],[],[],[[]],[[[]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 3
[[],[],[],[[],[]],[],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 2
[[],[],[],[[[]]],[],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 2
[[],[],[],[[],[]],[[]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 3
[[],[],[],[[[]]],[[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 3
[[],[],[],[[],[],[]],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2
[[],[],[],[[],[[]]],[]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 3
[[],[],[],[[[]],[]],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 3
[[],[],[],[[[],[]]],[]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2
[[],[],[],[[[[]]]],[]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2
[[],[],[],[[],[],[],[]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[[],[],[],[[],[],[[]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[[],[],[],[[],[[]],[]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3
[[],[],[],[[],[[],[]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[[],[],[],[[],[[[]]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[[],[],[],[[[]],[],[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 2
[[],[],[],[[[]],[[]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 3
[[],[],[],[[[],[]],[]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 2
[[],[],[],[[[[]]],[]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 2
[[],[],[],[[[],[],[]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1
[[],[],[],[[[],[[]]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 2
[[],[],[],[[[[]],[]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 2
[[],[],[],[[[[],[]]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1
[[],[],[],[[[[[]]]]]]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1
[[],[],[[]],[],[],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 2
[[],[],[[]],[],[],[[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 3
[[],[],[[]],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> ? = 4
[[],[],[[]],[],[[],[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 3
[[],[],[[]],[],[[[]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 3
[[],[],[[]],[[]],[],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> ? = 4
[[],[],[[]],[[]],[[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 5
[[],[],[[]],[[],[]],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 4
Description
The number of double up and double down steps of a Dyck path.
In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St001036
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 88%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 88%
Values
[[]]
=> [1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[[],[]]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[[[]]]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 2
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 2
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 3
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 2
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> 2
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 2
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 3
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 4
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 3
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 3
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 2
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 3
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 3
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> 3
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> 3
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 2
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2
[[],[],[],[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[[],[],[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[[],[],[],[],[],[[]],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[[],[],[],[],[],[[],[]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[[],[],[],[],[],[[[]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
[[],[],[],[],[[]],[],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
[[],[],[],[],[[]],[[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3
[[],[],[],[],[[],[]],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 2
[[],[],[],[],[[[]]],[]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
[[],[],[],[],[[],[],[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[[],[],[],[],[[],[[]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 2
[[],[],[],[],[[[]],[]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 2
[[],[],[],[],[[[],[]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 1
[[],[],[],[],[[[[]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[[],[],[],[[]],[],[],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[[],[],[],[[]],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3
[[],[],[],[[]],[[]],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ?
=> ? = 4
[[],[],[],[[]],[[],[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> ?
=> ? = 3
[[],[],[],[[]],[[[]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3
[[],[],[],[[],[]],[],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 2
[[],[],[],[[[]]],[],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2
[[],[],[],[[],[]],[[]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> ?
=> ? = 3
[[],[],[],[[[]]],[[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> ?
=> ? = 3
[[],[],[],[[],[],[]],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> ?
=> ? = 2
[[],[],[],[[],[[]]],[]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> ?
=> ? = 3
[[],[],[],[[[]],[]],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> ?
=> ? = 3
[[],[],[],[[[],[]]],[]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 2
[[],[],[],[[[[]]]],[]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
[[],[],[],[[],[],[],[]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
[[],[],[],[[],[],[[]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> ? = 2
[[],[],[],[[],[[]],[]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> ?
=> ? = 3
[[],[],[],[[],[[],[]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ?
=> ? = 2
[[],[],[],[[],[[[]]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 2
[[],[],[],[[[]],[],[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 2
[[],[],[],[[[]],[[]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> ?
=> ? = 3
[[],[],[],[[[],[]],[]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 2
[[],[],[],[[[[]]],[]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 2
[[],[],[],[[[],[],[]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 1
[[],[],[],[[[],[[]]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ?
=> ? = 2
[[],[],[],[[[[]],[]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ?
=> ? = 2
[[],[],[],[[[[],[]]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1
[[],[],[],[[[[[]]]]]]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[[],[],[[]],[],[],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2
[[],[],[[]],[],[],[[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 3
[[],[],[[]],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 4
[[],[],[[]],[],[[],[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 3
[[],[],[[]],[],[[[]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> ?
