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Your data matches 63 different statistics following compositions of up to 3 maps.
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Matching statistic: St001630
Values
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> ([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,1,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(2,5),(3,4),(3,5)],6)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,6),(1,3),(1,6),(3,5),(4,2),(5,4),(6,5)],7)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> ([(0,3),(1,5),(1,6),(3,5),(3,6),(4,2),(5,4),(6,4)],7)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(0,6),(1,5),(1,6),(3,4),(4,2),(5,3),(6,4)],7)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,3),(0,6),(1,5),(1,6),(3,5),(4,2),(5,4),(6,4)],7)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,6),(1,5),(2,3),(2,5),(3,6),(5,6),(6,4)],7)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> ([(0,6),(1,2),(2,6),(3,5),(4,5),(6,3),(6,4)],7)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(0,6),(1,5),(1,6),(3,2),(4,2),(5,3),(5,4),(6,3),(6,4)],7)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(4,6),(5,6)],7)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,6),(1,2),(1,6),(2,4),(2,5),(4,3),(5,3),(6,4),(6,5)],7)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> ([(0,6),(1,6),(2,5),(3,5),(4,3),(6,2),(6,4)],7)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1)],7)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> ([(0,3),(1,4),(1,6),(2,5),(3,4),(3,6),(4,2),(6,5)],7)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,2),(2,1)],3)
=> 1
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St001432
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00222: Dyck paths āpeaks-to-valleysā¶ Dyck paths
Mp00124: Dyck paths āAdin-Bagno-Roichman transformationā¶ Dyck paths
Mp00027: Dyck paths āto partitionā¶ Integer partitions
St001432: Integer partitions ā¶ ā¤Result quality: 33% āvalues known / values provided: 33%ādistinct values known / distinct values provided: 100%
Mp00124: Dyck paths āAdin-Bagno-Roichman transformationā¶ Dyck paths
Mp00027: Dyck paths āto partitionā¶ Integer partitions
St001432: Integer partitions ā¶ ā¤Result quality: 33% āvalues known / values provided: 33%ādistinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> []
=> ? = 1 + 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? = 2 + 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3 = 1 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 3 = 1 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 3 = 1 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 3 = 1 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 3 = 1 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 4 = 2 + 2
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 3 = 1 + 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 1 + 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> 3 = 1 + 2
[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]
=> [5,4,2]
=> 3 = 1 + 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,3,2]
=> 3 = 1 + 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> 3 = 1 + 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> ? = 2 + 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> ? = 1 + 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [5,4,1]
=> 3 = 1 + 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [5,3,1]
=> 3 = 1 + 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,3,1]
=> 3 = 1 + 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> ? = 2 + 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,2,1]
=> 3 = 1 + 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,2,1]
=> 3 = 1 + 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> 4 = 2 + 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> 3 = 1 + 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> 4 = 2 + 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> 4 = 2 + 2
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> ? = 1 + 2
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> ? = 1 + 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> ? = 1 + 2
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> ? = 1 + 2
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> 3 = 1 + 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> ? = 1 + 2
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> ? = 2 + 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> 3 = 1 + 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> ? = 1 + 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> ? = 1 + 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [6,5,3]
=> ? = 1 + 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [6,4,3]
=> ? = 1 + 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [5,4,3]
=> 3 = 1 + 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> ? = 2 + 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3]
=> ? = 1 + 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [6,5,2]
=> ? = 1 + 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [6,4,2]
=> 3 = 1 + 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [5,4,2]
=> 3 = 1 + 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2]
=> ? = 2 + 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [6,3,2]
=> 3 = 1 + 2
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [5,3,2]
=> 3 = 1 + 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2]
=> ? = 2 + 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [4,3,2]
=> 3 = 1 + 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2]
=> ? = 2 + 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2]
=> ? = 2 + 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> ? = 1 + 2
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> ? = 1 + 2
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2]
=> ? = 1 + 2
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2]
=> ? = 1 + 2
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2]
=> ? = 1 + 2
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2]
=> ? = 1 + 2
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2]
=> ? = 2 + 2
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2]
=> ? = 1 + 2
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [6,5,1]
=> 3 = 1 + 2
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [6,4,1]
=> 3 = 1 + 2
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [5,4,1]
=> 3 = 1 + 2
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1]
=> ? = 2 + 2
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [6,3,1]
=> 3 = 1 + 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [5,3,1]
=> 3 = 1 + 2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [6,5,3,1]
=> ? = 2 + 2
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [4,3,1]
=> 3 = 1 + 2
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [6,4,3,1]
=> ? = 2 + 2
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [5,4,3,1]
=> ? = 2 + 2
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1]
=> ? = 1 + 2
[1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,1]
=> ? = 1 + 2
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [6,2,1]
=> 3 = 1 + 2
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [5,2,1]
=> 3 = 1 + 2
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1]
=> ? = 2 + 2
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [4,2,1]
=> 3 = 1 + 2
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [6,4,2,1]
=> ? = 2 + 2
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [5,4,2,1]
=> 4 = 2 + 2
[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]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1]
=> ? = 1 + 2
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [3,2,1]
=> 3 = 1 + 2
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1]
=> 4 = 2 + 2
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1]
=> 4 = 2 + 2
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1]
=> ? = 1 + 2
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1]
=> 4 = 2 + 2
[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]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1]
=> ? = 1 + 2
[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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1]
=> ? = 1 + 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ? = 1 + 2
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> ? = 1 + 2
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,1,1]
=> ? = 1 + 2
[1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [5,3,3,2,2,1]
=> ? = 1 + 2
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,3,3,2,1]
=> ? = 1 + 2
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ? = 1 + 2
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,1]
=> 3 = 1 + 2
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,3,2,1]
=> 3 = 1 + 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
=> [5,4,3]
=> 3 = 1 + 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [6,4,2]
=> 3 = 1 + 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> [4,3,2]
=> 3 = 1 + 2
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [3,3,2,1]
=> 3 = 1 + 2
Description
The order dimension of the partition.
