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Your data matches 28 different statistics following compositions of up to 3 maps.
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Matching statistic: St000839
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(load all 51 compositions to match this statistic)
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000839: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000839: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> {{1}}
=> 1
[1,0,1,0]
=> {{1},{2}}
=> 2
[1,1,0,0]
=> {{1,2}}
=> 1
[1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> 2
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> 3
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> 2
[1,1,1,0,0,0]
=> {{1,2,3}}
=> 1
[1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 4
[1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 3
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 4
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 3
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
[1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 4
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 3
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 4
[1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 3
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 2
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 4
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
[1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 3
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 2
Description
The largest opener of a set partition.
An opener (or left hand endpoint) of a set partition is a number that is minimal in its block. For this statistic, singletons are considered as openers.
Matching statistic: St000476
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 76%●distinct values known / distinct values provided: 73%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 76%●distinct values known / distinct values provided: 73%
Values
[1,0]
=> [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 = 2 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 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]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 4 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 6 - 1
[1,1,0,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 7 - 1
[1,1,1,0,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 7 - 1
[1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 7 - 1
[1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 7 - 1
[1,1,1,0,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 7 - 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 8 - 1
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 4 - 1
[1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 7 - 1
[1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 6 - 1
[1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 4 - 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 8 - 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 8 - 1
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 7 - 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 8 - 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 8 - 1
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 8 - 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 6 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8 - 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 6 - 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 7 - 1
[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]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ?
=> ? = 9 - 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ?
=> ?
=> ?
=> ? = 2 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ?
=> ? = 10 - 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ?
=> ? = 9 - 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ?
=> ?
=> ?
=> ? = 8 - 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ?
=> ?
=> ?
=> ? = 2 - 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ?
=> ? = 11 - 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ?
=> ? = 10 - 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ?
=> ? = 9 - 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ?
=> ? = 9 - 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ?
=> ?
=> ?
=> ? = 7 - 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ?
=> ?
=> ?
=> ? = 7 - 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ?
=> ?
=> ?
=> ? = 9 - 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ?
=> ?
=> ?
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ?
=> ?
=> ? = 8 - 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ?
=> ?
=> ? = 8 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ?
=> ?
=> ?
=> ? = 9 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ?
=> ?
=> ?
=> ? = 8 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ?
=> ?
=> ?
=> ? = 8 - 1
Description
The sum of the semi-lengths of tunnels before a valley of a Dyck path.
For each valley $v$ in a Dyck path $D$ there is a corresponding tunnel, which
is the factor $T_v = s_i\dots s_j$ of $D$ where $s_i$ is the step after the first intersection of $D$ with the line $y = ht(v)$ to the left of $s_j$. This statistic is
$$
\sum_v (j_v-i_v)/2.
$$
Matching statistic: St000738
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 73%
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 73%
Values
[1,0]
=> {{1}}
=> [[1]]
=> 1
[1,0,1,0]
=> {{1},{2}}
=> [[1],[2]]
=> 2
[1,1,0,0]
=> {{1,2}}
=> [[1,2]]
=> 1
[1,0,1,0,1,0]
=> {{1},{2},{3}}
=> [[1],[2],[3]]
=> 3
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> [[1,3],[2]]
=> 2
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> [[1,2],[3]]
=> 3
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> [[1,3],[2]]
=> 2
[1,1,1,0,0,0]
=> {{1,2,3}}
=> [[1,2,3]]
=> 1
[1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> [[1],[2],[3],[4]]
=> 4
[1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> [[1,4],[2],[3]]
=> 3
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> [[1,3],[2],[4]]
=> 4
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> [[1,4],[2],[3]]
=> 3
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> [[1,3,4],[2]]
=> 2
[1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> [[1,2],[3],[4]]
=> 4
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> [[1,2],[3,4]]
=> 3
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> [[1,3],[2],[4]]
=> 4
[1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> [[1,4],[2],[3]]
=> 3
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> [[1,3,4],[2]]
=> 2
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> [[1,2,3],[4]]
=> 4
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> [[1,3],[2,4]]
=> 2
[1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> [[1,2,4],[3]]
=> 3
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> [[1,2,3,4]]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> [[1],[2],[3],[4],[5]]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> [[1,5],[2],[3],[4]]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> [[1,4],[2],[3],[5]]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> [[1,5],[2],[3],[4]]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> [[1,4,5],[2],[3]]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> [[1,3],[2],[4],[5]]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> [[1,3],[2,5],[4]]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> [[1,4],[2],[3],[5]]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> [[1,5],[2],[3],[4]]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> [[1,4,5],[2],[3]]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> [[1,3,4],[2],[5]]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> [[1,4],[2,5],[3]]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> [[1,3,5],[2],[4]]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> [[1,3,4,5],[2]]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> [[1,2],[3],[4],[5]]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> [[1,2],[3,5],[4]]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> [[1,2],[3,4],[5]]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> [[1,2],[3,5],[4]]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> [[1,2,5],[3,4]]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> [[1,3],[2],[4],[5]]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> [[1,3],[2,5],[4]]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> [[1,4],[2],[3],[5]]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> [[1,5],[2],[3],[4]]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> [[1,4,5],[2],[3]]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> [[1,3,4],[2],[5]]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> [[1,4],[2,5],[3]]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> [[1,3,5],[2],[4]]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> [[1,3,4,5],[2]]
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ?
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> ?
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3},{4},{5},{6,8},{7}}
=> ?
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4},{5},{6,7,8}}
=> ?
=> ? = 6
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4},{5,8},{6},{7}}
=> ?
=> ? = 7
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> {{1},{2},{3},{4},{5,8},{6,7}}
=> ?
