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Your data matches 303 different statistics following compositions of up to 3 maps.
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Matching statistic: St001068
(load all 52 compositions to match this statistic)
(load all 52 compositions to match this statistic)
Mp00222: Dyck paths āpeaks-to-valleysā¶ Dyck paths
St001068: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St001068: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 3
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 4
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 4
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 4
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 4
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 4
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 5
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 4
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 5
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 5
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 5
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 5
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 5
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> 6
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> 6
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
Matching statistic: St000053
(load all 52 compositions to match this statistic)
(load all 52 compositions to match this statistic)
Mp00222: Dyck paths āpeaks-to-valleysā¶ Dyck paths
St000053: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000053: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> 5 = 6 - 1
Description
The number of valleys of the Dyck path.
Matching statistic: St000105
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00222: Dyck paths āpeaks-to-valleysā¶ Dyck paths
Mp00138: Dyck paths āto noncrossing partitionā¶ Set partitions
St000105: Set partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00138: Dyck paths āto noncrossing partitionā¶ Set partitions
St000105: Set partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> {{1}}
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> {{1,2}}
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> {{1},{2}}
=> 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 2
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 2
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 3
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 3
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 4
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 3
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 3
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 5
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 4
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> 3
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> 4
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> 4
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> {{1,2},{3,4},{5},{6}}
=> 4
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> {{1,2},{3},{4,5},{6}}
=> 4
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> {{1,2},{3},{4},{5,6}}
=> 4
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6}}
=> 5
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> {{1},{2,3},{4,5},{6}}
=> 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> {{1},{2,3},{4},{5,6}}
=> 4
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> {{1},{2,3},{4},{5},{6}}
=> 5
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> {{1},{2},{3,4},{5,6}}
=> 4
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> {{1},{2},{3,4},{5},{6}}
=> 5
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3},{4,5},{6}}
=> 5
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5,6}}
=> 5
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6}}
=> 6
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3},{4,6},{5}}
=> 5
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> {{1},{2},{3,5},{4},{6}}
=> 5
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> {{1},{2,4},{3},{5},{6}}
=> 5
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> {{1},{2},{3,4},{5},{6},{7}}
=> 6
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3},{4},{5,6},{7}}
=> 6
Description
The number of blocks in the set partition.
The generating function of this statistic yields the famous [[wiki:Stirling numbers of the second kind|Stirling numbers of the second kind]] $S_2(n,k)$ given by the number of [[SetPartitions|set partitions]] of $\{ 1,\ldots,n\}$ into $k$ blocks, see [1].
Matching statistic: St000167
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00222: Dyck paths āpeaks-to-valleysā¶ Dyck paths
Mp00026: Dyck paths āto ordered treeā¶ Ordered trees
St000167: Ordered trees ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00026: Dyck paths āto ordered treeā¶ Ordered trees
St000167: Ordered trees ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [[]]
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> [[[]]]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [[],[]]
=> 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[[[]]]]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [[[]],[]]
=> 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [[],[[]]]
=> 2
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [[],[],[]]
=> 3
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [[[],[]]]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 3
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 3
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 3
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 4
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> 3
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[[],[]],[]]
=> 3
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[[]],[],[[]]]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[[]],[],[],[]]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[[]],[[],[]]]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[],[[]],[],[]]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[],[],[[]],[]]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> 5
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[],[],[[],[]]]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[],[[],[]],[]]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[],[[],[],[]]]
=> 4
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[[],[]],[[]]]
=> 3
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[[],[]],[],[]]
=> 4
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[[],[],[]],[]]
=> 4
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [[[]],[[]],[],[]]
=> 4
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [[[]],[],[[]],[]]
=> 4
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [[[]],[],[],[[]]]
=> 4
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [[[]],[],[],[],[]]
=> 5
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [[],[[]],[[]],[]]
=> 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[],[[]],[],[[]]]
=> 4
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [[],[[]],[],[],[]]
=> 5
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[],[],[[]],[[]]]
=> 4
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [[],[],[[]],[],[]]
=> 5
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [[],[],[],[[]],[]]
=> 5
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[],[[]]]
=> 5
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[]]
=> 6
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[],[],[],[[],[]]]
=> 5
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [[],[],[[],[]],[]]
=> 5
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [[],[[],[]],[],[]]
=> 5
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [[],[],[[]],[],[],[]]
=> 6
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [[],[],[],[],[[]],[]]
=> 6
Description
The number of leaves of an ordered tree.
