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Your data matches 6 different statistics following compositions of up to 3 maps.
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Matching statistic: St000249
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(load all 12 compositions to match this statistic)
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000249: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000249: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> {{1},{2}}
=> 2
[1,1,0,0]
=> {{1,2}}
=> 2
[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}}
=> 2
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> 2
[1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[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}}
=> 3
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 2
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 3
[1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 2
[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}}
=> 3
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 3
[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}}
=> 4
[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}}
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 3
[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}}
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> 2
[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}}
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> 3
[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}}
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> 3
[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}}
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 4
[1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 4
Description
The number of singletons ([[St000247]]) plus the number of antisingletons ([[St000248]]) of a set partition.
Matching statistic: St001499
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 96%●distinct values known / distinct values provided: 86%
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 96%●distinct values known / distinct values provided: 86%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 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]
=> [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7 - 1
[1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 6 - 1
[1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 6 - 1
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 6 - 1
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 6 - 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 7 - 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 6 - 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 7 - 1
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6 - 1
[1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 7 - 1
[1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 6 - 1
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 7 - 1
[1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 6 - 1
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 7 - 1
[1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 6 - 1
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 7 - 1
[1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 6 - 1
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 7 - 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> ? = 6 - 1
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 6 - 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 7 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 6 - 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 6 - 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 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]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8 - 1
Description
The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra.
We use the bijection in the code by Christian Stump to have a bijection to Dyck paths.
Matching statistic: St000619
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 71%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 71%
Values
[1,0,1,0]
=> [1,2] => [1,2] => [2,1] => 1 = 2 - 1
[1,1,0,0]
=> [2,1] => [2,1] => [1,2] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [3,2,1] => 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [3,1,2] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,3,1] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => [1,2,3] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,1,2] => [1,3,2] => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [4,1,2,3] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,2,3] => [4,1,3,2] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => [2,3,4,1] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,2,3,1] => [1,3,2,4] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,1,3] => [1,3,4,2] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,1,2,4] => [2,4,3,1] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,3,2,1] => [1,2,3,4] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4,1,3,2] => [1,4,2,3] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,1,2,3] => [1,4,3,2] => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,2,3] => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,3,4] => [5,4,1,3,2] => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [5,1,3,2,4] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,3,2,4] => [5,1,3,4,2] => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,2,3,5] => [5,2,4,3,1] => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,2,4,3] => [5,1,4,2,3] => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,2,3,4] => [5,1,4,3,2] => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [4,5,3,2,1] => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [4,5,3,1,2] => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [4,5,2,3,1] => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [4,5,1,2,3] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,3,4] => [4,5,1,3,2] => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [3,4,5,2,1] => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [3,4,5,1,2] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [2,4,3,5,1] => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [1,4,3,2,5] => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,3,1,4] => [1,4,3,5,2] => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,1,3,5] => [2,4,5,3,1] => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [5,2,4,3,1] => [1,4,2,3,5] => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [5,2,1,4,3] => [1,4,5,2,3] => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,2,1,3,4] => [1,4,5,3,2] => 