=> ? = 6
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> {{1},{2},{3},{4},{5,6,8},{7}}
=> ?
=> ? = 7
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> {{1},{2},{3},{4},{5,6,7,8}}
=> ?
=> ? = 5
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4,8},{5},{6},{7}}
=> ?
=> ? = 7
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> {{1},{2},{3},{4,8},{5,6},{7}}
=> ?
=> ? = 7
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> {{1},{2},{3},{4,8},{5,7},{6}}
=> ?
=> ? = 6
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> {{1},{2},{3},{4,8},{5,6,7}}
=> ?
=> ? = 5
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2},{3},{4,5,6,7,8}}
=> ?
=> ? = 4
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3,8},{4},{5},{6},{7}}
=> ?
=> ? = 7
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> {{1},{2},{3,8},{4,5},{6},{7}}
=> ?
=> ? = 7
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> {{1},{2},{3,8},{4,6},{5},{7}}
=> ?
=> ? = 7
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> {{1},{2},{3,8},{4,5,6},{7}}
=> ?
=> ? = 7
[1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> {{1},{2},{3,4,8},{5,6,7}}
=> ?
=> ? = 5
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1},{2},{3,4,5,6,7,8}}
=> ?
=> ? = 3
[1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> {{1},{2,3},{4,7,8},{5,6}}
=> ?
=> ? = 5
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2,8},{3},{4},{5},{6},{7}}
=> ?
=> ? = 7
[1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> {{1},{2,4,5,8},{3},{6,7}}
=> ?
=> ? = 6
[1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> {{1},{2,4,5,6,7,8},{3}}
=> ?
=> ? = 3
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2,8},{3,4},{5},{6},{7}}
=> ?
=> ? = 7
[1,0,1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> {{1},{2,5,8},{3,4},{6,7}}
=> ?
=> ? = 6
[1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> {{1},{2,8},{3,5},{4},{6},{7}}
=> ?
=> ? = 7
[1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> {{1},{2,3,5,6,7,8},{4}}
=> ?
=> ? = 4
[1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> {{1},{2,8},{3,4,5},{6},{7}}
=> ?
=> ? = 7
[1,0,1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> {{1},{2,8},{3,4,5},{6,7}}
=> ?
=> ? = 6
[1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> {{1},{2,3,4,7,8},{5},{6}}
=> ?
=> ? = 6
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> {{1},{2,3,4,6,7,8},{5}}
=> ?
=> ? = 5
[1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> {{1},{2,3,8},{4,5,7},{6}}
=> ?
=> ? = 6
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> {{1},{2,3,4,5,8},{6},{7}}
=> ?
=> ? = 7
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,7,8},{6}}
=> ?
=> ? = 6
[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> {{1},{2,3,4,5,6,7},{8}}
=> ?
=> ? = 8
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> {{1},{2,3,4,8},{5,6,7}}
=> ?
=> ? = 5
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> {{1},{2,3,4,5,8},{6,7}}
=> ?
=> ? = 6
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,8},{7}}
=> ?
=> ? = 7
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,8}}
=> ?
=> ? = 2
[1,1,0,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> {{1,2},{3},{4,6,7,8},{5}}
=> ?
=> ? = 5
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1,2},{3,4,5,6,7,8}}
=> ?
=> ? = 3
[1,1,0,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> {{1,3},{2},{4,7,8},{5,6}}
=> ?
=> ? = 5
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> ?
=> ? = 7
[1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> {{1,4,5,6,7,8},{2},{3}}
=> ?
=> ? = 3
[1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> {{1,3,5,6,7,8},{2},{4}}
=> ?
=> ? = 4
[1,1,0,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> {{1,8},{2},{3,4,5,6},{7}}
=> ?
=> ? = 7
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> {{1,3,4,5,6,7,8},{2}}
=> ?
=> ? = 2
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2,3},{4},{5},{6},{7}}
=> ?
=> ? = 7
[1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> {{1,4,5,6,7,8},{2,3}}
=> ?
=> ? = 2
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2,4},{3},{5},{6},{7}}
=> ?
=> ? = 7
Description
The first entry in the last row of a standard tableau.
For the last entry in the first row, see [[St000734]].
Matching statistic: St000734
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 73%
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 73%
Values
[1,0]
=> {{1}}
=> [[1]]
=> [[1]]
=> 1
[1,0,1,0]
=> {{1},{2}}
=> [[1],[2]]
=> [[1,2]]
=> 2
[1,1,0,0]
=> {{1,2}}
=> [[1,2]]
=> [[1],[2]]
=> 1
[1,0,1,0,1,0]
=> {{1},{2},{3}}
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 3
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 2
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> [[1,2],[3]]
=> [[1,3],[2]]
=> 3
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 2
[1,1,1,0,0,0]
=> {{1,2,3}}
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 1
[1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 4
[1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 3
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> [[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> 4
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 3
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2
[1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 4
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 3
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> [[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> 4
[1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 3
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 4
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> [[1,3],[2,4]]
=> [[1,2],[3,4]]
=> 2
[1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> [[1,2,4],[3]]
=> [[1,3],[2],[4]]
=> 3
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> [[1,4],[2],[3],[5]]
=> [[1,2,3,5],[4]]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> [[1,3],[2],[4],[5]]
=> [[1,2,4,5],[3]]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> [[1,4],[2],[3],[5]]
=> [[1,2,3,5],[4]]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> [[1,3,4],[2],[5]]
=> [[1,2,5],[3],[4]]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> [[1,4],[2,5],[3]]
=> [[1,2,3],[4,5]]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> [[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> [[1,2],[3,5],[4]]
=> [[1,3,4],[2,5]]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> [[1,2],[3,5],[4]]
=> [[1,3,4],[2,5]]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> [[1,3],[2],[4],[5]]
=> [[1,2,4,5],[3]]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> [[1,4],[2],[3],[5]]
=> [[1,2,3,5],[4]]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> [[1,3,4],[2],[5]]
=> [[1,2,5],[3],[4]]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> [[1,4],[2,5],[3]]
=> [[1,2,3],[4,5]]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> [[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ?