Given a partition $\lambda$, let $I(\lambda)$ be the principal order ideal in the Young lattice generated by $\lambda$. The order dimension of a partition is defined as the order dimension of the poset $I(\lambda)$.
Matching statistic: St000649
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00030: Dyck paths āzeta mapā¶ Dyck paths
Mp00119: Dyck paths āto 321-avoiding permutation (Krattenthaler)ā¶ Permutations
Mp00236: Permutations āClarke-Steingrimsson-Zeng inverseā¶ Permutations
St000649: Permutations ā¶ ā¤Result quality: 24% āvalues known / values provided: 24%ādistinct values known / distinct values provided: 100%
Mp00119: Dyck paths āto 321-avoiding permutation (Krattenthaler)ā¶ Permutations
Mp00236: Permutations āClarke-Steingrimsson-Zeng inverseā¶ Permutations
St000649: Permutations ā¶ ā¤Result quality: 24% āvalues known / values provided: 24%ādistinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => [3,1,2] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,1,2,4] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [5,1,2,3,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => [5,4,3,1,2] => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => [5,4,1,3,2] => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => [5,4,1,2,3] => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [4,1,2,3,5] => 1 = 2 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,1,2,3,4,5] => [6,1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => [3,1,2,6,4,5] => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,1,4,6,2,5] => [6,4,3,1,2,5] => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,4,1,6,2,5] => [6,4,1,3,2,5] => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,1,2,6,3,5] => [6,4,1,2,3,5] => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,1,2,3,6,5] => [4,1,2,3,6,5] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,2,4,6,5] => [3,1,2,4,6,5] => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,1,5,6,2,4] => [6,3,1,2,5,4] => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,1,2] => [6,1,5,4,3,2] => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [4,1,5,6,2,3] => [6,1,2,5,4,3] => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,1,2,5,6,3] => [6,5,4,1,2,3] => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,5,1,6,2,4] => [6,1,3,2,5,4] => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [4,5,1,6,2,3] => [6,1,5,4,2,3] => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,1,5,2,6,3] => [6,5,1,2,4,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,1,2,6,3,4] => [6,1,2,3,5,4] => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,5,1,2,6,3] => [6,5,1,4,2,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,6,4] => [6,5,1,2,3,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => [5,1,2,3,4,6] => 0 = 1 - 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,1,2,5,4,6] => [3,1,2,5,4,6] => 0 = 1 - 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [3,1,4,5,2,6] => [5,4,3,1,2,6] => 0 = 1 - 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [3,4,1,5,2,6] => [5,4,1,3,2,6] => 0 = 1 - 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,6,4] => [3,1,2,6,5,4] => 0 = 1 - 1
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,1,2,5,3,6] => [5,4,1,2,3,6] => 0 = 1 - 1
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,1,2,3,5,6] => [4,1,2,3,5,6] => 1 = 2 - 1
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6] => [3,1,2,4,5,6] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,2,7,4,5,6] => [3,1,2,7,4,5,6] => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [3,1,4,7,2,5,6] => [7,4,3,1,2,5,6] => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [3,4,1,7,2,5,6] => [7,4,1,3,2,5,6] => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [4,1,2,7,3,5,6] => [7,4,1,2,3,5,6] => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => [4,1,2,3,7,5,6] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,1,2,4,7,5,6] => [3,1,2,4,7,5,6] => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [3,1,5,7,2,4,6] => [7,3,1,2,5,4,6] => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,4,5,7,1,2,6] => [7,1,5,4,3,2,6] => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [4,1,5,7,2,3,6] => [7,1,2,5,4,3,6] => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [4,1,2,5,7,3,6] => [7,5,4,1,2,3,6] => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,5,1,7,2,4,6] => [7,1,3,2,5,4,6] => ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [4,5,1,7,2,3,6] => [7,1,5,4,2,3,6] => ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [4,1,5,2,7,3,6] => [7,5,1,2,4,3,6] => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [5,1,2,7,3,4,6] => [7,1,2,3,5,4,6] => ? = 1 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [4,5,1,2,7,3,6] => [7,5,1,4,2,3,6] => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [5,1,2,3,7,4,6] => [7,5,1,2,3,4,6] => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,1,2,3,4,7,6] => [5,1,2,3,4,7,6] => ? = 1 - 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,1,2,5,4,7,6] => [3,1,2,5,4,7,6] => ? = 1 - 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [3,1,4,5,2,7,6] => [5,4,3,1,2,7,6] => ? = 1 - 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [3,4,1,5,2,7,6] => [5,4,1,3,2,7,6] => ? = 1 - 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [3,1,2,5,7,4,6] => [3,1,2,7,5,4,6] => ? = 1 - 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [4,1,2,5,3,7,6] => [5,4,1,2,3,7,6] => ? = 1 - 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,1,2,3,5,7,6] => [4,1,2,3,5,7,6] => ? = 2 - 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [3,1,2,4,5,7,6] => [3,1,2,4,5,7,6] => ? = 1 - 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [3,1,6,7,2,4,5] => [7,3,1,2,6,4,5] => ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [3,4,6,7,1,2,5] => [7,1,6,4,3,2,5] => ? = 1 - 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [4,1,6,7,2,3,5] => [7,1,2,6,4,3,5] => ? = 1 - 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [4,1,2,6,7,3,5] => [7,4,1,2,3,6,5] => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [3,5,6,7,1,2,4] => [7,1,6,3,2,5,4] => ? = 1 - 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,5,6,7,1,2,3] => [7,1,6,2,5,4,3] => ? = 1 - 1
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [4,1,5,6,7,2,3] => [7,1,2,6,5,4,3] => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [5,1,6,7,2,3,4] => [7,1,2,6,3,5,4] => ? = 1 - 1
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [4,5,1,6,7,2,3] => [7,1,6,5,4,2,3] => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [5,1,2,6,7,3,4] => [7,1,2,3,6,5,4] => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [5,1,2,3,6,7,4] => [7,6,5,1,2,3,4] => ? = 1 - 1
[1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [3,1,2,5,6,7,4] => [3,1,2,7,6,5,4] => ? = 1 - 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [3,6,1,7,2,4,5] => [7,1,3,2,6,4,5] => ? = 1 - 1
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [4,6,1,7,2,3,5] => [7,1,6,4,2,3,5] => ? = 1 - 1
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,1,6,2,7,3,5] => [7,1,2,4,3,6,5] => ? = 2 - 1
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [5,6,1,7,2,3,4] => [7,1,6,2,3,5,4] => ? = 1 - 1
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [4,5,6,1,7,2,3] => [7,1,6,5,2,4,3] => ? = 2 - 1
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [5,1,6,2,7,3,4] => [7,1,2,6,5,3,4] => ? = 2 - 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [5,1,2,6,3,7,4] => [7,6,1,2,3,5,4] => ? = 1 - 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [6,1,2,7,3,4,5] => [7,1,2,3,6,4,5] => ? = 1 - 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [4,6,1,2,7,3,5] => [7,1,4,2,3,6,5] => ? = 2 - 1
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,6,1,2,7,3,4] => [7,1,6,5,2,3,4] => ? = 2 - 1
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [5,1,6,2,3,7,4] => [7,6,1,2,5,3,4] => ? = 1 - 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [6,1,2,3,7,4,5] => [7,1,2,3,4,6,5] => ? = 2 - 1
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [5,6,1,2,3,7,4] => [7,6,1,5,2,3,4] => ? = 1 - 1
Description
The number of 3-excedences of a permutation.
This is the number of positions $1\leq i\leq n$ such that $\sigma(i)=i+3$.