3 = 4 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,1,2,4,5] => [3,5,4,2,1] => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7 - 1
[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] => [7,6,5,4,3,1,2] => ? = 6 - 1
[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] => [7,6,5,4,2,3,1] => ? = 6 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => ? = 5 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 6 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 6 - 1
[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] => [7,6,5,3,4,1,2] => ? = 5 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => [7,6,5,2,3,4,1] => ? = 5 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [1,2,3,7,5,6,4] => [7,6,5,1,3,2,4] => ? = 5 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,6,4] => [1,2,3,7,5,4,6] => [7,6,5,1,3,4,2] => ? = 5 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 6 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => [1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => ? = 4 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,6,7,5,4] => [1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => ? = 5 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => ? = 6 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => ? = 6 - 1
[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] => [7,6,4,5,3,1,2] => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [7,6,4,5,2,3,1] => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => [1,2,4,3,7,6,5] => [7,6,4,5,1,2,3] => ? = 4 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [1,2,4,3,7,5,6] => [7,6,4,5,1,3,2] => ? = 5 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => [1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => ? = 5 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [1,2,5,4,3,7,6] => [7,6,3,4,5,1,2] => ? = 4 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [1,2,6,4,5,3,7] => [7,6,2,4,3,5,1] => ? = 5 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [1,2,7,4,5,6,3] => [7,6,1,4,3,2,5] => ? = 5 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,6,3] => [1,2,7,4,5,3,6] => [7,6,1,4,3,5,2] => ? = 5 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,5,3,7] => [1,2,6,4,3,5,7] => [7,6,2,4,5,3,1] => ? = 5 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,5,7,3] => [1,2,7,4,6,5,3] => [7,6,1,4,2,3,5] => ? = 4 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,6,7,5,3] => [1,2,7,4,3,6,5] => [7,6,1,4,5,2,3] => ? = 4 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => [1,2,7,4,3,5,6] => [7,6,1,4,5,3,2] => ? = 5 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,4,3,6,7] => [1,2,5,3,4,6,7] => [7,6,3,5,4,2,1] => ? = 6 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => [1,2,5,3,4,7,6] => [7,6,3,5,4,1,2] => ? = 5 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,4,6,3,7] => [1,2,6,5,4,3,7] => [7,6,2,3,4,5,1] => ? = 4 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => [1,2,7,5,4,6,3] => [7,6,1,3,4,2,5] => ? = 4 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,4,7,6,3] => [1,2,7,5,4,3,6] => [7,6,1,3,4,5,2] => ? = 4 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,4,3,7] => [1,2,6,3,5,4,7] => [7,6,2,5,3,4,1] => ? = 5 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,4,7,3] => [1,2,7,6,5,4,3] => [7,6,1,2,3,4,5] => ? = 3 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,4,3] => [1,2,7,3,5,6,4] => [7,6,1,5,3,2,4] => ? = 5 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,6,4,3] => [1,2,7,3,5,4,6] => [7,6,1,5,3,4,2] => ? = 5 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => [1,2,6,3,4,5,7] => [7,6,2,5,4,3,1] => ? = 6 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,5,4,7,3] => [1,2,7,6,4,5,3] => [7,6,1,2,4,3,5] => ? = 4 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,5,7,4,3] => [1,2,7,3,6,5,4] => [7,6,1,5,2,3,4] => ? = 4 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,5,4,3] => [1,2,7,3,4,6,5] => [7,6,1,5,4,2,3] => ? = 5 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => [1,2,7,3,4,5,6] => [7,6,1,5,4,3,2] => ? = 6 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => [7,5,6,4,3,2,1] => ? = 6 - 1
[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] => [7,5,6,4,3,1,2] => ? = 5 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => [7,5,6,4,2,3,1] => ? = 5 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [1,3,2,4,7,6,5] => [7,5,6,4,1,2,3] => ? = 4 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,6,5] => [1,3,2,4,7,5,6] => [7,5,6,4,1,3,2] => ? = 5 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => [7,5,6,3,4,2,1] => ? = 5 - 1
[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] => [7,5,6,3,4,1,2] => ? = 4 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => [1,3,2,6,5,4,7] => [7,5,6,2,3,4,1] => ? = 4 - 1
Description
The number of cyclic descents of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is given by the number of indices $1 \leq i \leq n$ such that $\pi(i) > \pi(i+1)$ where we set $\pi(n+1) = \pi(1)$.