=> ?
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> ?
=> ?
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3},{4},{5},{6,8},{7}}
=> ?
=> ?
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4},{5},{6,7,8}}
=> ?
=> ?
=> ? = 6
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4},{5,8},{6},{7}}
=> ?
=> ?
=> ? = 7
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> {{1},{2},{3},{4},{5,8},{6,7}}
=> ?
=> ?
=> ? = 6
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> {{1},{2},{3},{4},{5,6,8},{7}}
=> ?
=> ?
=> ? = 7
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> {{1},{2},{3},{4},{5,6,7,8}}
=> ?
=> ?
=> ? = 5
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4,8},{5},{6},{7}}
=> ?
=> ?
=> ? = 7
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> {{1},{2},{3},{4,8},{5,6},{7}}
=> ?
=> ?
=> ? = 7
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> {{1},{2},{3},{4,8},{5,7},{6}}
=> ?
=> ?
=> ? = 6
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> {{1},{2},{3},{4,8},{5,6,7}}
=> ?
=> ?
=> ? = 5
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2},{3},{4,5,6,7,8}}
=> ?
=> ?
=> ? = 4
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3,8},{4},{5},{6},{7}}
=> ?
=> ?
=> ? = 7
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> {{1},{2},{3,8},{4,5},{6},{7}}
=> ?
=> ?
=> ? = 7
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> {{1},{2},{3,8},{4,6},{5},{7}}
=> ?
=> ?
=> ? = 7
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> {{1},{2},{3,8},{4,5,6},{7}}
=> ?
=> ?
=> ? = 7
[1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> {{1},{2},{3,4,8},{5,6,7}}
=> ?
=> ?
=> ? = 5
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1},{2},{3,4,5,6,7,8}}
=> ?
=> ?
=> ? = 3
[1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> {{1},{2,3},{4,7,8},{5,6}}
=> ?
=> ?
=> ? = 5
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2,8},{3},{4},{5},{6},{7}}
=> ?
=> ?
=> ? = 7
[1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> {{1},{2,4,5,8},{3},{6,7}}
=> ?
=> ?
=> ? = 6
[1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> {{1},{2,4,5,6,7,8},{3}}
=> ?
=> ?
=> ? = 3
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2,8},{3,4},{5},{6},{7}}
=> ?
=> ?
=> ? = 7
[1,0,1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> {{1},{2,5,8},{3,4},{6,7}}
=> ?
=> ?
=> ? = 6
[1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> {{1},{2,8},{3,5},{4},{6},{7}}
=> ?
=> ?
=> ? = 7
[1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> {{1},{2,3,5,6,7,8},{4}}
=> ?
=> ?
=> ? = 4
[1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> {{1},{2,8},{3,4,5},{6},{7}}
=> ?
=> ?
=> ? = 7
[1,0,1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> {{1},{2,8},{3,4,5},{6,7}}
=> ?
=> ?
=> ? = 6
[1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> {{1},{2,3,4,7,8},{5},{6}}
=> ?
=> ?
=> ? = 6
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> {{1},{2,3,4,6,7,8},{5}}
=> ?
=> ?
=> ? = 5
[1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> {{1},{2,3,8},{4,5,7},{6}}
=> ?
=> ?
=> ? = 6
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> {{1},{2,3,4,5,8},{6},{7}}
=> ?
=> ?
=> ? = 7
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,7,8},{6}}
=> ?
=> ?
=> ? = 6
[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> {{1},{2,3,4,5,6,7},{8}}
=> ?
=> ?
=> ? = 8
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> {{1},{2,3,4,8},{5,6,7}}
=> ?
=> ?
=> ? = 5
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> {{1},{2,3,4,5,8},{6,7}}
=> ?
=> ?
=> ? = 6
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,8},{7}}
=> ?
=> ?
=> ? = 7
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,8}}
=> ?
=> ?
=> ? = 2
[1,1,0,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> {{1,2},{3},{4,6,7,8},{5}}
=> ?
=> ?
=> ? = 5
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1,2},{3,4,5,6,7,8}}
=> ?
=> ?
=> ? = 3
[1,1,0,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> {{1,3},{2},{4,7,8},{5,6}}
=> ?
=> ?
=> ? = 5
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> ?
=> ?
=> ? = 7
[1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> {{1,4,5,6,7,8},{2},{3}}
=> ?
=> ?
=> ? = 3
[1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> {{1,3,5,6,7,8},{2},{4}}
=> ?
=> ?
=> ? = 4
[1,1,0,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> {{1,8},{2},{3,4,5,6},{7}}
=> ?
=> ?
=> ? = 7
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> {{1,3,4,5,6,7,8},{2}}
=> ?
=> ?
=> ? = 2
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2,3},{4},{5},{6},{7}}
=> ?
=> ?
=> ? = 7
[1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> {{1,4,5,6,7,8},{2,3}}
=> ?
=> ?
=> ? = 2
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2,4},{3},{5},{6},{7}}
=> ?
=> ?
