Processing math: 100%

Your data matches 22 different statistics following compositions of up to 3 maps.
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Matching statistic: St000944
St000944: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 0
[1,1]
=> 0
[3]
=> 1
[2,1]
=> 1
[1,1,1]
=> 0
[4]
=> 1
[3,1]
=> 0
[2,2]
=> 1
[2,1,1]
=> 0
[1,1,1,1]
=> 0
[5]
=> 1
[4,1]
=> 4
[3,2]
=> 1
[3,1,1]
=> 0
[2,2,1]
=> 4
[2,1,1,1]
=> 0
[1,1,1,1,1]
=> 0
[6]
=> 2
[5,1]
=> 9
[4,2]
=> 0
[4,1,1]
=> 16
[3,3]
=> 6
[3,2,1]
=> 16
[3,1,1,1]
=> 4
[2,2,2]
=> 4
[2,2,1,1]
=> 0
[2,1,1,1,1]
=> 1
[1,1,1,1,1,1]
=> 0
[7]
=> 2
[6,1]
=> 6
[5,2]
=> 27
[5,1,1]
=> 15
[4,3]
=> 15
[4,2,1]
=> 42
[4,1,1,1]
=> 20
[3,3,1]
=> 6
[3,2,2]
=> 15
[3,2,1,1]
=> 28
[3,1,1,1,1]
=> 0
[2,2,2,1]
=> 13
[2,2,1,1,1]
=> 1
[2,1,1,1,1,1]
=> 0
[1,1,1,1,1,1,1]
=> 0
[8]
=> 2
[7,1]
=> 14
[6,2]
=> 33
[6,1,1]
=> 21
[5,3]
=> 56
[5,2,1]
=> 99
Description
The 3-degree of an integer partition. For an integer partition λ, this is given by the exponent of 3 in the Gram determinant of the integal Specht module of the symmetric group indexed by λ. This stupid comment should not be accepted as an edit!
Matching statistic: St001964
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00132: Dyck paths switch returns and last double riseDyck paths
Mp00232: Dyck paths parallelogram posetPosets
St001964: Posets ⟶ ℤResult quality: 3% values known / values provided: 20%distinct values known / distinct values provided: 3%
Values
[1]
=> [1,0,1,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 0
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 0
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? ∊ {0,1}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {0,1}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? ∊ {1,1,4,4}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? ∊ {1,1,4,4}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? ∊ {1,1,4,4}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? ∊ {1,1,4,4}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? ∊ {0,1,2,4,4,6,9,16,16}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? ∊ {0,1,2,4,4,6,9,16,16}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? ∊ {0,1,2,4,4,6,9,16,16}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? ∊ {0,1,2,4,4,6,9,16,16}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? ∊ {0,1,2,4,4,6,9,16,16}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? ∊ {0,1,2,4,4,6,9,16,16}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? ∊ {0,1,2,4,4,6,9,16,16}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? ∊ {0,1,2,4,4,6,9,16,16}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ? ∊ {0,1,2,4,4,6,9,16,16}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ([(0,6),(0,7),(2,5),(2,16),(3,4),(3,15),(4,9),(5,10),(6,3),(6,14),(7,2),(7,14),(8,1),(9,11),(10,12),(11,8),(12,8),(13,11),(13,12),(14,15),(14,16),(15,9),(15,13),(16,10),(16,13)],17)
=> ? ∊ {1,2,6,6,13,15,15,15,20,27,28,42}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,6),(2,9),(3,8),(4,3),(4,7),(5,2),(5,7),(6,4),(6,5),(7,8),(7,9),(8,10),(9,10),(10,1)],11)
=> ? ∊ {1,2,6,6,13,15,15,15,20,27,28,42}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? ∊ {1,2,6,6,13,15,15,15,20,27,28,42}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? ∊ {1,2,6,6,13,15,15,15,20,27,28,42}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? ∊ {1,2,6,6,13,15,15,15,20,27,28,42}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? ∊ {1,2,6,6,13,15,15,15,20,27,28,42}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? ∊ {1,2,6,6,13,15,15,15,20,27,28,42}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? ∊ {1,2,6,6,13,15,15,15,20,27,28,42}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {1,2,6,6,13,15,15,15,20,27,28,42}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? ∊ {1,2,6,6,13,15,15,15,20,27,28,42}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ? ∊ {1,2,6,6,13,15,15,15,20,27,28,42}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ([(0,6),(0,7),(1,4),(1,16),(2,5),(2,15),(3,13),(4,12),(5,3),(5,19),(6,1),(6,17),(7,2),(7,17),(9,11),(10,8),(11,8),(12,9),(13,10),(14,9),(14,18),(15,14),(15,19),(16,12),(16,14),(17,15),(17,16),(18,10),(18,11),(19,13),(19,18)],20)
=> ? ∊ {1,2,6,6,13,15,15,15,20,27,28,42}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ([(0,7),(0,8),(2,5),(2,17),(3,6),(3,16),(4,14),(5,13),(6,4),(6,20),(7,2),(7,18),(8,3),(8,18),(9,1),(10,12),(11,9),(12,9),(13,10),(14,11),(15,10),(15,19),(16,15),(16,20),(17,13),(17,15),(18,16),(18,17),(19,11),(19,12),(20,14),(20,19)],21)
=> ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ([(0,7),(2,10),(3,9),(4,3),(4,8),(5,6),(5,8),(6,2),(6,11),(7,4),(7,5),(8,9),(8,11),(9,12),(10,13),(11,10),(11,12),(12,13),(13,1)],14)
=> ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ([(0,4),(0,6),(2,8),(3,9),(4,10),(5,3),(5,7),(6,5),(6,10),(7,8),(7,9),(8,11),(9,11),(10,2),(10,7),(11,1)],12)
=> ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,6),(1,11),(2,8),(3,9),(4,3),(4,7),(5,1),(5,7),(6,4),(6,5),(7,9),(7,11),(9,10),(10,8),(11,2),(11,10)],12)
=> ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ([(0,1),(1,2),(1,3),(2,5),(2,13),(3,7),(3,13),(4,12),(5,11),(6,4),(6,15),(7,6),(7,14),(9,10),(10,8),(11,9),(12,8),(13,11),(13,14),(14,9),(14,15),(15,10),(15,12)],16)
=> ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ([(0,7),(0,8),(1,6),(1,19),(2,4),(2,18),(3,14),(4,15),(5,3),(5,23),(6,5),(6,22),(7,1),(7,20),(8,2),(8,20),(10,11),(11,12),(12,9),(13,9),(14,13),(15,10),(16,12),(16,13),(17,11),(17,16),(18,15),(18,21),(19,21),(19,22),(20,18),(20,19),(21,10),(21,17),(22,17),(22,23),(23,14),(23,16)],24)
=> ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ([(0,8),(0,9),(2,7),(2,20),(3,6),(3,19),(4,15),(5,16),(6,4),(6,24),(7,5),(7,25),(8,3),(8,21),(9,2),(9,21),(10,1),(11,13),(12,14),(13,10),(14,10),(15,11),(16,12),(17,22),(17,23),(18,13),(18,14),(19,17),(19,24),(20,17),(20,25),(21,19),(21,20),(22,11),(22,18),(23,12),(23,18),(24,15),(24,22),(25,16),(25,23)],26)
=> ? ∊ {1,3,7,7,23,27,27,41,63,77,77,77,85,91,105,118,127,134,137,189,189,232,233,238,272}
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ([(0,2),(2,3),(2,4),(3,8),(3,15),(4,7),(4,15),(5,12),(6,13),(7,5),(7,16),(8,6),(8,17),(9,1),(10,9),(11,9),(12,10),(13,11),(14,10),(14,11),(15,16),(15,17),(16,12),(16,14),(17,13),(17,14)],18)
=> ? ∊ {1,3,7,7,23,27,27,41,63,77,77,77,85,91,105,118,127,134,137,189,189,232,233,238,272}
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,6),(2,10),(3,9),(4,3),(4,8),(5,2),(5,8),(6,7),(7,4),(7,5),(8,9),(8,10),(9,11),(10,11),(11,1)],12)
=> ? ∊ {1,3,7,7,23,27,27,41,63,77,77,77,85,91,105,118,127,134,137,189,189,232,233,238,272}
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
Description
The interval resolution global dimension of a poset. This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
Matching statistic: St001096
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00199: Dyck paths prime Dyck pathDyck paths
Mp00201: Dyck paths RingelPermutations
St001096: Permutations ⟶ ℤResult quality: 4% values known / values provided: 19%distinct values known / distinct values provided: 4%
Values
[1]
=> [1,0]
=> [1,1,0,0]
=> [2,3,1] => 1 = 0 + 1
[2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 0 + 1
[1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 0 + 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 1 = 0 + 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 2 = 1 + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 2 = 1 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 2 = 1 + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 2 = 1 + 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 0 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => 1 = 0 + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => 1 = 0 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => 1 = 0 + 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => 1 = 0 + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 2 = 1 + 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => ? ∊ {0,4,4} + 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => 2 = 1 + 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => ? ∊ {0,4,4} + 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,7,5,1,2,3,4] => ? ∊ {0,4,4} + 1
[6]
=> [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]
=> [8,7,1,2,3,4,5,6] => 1 = 0 + 1
[5,1]
=> [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]
=> [7,6,1,2,3,4,8,5] => ? ∊ {0,4,4,6,9,16,16} + 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => ? ∊ {0,4,4,6,9,16,16} + 1
[4,1,1]
=> [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]
=> [8,6,1,2,3,7,4,5] => ? ∊ {0,4,4,6,9,16,16} + 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => 2 = 1 + 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [7,3,1,5,6,2,4] => ? ∊ {0,4,4,6,9,16,16} + 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [8,7,1,2,6,3,4,5] => ? ∊ {0,4,4,6,9,16,16} + 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 1 = 0 + 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => ? ∊ {0,4,4,6,9,16,16} + 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [8,7,1,6,2,3,4,5] => ? ∊ {0,4,4,6,9,16,16} + 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [6,7,8,1,2,3,4,5] => 3 = 2 + 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,9,1,2,3,4,5,6,7] => 2 = 1 + 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [8,7,1,2,3,4,5,9,6] => ? ∊ {0,0,2,6,6,13,15,15,15,20,27,28,42} + 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [5,6,1,2,3,7,8,4] => ? ∊ {0,0,2,6,6,13,15,15,15,20,27,28,42} + 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [9,7,1,2,3,4,8,5,6] => ? ∊ {0,0,2,6,6,13,15,15,15,20,27,28,42} + 