Matching statistic: St000236
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000236: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 71%
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000236: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 71%
Values
[1,0,1,0]
=> [1,2] => [1,2] => 2
[1,1,0,0]
=> [2,1] => [2,1] => 2
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 3
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 2
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => 2
[1,1,1,0,0,0]
=> [3,2,1] => [2,3,1] => 3
[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]
=> [1,2,4,3] => [1,2,4,3] => 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 3
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,4,2] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,2,3,1] => 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,2,4,1] => 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,1,4] => 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,3,2,1] => 2
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [2,4,3,1] => 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [2,3,4,1] => 4
[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]
=> [1,2,3,5,4] => [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] => 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,5,3] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,4,3,5,2] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,3,4,2,5] => 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,3,5,4,2] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,3,4,5,2] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,5,3] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,2,3,5,1] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,2,4,1,5] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [5,2,4,3,1] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [3,2,5,4,1] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [3,2,4,5,1] => 4
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [2,3,1,4,5] => 4
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 7
[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] => ? = 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] => ? = 6
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => ? = 5
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => ? = 6
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 6
[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
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => ? = 5
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [1,2,3,7,5,6,4] => ? = 5
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,6,4] => [1,2,3,6,5,7,4] => ? = 5
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => ? = 6
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => [1,2,3,7,6,5,4] => ? = 4
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,6,7,5,4] => [1,2,3,5,7,6,4] => ? = 5
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [1,2,3,5,6,7,4] => ? = 6
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 6
[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] => ? = 5
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => ? = 5
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => [1,2,4,3,7,6,5] => ? = 4
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => ? = 5
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => [1,2,5,4,3,6,7] => ? = 5
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [1,2,5,4,3,7,6] => ? = 4
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [1,2,6,4,5,3,7] => ? = 5
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [1,2,7,4,5,6,3] => ? = 5
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,6,3] => [1,2,6,4,5,7,3] => ? = 5
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,5,3,7] => [1,2,5,4,6,3,7] => ? = 5
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,5,7,3] => [1,2,7,4,6,5,3] => ? = 4
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,6,7,5,3] => [1,2,5,4,7,6,3] => ? = 4
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => [1,2,5,4,6,7,3] => ? = 5
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,4,3,6,7] => [1,2,4,5,3,6,7] => ? = 6
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => [1,2,4,5,3,7,6] => ? = 5
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,4,6,3,7] => [1,2,6,5,4,3,7] => ? = 4
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => [1,2,7,5,4,6,3] => ? = 4
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,4,7,6,3] => [1,2,6,5,4,7,3] => ? = 4
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,4,3,7] => [1,2,4,6,5,3,7] => ? = 5
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,4,7,3] => [1,2,7,6,5,4,3] => ? = 3
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,4,3] => [1,2,4,7,5,6,3] => ? = 5
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,6,4,3] => [1,2,4,6,5,7,3] => ? = 5
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => [1,2,4,5,6,3,7] => ? = 6
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,5,4,7,3] => [1,2,7,5,6,4,3] => ? = 4
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,5,7,4,3] => [1,2,4,7,6,5,3] => ? = 4
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,5,4,3] => [1,2,4,5,7,6,3] => ? = 5
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => [1,2,4,5,6,7,3] => ? = 6
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => ? = 6
[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] => ? = 5
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => ? = 5
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [1,3,2,4,7,6,5] => ? = 4
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,6,5] => [1,3,2,4,6,7,5] => ? = 5
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => ? = 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] => ? = 4
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => [1,3,2,6,5,4,7] => ? = 4
Description
The number of cyclical small weak excedances.
A cyclical small weak excedance is an index $i$ such that $\pi_i \in \{ i,i+1 \}$ considered cyclically.
Matching statistic: St001960
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 57%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 57%
Values
[1,0,1,0]