=> ? = 3
[[],[],[[]],[[]],[],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 4
[[],[],[[]],[[]],[[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ?
=> ? = 5
[[],[],[[]],[[],[]],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 4
Description
The number of inner corners of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000105
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000105: Set partitions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 88%
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000105: Set partitions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 88%
Values
[[]]
=> [1,0]
=> [1,0]
=> {{1}}
=> 1 = 0 + 1
[[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> {{1,2}}
=> 1 = 0 + 1
[[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> {{1},{2}}
=> 2 = 1 + 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 1 = 0 + 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 2 = 1 + 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3 = 2 + 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 2 = 1 + 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2 = 1 + 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1 = 0 + 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 2 = 1 + 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 3 = 2 + 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 2 = 1 + 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2 = 1 + 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 3 = 2 + 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 4 = 3 + 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 3 = 2 + 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 3 = 2 + 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2 = 1 + 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 3 = 2 + 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3 = 2 + 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2 = 1 + 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2 = 1 + 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1 = 0 + 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 2 = 1 + 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> {{1,2,5},{3},{4}}
=> 3 = 2 + 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> 2 = 1 + 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> 2 = 1 + 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> 3 = 2 + 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> 4 = 3 + 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> 3 = 2 + 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> {{1,5},{2,4},{3}}
=> 3 = 2 + 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 2 = 1 + 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> 3 = 2 + 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> {{1,5},{2,3},{4}}
=> 3 = 2 + 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> {{1,4,5},{2,3}}
=> 2 = 1 + 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> {{1,5},{2,3,4}}
=> 2 = 1 + 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 3 = 2 + 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 4 = 3 + 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 5 = 4 + 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 4 = 3 + 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 4 = 3 + 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 3 = 2 + 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> 3 = 2 + 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 4 = 3 + 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> 4 = 3 + 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 3 = 2 + 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 4 = 3 + 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 4 = 3 + 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 3 = 2 + 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 3 = 2 + 1
[[],[],[],[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8}}
=> ? = 0 + 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]
=> {{1,2,3,4,5,6,8},{7}}
=> ? = 1 + 1
[[],[],[],[],[],[[]],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> {{1,2,3,4,5,8},{6},{7}}
=> ? = 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]
=> {{1,2,3,4,5,7,8},{6}}
=> ? = 1 + 1
[[],[],[],[],[],[[[]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> {{1,2,3,4,5,8},{6,7}}
=> ? = 1 + 1
[[],[],[],[],[[]],[],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5},{6,7}}
=> ? = 2 + 1
[[],[],[],[],[[]],[[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5},{6},{7}}
=> ? = 3 + 1
[[],[],[],[],[[],[]],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> {{1,2,3,4,6,8},{5},{7}}
=> ? = 2 + 1
[[],[],[],[],[[[]]],[]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,7},{6}}