Matching statistic: St001162
Mp00027: Dyck paths āto partitionā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00119: Dyck paths āto 321-avoiding permutation (Krattenthaler)ā¶ Permutations
St001162: Permutations ā¶ ā¤Result quality: 21% āvalues known / values provided: 21%ādistinct values known / distinct values provided: 100%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00119: Dyck paths āto 321-avoiding permutation (Krattenthaler)ā¶ Permutations
St001162: Permutations ā¶ ā¤Result quality: 21% āvalues known / values provided: 21%ādistinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 2
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,6,2,3,7,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,6,3] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,3,6,4] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 2
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,4,7,2,8,3,5,9,6] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,4,6,7,2,3,8,5] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,6,2,3,4,7,8,5] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [3,4,6,1,2,7,8,5] => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,2,5,7,3,4,8,9,6] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,4,6,2,3,7,8,5] => ? = 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,2,5,6,7,3] => 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,2,7,3,4,5] => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [3,4,6,7,1,2,5] => ? = 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,2,5,7,8,3,4,6] => ? = 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,7,2,3,5] => ? = 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [3,6,1,2,4,7,5] => ? = 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,2,7,3,4,5,8,6] => ? = 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,2,3,4,7,5] => ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,4,5,7,2,3,8,6] => ? = 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [3,4,6,1,2,7,5] => ? = 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,2,5,7,3,4,8,6] => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,6,2,3,7,5] => 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,6,3] => 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,4,7,5] => 1
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,3,6,4] => 1
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0,0]
=> ? => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> ? => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0,0]
=> ? => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2,1]
=> [1,1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> ? => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> ? => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> ? => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1,1]
=> [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? => ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,1,1]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0,0]
=> [3,6,7,1,8,2,4,9,10,5] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1,1]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,1,1]
=> [1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? => ? = 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? => ? = 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [4,3,3,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? => ? = 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,4,2,1,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> [3,4,6,1,2,7,8,9,5] => ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? => ? = 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? => ? = 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? => ? = 2
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,4,2,5,6,7,8,3] => ? = 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> ? => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? => ? = 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? => ? = 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? => ? = 2
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> ? => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> ? => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> ? => ? = 2
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? => ? = 1
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2]
=> [1,1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? => ? = 2
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,2,5,6,7,3] => 1
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,6,2,3,7,5] => 1
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,6,3] => 1
[1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 1
[1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => 1
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,4,7,5] => 1
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,3,6,4] => 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 2
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,6,2,3,7,5] => 1
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
Description
The minimum jump of a permutation.
This is $\min_i |\pi_{i+1}-\pi_i|$, see [1].
Matching statistic: St001344
Mp00027: Dyck paths āto partitionā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00119: Dyck paths āto 321-avoiding permutation (Krattenthaler)ā¶ Permutations
St001344: Permutations ā¶ ā¤Result quality: 21% āvalues known / values provided: 21%ādistinct values known / distinct values provided: 100%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00119: Dyck paths āto 321-avoiding permutation (Krattenthaler)ā¶ Permutations
St001344: Permutations ā¶ ā¤Result quality: 21% āvalues known / values provided: 21%ādistinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 2
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,6,2,3,7,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,6,3] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,3,6,4] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 2
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,4,7,2,8,3,5,9,6] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,4,6,7,2,3,8,5] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,6,2,3,4,7,8,5] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [3,4,6,1,2,7,8,5] => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,2,5,7,3,4,8,9,6] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,4,6,2,3,7,8,5] => ? = 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,2,5,6,7,3] => 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,2,7,3,4,5] => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [3,4,6,7,1,2,5] => ? = 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,2,5,7,8,3,4,6] => ? = 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,7,2,3,5] => ? = 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [3,6,1,2,4,7,5] => ? = 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,2,7,3,4,5,8,6] => ? = 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,2,3,4,7,5] => ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,4,5,7,2,3,8,6] => ? = 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [3,4,6,1,2,7,5] => ? = 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,2,5,7,3,4,8,6] => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,6,2,3,7,5] => 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,6,3] => 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,4,7,5] => 1
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,3,6,4] => 1
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0,0]
=> ? => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> ? => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0,0]
=> ? => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2,1]
=> [1,1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> ? => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> ? => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> ? => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1,1]
=> [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? => ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,1,1]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0,0]
=> [3,6,7,1,8,2,4,9,10,5] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1,1]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,1,1]
=> [1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? => ? = 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? => ? = 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [4,3,3,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? => ? = 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,4,2,1,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> [3,4,6,1,2,7,8,9,5] => ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? => ? = 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? => ? = 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? => ? = 2
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,4,2,5,6,7,8,3] => ? = 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> ? => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? => ? = 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? => ? = 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? => ? = 2
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> ? => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> ? => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> ? => ? = 2
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? => ? = 1
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2]
=> [1,1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? => ? = 2
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,2,5,6,7,3] => 1
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,6,2,3,7,5] => 1
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,6,3] => 1
[1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 1
[1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => 1
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,4,7,5] => 1
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,3,6,4] => 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 2
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,6,2,3,7,5] => 1
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
Description
The neighbouring number of a permutation.