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => ? ∊ {0,0,2,6,6,13,15,15,15,20,27,28,42} + 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [8,4,1,2,6,7,3,5] => ? ∊ {0,0,2,6,6,13,15,15,15,20,27,28,42} + 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [9,8,1,2,3,7,4,5,6] => ? ∊ {0,0,2,6,6,13,15,15,15,20,27,28,42} + 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => ? ∊ {0,0,2,6,6,13,15,15,15,20,27,28,42} + 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => ? ∊ {0,0,2,6,6,13,15,15,15,20,27,28,42} + 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [8,3,1,7,6,2,4,5] => ? ∊ {0,0,2,6,6,13,15,15,15,20,27,28,42} + 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [9,8,1,2,7,3,4,5,6] => ? ∊ {0,0,2,6,6,13,15,15,15,20,27,28,42} + 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,7,5,6,1,4] => ? ∊ {0,0,2,6,6,13,15,15,15,20,27,28,42} + 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [2,7,8,6,1,3,4,5] => ? ∊ {0,0,2,6,6,13,15,15,15,20,27,28,42} + 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [8,7,1,9,2,3,4,5,6] => ? ∊ {0,0,2,6,6,13,15,15,15,20,27,28,42} + 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [9,7,8,1,2,3,4,5,6] => 1 = 0 + 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10,9,1,2,3,4,5,6,7,8] => 1 = 0 + 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [8,9,1,2,3,4,5,6,10,7] => ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [7,6,1,2,3,4,8,9,5] => ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [8,10,1,2,3,4,5,9,6,7] => ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [5,8,1,2,6,7,3,4] => ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [5,9,1,2,3,7,8,4,6] => ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [9,10,1,2,3,4,8,5,6,7] => ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => 1 = 0 + 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [8,4,1,7,6,2,3,5] => ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [5,4,1,2,6,7,8,3] => ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [9,4,1,2,8,7,3,5,6] => ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [9,10,1,2,3,8,4,5,6,7] => ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,4,5,1,7,2] => ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [7,3,8,6,1,2,4,5] => ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [4,3,1,8,6,7,2,5] => ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [8,3,1,9,7,2,4,5,6] => ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [9,8,1,2,10,3,4,5,6,7] => ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [7,3,4,5,6,1,2] => ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [2,3,8,7,6,1,4,5] => ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [2,7,8,9,1,3,4,5,6] => ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [10,8,1,9,2,3,4,5,6,7] => ? ∊ {0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [10,9,8,1,2,3,4,5,6,7] => 1 = 0 + 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [11,10,1,2,3,4,5,6,7,8,9] => ? ∊ {0,0,0,0,1,3,7,7,23,27,27,41,63,77,77,77,85,91,105,118,127,134,137,189,189,232,233,238,272} + 1
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [10,9,1,2,3,4,5,6,7,11,8] => ? ∊ {0,0,0,0,1,3,7,7,23,27,27,41,63,77,77,77,85,91,105,118,127,134,137,189,189,232,233,238,272} + 1
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [8,7,1,2,3,4,5,9,10,6] => ? ∊ {0,0,0,0,1,3,7,7,23,27,27,41,63,77,77,77,85,91,105,118,127,134,137,189,189,232,233,238,272} + 1
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [11,9,1,2,3,4,5,6,10,7,8] => ? ∊ {0,0,0,0,1,3,7,7,23,27,27,41,63,77,77,77,85,91,105,118,127,134,137,189,189,232,233,238,272} + 1
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [9,6,1,2,3,7,8,4,5] => ? ∊ {0,0,0,0,1,3,7,7,23,27,27,41,63,77,77,77,85,91,105,118,127,134,137,189,189,232,233,238,272} + 1
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [10,6,1,2,3,4,8,9,5,7] => ? ∊ {0,0,0,0,1,3,7,7,23,27,27,41,63,77,77,77,85,91,105,118,127,134,137,189,189,232,233,238,272} + 1
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [11,10,1,2,3,4,5,9,6,7,8] => ? ∊ {0,0,0,0,1,3,7,7,23,27,27,41,63,77,77,77,85,91,105,118,127,134,137,189,189,232,233,238,272} + 1
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [7,8,1,5,6,2,3,4] => ? ∊ {0,0,0,0,1,3,7,7,23,27,27,41,63,77,77,77,85,91,105,118,127,134,137,189,189,232,233,238,272} + 1
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1 = 0 + 1
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [8,7,4,5,6,1,2,3] => 1 = 0 + 1
Description
The size of the overlap set of a permutation. For a permutation πSn this is the number of indices i<n such that the standardisation of π1πni equals the standardisation of πi+1πn. In particular, for n>1, the statistic is at least one, because the standardisations of π1 and πn are both 1. For example, for π=2143, the standardisations of 21 and 43 are equal, and so are the standardisations of 2 and 3. Thus, the statistic on π is 2.