=> ? = 7
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000147
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 73%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 73%
Values
[1,0]
=> [1,0]
=> [1,0]
=> []
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1]
=> 1 = 2 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> 2 = 3 - 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 3 = 4 - 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]
=> [2,2,1]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [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 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ? = 7 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,2,1]
=> ? = 6 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1]
=> ? = 7 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,1]
=> ? = 7 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> ? = 7 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> ? = 6 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1]
=> ? = 7 - 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,1]
=> ? = 7 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2,1]
=> ? = 7 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ? = 7 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2,1]
=> ? = 6 - 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,1,1]
=> ? = 6 - 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,1]
=> ? = 6 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> ? = 7 - 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2]
=> ? = 6 - 1
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2]
=> ? = 7 - 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> ? = 7 - 1
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> ? = 6 - 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,1]
=> ? = 7 - 1
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,1,1]
=> ? = 6 - 1
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1,1]
=> ? = 7 - 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,1,1]
=> ? = 6 - 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> ? = 7 - 1
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2,1]
=> ? = 6 - 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> ? = 7 - 1
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,2,1]
=> ? = 6 - 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2,1]
=> ? = 7 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> ? = 6 - 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> ? = 5 - 1
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2,1]
=> ? = 7 - 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> ? = 6 - 1
[1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,2,1]
=> ? = 7 - 1
[1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> ? = 7 - 1
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2,1]
=> ? = 6 - 1
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,2,1]
=> ? = 5 - 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,2,1]
=> ? = 6 - 1
[1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,1,1]
=> ? = 7 - 1
[1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1,1]
=> ? = 7 - 1
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1,1]
=> ? = 7 - 1
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,1,1]
=> ? = 6 - 1
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,1,1]
=> ? = 5 - 1
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> ? = 6 - 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> ? = 7 - 1
[1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1]
=> ? = 7 - 1
[1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1]
=> ? = 7 - 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,1]
=> ? = 6 - 1
[1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2]
=> ? = 7 - 1
[1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2]
=> ? = 6 - 1
[1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1,1]
=> ? = 7 - 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,1,1]
=> ? = 6 - 1
Description
The largest part of an integer partition.
Matching statistic: St000727
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000727: Permutations ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 64%
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000727: Permutations ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 64%
Values
[1,0]
=> [1] => [1] => [1] => ? = 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 2
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 3
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [3,1,2] => 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,3,1] => 3
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => [1,3,2] => 2
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [3,2,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 4
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 3
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => [4,3,1,2] => 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,3,4,1] => 4
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 3
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => [1,3,4,2] => 4
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => [1,2,4,3] => 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,3,1,2] => [1,4,3,2] => 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [3,4,2,1] => 4
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,1,3,2] => [4,1,3,2] => 2
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4,2,1,3] => [2,4,3,1] => 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,4,2,3,5] => 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => [1,5,2,3,4] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,4,2,5] => 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [3,1,5,2,4] => 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => [1,2,4,3,5] => 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => [1,2,5,3,4] => 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,2,3] => [1,5,4,2,3] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 5
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,2,4,3] => [5,1,4,2,3] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,3,2,4] => [1,5,3,2,4] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,1,2] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,3,4,5,1] => 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,4,1,5,3] => 5
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => [2,3,1,5,4] => 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => [5,2,1,4,3] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [1,3,4,5,2] => 5
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => 4
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [1,2,4,5,3] => 5
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,2,3,5,4] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,4,1,2,3] => [1,2,5,4,3] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,3,1,2,5] => [1,4,5,3,2] => 5
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [5,1,2,4,3] => [5,1,2,4,3] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [5,3,1,2,4] => [1,3,5,4,2] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,3,1,2] => [1,5,4,3,2] => 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4,5] => [3,4,5,2,1] => 5
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 6
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,6,2,3,4,5,7] => ? = 7
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [1,7,2,3,4,5,6] => ? = 6
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => ? = 5
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [5,1,7,2,3,4,6] => ? = 6
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,6,4] => [1,2,3,7,6,4,5] => [1,7,6,2,3,4,5] => ? = 5
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,6,5,2,3,4,7] => ? = 7
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => [1,2,3,7,4,6,5] => [7,1,6,2,3,4,5] => ? = 5
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,6,7,5,4] => [1,2,3,7,5,4,6] => [1,7,5,2,3,4,6] => ? = 6
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => ? = 4
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [4,1,2,7,3,5,6] => ? = 6
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [7,4,1,6,2,3,5] => ? = 5