For a permutation $\pi$, this is
$$\min \big(\big\{|\pi(k)-\pi(k+1)|:k\in\{1,\ldots,n-1\}\big\}\cup \big\{|\pi(1) - \pi(n)|\big\}\big).$$
Matching statistic: St000900
Mp00027: Dyck paths āto partitionā¶ Integer partitions
Mp00045: Integer partitions āreading tableauā¶ Standard tableaux
Mp00294: Standard tableaux āpeak compositionā¶ Integer compositions
St000900: Integer compositions ā¶ ā¤Result quality: 19% āvalues known / values provided: 19%ādistinct values known / distinct values provided: 100%
Mp00045: Integer partitions āreading tableauā¶ Standard tableaux
Mp00294: Standard tableaux āpeak compositionā¶ Integer compositions
St000900: Integer compositions ā¶ ā¤Result quality: 19% āvalues known / values provided: 19%ādistinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
=> [2,1]
=> [[1,3],[2]]
=> [3] => 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [3,3] => 2
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,3],[2]]
=> [3] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,3,6,10],[2,5,9],[4,8],[7]]
=> [3,3,4] => ? = 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [4,3] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [2,2,3] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [4,3] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [3,4] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [3,3] => 2
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,3],[2]]
=> [3] => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
=> [3,3,4,5] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
=> [3,2,3,4] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[1,4,5,12],[2,7,8],[3,10,11],[6],[9]]
=> [5,3,4] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,4,7,8],[2,6,11,12],[3,10],[5],[9]]
=> [4,4,4] => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,4,7,11,12],[2,6,10],[3,9],[5],[8]]
=> [4,3,5] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
=> [4,3,4] => ? = 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [5,3] => 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [[1,2,5,12],[3,4,8],[6,7,11],[9,10]]
=> [2,3,3,4] => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [2,2,4,4] => ? = 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> [2,2,3,5] => ? = 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [[1,2,7,11],[3,4,10],[5,6],[8,9]]
=> [2,2,3,4] => ? = 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [[1,3,4,8],[2,6,7,12],[5,10,11],[9]]
=> [4,4,4] => ? = 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [[1,3,4,11,12],[2,6,7],[5,9,10],[8]]
=> [4,3,5] => ? = 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [[1,3,4,11],[2,6,7],[5,9,10],[8]]
=> [4,3,4] => ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> [3,4,5] => ? = 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [[1,3,6,7],[2,5,10,11],[4,9],[8]]
=> [3,4,4] => ? = 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> [3,3,5] => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [[1,3,6,10],[2,5,9],[4,8],[7]]
=> [3,3,4] => ? = 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [4,3] => 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [2,2,3] => 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [4,3] => 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [3,5] => 1
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [3,4] => 1
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [3,3] => 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [[1,3],[2]]
=> [3] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> [[1,3,6,10,15,21],[2,5,9,14,20],[4,8,13,19],[7,12,18],[11,17],[16]]
=> ? => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> [[1,3,6,13,18],[2,5,9,17],[4,8,12],[7,11,16],[10,15],[14]]
=> ? => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2,1]
=> [[1,3,8,9,18],[2,5,12,13],[4,7,16,17],[6,11],[10,15],[14]]
=> ? => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2,1]
=> [[1,3,8,12,13],[2,5,11,17,18],[4,7,16],[6,10],[9,15],[14]]
=> ? => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> [[1,3,8,12,17,18],[2,5,11,16],[4,7,15],[6,10],[9,14],[13]]
=> ? => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,2,1]
=> [[1,3,8,12,17],[2,5,11,16],[4,7,15],[6,10],[9,14],[13]]
=> ? => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,2,1]
=> [[1,3,10,14],[2,5,13],[4,7],[6,9],[8,12],[11]]
=> ? => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,1,1]
=> [[1,4,5,9,18],[2,7,8,13],[3,11,12,17],[6,15,16],[10],[14]]
=> ? => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> [[1,4,5,12,13],[2,7,8,17,18],[3,10,11],[6,15,16],[9],[14]]
=> ? => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1,1]
=> [[1,4,5,12,17,18],[2,7,8,16],[3,10,11],[6,14,15],[9],[13]]
=> ? => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,1,1]
=> [[1,4,5,12,17],[2,7,8,16],[3,10,11],[6,14,15],[9],[13]]
=> ? => ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,1,1]
=> [[1,4,7,8,13],[2,6,11,12,18],[3,10,16,17],[5,15],[9],[14]]
=> ? => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1,1]
=> [[1,4,7,8,17,18],[2,6,11,12],[3,10,15,16],[5,14],[9],[13]]
=> ? => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,1,1]
=> [[1,4,7,8,17],[2,6,11,12],[3,10,15,16],[5,14],[9],[13]]
=> ? => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,1]
=> [[1,4,7,11,12,18],[2,6,10,16,17],[3,9,15],[5,14],[8],[13]]
=> ? => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,1,1]
=> [[1,4,7,11,12],[2,6,10,16,17],[3,9,15],[5,14],[8],[13]]
=> ? => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,1]