Matching statistic: St001087
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00199: Dyck paths prime Dyck pathDyck paths
Mp00201: Dyck paths RingelPermutations
St001087: Permutations ⟶ ℤResult quality: 4% values known / values provided: 12%distinct values known / distinct values provided: 4%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 0
[2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 0
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,4,7,6,1,5] => ? ∊ {1,1}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => 0
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => ? ∊ {1,1}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [2,3,4,5,8,7,1,6] => ? ∊ {0,1,4,4}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => ? ∊ {0,1,4,4}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 0
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 0
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => ? ∊ {0,1,4,4}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [4,3,1,5,6,7,8,2] => ? ∊ {0,1,4,4}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [2,3,4,5,6,9,8,1,7] => ? ∊ {0,0,1,2,4,4,6,9,16,16}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [8,3,4,5,7,1,2,6] => ? ∊ {0,0,1,2,4,4,6,9,16,16}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => ? ∊ {0,0,1,2,4,4,6,9,16,16}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,3,4,1,6,2,5] => ? ∊ {0,0,1,2,4,4,6,9,16,16}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => ? ∊ {0,0,1,2,4,4,6,9,16,16}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [7,3,1,5,6,2,4] => ? ∊ {0,0,1,2,4,4,6,9,16,16}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => ? ∊ {0,0,1,2,4,4,6,9,16,16}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => ? ∊ {0,0,1,2,4,4,6,9,16,16}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [8,4,1,5,6,7,2,3] => ? ∊ {0,0,1,2,4,4,6,9,16,16}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [4,3,1,5,6,7,8,9,2] => ? ∊ {0,0,1,2,4,4,6,9,16,16}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [2,3,4,5,6,7,10,9,1,8] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> [9,3,4,5,6,8,1,2,7] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [2,8,4,5,7,1,3,6] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [8,3,4,5,1,7,2,6] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,3,1,7,6,2,5] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [8,3,1,5,6,7,2,4] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [7,4,1,5,6,2,8,3] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [9,4,1,5,6,7,8,2,3] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [4,3,1,5,6,7,8,9,10,2] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0]
=> [2,3,4,5,6,7,8,11,10,1,9] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0]
=> [10,3,4,5,6,7,9,1,2,8] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [2,9,4,5,6,8,1,3,7] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> [9,3,4,5,6,1,8,2,7] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [2,3,8,5,7,1,4,6] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [7,8,4,5,1,2,3,6] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [5,3,4,1,8,7,2,6] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [2,3,4,7,6,1,8,5] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,7,4,1,6,3,5] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [7,4,1,6,2,3,5] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [4,3,1,8,6,7,2,5] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [6,3,1,5,2,7,4] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [7,8,1,5,6,2,3,4] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [9,3,1,5,6,7,8,2,4] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [2,5,4,1,6,7,8,3] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [6,4,1,5,2,7,8,3] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => 3
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => 0
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => 0
Description
The number of occurrences of the vincular pattern |12-3 in a permutation. This is the number of occurrences of the pattern 123, where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive. In other words, this is the number of ascents whose bottom value is strictly larger than the first entry of the permutation.