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => [5,1,2,7,3,4,6] => ? = 6
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,5,7,3] => [1,2,7,3,4,6,5] => [7,1,2,6,3,4,5] => ? = 5
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => [1,2,7,6,5,3,4] => [1,7,6,5,2,3,4] => ? = 4
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => [5,1,7,4,2,3,6] => ? = 6
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,4,6,3,7] => [1,2,6,3,5,4,7] => [1,6,2,5,3,4,7] => ? = 7
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => [1,2,7,3,5,4,6] => [1,7,2,5,3,4,6] => ? = 6
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,4,7,6,3] => [1,2,7,6,3,5,4] => [7,1,6,5,2,3,4] => ? = 4
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,4,7,3] => [1,2,7,3,6,4,5] => [1,7,2,6,3,4,5] => ? = 5
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,6,4,3] => [1,2,7,6,4,3,5] => [1,7,6,4,2,3,5] => ? = 5
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => [1,6,5,4,2,3,7] => ? = 7
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,5,4,7,3] => [1,2,7,3,6,5,4] => [7,6,1,5,2,3,4] => ? = 4
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,5,7,4,3] => [1,2,7,4,3,6,5] => [4,1,7,6,2,3,5] => ? = 5
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,5,4,3] => [1,2,7,5,4,3,6] => [1,7,5,4,2,3,6] => ? = 6
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => [7,6,5,4,1,2,3] => ? = 3
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [1,3,2,4,5,7,6] => [3,1,2,4,7,5,6] => ? = 6
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,6,5] => [1,3,2,4,7,6,5] => [7,3,1,2,6,4,5] => ? = 5
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => [3,1,5,2,7,4,6] => ? = 6
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,3,2,5,7,6,4] => [1,3,2,7,6,4,5] => [1,7,3,2,6,4,5] => ? = 5
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,6,5,4,7] => [1,3,2,6,5,4,7] => [1,6,3,2,5,4,7] => ? = 7
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,2,6,5,7,4] => [1,3,2,7,4,6,5] => [7,1,3,2,6,4,5] => ? = 5
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,3,2,6,7,5,4] => [1,3,2,7,5,4,6] => [1,7,3,2,5,4,6] => ? = 6
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,7,6,5,4] => [1,3,2,7,6,5,4] => [7,6,3,1,5,2,4] => ? = 4
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5,7,6] => [1,4,2,3,5,7,6] => [4,1,2,3,7,5,6] => ? = 6
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,3,4,2,7,6,5] => [1,4,2,3,7,6,5] => [7,4,1,2,6,3,5] => ? = 5
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,3,4,5,2,7,6] => [1,5,2,3,4,7,6] => [5,1,2,3,7,4,6] => ? = 6
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,4,6,5,7,2] => [1,7,2,3,4,6,5] => [7,1,2,3,6,4,5] => ? = 5
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,3,5,4,2,7,6] => [1,5,4,2,3,7,6] => [5,1,2,7,4,3,6] => ? = 6
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,3,5,4,6,2,7] => [1,6,2,3,5,4,7] => [1,6,2,3,5,4,7] => ? = 7
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,5,4,6,7,2] => [1,7,2,3,5,4,6] => [1,7,2,3,5,4,6] => ? = 6
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,5,4,7,6,2] => [1,7,6,2,3,5,4] => [7,1,2,6,5,3,4] => ? = 4
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,3,5,6,4,7,2] => [1,7,2,3,6,4,5] => [1,7,2,3,6,4,5] => ? = 5
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,6,5,4,7,2] => [1,7,2,3,6,5,4] => [7,6,1,2,5,3,4] => ? = 4
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,6,5,7,4,2] => [1,7,4,2,3,6,5] => [4,1,2,7,6,3,5] => ? = 5
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,7,6,5,4,2] => [1,7,6,5,4,2,3] => [1,7,6,5,4,2,3] => ? = 3
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,4,3,2,5,7,6] => [1,4,3,2,5,7,6] => [4,1,3,7,5,2,6] => ? = 6
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,4,3,2,7,6,5] => [1,4,3,2,7,6,5] => [4,3,1,7,6,2,5] => ? = 5
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,4,3,5,2,7,6] => [1,5,2,4,3,7,6] => [5,1,7,2,4,3,6] => ? = 6
Description
The largest label of a leaf in the binary search tree associated with the permutation.
Alternatively, this is 1 plus the position of the last descent of the inverse of the reversal of the permutation, and 1 if there is no descent.
Matching statistic: St000019
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 73%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 73%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> [2,1] => 1 = 2 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2 = 3 - 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [5,7,1,2,3,4,6] => ? = 7 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,3,4,5,6] => ? = 5 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,7,1,2,3,5,6] => ? = 7 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [5,1,7,2,3,4,6] => ? = 7 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,2,7,4,5,6] => ? = 6 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [4,5,7,1,2,3,6] => ? = 7 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,7,4,5,6] => ? = 5 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [2,3,1,7,4,5,6] => ? = 6 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,4,5,6] => ? = 4 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,7,1,2,4,5,6] => ? = 7 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,5,7,1,2,4,6] => ? = 7 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,2,7,3,5,6] => ? = 6 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,3,4,6] => ? = 5 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [4,1,7,2,3,5,6] => ? = 7 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [2,1,4,7,3,5,6] => ? = 6 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [5,1,2,7,3,4,6] => ? = 7 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 6 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,1,2,4,7,5,6] => ? = 5 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [4,1,5,7,2,3,6] => ? = 7 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,4,2,3,7,5,6] => ? = 5 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [3,1,4,2,7,5,6] => ? = 6 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,7,1,2,5,6] => ? = 7 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,4,7,2,5,6] => ? = 6 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [4,5,1,7,2,3,6] => ? = 7 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [2,4,1,3,7,5,6] => ? = 6 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,5,6] => ? = 4 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,5,1,7,2,4,6] => ? = 7 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,5,6] => ? = 5 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [3,4,1,2,7,5,6] => ? = 6 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,7,5,6] => ? = 5 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,4,5,7,1,2,6] => ? = 7 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,5,6] => ? = 4 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,3,7,4,6] => ? = 5 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,4,1,7,5,6] => ? = 6 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,5,6] => ? = 3 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,3,5,6] => ? = 5 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,4,7,1,3,5,6] => ? = 7 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,5,7,2,4,6] => ? = 6 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [2,5,1,7,3,4,6] => ? = 7 - 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,5,2,3,7,4,6] => ? = 6 - 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,3,2,5,7,4,6] => ? = 5 - 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [2,4,5,7,1,3,6] => ? = 7 - 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,3,4,2,7,5,6] => ? = 5 - 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,5,2,7,4,6] => ? = 6 - 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,4,6] => ? = 4 - 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,1,7,2,4,5,6] => ? = 7 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [2,1,5,7,3,4,6] => ? = 6 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [3,1,5,7,2,4,6] => ? = 7 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [2,1,5,3,7,4,6] => ? = 6 - 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [2,1,3,5,7,4,6] => ? = 5 - 1
Description
The cardinality of the support of a permutation.