=> [[1,4,7,11,16,17],[2,6,10,15],[3,9,14],[5,13],[8],[12]]
=> ? => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,1]
=> [[1,4,7,11,16],[2,6,10,15],[3,9,14],[5,13],[8],[12]]
=> ? => ? = 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,1,1]
=> [[1,4,9,13],[2,6,12],[3,8],[5,11],[7],[10]]
=> ? => ? = 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [4,3,3,1,1,1]
=> [[1,5,6,13],[2,8,9],[3,11,12],[4],[7],[10]]
=> ? => ? = 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,4,2,1,1,1]
=> [[1,5,8,9],[2,7,12,13],[3,11],[4],[6],[10]]
=> ? => ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,1,1]
=> [[1,5,8,12,13,14],[2,7,11],[3,10],[4],[6],[9]]
=> ? => ? = 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1,1,1]
=> [[1,5,8,12,13],[2,7,11],[3,10],[4],[6],[9]]
=> ? => ? = 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1,1,1]
=> [[1,5,8,12],[2,7,11],[3,10],[4],[6],[9]]
=> ? => ? = 2
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> [6,3] => 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> [[1,2,5,9,18],[3,4,8,13],[6,7,12,17],[10,11,16],[14,15]]
=> ? => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2]
=> [[1,2,5,12,13],[3,4,8,17,18],[6,7,11],[9,10,16],[14,15]]
=> ? => ? = 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> [[1,2,5,12,17,18],[3,4,8,16],[6,7,11],[9,10,15],[13,14]]
=> ? => ? = 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2]
=> [[1,2,5,12,17],[3,4,8,16],[6,7,11],[9,10,15],[13,14]]
=> ? => ? = 2
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2]
=> [[1,2,7,8,13],[3,4,11,12,18],[5,6,16,17],[9,10],[14,15]]
=> ? => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2]
=> [[1,2,7,8,17,18],[3,4,11,12],[5,6,15,16],[9,10],[13,14]]
=> ? => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2]
=> [[1,2,7,8,17],[3,4,11,12],[5,6,15,16],[9,10],[13,14]]
=> ? => ? = 2
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2]
=> [[1,2,7,11,12,18],[3,4,10,16,17],[5,6,15],[8,9],[13,14]]
=> ? => ? = 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [5,3] => 1
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [4,3] => 1
[1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [2,2,3] => 1
[1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [4,3] => 1
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> [3,6] => 1
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [3,5] => 1
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [3,4] => 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [3,3] => 2
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,1]
=> [[1,3],[2]]
=> [3] => 1
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [2,1]
=> [[1,3],[2]]
=> [3] => 1
Description
The minimal number of repetitions of a part in an integer composition.
This is the smallest letter in the word obtained by applying the delta morphism.
Matching statistic: St000902
Mp00027: Dyck paths āto partitionā¶ Integer partitions
Mp00045: Integer partitions āreading tableauā¶ Standard tableaux
Mp00294: Standard tableaux āpeak compositionā¶ Integer compositions
St000902: Integer compositions ā¶ ā¤Result quality: 19% āvalues known / values provided: 19%ādistinct values known / distinct values provided: 100%
Mp00045: Integer partitions āreading tableauā¶ Standard tableaux
Mp00294: Standard tableaux āpeak compositionā¶ Integer compositions
St000902: Integer compositions ā¶ ā¤Result quality: 19% āvalues known / values provided: 19%ādistinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
=> [2,1]
=> [[1,3],[2]]
=> [3] => 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [3,3] => 2
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,3],[2]]
=> [3] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,3,6,10],[2,5,9],[4,8],[7]]
=> [3,3,4] => ? = 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [4,3] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [2,2,3] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [4,3] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [3,4] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [3,3] => 2
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,3],[2]]
=> [3] => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
=> [3,3,4,5] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
=> [3,2,3,4] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[1,4,5,12],[2,7,8],[3,10,11],[6],[9]]
=> [5,3,4] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,4,7,8],[2,6,11,12],[3,10],[5],[9]]
=> [4,4,4] => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,4,7,11,12],[2,6,10],[3,9],[5],[8]]
=> [4,3,5] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
=> [4,3,4] => ? = 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [5,3] => 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [[1,2,5,12],[3,4,8],[6,7,11],[9,10]]
=> [2,3,3,4] => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [2,2,4,4] => ? = 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> [2,2,3,5] => ? = 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [[1,2,7,11],[3,4,10],[5,6],[8,9]]
=> [2,2,3,4] => ? = 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [[1,3,4,8],[2,6,7,12],[5,10,11],[9]]
=> [4,4,4] => ? = 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [[1,3,4,11,12],[2,6,7],[5,9,10],[8]]
=> [4,3,5] => ? = 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [[1,3,4,11],[2,6,7],[5,9,10],[8]]
=> [4,3,4] => ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> [3,4,5] => ? = 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [[1,3,6,7],[2,5,10,11],[4,9],[8]]