=> ? = 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]
=> {{1,2,3,4,6,7,8},{5}}
=> ? = 1 + 1
[[],[],[],[],[[],[[]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> {{1,2,3,4,7,8},{5},{6}}
=> ? = 2 + 1
[[],[],[],[],[[[]],[]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6},{7}}
=> ? = 2 + 1
[[],[],[],[],[[[],[]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> {{1,2,3,4,7,8},{5,6}}
=> ? = 1 + 1
[[],[],[],[],[[[[]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6,7}}
=> ? = 1 + 1
[[],[],[],[[]],[],[],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4},{5,6,7}}
=> ? = 2 + 1
[[],[],[],[[]],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4},{5,7},{6}}
=> ? = 3 + 1
[[],[],[],[[]],[[]],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> {{1,2,3,8},{4},{5},{6},{7}}
=> ? = 4 + 1
[[],[],[],[[]],[[],[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4},{5},{6,7}}
=> ? = 3 + 1
[[],[],[],[[]],[[[]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,1,0,0,0,0]
=> {{1,2,3,8},{4},{5,6},{7}}
=> ? = 3 + 1
[[],[],[],[[],[]],[],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> {{1,2,3,5,8},{4},{6,7}}
=> ? = 2 + 1
[[],[],[],[[[]]],[],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,6,7},{5}}
=> ? = 2 + 1
[[],[],[],[[],[]],[[]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,5,8},{4},{6},{7}}
=> ? = 3 + 1
[[],[],[],[[[]]],[[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,7},{5},{6}}
=> ? = 3 + 1
[[],[],[],[[],[],[]],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> {{1,2,3,5,6,8},{4},{7}}
=> ? = 2 + 1
[[],[],[],[[],[[]]],[]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> {{1,2,3,7,8},{4},{5},{6}}
=> ? = 3 + 1
[[],[],[],[[[]],[]],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5},{6},{7}}
=> ? = 3 + 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]
=> {{1,2,3,7,8},{4,5},{6}}
=> ? = 2 + 1
[[],[],[],[[[[]]]],[]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5,7},{6}}
=> ? = 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]
=> {{1,2,3,5,6,7,8},{4}}
=> ? = 1 + 1
[[],[],[],[[],[],[[]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> {{1,2,3,5,7,8},{4},{6}}
=> ? = 2 + 1
[[],[],[],[[],[[]],[]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> {{1,2,3,6,8},{4},{5},{7}}
=> ? = 3 + 1
[[],[],[],[[],[[],[]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> {{1,2,3,6,7,8},{4},{5}}
=> ? = 2 + 1
[[],[],[],[[],[[[]]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
=> {{1,2,3,7,8},{4},{5,6}}
=> ? = 2 + 1
[[],[],[],[[[]],[],[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5},{6,7}}
=> ? = 2 + 1
[[],[],[],[[[]],[[]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> {{1,2,3,8},{4,6},{5},{7}}
=> ? = 3 + 1
[[],[],[],[[[],[]],[]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
=> {{1,2,3,6,8},{4,5},{7}}
=> ? = 2 + 1
[[],[],[],[[[[]]],[]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,7},{5,6}}
=> ? = 2 + 1
[[],[],[],[[[],[],[]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> {{1,2,3,6,7,8},{4,5}}
=> ? = 1 + 1
[[],[],[],[[[],[[]]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> {{1,2,3,7,8},{4,6},{5}}
=> ? = 2 + 1
[[],[],[],[[[[]],[]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5,6},{7}}
=> ? = 2 + 1
[[],[],[],[[[[],[]]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 1 + 1
[[],[],[],[[[[[]]]]]]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5,6,7}}
=> ? = 1 + 1
[[],[],[[]],[],[],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> {{1,2,8},{3},{4,5,6,7}}
=> ? = 2 + 1
[[],[],[[]],[],[],[[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0]
=> {{1,2,8},{3},{4,5,7},{6}}
=> ? = 3 + 1
[[],[],[[]],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
=> {{1,2,8},{3},{4,7},{5},{6}}
=> ? = 4 + 1
[[],[],[[]],[],[[],[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,1,0,0,0]
=> {{1,2,8},{3},{4,6,7},{5}}
=> ? = 3 + 1
[[],[],[[]],[],[[[]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,1,0,0,0]
=> {{1,2,8},{3},{4,7},{5,6}}
=> ? = 3 + 1
[[],[],[[]],[[]],[],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,1,0,0,0]
=> {{1,2,8},{3},{4},{5,6},{7}}
=> ? = 4 + 1
[[],[],[[]],[[]],[[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> {{1,2,8},{3},{4},{5},{6},{7}}
=> ? = 5 + 1
[[],[],[[]],[[],[]],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,1,0,0,0]
=> {{1,2,8},{3},{4},{5,7},{6}}
=> ? = 4 + 1
Description
The number of blocks in the set partition.
The generating function of this statistic yields the famous [[wiki:Stirling numbers of the second kind|Stirling numbers of the second kind]] $S_2(n,k)$ given by the number of [[SetPartitions|set partitions]] of $\{ 1,\ldots,n\}$ into $k$ blocks, see [1].