A permutation $\sigma$ may be written as a product $\sigma = s_{i_1}\dots s_{i_k}$ with $k$ minimal, where $s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions $i$ and $i+1$.
The set of indices $\{i_1,\dots,i_k\}$ is the '''support''' of $\sigma$ and independent of the chosen way to write $\sigma$ as such a product.
See [2], Definition 1 and Proposition 10.
The '''connectivity set''' of $\sigma$ of length $n$ is the set of indices $1 \leq i < n$ such that $\sigma(k) < i$ for all $k < i$.
Thus, the connectivity set is the complement of the support.
Matching statistic: St000740
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 64%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 64%
Values
[1,0]
=> [1,0]
=> [1] => 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 2
[1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 3
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 3
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 4
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 3
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 4
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 3
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 4
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 3
[1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 2
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 4
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 3
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 5
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 5
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => 5
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 4
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 5
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => 4
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 5
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 5
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [7,1,2,3,4,5,6] => ? = 6
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [6,1,2,3,4,5,7] => ? = 7
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,7,2,3,4,5,6] => ? = 6
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [6,7,1,2,3,4,5] => ? = 5
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [5,1,2,3,4,6,7] => ? = 7
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [5,7,1,2,3,4,6] => ? = 6
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5,7] => ? = 7
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,6,7,2,3,4,5] => ? = 5
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [5,6,1,2,3,4,7] => ? = 7
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [6,1,7,2,3,4,5] => ? = 5
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [5,1,7,2,3,4,6] => ? = 6
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [5,6,7,1,2,3,4] => ? = 4
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [4,1,2,3,5,6,7] => ? = 7
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [4,7,1,2,3,5,6] => ? = 6
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [4,6,1,2,3,5,7] => ? = 7
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [4,1,7,2,3,5,6] => ? = 6
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [4,6,7,1,2,3,5] => ? = 5
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,5,2,3,4,6,7] => ? = 7
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,5,7,2,3,4,6] => ? = 6
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,5,6,2,3,4,7] => ? = 7
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [6,1,2,7,3,4,5] => ? = 5
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,5,2,7,3,4,6] => ? = 6
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,5,6,7,2,3,4] => ? = 4
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [4,5,1,2,3,6,7] => ? = 7
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [4,5,7,1,2,3,6] => ? = 6
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [5,1,6,2,3,4,7] => ? = 7
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [5,1,2,7,3,4,6] => ? = 6
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [5,1,6,7,2,3,4] => ? = 4
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [4,1,6,2,3,5,7] => ? = 7
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,6,2,7,3,4,5] => ? = 5
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [4,1,2,7,3,5,6] => ? = 6
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [4,1,6,7,2,3,5] => ? = 5
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [4,5,6,1,2,3,7] => ? = 7
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [5,6,1,7,2,3,4] => ? = 4
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,6,1,7,2,3,5] => ? = 5
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [4,5,1,7,2,3,6] => ? = 6
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [4,5,6,7,1,2,3] => ? = 3
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,2,4,5,6,7] => ? = 7
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [3,7,1,2,4,5,6] => ? = 6
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [3,6,1,2,4,5,7] => ? = 7
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,7,2,4,5,6] => ? = 6
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [3,6,7,1,2,4,5] => ? = 5
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [3,5,1,2,4,6,7] => ? = 7
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [3,5,7,1,2,4,6] => ? = 6
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [3,1,6,2,4,5,7] => ? = 7
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [3,1,2,7,4,5,6] => ? = 6
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [3,1,6,7,2,4,5] => ? = 5
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [3,5,6,1,2,4,7] => ? = 7
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [3,6,1,7,2,4,5] => ? = 5
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [3,5,1,7,2,4,6] => ? = 6
Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
Matching statistic: St001497
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St001497: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 64%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St001497: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 64%
Values
[1,0]
=> [1] => [1] => [1] => 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 2
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 3
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 3
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => [2,3,1] => 2
[1,1,1,0,0,0]
=> [3,2,1] => [3,1,2] => [3,1,2] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 4
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [1,3,4,2] => 3
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 4
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 3
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => [2,3,1,4] => 4