=> [3,4,4] => ? = 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> [3,3,5] => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [[1,3,6,10],[2,5,9],[4,8],[7]]
=> [3,3,4] => ? = 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [4,3] => 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [2,2,3] => 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [4,3] => 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [3,5] => 1
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [3,4] => 1
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [3,3] => 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [[1,3],[2]]
=> [3] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> [[1,3,6,10,15,21],[2,5,9,14,20],[4,8,13,19],[7,12,18],[11,17],[16]]
=> ? => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> [[1,3,6,13,18],[2,5,9,17],[4,8,12],[7,11,16],[10,15],[14]]
=> ? => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2,1]
=> [[1,3,8,9,18],[2,5,12,13],[4,7,16,17],[6,11],[10,15],[14]]
=> ? => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2,1]
=> [[1,3,8,12,13],[2,5,11,17,18],[4,7,16],[6,10],[9,15],[14]]
=> ? => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> [[1,3,8,12,17,18],[2,5,11,16],[4,7,15],[6,10],[9,14],[13]]
=> ? => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,2,1]
=> [[1,3,8,12,17],[2,5,11,16],[4,7,15],[6,10],[9,14],[13]]
=> ? => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,2,1]
=> [[1,3,10,14],[2,5,13],[4,7],[6,9],[8,12],[11]]
=> ? => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,1,1]
=> [[1,4,5,9,18],[2,7,8,13],[3,11,12,17],[6,15,16],[10],[14]]
=> ? => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> [[1,4,5,12,13],[2,7,8,17,18],[3,10,11],[6,15,16],[9],[14]]
=> ? => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1,1]
=> [[1,4,5,12,17,18],[2,7,8,16],[3,10,11],[6,14,15],[9],[13]]
=> ? => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,1,1]
=> [[1,4,5,12,17],[2,7,8,16],[3,10,11],[6,14,15],[9],[13]]
=> ? => ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,1,1]
=> [[1,4,7,8,13],[2,6,11,12,18],[3,10,16,17],[5,15],[9],[14]]
=> ? => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1,1]
=> [[1,4,7,8,17,18],[2,6,11,12],[3,10,15,16],[5,14],[9],[13]]
=> ? => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,1,1]
=> [[1,4,7,8,17],[2,6,11,12],[3,10,15,16],[5,14],[9],[13]]
=> ? => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,1]
=> [[1,4,7,11,12,18],[2,6,10,16,17],[3,9,15],[5,14],[8],[13]]
=> ? => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,1,1]
=> [[1,4,7,11,12],[2,6,10,16,17],[3,9,15],[5,14],[8],[13]]
=> ? => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,1]
=> [[1,4,7,11,16,17],[2,6,10,15],[3,9,14],[5,13],[8],[12]]
=> ? => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,1]
=> [[1,4,7,11,16],[2,6,10,15],[3,9,14],[5,13],[8],[12]]
=> ? => ? = 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,1,1]
=> [[1,4,9,13],[2,6,12],[3,8],[5,11],[7],[10]]
=> ? => ? = 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [4,3,3,1,1,1]
=> [[1,5,6,13],[2,8,9],[3,11,12],[4],[7],[10]]
=> ? => ? = 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,4,2,1,1,1]
=> [[1,5,8,9],[2,7,12,13],[3,11],[4],[6],[10]]
=> ? => ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,1,1]
=> [[1,5,8,12,13,14],[2,7,11],[3,10],[4],[6],[9]]
=> ? => ? = 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1,1,1]
=> [[1,5,8,12,13],[2,7,11],[3,10],[4],[6],[9]]
=> ? => ? = 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1,1,1]
=> [[1,5,8,12],[2,7,11],[3,10],[4],[6],[9]]
=> ? => ? = 2
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> [6,3] => 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> [[1,2,5,9,18],[3,4,8,13],[6,7,12,17],[10,11,16],[14,15]]
=> ? => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2]
=> [[1,2,5,12,13],[3,4,8,17,18],[6,7,11],[9,10,16],[14,15]]
=> ? => ? = 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> [[1,2,5,12,17,18],[3,4,8,16],[6,7,11],[9,10,15],[13,14]]
=> ? => ? = 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2]
=> [[1,2,5,12,17],[3,4,8,16],[6,7,11],[9,10,15],[13,14]]
=> ? => ? = 2
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2]
=> [[1,2,7,8,13],[3,4,11,12,18],[5,6,16,17],[9,10],[14,15]]
=> ? => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2]
=> [[1,2,7,8,17,18],[3,4,11,12],[5,6,15,16],[9,10],[13,14]]
=> ? => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2]
=> [[1,2,7,8,17],[3,4,11,12],[5,6,15,16],[9,10],[13,14]]
=> ? => ? = 2
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2]
=> [[1,2,7,11,12,18],[3,4,10,16,17],[5,6,15],[8,9],[13,14]]
=> ? => ? = 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [5,3] => 1
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [4,3] => 1
[1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [2,2,3] => 1
[1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [4,3] => 1
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> [3,6] => 1
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [3,5] => 1
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [3,4] => 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [3,3] => 2
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,1]
=> [[1,3],[2]]
=> [3] => 1
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [2,1]
=> [[1,3],[2]]
=> [3] => 1
Description
The minimal number of repetitions of an integer composition.