Matching statistic: St001007
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 88%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 88%
Values
[[]]
=> [1,0]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
[[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2 = 1 + 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3 = 2 + 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4 = 3 + 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3 = 2 + 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4 = 3 + 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 3 + 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4 = 3 + 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[[],[],[],[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 1
[[],[],[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[],[],[],[],[],[[]],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 1
[[],[],[],[],[],[[],[]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[],[],[],[],[],[[[]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[],[],[],[],[[]],[],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 1
[[],[],[],[],[[]],[[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 3 + 1
[[],[],[],[],[[],[]],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 1
[[],[],[],[],[[[]]],[]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 1
[[],[],[],[],[[],[],[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[],[],[],[],[[],[[]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 1
[[],[],[],[],[[[]],[]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 1
[[],[],[],[],[[[],[]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[],[],[],[],[[[[]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[],[],[],[[]],[],[],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[[],[],[],[[]],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[[],[],[],[[]],[[]],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 4 + 1
[[],[],[],[[]],[[],[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[[],[],[],[[]],[[[]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[[],[],[],[[],[]],[],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[[],[],[],[[[]]],[],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[[],[],[],[[],[]],[[]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[[],[],[],[[[]]],[[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[[],[],[],[[],[],[]],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[[],[],[],[[],[[]]],[]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[[],[],[],[[[]],[]],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[[],[],[],[[[],[]]],[]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[[],[],[],[[[[]]]],[]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[[],[],[],[[],[],[],[]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[],[],[],[[],[],[[]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[[],[],[],[[],[[]],[]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[[],[],[],[[],[[],[]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[[],[],[],[[],[[[]]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[[],[],[],[[[]],[],[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[[],[],[],[[[]],[[]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[[],[],[],[[[],[]],[]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[[],[],[],[[[[]]],[]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[[],[],[],[[[],[],[]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[],[],[],[[[],[[]]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[[],[],[],[[[[]],[]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[[],[],[],[[[[],[]]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[],[],[],[[[[[]]]]]]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[],[],[[]],[],[],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 2 + 1
[[],[],[[]],[],[],[[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 3 + 1
[[],[],[[]],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> ? = 4 + 1
[[],[],[[]],[],[[],[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 3 + 1
[[],[],[[]],[],[[[]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 3 + 1
[[],[],[[]],[[]],[],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> ? = 4 + 1
[[],[],[[]],[[]],[[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 5 + 1
[[],[],[[]],[[],[]],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 4 + 1
Description
Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000159
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 62%
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 62%
Values
[[]]
=> [1,0]
=> [1,0]
=> []
=> 0
[[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> 0
[[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1]
=> 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> 0
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> 0
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 2
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 2
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 2
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 3
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 2
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 2
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 2
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 3
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 4
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 3
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 3
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 2
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 3
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 3
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 3
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 3
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 2
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 2
[[[]],[[]],[],[]]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> ? = 4
[[[]],[[]],[[]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> ? = 5
[[[]],[[],[]],[]]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> ? = 4
[[[]],[[[]]],[]]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> ? = 4
[[[]],[[],[[]]]]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> ? = 4
[[[]],[[[]],[]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> ? = 4
[[[]],[[[],[]]]]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> ? = 3