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,2,3,1] => [2,3,4,1] => 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,1,3] => [2,4,1,3] => 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,1,2,4] => [3,1,2,4] => 4
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,3,2,1] => [3,4,1,2] => 2
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4,1,3,2] => [3,1,4,2] => 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,5,3] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,3,4] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,3,2,4] => [1,3,5,2,4] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 5
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => [1,4,5,2,3] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,2,4,3] => [1,4,2,5,3] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,2,3,4] => [1,5,2,3,4] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 5
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,5,3] => 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,3,4] => [2,1,5,3,4] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => 5
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 4
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [2,3,4,1,5] => 5
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [2,3,4,5,1] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,3,1,4] => [2,3,5,1,4] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,1,3,5] => [2,4,1,3,5] => 5
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [5,2,4,3,1] => [2,4,5,1,3] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [5,2,1,4,3] => [2,4,1,5,3] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,2,1,3,4] => [2,5,1,3,4] => 2
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,3,7,6,5,2] => [1,7,4,3,2,5,6] => [1,4,7,2,3,5,6] => ? = 3
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,6,5,3,7,2] => [1,7,6,4,3,5,2] => [1,4,6,7,2,3,5] => ? = 4
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,4,3,2,6,7] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? = 7
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,5,4,3,2,7,6] => [1,5,2,3,4,7,6] => [1,5,2,3,4,7,6] => ? = 6
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,5,4,3,6,2,7] => [1,6,5,3,4,2,7] => [1,5,6,2,3,4,7] => ? = 7
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,4,3,6,7,2] => [1,7,5,3,4,6,2] => [1,5,6,2,3,7,4] => ? = 6
[1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,5,4,3,7,6,2] => [1,7,5,3,4,2,6] => [1,5,7,2,3,4,6] => ? = 3
[1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,5,4,6,3,2,7] => [1,6,2,5,4,3,7] => [1,5,2,6,3,4,7] => ? = 7
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,5,4,6,3,7,2] => [1,7,6,5,4,3,2] => [1,5,6,7,2,3,4] => ? = 4
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,5,4,6,7,3,2] => [1,7,2,5,4,6,3] => [1,5,2,6,3,7,4] => ? = 6
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,5,4,7,6,3,2] => [1,7,2,5,4,3,6] => [1,5,2,7,3,4,6] => ? = 4
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,5,6,4,3,2,7] => [1,6,2,3,5,4,7] => [1,5,2,3,6,4,7] => ? = 7
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,5,6,4,3,7,2] => [1,7,6,3,5,4,2] => [1,5,6,2,7,3,4] => ? = 5
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,5,6,4,7,3,2] => [1,7,2,6,5,4,3] => [1,5,2,6,7,3,4] => ? = 5
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,5,6,7,4,3,2] => [1,7,2,3,5,6,4] => [1,5,2,3,6,7,4] => ? = 6
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => [1,7,2,3,5,4,6] => [1,5,2,3,7,4,6] => ? = 5
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,5,4,3,2,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 7
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,5,4,3,7,2] => [1,7,6,3,4,5,2] => [1,6,7,2,3,4,5] => ? = 3
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,6,5,4,7,3,2] => [1,7,2,6,4,5,3] => [1,6,2,7,3,4,5] => ? = 4
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,5,7,4,3,2] => [1,7,2,3,6,5,4] => [1,6,2,3,7,4,5] => ? = 5
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => [1,7,2,3,4,6,5] => [1,6,2,3,4,7,5] => ? = 6
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 7
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => ? = 6
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => ? = 7
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [2,1,3,4,6,7,5] => [2,1,3,4,7,6,5] => [2,1,3,4,6,7,5] => ? = 6
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [2,1,3,4,7,6,5] => [2,1,3,4,7,5,6] => [2,1,3,4,7,5,6] => ? = 5
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => ? = 7
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => ? = 6
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,6,4,7] => [2,1,3,6,5,4,7] => [2,1,3,5,6,4,7] => ? = 7
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [2,1,3,5,6,7,4] => [2,1,3,7,5,6,4] => [2,1,3,5,6,7,4] => ? = 6
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [2,1,3,5,7,6,4] => [2,1,3,7,5,4,6] => [2,1,3,5,7,4,6] => ? = 5
[1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,6,5,4,7] => [2,1,3,6,4,5,7] => [2,1,3,6,4,5,7] => ? = 7
[1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [2,1,3,6,5,7,4] => [2,1,3,7,6,5,4] => [2,1,3,6,7,4,5] => ? = 5
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [2,1,3,6,7,5,4] => [2,1,3,7,4,6,5] => [2,1,3,6,4,7,5] => ? = 6
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,7,6,5,4] => [2,1,3,7,4,5,6] => [2,1,3,7,4,5,6] => ? = 4
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 7
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => ? = 6
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => ? = 7
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,3,6,7,5] => [2,1,4,3,7,6,5] => [2,1,4,3,6,7,5] => ? = 6
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,6,5] => [2,1,4,3,7,5,6] => [2,1,4,3,7,5,6] => ? = 5
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,5,3,6,7] => [2,1,5,4,3,6,7] => [2,1,4,5,3,6,7] => ? = 7
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [2,1,4,5,3,7,6] => [2,1,5,4,3,7,6] => [2,1,4,5,3,7,6] => ? = 6
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,5,6,3,7] => [2,1,6,4,5,3,7] => [2,1,4,5,6,3,7] => ? = 7
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [2,1,4,5,6,7,3] => [2,1,7,4,5,6,3] => [2,1,4,5,6,7,3] => ? = 6
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [2,1,4,5,7,6,3] => [2,1,7,4,5,3,6] => [2,1,4,5,7,3,6] => ? = 5
[1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,6,5,3,7] => [2,1,6,4,3,5,7] => [2,1,4,6,3,5,7] => ? = 7
[1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [2,1,4,6,5,7,3] => [2,1,7,4,6,5,3] => [2,1,4,6,7,3,5] => ? = 5
[1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [2,1,4,6,7,5,3] => [2,1,7,4,3,6,5] => [2,1,4,6,3,7,5] => ? = 6
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [2,1,4,7,6,5,3] => [2,1,7,4,3,5,6] => [2,1,4,7,3,5,6] => ? = 4
Description
The position of the largest weak excedence of a permutation.