=> [1,2] => [1,2] => [2,1] => 0 = 2 - 2
[1,1,0,0]
=> [2,1] => [2,1] => [1,2] => 0 = 2 - 2
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [3,2,1] => 1 = 3 - 2
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [3,1,2] => 0 = 2 - 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,3,1] => 0 = 2 - 2
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => [1,2,3] => 0 = 2 - 2
[1,1,1,0,0,0]
=> [3,2,1] => [3,1,2] => [1,3,2] => 1 = 3 - 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 2 = 4 - 2
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 1 = 3 - 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1 = 3 - 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [4,1,2,3] => 0 = 2 - 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,2,3] => [4,1,3,2] => 1 = 3 - 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 1 = 3 - 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 0 = 2 - 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => [2,3,4,1] => 0 = 2 - 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,2,3,1] => [1,3,2,4] => 1 = 3 - 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,1,3] => [1,3,4,2] => 1 = 3 - 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,1,2,4] => [2,4,3,1] => 1 = 3 - 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,3,2,1] => [1,2,3,4] => 0 = 2 - 2
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4,1,3,2] => [1,4,2,3] => 1 = 3 - 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,1,2,3] => [1,4,3,2] => 2 = 4 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 3 = 5 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 2 = 4 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,2,3] => 1 = 3 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,3,4] => [5,4,1,3,2] => 2 = 4 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 2 = 4 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 1 = 3 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => 1 = 3 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [5,1,3,2,4] => 1 = 3 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,3,2,4] => [5,1,3,4,2] => 1 = 3 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,2,3,5] => [5,2,4,3,1] => 2 = 4 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => 0 = 2 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,2,4,3] => [5,1,4,2,3] => 1 = 3 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,2,3,4] => [5,1,4,3,2] => 2 = 4 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [4,5,3,2,1] => 2 = 4 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [4,5,3,1,2] => 1 = 3 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [4,5,2,3,1] => 1 = 3 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [4,5,1,2,3] => 0 = 2 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,3,4] => [4,5,1,3,2] => 1 = 3 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [3,4,5,2,1] => 1 = 3 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [3,4,5,1,2] => 0 = 2 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [2,4,3,5,1] => 1 = 3 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [1,4,3,2,5] => 2 = 4 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,3,1,4] => [1,4,3,5,2] => 2 = 4 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,1,3,5] => [2,4,5,3,1] => 1 = 3 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [5,2,4,3,1] => [1,4,2,3,5] => 1 = 3 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [5,2,1,4,3] => [1,4,5,2,3] => 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,2,1,3,4] => [1,4,5,3,2] => 2 = 4 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,1,2,4,5] => [3,5,4,2,1] => 2 = 4 - 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ? = 6 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [1,2,3,4,6,5] => [6,5,4,3,1,2] => ? = 5 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [1,2,3,5,4,6] => [6,5,4,2,3,1] => ? = 5 - 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [1,2,3,6,5,4] => [6,5,4,1,2,3] => ? = 4 - 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => [1,2,3,6,4,5] => [6,5,4,1,3,2] => ? = 5 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => [1,2,4,3,5,6] => [6,5,3,4,2,1] => ? = 5 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => [1,2,4,3,6,5] => [6,5,3,4,1,2] => ? = 4 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => [1,2,5,4,3,6] => [6,5,2,3,4,1] => ? = 4 - 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => [1,2,6,4,5,3] => [6,5,1,3,2,4] => ? = 4 - 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,5,3] => [1,2,6,4,3,5] => [6,5,1,3,4,2] => ? = 4 - 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,4,3,6] => [1,2,5,3,4,6] => [6,5,2,4,3,1] => ? = 5 - 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => [1,2,6,5,4,3] => [6,5,1,2,3,4] => ? = 3 - 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,5,6,4,3] => [1,2,6,3,5,4] => [6,5,1,4,2,3] => ? = 4 - 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => [1,2,6,3,4,5] => [6,5,1,4,3,2] => ? = 5 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => [1,3,2,4,5,6] => [6,4,5,3,2,1] => ? = 5 - 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => [1,3,2,4,6,5] => [6,4,5,3,1,2] => ? = 4 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => [1,3,2,5,4,6] => [6,4,5,2,3,1] => ? = 4 - 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,3,2,5,6,4] => [1,3,2,6,5,4] => [6,4,5,1,2,3] => ? = 3 - 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,5,4] => [1,3,2,6,4,5] => [6,4,5,1,3,2] => ? = 4 - 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,4,2,5,6] => [1,4,3,2,5,6] => [6,3,4,5,2,1] => ? = 4 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,4,2,6,5] => [1,4,3,2,6,5] => [6,3,4,5,1,2] => ? = 3 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,2,6] => [1,5,3,4,2,6] => [6,2,4,3,5,1] => ? = 4 - 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => [1,6,3,4,5,2] => [6,1,4,3,2,5] => ? = 4 - 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,4,6,5,2] => [1,6,3,4,2,5] => [6,1,4,3,5,2] => ? = 4 - 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,5,4,2,6] => [1,5,3,2,4,6] => [6,2,4,5,3,1] => ? = 4 - 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,5,4,6,2] => [1,6,3,5,4,2] => [6,1,4,2,3,5] => ? = 3 - 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,3,5,6,4,2] => [1,6,3,2,5,4] => [6,1,4,5,2,3] => ? = 3 - 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,6,5,4,2] => [1,6,3,2,4,5] => [6,1,4,5,3,2] => ? = 4 - 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => [1,4,2,3,5,6] => [6,3,5,4,2,1] => ? = 5 - 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,3,2,6,5] => [1,4,2,3,6,5] => [6,3,5,4,1,2] => ? = 4 - 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,3,5,2,6] => [1,5,4,3,2,6] => [6,2,3,4,5,1] => ? = 3 - 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,3,5,6,2] => [1,6,4,3,5,2] => [6,1,3,4,2,5] => ? = 3 - 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,4,3,6,5,2] => [1,6,4,3,2,5] => [6,1,3,4,5,2] => ? = 3 - 2