This is the number of nodes which do not have any children.
Matching statistic: St001007
(load all 52 compositions to match this statistic)
(load all 52 compositions to match this statistic)
Mp00222: Dyck paths āpeaks-to-valleysā¶ Dyck paths
Mp00120: Dyck paths āLalanne-Kreweras involutionā¶ Dyck paths
St001007: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00120: Dyck paths āLalanne-Kreweras involutionā¶ Dyck paths
St001007: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 4
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 4
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 4
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 4
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 5
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 4
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 5
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 5
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 5
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 5
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 5
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 6
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 6
Description
Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000024
(load all 52 compositions to match this statistic)
(load all 52 compositions to match this statistic)
Mp00222: Dyck paths āpeaks-to-valleysā¶ Dyck paths
Mp00120: Dyck paths āLalanne-Kreweras involutionā¶ Dyck paths
St000024: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00120: Dyck paths āLalanne-Kreweras involutionā¶ Dyck paths
St000024: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [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]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 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 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 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]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 5 = 6 - 1
Description
The number of double up and double down steps of a Dyck path.
In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St000052
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00222: Dyck paths āpeaks-to-valleysā¶ Dyck paths
Mp00199: Dyck paths āprime Dyck pathā¶ Dyck paths
St000052: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00199: Dyck paths āprime Dyck pathā¶ Dyck paths
St000052: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 5 = 6 - 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> 5 = 6 - 1
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> 5 = 6 - 1
Description
The number of valleys of a Dyck path not on the x-axis.
That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St000211
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00031: Dyck paths āto 312-avoiding permutationā¶ Permutations
Mp00240: Permutations āweak exceedance partitionā¶ Set partitions
St000211: Set partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00240: Permutations āweak exceedance partitionā¶ Set partitions
St000211: Set partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => {{1}}
=> 0 = 1 - 1
[1,0,1,0]
=> [1,2] => {{1},{2}}
=> 0 = 1 - 1
[1,1,0,0]
=> [2,1] => {{1,2}}
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,2,3] => {{1},{2},{3}}
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => {{1,2},{3}}
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> [3,2,1] => {{1,3},{2}}
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 3 = 4 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => {{1,2,4},{3}}
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => {{1,3,4},{2}}
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => {{1,3},{2,4}}
=> 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => {{1,2},{3,4,5}}
=> 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => {{1,2,3,5},{4}}
=> 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => {{1,2,4,5},{3}}
=> 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => {{1,2,4},{3,5}}
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => {{1,3,4},{2},{5}}
=> 2 = 3 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => {{1,3,4,5},{2}}
=> 3 = 4 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => {{1,3},{2,4,5}}
=> 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,3,2,5,6,4] => {{1},{2,3},{4,5,6}}
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,4,2,6,5] => {{1},{2,3,4},{5,6}}
=> 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,2,6] => {{1},{2,3,4,5},{6}}
=> 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => {{1},{2,3,4,5,6}}
=> 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5] => {{1,2},{3,4},{5,6}}
=> 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => {{1,2},{3,4,5},{6}}
=> 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,1,4,5,6,3] => {{1,2},{3,4,5,6}}
=> 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,3,1,5,4,6] => {{1,2,3},{4,5},{6}}
=> 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,3,1,5,6,4] => {{1,2,3},{4,5,6}}
=> 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [2,3,4,1,6,5] => {{1,2,3,4},{5,6}}
=> 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,1,6] => {{1,2,3,4,5},{6}}
=> 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => {{1,2,3,4,5,6}}
=> 5 = 6 - 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,5,1] => {{1,2,3,4,6},{5}}
=> 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,4,6,1] => {{1,2,3,5,6},{4}}
=> 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,3,5,6,1] => {{1,2,4,5,6},{3}}
=> 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,6,7,4] => {{1,2,3},{4,5,6,7}}
=> 5 = 6 - 1
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [2,3,4,5,1,7,6] => {{1,2,3,4,5},{6,7}}
=> 5 = 6 - 1
Description
The rank of the set partition.