Matching statistic: St001673
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00093: Dyck paths to binary wordBinary words
Mp00178: Binary words to compositionInteger compositions
St001673: Integer compositions ⟶ ℤResult quality: 3% values known / values provided: 11%distinct values known / distinct values provided: 3%
Values
[1]
=> [1,0]
=> 10 => [1,2] => 1 = 0 + 1
[2]
=> [1,0,1,0]
=> 1010 => [1,2,2] => 1 = 0 + 1
[1,1]
=> [1,1,0,0]
=> 1100 => [1,1,3] => 1 = 0 + 1
[3]
=> [1,0,1,0,1,0]
=> 101010 => [1,2,2,2] => 1 = 0 + 1
[2,1]
=> [1,0,1,1,0,0]
=> 101100 => [1,2,1,3] => 2 = 1 + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => [1,1,2,3] => 2 = 1 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => [1,2,2,2,2] => 1 = 0 + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => [1,2,2,1,3] => 2 = 1 + 1
[2,2]
=> [1,1,1,0,0,0]
=> 111000 => [1,1,1,4] => 1 = 0 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => [1,2,1,2,3] => 1 = 0 + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => [1,1,2,2,3] => 2 = 1 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => [1,2,2,2,2,2] => ? ∊ {0,0,0,4,4} + 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => [1,2,2,2,1,3] => ? ∊ {0,0,0,4,4} + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => [1,2,1,1,4] => 2 = 1 + 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => [1,2,2,1,2,3] => ? ∊ {0,0,0,4,4} + 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => [1,1,1,3,3] => 2 = 1 + 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => [1,2,1,2,2,3] => ? ∊ {0,0,0,4,4} + 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => [1,1,2,2,2,3] => ? ∊ {0,0,0,4,4} + 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => [1,2,2,2,2,2,2] => ? ∊ {0,0,2,4,4,6,9,16,16} + 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => [1,2,2,2,2,1,3] => ? ∊ {0,0,2,4,4,6,9,16,16} + 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => [1,2,2,1,1,4] => ? ∊ {0,0,2,4,4,6,9,16,16} + 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => [1,2,2,2,1,2,3] => ? ∊ {0,0,2,4,4,6,9,16,16} + 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => [1,1,1,2,4] => 2 = 1 + 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => [1,2,1,1,3,3] => ? ∊ {0,0,2,4,4,6,9,16,16} + 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => [1,2,2,1,2,2,3] => ? ∊ {0,0,2,4,4,6,9,16,16} + 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => [1,1,1,1,5] => 1 = 0 + 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => [1,1,1,3,2,3] => ? ∊ {0,0,2,4,4,6,9,16,16} + 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 101101010100 => [1,2,1,2,2,2,3] => ? ∊ {0,0,2,4,4,6,9,16,16} + 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 110101010100 => [1,1,2,2,2,2,3] => ? ∊ {0,0,2,4,4,6,9,16,16} + 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 10101010101010 => [1,2,2,2,2,2,2,2] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 10101010101100 => [1,2,2,2,2,2,1,3] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 101010111000 => [1,2,2,2,1,1,4] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 10101010110100 => [1,2,2,2,2,1,2,3] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => [1,2,1,1,2,4] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 101011100100 => [1,2,2,1,1,3,3] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 10101011010100 => [1,2,2,2,1,2,2,3] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1110100100 => [1,1,1,2,3,3] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => [1,2,1,1,1,5] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 101110010100 => [1,2,1,1,3,2,3] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 10101101010100 => [1,2,2,1,2,2,2,3] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => [1,1,1,1,4,3] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 111001010100 => [1,1,1,3,2,2,3] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 10110101010100 => [1,2,1,2,2,2,2,3] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 11010101010100 => [1,1,2,2,2,2,2,3] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 1010101010101010 => [1,2,2,2,2,2,2,2,2] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 1010101010101100 => [1,2,2,2,2,2,2,1,3] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 10101010111000 => [1,2,2,2,2,1,1,4] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 1010101010110100 => [1,2,2,2,2,2,1,2,3] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 101011101000 => [1,2,2,1,1,2,4] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> 10101011100100 => [1,2,2,2,1,1,3,3] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 1010101011010100 => [1,2,2,2,2,1,2,2,3] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => [1,1,1,2,2,4] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 101110100100 => [1,2,1,1,2,3,3] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 101011110000 => [1,2,2,1,1,1,5] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> 10101110010100 => [1,2,2,1,1,3,2,3] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 1010101101010100 => [1,2,2,2,1,2,2,2,3] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => [1,1,1,2,1,5] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 111010010100 => [1,1,1,2,3,2,3] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 101111000100 => [1,2,1,1,1,4,3] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> 10111001010100 => [1,2,1,1,3,2,2,3] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 1010110101010100 => [1,2,2,1,2,2,2,2,3] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1111010000 => [1,1,1,1,2,5] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 111100010100 => [1,1,1,1,4,2,3] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 11100101010100 => [1,1,1,3,2,2,2,3] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 1011010101010100 => [1,2,1,2,2,2,2,2,3] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 1
Description
The degree of asymmetry of an integer composition. This is the number of pairs of symmetrically positioned distinct entries.