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,5,3,2,6] => [1,5,2,4,3,6] => [6,2,5,3,4,1] => ? = 4 - 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,4,5,3,6,2] => [1,6,5,4,3,2] => [6,1,2,3,4,5] => ? = 2 - 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,4,5,6,3,2] => [1,6,2,4,5,3] => [6,1,5,3,2,4] => ? = 4 - 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,5,3,2] => [1,6,2,4,3,5] => [6,1,5,3,4,2] => ? = 4 - 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => [1,5,2,3,4,6] => [6,2,5,4,3,1] => ? = 5 - 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,4,3,6,2] => [1,6,5,3,4,2] => [6,1,2,4,3,5] => ? = 3 - 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,4,6,3,2] => [1,6,2,5,4,3] => [6,1,5,2,3,4] => ? = 3 - 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => [1,6,2,3,5,4] => [6,1,5,4,2,3] => ? = 4 - 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => [1,6,2,3,4,5] => [6,1,5,4,3,2] => ? = 5 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => [5,6,4,3,2,1] => ? = 5 - 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => [2,1,3,4,6,5] => [5,6,4,3,1,2] => ? = 4 - 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => [2,1,3,5,4,6] => [5,6,4,2,3,1] => ? = 4 - 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,3,5,6,4] => [2,1,3,6,5,4] => [5,6,4,1,2,3] => ? = 3 - 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,5,4] => [2,1,3,6,4,5] => [5,6,4,1,3,2] => ? = 4 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => [2,1,4,3,5,6] => [5,6,3,4,2,1] => ? = 4 - 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => [5,6,3,4,1,2] => ? = 3 - 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => [2,1,5,4,3,6] => [5,6,2,3,4,1] => ? = 3 - 2
Description
The number of descents of a permutation minus one if its first entry is not one.
This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
Matching statistic: St001060
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 57%
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 57%
Values
[1,0,1,0]
=> [[1,0],[0,1]]
=> [1,2] => ([],2)
=> ? = 2 - 1
[1,1,0,0]
=> [[0,1],[1,0]]
=> [2,1] => ([(0,1)],2)
=> ? = 2 - 1
[1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => ([],3)
=> ? = 3 - 1
[1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => ([(1,2)],3)
=> ? = 2 - 1
[1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => ([(1,2)],3)
=> ? = 2 - 1
[1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => ([(1,2)],3)
=> ? = 2 - 1
[1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => ([],4)
=> ? = 4 - 1
[1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => ([(2,3)],4)
=> ? = 3 - 1
[1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => ([(2,3)],4)
=> ? = 3 - 1
[1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => ([(2,3)],4)
=> ? = 2 - 1
[1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => ([(2,3)],4)
=> ? = 3 - 1
[1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 2 - 1
[1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => ([(2,3)],4)
=> ? = 2 - 1
[1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => ([(2,3)],4)
=> ? = 3 - 1
[1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 2 - 1
[1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => ([],5)
=> ? = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ? = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => ([(3,4)],5)
=> ? = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => ([(3,4)],5)
=> ? = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => ([(3,4)],5)
=> ? = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ? = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => ([(3,4)],5)
=> ? = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => ([(3,4)],5)
=> ? = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ? = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? = 4 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 3 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? = 2 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? = 3 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 3 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? = 3 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 2 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 4 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 3 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 5 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 5 - 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [3,1,2,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 4 - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [[0,0,1,0,0,0],[1,0,-1,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 5 - 1
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [3,1,2,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 4 - 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [[0,0,0,1,0,0],[1,0,0,-1,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 5 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 5 - 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [3,1,2,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 4 - 1
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [[0,0,0,0,1,0],[1,0,0,0,-1,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 5 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5 = 6 - 1
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Sorry, this statistic was not found in the database
or
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