This is defined as the number of elements in the set partition minus the number of blocks, or, equivalently, the number of arcs in the one-line diagram associated to the set partition.
Matching statistic: St000245
(load all 40 compositions to match this statistic)
(load all 40 compositions to match this statistic)
Mp00222: Dyck paths āpeaks-to-valleysā¶ Dyck paths
Mp00031: Dyck paths āto 312-avoiding permutationā¶ Permutations
St000245: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00031: Dyck paths āto 312-avoiding permutationā¶ Permutations
St000245: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 0 = 1 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 2 = 3 - 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 3 = 4 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 3 = 4 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 2 = 3 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 3 = 4 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 5 = 6 - 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,4,2,5,6] => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => 5 = 6 - 1
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => 5 = 6 - 1
Description
The number of ascents of a permutation.
Matching statistic: St000340
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00222: Dyck paths āpeaks-to-valleysā¶ Dyck paths
Mp00122: Dyck paths āElizalde-Deutsch bijectionā¶ Dyck paths
St000340: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00122: Dyck paths āElizalde-Deutsch bijectionā¶ Dyck paths
St000340: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 5 = 6 - 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> 5 = 6 - 1
Description
The number of non-final maximal constant sub-paths of length greater than one.
This is the total number of occurrences of the patterns $110$ and $001$.
The following 293 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001512The minimum rank of a graph. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000147The largest part of an integer partition. St000172The Grundy number of a graph. St000288The number of ones in a binary word. St000451The length of the longest pattern of the form k 1 2. St000482The (zero)-forcing number of a graph. St000507The number of ascents of a standard tableau. St000527The width of the poset. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000676The number of odd rises of a Dyck path. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001029The size of the core of a graph. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001116The game chromatic number of a graph. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001286The annihilation number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001389The number of partitions of the same length below the given integer partition. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001616The number of neutral elements in a lattice. St001670The connected partition number of a graph. St001720The minimal length of a chain of small intervals in a lattice. St001820The size of the image of the pop stack sorting operator. St000012The area of a Dyck path. St000141The maximum drop size of a permutation. St000157The number of descents of a standard tableau. St000234The number of global ascents of a permutation. St000272The treewidth of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000362The size of a minimal vertex cover of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000536The pathwidth of a graph. St000632The jump number of the poset. St000672The number of minimal elements in Bruhat order not less than the permutation. St000703The number of deficiencies of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001176The size of a partition minus its first part. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001298The number of repeated entries in the Lehmer code of a permutation. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001619The number of non-isomorphic sublattices of a lattice. St001622The number of join-irreducible elements of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000702The number of weak deficiencies of a permutation. St000912The number of maximal antichains in a poset. St000925The number of topologically connected components of a set partition. St000031The number of cycles in the cycle decomposition of a permutation. St000153The number of adjacent cycles of a permutation. St000489The number of cycles of a permutation of length at most 3. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000984The number of boxes below precisely one peak. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000007The number of saliances of the permutation. St000528The height of a poset. St000161The sum of the sizes of the right subtrees of a binary tree. St000306The bounce count of a Dyck path. St000332The positive inversions of an alternating sign matrix. St000662The staircase size of the code of a permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000308The height of the tree associated to a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000065The number of entries equal to -1 in an alternating sign matrix. St000246The number of non-inversions of a permutation. St000783The side length of the largest staircase partition fitting into a partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001651The Frankl number of a lattice. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St000015The number of peaks of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000062The length of the longest increasing subsequence of the permutation. St000164The number of short pairs. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000291The number of descents of a binary word. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000390The number of runs of ones in a binary word. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000991The number of right-to-left minima of a permutation. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000021The number of descents of a permutation. St000120The number of left tunnels of a Dyck path. St000292The number of ascents of a binary word. St000316The number of non-left-to-right-maxima of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St001180Number of indecomposable injective modules with projective dimension at most 1. St001489The maximum of the number of descents and the number of inverse descents. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000035The number of left outer peaks of a permutation. St000056The decomposition (or block) number of a permutation. St000237The number of small exceedances. St000389The number of runs of ones of odd length in a binary word. St000553The number of blocks of a graph. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000740The last entry of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000822The Hadwiger number of the graph. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001963The tree-depth of a graph. St000080The rank of the poset. St000168The number of internal nodes of an ordered tree. St000238The number of indices that are not small weak excedances. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000834The number of right outer peaks of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000871The number of very big ascents of a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001358The largest degree of a regular subgraph of a graph. St001427The number of descents of a signed permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St000083The number of left oriented leafs of a binary tree except the first one. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000829The Ulam distance of a permutation to the identity permutation. St000061The number of nodes on the left branch of a binary tree. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001346The number of parking functions that give the same permutation. St000216The absolute length of a permutation. St000809The reduced reflection length of the permutation. St001480The number of simple summands of the module J^2/J^3. St001812The biclique partition number of a graph. St000159The number of distinct parts of the integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001875The number of simple modules with projective dimension at most 1. St001432The order dimension of the partition. St000646The number of big ascents of a permutation. St000359The number of occurrences of the pattern 23-1. St000711The number of big exceedences of a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000028The number of stack-sorts needed to sort a permutation. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000710The number of big deficiencies of a permutation. St000647The number of big descents of a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St000619The number of cyclic descents of a permutation. St000039The number of crossings of a permutation. St000236The number of cyclical small weak excedances. St000317The cycle descent number of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001152The number of pairs with even minimum in a perfect matching. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001668The number of points of the poset minus the width of the poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001782The order of rowmotion on the set of order ideals of a poset. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St001863The number of weak excedances of a signed permutation. St001866The nesting alignments of a signed permutation. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001637The number of (upper) dissectors of a poset. St000259The diameter of a connected graph. St001330The hat guessing number of a graph. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St001712The number of natural descents of a standard Young tableau. St000942The number of critical left to right maxima of the parking functions. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St000872The number of very big descents of a permutation. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000454The largest eigenvalue of a graph if it is integral. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001435The number of missing boxes in the first row. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001115The number of even descents of a permutation. St000665The number of rafts of a permutation. St001394The genus of a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000502The number of successions of a set partitions. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001061The number of indices that are both descents and recoils of a permutation. St001114The number of odd descents of a permutation. St001151The number of blocks with odd minimum. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001737The number of descents of type 2 in a permutation. St000022The number of fixed points of a permutation. St000023The number of inner peaks of a permutation. St000091The descent variation of a composition. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000353The number of inner valleys of a permutation. St000360The number of occurrences of the pattern 32-1. St000485The length of the longest cycle of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000663The number of right floats of a permutation. St000732The number of double deficiencies of a permutation. St000779The tier of a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000836The number of descents of distance 2 of a permutation. St001153The number of blocks with even minimum in a set partition. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001665The number of pure excedances of a permutation. St001728The number of invisible descents of a permutation. St001729The number of visible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001871The number of triconnected components of a graph. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000735The last entry on the main diagonal of a standard tableau. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000307The number of rowmotion orbits of a poset. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001644The dimension of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000893The number of distinct diagonal sums of an alternating sign matrix. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.
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