Matching statistic: St000653
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000653: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 64%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000653: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 64%
Values
[1,0]
=> [1,0]
=> [1] => [1] => ? = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => [2,1] => 1 = 2 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => [1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => [2,3,1] => 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 2 = 3 - 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [3,4,1,2] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [3,1,4,2] => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,3,4,2] => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [2,3,4,5,1] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => [3,4,5,1,2] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => [3,4,1,5,2] => 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [2,4,5,1,3] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => [4,5,1,2,3] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [3,1,4,5,2] => 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [4,1,5,2,3] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => [2,4,1,5,3] => 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => [2,3,5,1,4] => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => [2,5,1,3,4] => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,1,2,5,3] => 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => [3,5,1,2,4] => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [3,1,5,2,4] => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,3,4,5,2] => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,4,5,2,3] => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,4,2,5,3] => 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,3,5,2,4] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [2,3,1,5,4] => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [2,3,4,1,5] => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [2,3,1,4,5] => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [3,4,1,2,5] => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,4,5,3] => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? = 7 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,1,2,3,4,5] => [3,4,5,6,7,1,2] => ? = 6 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [5,7,1,2,3,4,6] => [3,4,5,6,1,7,2] => ? = 7 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,1,7,2,3,4,5] => [2,4,5,6,7,1,3] => ? = 6 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [5,6,7,1,2,3,4] => [4,5,6,7,1,2,3] => ? = 5 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,7,1,2,3,5,6] => [3,4,5,1,6,7,2] => ? = 7 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [4,6,7,1,2,3,5] => [4,5,6,1,7,2,3] => ? = 6 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [5,1,7,2,3,4,6] => [2,4,5,6,1,7,3] => ? = 7 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [6,1,2,7,3,4,5] => [2,3,5,6,7,1,4] => ? = 6 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [5,1,6,7,2,3,4] => [2,5,6,7,1,3,4] => ? = 5 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [4,5,7,1,2,3,6] => [4,5,6,1,2,7,3] => ? = 7 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [5,6,1,7,2,3,4] => [3,5,6,7,1,2,4] => ? = 5 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [4,6,1,7,2,3,5] => [3,5,6,1,7,2,4] => ? = 6 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,5,6,7,1,2,3] => [5,6,7,1,2,3,4] => ? = 4 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,7,1,2,4,5,6] => [3,4,1,5,6,7,2] => ? = 7 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,6,7,1,2,4,5] => [4,5,1,6,7,2,3] => ? = 6 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,5,7,1,2,4,6] => [4,5,1,6,2,7,3] => ? = 7 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [3,6,1,7,2,4,5] => [3,5,1,6,7,2,4] => ? = 6 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [3,5,6,7,1,2,4] => [5,6,1,7,2,3,4] => ? = 5 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [4,1,7,2,3,5,6] => [2,4,5,1,6,7,3] => ? = 7 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [4,1,6,7,2,3,5] => [2,5,6,1,7,3,4] => ? = 6 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [5,1,2,7,3,4,6] => [2,3,5,6,1,7,4] => ? = 7 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [6,1,2,3,7,4,5] => [2,3,4,6,7,1,5] => ? = 6 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [5,1,2,6,7,3,4] => [2,3,6,7,1,4,5] => ? = 5 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [4,1,5,7,2,3,6] => [2,5,6,1,3,7,4] => ? = 7 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,6,1,2,7,3,4] => [3,4,6,7,1,2,5] => ? = 5 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,1,6,2,7,3,5] => [2,4,6,1,7,3,5] => ? = 6 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [4,1,5,6,7,2,3] => [2,6,7,1,3,4,5] => ? = 4 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,7,1,2,5,6] => [4,5,1,2,6,7,3] => ? = 7 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [3,4,6,7,1,2,5] => [5,6,1,2,7,3,4] => ? = 6 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [4,5,1,7,2,3,6] => [3,5,6,1,2,7,4] => ? = 7 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [4,6,1,2,7,3,5] => [3,4,6,1,7,2,5] => ? = 6 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [4,5,1,6,7,2,3] => [3,6,7,1,2,4,5] => ? = 4 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,5,1,7,2,4,6] => [3,5,1,6,2,7,4] => ? = 7 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [5,1,6,2,7,3,4] => [2,4,6,7,1,3,5] => ? = 5 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [3,6,1,2,7,4,5] => [3,4,1,6,7,2,5] => ? = 6 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [3,5,1,6,7,2,4] => [3,6,1,7,2,4,5] => ? = 5 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,4,5,7,1,2,6] => [5,6,1,2,3,7,4] => ? = 7 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [4,5,6,1,7,2,3] => [4,6,7,1,2,3,5] => ? = 4 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [3,5,6,1,7,2,4] => [4,6,1,7,2,3,5] => ? = 5 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [3,4,6,1,7,2,5] => [4,6,1,2,7,3,5] => ? = 6 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,7,1,2] => [6,7,1,2,3,4,5] => ? = 3 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,1,3,4,5,6] => [3,1,4,5,6,7,2] => ? = 7 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,6,7,1,3,4,5] => [4,1,5,6,7,2,3] => ? = 6 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,5,7,1,3,4,6] => [4,1,5,6,2,7,3] => ? = 7 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,6,1,7,3,4,5] => [3,1,5,6,7,2,4] => ? = 6 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [2,5,6,7,1,3,4] => [5,1,6,7,2,3,4] => ? = 5 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,4,7,1,3,5,6] => [4,1,5,2,6,7,3] => ? = 7 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [2,4,6,7,1,3,5] => [5,1,6,2,7,3,4] => ? = 6 - 1
Description
The last descent of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the largest index $0 \leq i < n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(0) = n+1$.
The following 18 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000957The number of Bruhat lower covers of a permutation. St000054The first entry of the permutation. St000141The maximum drop size of a permutation. St000067The inversion number of the alternating sign matrix. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000316The number of non-left-to-right-maxima of a permutation. St000240The number of indices that are not small excedances. St000809The reduced reflection length of the permutation. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St000216The absolute length of a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000840The number of closers smaller than the largest opener in a perfect matching. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
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