Matching statistic: St000982
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00093: Dyck paths to binary wordBinary words
Mp00200: Binary words twistBinary words
St000982: Binary words ⟶ ℤResult quality: 3% values known / values provided: 11%distinct values known / distinct values provided: 3%
Values
[1]
=> [1,0,1,0]
=> 1010 => 0010 => 2 = 0 + 2
[2]
=> [1,1,0,0,1,0]
=> 110010 => 010010 => 2 = 0 + 2
[1,1]
=> [1,0,1,1,0,0]
=> 101100 => 001100 => 2 = 0 + 2
[3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 01100010 => 3 = 1 + 2
[2,1]
=> [1,0,1,0,1,0]
=> 101010 => 001010 => 2 = 0 + 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 00111000 => 3 = 1 + 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => 0111000010 => ? ∊ {1,1} + 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 01010010 => 2 = 0 + 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 01001100 => 2 = 0 + 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 00110100 => 2 = 0 + 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 0011110000 => ? ∊ {1,1} + 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 111110000010 => 011110000010 => ? ∊ {1,1,4,4} + 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => 0110100010 => ? ∊ {1,1,4,4} + 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 01001010 => 2 = 0 + 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 00110010 => 2 = 0 + 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 00101100 => 2 = 0 + 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 0011101000 => ? ∊ {1,1,4,4} + 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => 001111100000 => ? ∊ {1,1,4,4} + 2
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 11111100000010 => 01111100000010 => ? ∊ {0,0,1,2,4,4,6,9,16,16} + 2
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => 011101000010 => ? ∊ {0,0,1,2,4,4,6,9,16,16} + 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => 0110010010 => ? ∊ {0,0,1,2,4,4,6,9,16,16} + 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 0101100010 => ? ∊ {0,0,1,2,4,4,6,9,16,16} + 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => 0110001100 => ? ∊ {0,0,1,2,4,4,6,9,16,16} + 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 00101010 => 2 = 0 + 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 0011100100 => ? ∊ {0,0,1,2,4,4,6,9,16,16} + 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 0100111000 => ? ∊ {0,0,1,2,4,4,6,9,16,16} + 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 0011011000 => ? ∊ {0,0,1,2,4,4,6,9,16,16} + 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => 001111010000 => ? ∊ {0,0,1,2,4,4,6,9,16,16} + 2
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 10111111000000 => 00111111000000 => ? ∊ {0,0,1,2,4,4,6,9,16,16} + 2
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> 1111111000000010 => 0111111000000010 => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 2
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> 11111010000010 => 01111010000010 => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 111100100010 => 011100100010 => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 111011000010 => 011011000010 => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => 0110001010 => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => 0101010010 => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 0011100010 => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => 0101001100 => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 0100110100 => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 0011010100 => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 101111001000 => 001111001000 => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 0010111000 => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 101110110000 => 001110110000 => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 2
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 10111110100000 => 00111110100000 => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 2
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1011111110000000 => 0011111110000000 => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 2
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> 111111110000000010 => 011111110000000010 => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 2
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> 1111110100000010 => 0111110100000010 => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 2
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> 11111001000010 => 01111001000010 => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 2
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> 11110110000010 => 01110110000010 => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 111100010010 => 011100010010 => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 2
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 111010100010 => 011010100010 => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 110111000010 => 010111000010 => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 111100001100 => 011100001100 => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 0101001010 => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 0100110010 => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 2
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => 0011010010 => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 2
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 101111000100 => 001111000100 => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 2
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 0100101100 => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 2
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 0011001100 => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 2
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => 0010110100 => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 2
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 101110101000 => 001110101000 => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 2
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> 10111110010000 => 00111110010000 => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 2
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 110011110000 => 010011110000 => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 2
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 101101110000 => 001101110000 => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 2
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 0010101100 => 2 = 0 + 2
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 0010101010 => 2 = 0 + 2
Description
The length of the longest constant subword.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00199: Dyck paths prime Dyck pathDyck paths
St001526: Dyck paths ⟶ ℤResult quality: 3% values known / values provided: 9%distinct values known / distinct values provided: 3%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 3 = 0 + 3
[2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 0 + 3
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4 = 1 + 3
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 0 + 3
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 1 + 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? ∊ {0,0} + 3
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4 = 1 + 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 0 + 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4 = 1 + 3
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0} + 3
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? ∊ {1,1,4,4} + 3
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? ∊ {1,1,4,4} + 3