Matching statistic: St001006
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00027: Dyck paths āto partitionā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
St001006: Dyck paths ā¶ ā¤Result quality: 15% āvalues known / values provided: 15%ādistinct values known / distinct values provided: 100%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
St001006: Dyck paths ā¶ ā¤Result quality: 15% āvalues known / values provided: 15%ādistinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 1
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2,1]
=> [1,1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1,1]
=> [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,1,1]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1,1]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,1,1]
=> [1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [4,3,3,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,4,2,1,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 2
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> ? = 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 2
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> ? = 1
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
Description
Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001191
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00027: Dyck paths āto partitionā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
St001191: Dyck paths ā¶ ā¤Result quality: 15% āvalues known / values provided: 15%ādistinct values known / distinct values provided: 100%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
St001191: Dyck paths ā¶ ā¤Result quality: 15% āvalues known / values provided: 15%ādistinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 1
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2,1]
=> [1,1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1,1]
=> [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,1,1]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1,1]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,1,1]
=> [1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [4,3,3,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,4,2,1,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 2
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> ? = 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 2
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> ? = 1
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
Description
Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$.
Matching statistic: St000772
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00227: Dyck paths āDelest-Viennot-inverseā¶ Dyck paths
Mp00024: Dyck paths āto 321-avoiding permutationā¶ Permutations
Mp00160: Permutations āgraph of inversionsā¶ Graphs
St000772: Graphs ā¶ ā¤Result quality: 15% āvalues known / values provided: 15%ādistinct values known / distinct values provided: 50%
Mp00024: Dyck paths āto 321-avoiding permutationā¶ Permutations
Mp00160: Permutations āgraph of inversionsā¶ Graphs
St000772: Graphs ā¶ ā¤Result quality: 15% āvalues known / values provided: 15%ādistinct values known / distinct values provided: 50%
Values
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ? = 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ? = 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([],5)
=> ? = 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ? = 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? = 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? = 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? = 2
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ([],6)
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [4,1,5,2,6,3] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 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]
=> [3,1,4,2,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ? = 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]
=> [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 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]
=> [3,4,1,2,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 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]
=> [3,1,4,2,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ? = 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ? = 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,6,3,4,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ? = 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]
=> [2,1,5,6,3,4] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 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]
=> [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 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]
=> [2,1,5,3,6,4] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ? = 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,5,6,3] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ? = 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]
=> [2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ? = 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]
=> [2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 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]
=> [2,1,4,6,3,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ? = 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6)
=> ? = 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> ? = 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,2,3,5,6] => ([(3,5),(4,5)],6)
=> ? = 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ? = 1
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,2,3,4,6,5] => ([(4,5)],6)
=> ? = 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => ([],7)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [5,1,6,2,7,3,4] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [4,1,5,2,6,7,3] => ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [4,1,5,6,2,3,7] => ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [4,5,1,2,6,3,7] => ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [4,1,5,2,6,3,7] => ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,2,3,7,4,5,6] => ([(3,6),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,2,5,6,7] => ([(3,6),(4,5),(5,6)],7)
=> ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,7,2,5,6] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,7,5,6] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,7,5,6] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,7,2,5] => ([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [3,4,1,2,6,7,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [3,1,4,2,6,7,5] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [3,4,6,1,2,5,7] => ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [3,1,4,6,2,5,7] => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [3,4,1,2,6,5,7] => ([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,4,2,6,5,7] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> ? = 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [3,4,5,6,1,7,2] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,4,5,2,6,7,3] => ([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,5,2,3,6,7,4] => ([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,2,3,6,7,4,5] => ([(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [6,1,2,3,4,7,5] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> ? = 2
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,4,2,5,3,7,6] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> ? = 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,6,1,3,7,4,5] => ([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,6,7,4] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 1
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,4,7,1,3,5,6] => ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,6,7,1,3,5] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 1
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,4,5,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 1
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,3,5,1,6,7,4] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 1
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,4,5,7,1,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 1
[1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [3,5,6,1,7,2,4] => ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 1
[1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [4,6,1,7,2,3,5] => ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 1
[1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [4,1,2,7,3,5,6] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 1
[1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [5,7,1,2,3,4,6] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 1
[1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [5,1,7,2,3,4,6] => ([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $1$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$.
The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
The following 53 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{nā1}]$ by adding $c_0$ to $c_{nā1}$. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000260The radius of a connected graph. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000317The cycle descent number of a permutation. St000624The normalized sum of the minimal distances to a greater element. 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 $eA$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001394The genus of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000021The number of descents of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000366The number of double descents of a permutation. St000367The number of simsun double descents of a permutation. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001115The number of even descents of a permutation. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. 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 $eA$. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001778The largest greatest common divisor of an element and its image in a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000670The reversal length of a permutation. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000714The number of semistandard Young tableau of given shape, with entries at most 2.
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