[[[],[[]]],[],[]]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> ? = 3
[[[[]],[]],[],[]]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> ? = 3
[[[],[[]]],[[]]]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> ? = 4
[[[[]],[]],[[]]]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> ? = 4
[[[[],[]]],[[]]]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> ? = 3
[[[],[[]],[[]]]]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> ? = 4
[[[],[[[]]],[]]]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> ? = 3
[[[[]],[],[[]]]]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> ? = 3
[[[[]],[[]],[]]]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> ? = 4
[[],[[]],[[]],[],[]]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [5,4,2,2,1]
=> ? = 4
[[],[[]],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1]
=> ? = 5
[[],[[]],[[],[]],[]]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [5,3,2,2,1]
=> ? = 4
[[],[[]],[[[]]],[]]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [5,4,2,1,1]
=> ? = 4
[[],[[]],[[],[[]]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [5,3,3,2,1]
=> ? = 4
[[],[[]],[[[]],[]]]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [5,4,3,1,1]
=> ? = 4
[[],[[]],[[[],[]]]]
=> [1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [5,3,3,1,1]
=> ? = 3
[[],[[],[[]]],[],[]]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [4,4,2,2,1]
=> ? = 3
[[],[[[]],[]],[],[]]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [5,4,2,2]
=> ? = 3
[[],[[],[[]]],[[]]]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,1]
=> ? = 4
[[],[[[]],[]],[[]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2]
=> ? = 4
[[],[[[],[]]],[[]]]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2]
=> ? = 3
[[],[[],[[]],[[]]]]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [4,3,3,2,1]
=> ? = 4
[[],[[],[[[]]],[]]]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [4,4,3,1,1]
=> ? = 3
[[],[[[]],[],[[]]]]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [5,3,3,2]
=> ? = 3
[[],[[[]],[[]],[]]]
=> [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [5,4,3,1]
=> ? = 4
[[[]],[],[],[[]],[]]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,1,1]
=> ? = 4
[[[]],[],[],[[],[]]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,2,1,1,1]
=> ? = 3
[[[]],[],[],[[[]]]]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,1,1,1]
=> ? = 3
[[[]],[],[[]],[],[]]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [6,4,2,2,1,1]
=> ? = 4
[[[]],[],[[]],[[]]]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,1]
=> ? = 5
[[[]],[],[[],[]],[]]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [6,3,2,2,1,1]
=> ? = 4
[[[]],[],[[[]]],[]]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [6,4,2,1,1,1]
=> ? = 4
[[[]],[],[[],[],[]]]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,1,1]
=> ? = 3
[[[]],[],[[],[[]]]]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [6,3,3,2,1,1]
=> ? = 4
[[[]],[],[[[]],[]]]
=> [1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [6,4,3,1,1,1]
=> ? = 4
[[[]],[],[[[],[]]]]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [6,3,3,1,1,1]
=> ? = 3
[[[]],[],[[[[]]]]]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,1,1,1]
=> ? = 3
[[[]],[[]],[],[],[]]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,2,1]
=> ? = 4
[[[]],[[]],[],[[]]]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2,1]
=> ? = 5
[[[]],[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ? = 6
[[[]],[[]],[[],[]]]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> ? = 5
[[[]],[[]],[[[]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> ? = 5
[[[]],[[],[]],[],[]]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [6,4,2,2,2,1]
=> ? = 4
Description
The number of distinct parts of the integer partition.
This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
Matching statistic: St000318
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 75%
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 75%
Values
[[]]
=> [1,0]
=> [1,0]
=> []
=> 1 = 0 + 1
[[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> 1 = 0 + 1
[[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1]
=> 2 = 1 + 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> 1 = 0 + 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> 2 = 1 + 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 3 = 2 + 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2 = 1 + 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> 2 = 1 + 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> 1 = 0 + 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 2 = 1 + 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 3 = 2 + 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2 = 1 + 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 2 = 1 + 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 3 = 2 + 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 4 = 3 + 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 3 = 2 + 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 3 = 2 + 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 2 = 1 + 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 3 = 2 + 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 3 = 2 + 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2 = 1 + 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 2 = 1 + 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> 1 = 0 + 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 2 = 1 + 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 3 = 2 + 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 2 = 1 + 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 2 = 1 + 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 3 = 2 + 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 4 = 3 + 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 3 = 2 + 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 3 = 2 + 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 2 = 1 + 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 3 = 2 + 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 3 = 2 + 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 2 = 1 + 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 2 = 1 + 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 3 = 2 + 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 4 = 3 + 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 5 = 4 + 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 4 = 3 + 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 4 = 3 + 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 3 = 2 + 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 3 = 2 + 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 4 = 3 + 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 4 = 3 + 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 3 = 2 + 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 4 = 3 + 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 4 = 3 + 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 3 = 2 + 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 3 = 2 + 1