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 0 + 3
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 0 + 3
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? ∊ {1,1,4,4} + 3
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,4,4} + 3
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? ∊ {0,0,1,2,4,4,6,9,16,16} + 3
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? ∊ {0,0,1,2,4,4,6,9,16,16} + 3
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? ∊ {0,0,1,2,4,4,6,9,16,16} + 3
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {0,0,1,2,4,4,6,9,16,16} + 3
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? ∊ {0,0,1,2,4,4,6,9,16,16} + 3
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 0 + 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? ∊ {0,0,1,2,4,4,6,9,16,16} + 3
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> ? ∊ {0,0,1,2,4,4,6,9,16,16} + 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> ? ∊ {0,0,1,2,4,4,6,9,16,16} + 3
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {0,0,1,2,4,4,6,9,16,16} + 3
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,2,4,4,6,9,16,16} + 3
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 3
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 3
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 3
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 3
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 3
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 3
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 3
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 3
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 3
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 3
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 3
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 3
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42} + 3
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 3
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 3
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 3
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 3
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 3
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 3
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 3
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 3
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 3
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 3
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 3
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 3
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 3
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 3
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 3
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 3
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 3
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 3
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99} + 3
Description
The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00023: Dyck paths to non-crossing permutationPermutations
Mp00090: Permutations cycle-as-one-line notationPermutations
St001171: Permutations ⟶ ℤResult quality: 3% values known / values provided: 9%distinct values known / distinct values provided: 3%
Values
[1]
=> [1,0,1,0]
=> [1,2] => [1,2] => 0
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => [1,2,3] => 0
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,3,2,4] => 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [1,4,2,3,5] => ? ∊ {1,1}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,2,3,4] => 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,3,4] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,3,4] => 0
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => ? ∊ {1,1}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => [1,5,2,4,3,6] => ? ∊ {1,1,4,4}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => [1,4,2,3,5] => ? ∊ {1,1,4,4}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,2,3,4] => 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,2,3,4] => 0
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => 0
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,2,5,3,4] => ? ∊ {1,1,4,4}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => [1,2,6,3,5,4] => ? ∊ {1,1,4,4}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => [1,6,2,5,3,4,7] => ? ∊ {0,0,1,2,4,4,6,9,16,16}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,3,4,2,1,6] => [1,5,2,3,4,6] => ? ∊ {0,0,1,2,4,4,6,9,16,16}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [1,3,4,2,5] => ? ∊ {0,0,1,2,4,4,6,9,16,16}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,2,4,3,5] => ? ∊ {0,0,1,2,4,4,6,9,16,16}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [1,3,2,4,5] => ? ∊ {0,0,1,2,4,4,6,9,16,16}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,2,4,5,3] => ? ∊ {0,0,1,2,4,4,6,9,16,16}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,2,3,5,4] => ? ∊ {0,0,1,2,4,4,6,9,16,16}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,2,3,5,4] => ? ∊ {0,0,1,2,4,4,6,9,16,16}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => [1,2,6,3,4,5] => ? ∊ {0,0,1,2,4,4,6,9,16,16}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => [1,2,7,3,6,4,5] => ? ∊ {0,0,1,2,4,4,6,9,16,16}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => [1,7,2,6,3,5,4,8] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,5,3,4,2,1,7] => [1,6,2,5,3,4,7] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,3,2,4,1,6] => [1,5,2,3,4,6] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,2,4,3,1,6] => [1,5,2,3,4,6] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [1,3,2,4,5] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,2,4,3,5] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,3,4,5] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,6,4,3,5,2] => [1,2,6,3,4,5] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,6,3,5,4,2] => [1,2,6,3,4,5] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,7,6,4,5,3,2] => [1,2,7,3,6,4,5] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => [1,2,8,3,7,4,6,5] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,7,6,5,4,3,2,1,9] => [1,8,2,7,3,6,4,5,9] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,6,4,5,3,2,1,8] => [1,7,2,6,3,4,5,8] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [6,4,3,5,2,1,7] => [1,6,2,4,5,3,7] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,5,4,2,1,7] => [1,6,2,3,5,4,7] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,2,5,1,6] => [1,4,5,2,3,6] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,2,3,4,1,6] => [1,5,2,3,4,6] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,4,3,1,6] => [1,2,5,3,4,6] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => [1,4,2,3,5,6] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,2,3,4,5] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,2,3,4,5] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,4,3,6,2] => [1,2,5,6,3,4] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,2,3,4,5] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,2,3,4,5] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,6,3,4,5,2] => [1,2,6,3,4,5] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,7,5,4,6,3,2] => [1,2,7,3,5,6,4] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,5,4,3] => [1,2,3,6,4,5] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,6,5,4,2] => [1,2,3,6,4,5] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
Description
The vector space dimension of Ext1A(Io,A) when Io is the tilting module corresponding to the permutation o in the Auslander algebra A of K[x]/(xn).
Matching statistic: St001195
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00199: Dyck paths prime Dyck pathDyck paths
Mp00143: Dyck paths inverse promotionDyck paths
St001195: Dyck paths ⟶ ℤResult quality: 3% values known / values provided: 9%distinct values known / distinct values provided: 3%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? ∊ {1,1}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> ? ∊ {1,1}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? ∊ {1,1,4,4}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> ? ∊ {1,1,4,4}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {1,1,4,4}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? ∊ {1,1,4,4}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? ∊ {0,0,1,2,4,4,6,9,16,16}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> ? ∊ {0,0,1,2,4,4,6,9,16,16}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ? ∊ {0,0,1,2,4,4,6,9,16,16}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> ? ∊ {0,0,1,2,4,4,6,9,16,16}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> ? ∊ {0,0,1,2,4,4,6,9,16,16}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {0,0,1,2,4,4,6,9,16,16}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> ? ∊ {0,0,1,2,4,4,6,9,16,16}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> ? ∊ {0,0,1,2,4,4,6,9,16,16}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> ? ∊ {0,0,1,2,4,4,6,9,16,16}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,0,1,2,4,4,6,9,16,16}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
Description
The global dimension of the algebra A/AfA of the corresponding Nakayama algebra A with minimal left faithful projective-injective module Af.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00023: Dyck paths to non-crossing permutationPermutations
Mp00090: Permutations cycle-as-one-line notationPermutations
St001207: Permutations ⟶ ℤResult quality: 3% values known / values provided: 9%distinct values known / distinct values provided: 3%
Values
[1]
=> [1,0,1,0]
=> [1,2] => [1,2] => 0
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => [1,2,3] => 0
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,3,2,4] => 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [1,4,2,3,5] => ? ∊ {1,1}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,2,3,4] => 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,3,4] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,3,4] => 0
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => ? ∊ {1,1}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => [1,5,2,4,3,6] => ? ∊ {1,1,4,4}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => [1,4,2,3,5] => ? ∊ {1,1,4,4}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,2,3,4] => 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,2,3,4] => 0
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => 0
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,2,5,3,4] => ? ∊ {1,1,4,4}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => [1,2,6,3,5,4] => ? ∊ {1,1,4,4}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => [1,6,2,5,3,4,7] => ? ∊ {0,0,1,2,4,4,6,9,16,16}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,3,4,2,1,6] => [1,5,2,3,4,6] => ? ∊ {0,0,1,2,4,4,6,9,16,16}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [1,3,4,2,5] => ? ∊ {0,0,1,2,4,4,6,9,16,16}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,2,4,3,5] => ? ∊ {0,0,1,2,4,4,6,9,16,16}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [1,3,2,4,5] => ? ∊ {0,0,1,2,4,4,6,9,16,16}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,2,4,5,3] => ? ∊ {0,0,1,2,4,4,6,9,16,16}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,2,3,5,4] => ? ∊ {0,0,1,2,4,4,6,9,16,16}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,2,3,5,4] => ? ∊ {0,0,1,2,4,4,6,9,16,16}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => [1,2,6,3,4,5] => ? ∊ {0,0,1,2,4,4,6,9,16,16}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => [1,2,7,3,6,4,5] => ? ∊ {0,0,1,2,4,4,6,9,16,16}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => [1,7,2,6,3,5,4,8] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,5,3,4,2,1,7] => [1,6,2,5,3,4,7] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,3,2,4,1,6] => [1,5,2,3,4,6] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,2,4,3,1,6] => [1,5,2,3,4,6] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [1,3,2,4,5] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,2,4,3,5] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,3,4,5] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,6,4,3,5,2] => [1,2,6,3,4,5] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,6,3,5,4,2] => [1,2,6,3,4,5] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,7,6,4,5,3,2] => [1,2,7,3,6,4,5] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => [1,2,8,3,7,4,6,5] => ? ∊ {0,0,0,1,2,6,6,13,15,15,15,20,27,28,42}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,7,6,5,4,3,2,1,9] => [1,8,2,7,3,6,4,5,9] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,6,4,5,3,2,1,8] => [1,7,2,6,3,4,5,8] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [6,4,3,5,2,1,7] => [1,6,2,4,5,3,7] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,5,4,2,1,7] => [1,6,2,3,5,4,7] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,2,5,1,6] => [1,4,5,2,3,6] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,2,3,4,1,6] => [1,5,2,3,4,6] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,4,3,1,6] => [1,2,5,3,4,6] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => [1,4,2,3,5,6] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,2,3,4,5] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,2,3,4,5] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,4,3,6,2] => [1,2,5,6,3,4] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,2,3,4,5] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,2,3,4,5] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,6,3,4,5,2] => [1,2,6,3,4,5] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,7,5,4,6,3,2] => [1,2,7,3,5,6,4] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,5,4,3] => [1,2,3,6,4,5] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,6,5,4,2] => [1,2,3,6,4,5] => ? ∊ {0,0,0,0,0,2,7,13,14,15,21,21,29,33,34,35,35,49,56,78,91,99}
Description
The Lowey length of the algebra A/T when T is the 1-tilting module corresponding to the permutation in the Auslander algebra of K[x]/(xn).
The following 12 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001423The number of distinct cubes in a binary word. St001520The number of strict 3-descents. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001846The number of elements which do not have a complement in the lattice. St001948The number of augmented double ascents of a permutation. St000628The balance of a binary word. St001413Half the length of the longest even length palindromic prefix of a binary word. St001768The number of reduced words of a signed permutation. St001820The size of the image of the pop stack sorting operator. St000630The length of the shortest palindromic decomposition of a binary word. 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.