[[[]],[],[],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,1,1,1,1]
=> ? = 2 + 1
[[[]],[],[],[],[[]]]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,1,1,1]
=> ? = 3 + 1
[[[]],[],[],[[]],[]]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,1,1]
=> ? = 4 + 1
[[[]],[],[],[[],[]]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,2,1,1,1]
=> ? = 3 + 1
[[[]],[],[],[[[]]]]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,1,1,1]
=> ? = 3 + 1
[[[]],[],[[]],[],[]]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [6,4,2,2,1,1]
=> ? = 4 + 1
[[[]],[],[[]],[[]]]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,1]
=> ? = 5 + 1
[[[]],[],[[],[]],[]]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [6,3,2,2,1,1]
=> ? = 4 + 1
[[[]],[],[[[]]],[]]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [6,4,2,1,1,1]
=> ? = 4 + 1
[[[]],[],[[],[],[]]]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,1,1]
=> ? = 3 + 1
[[[]],[],[[],[[]]]]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [6,3,3,2,1,1]
=> ? = 4 + 1
[[[]],[],[[[]],[]]]
=> [1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [6,4,3,1,1,1]
=> ? = 4 + 1
[[[]],[],[[[],[]]]]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [6,3,3,1,1,1]
=> ? = 3 + 1
[[[]],[],[[[[]]]]]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,1,1,1]
=> ? = 3 + 1
[[[]],[[]],[],[],[]]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,2,1]
=> ? = 4 + 1
[[[]],[[]],[],[[]]]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2,1]
=> ? = 5 + 1
[[[]],[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ? = 6 + 1
[[[]],[[]],[[],[]]]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> ? = 5 + 1
[[[]],[[]],[[[]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> ? = 5 + 1
[[[]],[[],[]],[],[]]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [6,4,2,2,2,1]
=> ? = 4 + 1
[[[]],[[[]]],[],[]]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,1,1]
=> ? = 4 + 1
[[[]],[[],[]],[[]]]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> ? = 5 + 1
[[[]],[[[]]],[[]]]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,1]
=> ? = 5 + 1
[[[]],[[],[],[]],[]]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [6,3,2,2,2,1]
=> ? = 4 + 1
[[[]],[[],[[]]],[]]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2,1]
=> ? = 5 + 1
[[[]],[[[]],[]],[]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,1]
=> ? = 5 + 1
[[[]],[[[],[]]],[]]
=> [1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1,1]
=> ? = 4 + 1
[[[]],[[[[]]]],[]]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1,1,1]
=> ? = 4 + 1
[[[]],[[],[],[],[]]]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,2,1]
=> ? = 3 + 1
[[[]],[[],[],[[]]]]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,3,2,2,1]
=> ? = 4 + 1
[[[]],[[],[[]],[]]]
=> [1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ? = 5 + 1
[[[]],[[],[[],[]]]]
=> [1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2,1]
=> ? = 4 + 1
[[[]],[[],[[[]]]]]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> ? = 4 + 1
[[[]],[[[]],[],[]]]
=> [1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1,1]
=> ? = 4 + 1
[[[]],[[[]],[[]]]]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,1]
=> ? = 5 + 1
[[[]],[[[],[]],[]]]
=> [1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1,1]
=> ? = 4 + 1
[[[]],[[[[]]],[]]]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [6,5,3,1,1,1]
=> ? = 4 + 1
[[[]],[[[],[],[]]]]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,1,1]
=> ? = 3 + 1
[[[]],[[[],[[]]]]]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1,1]
=> ? = 4 + 1
[[[]],[[[[]],[]]]]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,1,1]
=> ? = 4 + 1
[[[]],[[[[],[]]]]]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [6,4,4,1,1,1]
=> ? = 3 + 1
[[[]],[[[[[]]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,1,1,1]
=> ? = 3 + 1
[[[],[]],[],[],[[]]]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,2,1,1,1,1]
=> ? = 3 + 1
[[[[]]],[],[],[[]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,1,1]
=> ? = 3 + 1
[[[],[]],[],[[]],[]]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1,1,1]
=> ? = 4 + 1
[[[[]]],[],[[]],[]]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,1]
=> ? = 4 + 1
[[[],[]],[],[[],[]]]
=> [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,2,1,1,1]
=> ? = 3 + 1
[[[],[]],[],[[[]]]]
=> [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [5,3,1,1,1,1]
=> ? = 3 + 1
[[[[]]],[],[[],[]]]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,2,1,1]
=> ? = 3 + 1
[[[[]]],[],[[[]]]]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,1,1]
=> ? = 3 + 1
Description
The number of addable cells of the Ferrers diagram of an integer partition.
The following 38 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000996The number of exclusive left-to-right maxima of a permutation. St000167The number of leaves of an ordered tree. St000245The number of ascents of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000672The number of minimal elements in Bruhat order not less than the permutation. St000292The number of ascents of a binary word. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St001489The maximum of the number of descents and the number of inverse descents. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000021The number of descents of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St000083The number of left oriented leafs of a binary tree except the first one. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000702The number of weak deficiencies of a permutation. St000991The number of right-to-left minima of a permutation. St001712The number of natural descents of a standard Young tableau.
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