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Your data matches 42 different statistics following compositions of up to 3 maps.
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Matching statistic: St000063
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Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000063: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000063: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
=> []
=> []
=> 1
[[]]
=> [1,0]
=> []
=> 1
[[],[]]
=> [1,0,1,0]
=> [1]
=> 2
[[[]]]
=> [1,1,0,0]
=> []
=> 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [2,1]
=> 6
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1]
=> 3
[[[]],[]]
=> [1,1,0,0,1,0]
=> [2]
=> 3
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1]
=> 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> []
=> 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 24
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 12
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 12
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 8
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 4
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 12
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 6
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 8
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 4
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 6
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 3
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 3
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 2
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 5
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 10
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 10
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 5
[[[],[[[]]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 4
[[[[]],[[]]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 6
[[[[[]]],[]]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 4
[[[[],[[]]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 3
[[[[[]],[]]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 3
[[[[[],[]]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 2
[[[[[[]]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> 1
[[[]],[[[[]]]]]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 15
[[[[]]],[[[]]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 20
[[[[[]]]],[[]]]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 15
[[[[[],[[]]]]]]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 3
[[[[[[]],[]]]]]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 3
[[[[[[],[]]]]]]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 2
[[[[[[[]]]]]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> 1
[[[[[[[[]]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> 1
[[[[[[[[[]]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> []
=> 1
Description
The number of linear extensions of a certain poset defined for an integer partition.
The poset is constructed in David Speyer's answer to Matt Fayers' question [3].
The value at the partition $\lambda$ also counts cover-inclusive Dyck tilings of $\lambda\setminus\mu$, summed over all $\mu$, as noticed by Philippe Nadeau in a comment.
This statistic arises in the homogeneous Garnir relations for the universal graded Specht modules for cyclotomic quiver Hecke algebras.
Matching statistic: St000071
Values
[]
=> ([],1)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[]]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 6
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 3
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 3
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> 24
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> 12
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> 12
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> 8
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> 4
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> 12
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(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)
=> ([(0,4),(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)
=> 6
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> 8
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> 4
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> 6
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> 3
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> 3
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> 5
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(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)
=> ([(0,5),(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)
=> 10
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(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)
=> ([(0,5),(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)
=> 10
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> 5
[[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> 4
[[[[]],[[]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> 6
[[[[[]]],[]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> 4
[[[[],[[]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> 3
[[[[[]],[]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> 3
[[[[[],[]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> 2
[[[[[[]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(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
[[[]],[[[[]]]]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> ([(0,6),(0,7),(2,5),(2,15),(3,12),(4,11),(5,4),(5,13),(6,2),(6,14),(7,3),(7,14),(8,1),(9,10),(10,8),(11,8),(12,9),(13,10),(13,11),(14,12),(14,15),(15,9),(15,13)],16)
=> ([(0,6),(0,7),(2,5),(2,15),(3,12),(4,11),(5,4),(5,13),(6,2),(6,14),(7,3),(7,14),(8,1),(9,10),(10,8),(11,8),(12,9),(13,10),(13,11),(14,12),(14,15),(15,9),(15,13)],16)
=> 15
[[[[]]],[[[]]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(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)
=> ([(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)
=> 20
[[[[[]]]],[[]]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> ([(0,6),(0,7),(2,5),(2,15),(3,12),(4,11),(5,4),(5,13),(6,2),(6,14),(7,3),(7,14),(8,1),(9,10),(10,8),(11,8),(12,9),(13,10),(13,11),(14,12),(14,15),(15,9),(15,13)],16)
=> ([(0,6),(0,7),(2,5),(2,15),(3,12),(4,11),(5,4),(5,13),(6,2),(6,14),(7,3),(7,14),(8,1),(9,10),(10,8),(11,8),(12,9),(13,10),(13,11),(14,12),(14,15),(15,9),(15,13)],16)
=> 15
[[[[[],[[]]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,7),(2,9),(3,8),(4,6),(5,4),(6,1),(7,2),(7,8),(8,9),(9,5)],10)
=> ([(0,3),(0,7),(2,9),(3,8),(4,6),(5,4),(6,1),(7,2),(7,8),(8,9),(9,5)],10)
=> 3
[[[[[[]],[]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,7),(2,9),(3,8),(4,6),(5,4),(6,1),(7,2),(7,8),(8,9),(9,5)],10)
=> ([(0,3),(0,7),(2,9),(3,8),(4,6),(5,4),(6,1),(7,2),(7,8),(8,9),(9,5)],10)
=> 3
[[[[[[],[]]]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(0,2),(0,3),(2,8),(3,8),(4,6),(5,4),(6,1),(7,5),(8,7)],9)
=> ([(0,2),(0,3),(2,8),(3,8),(4,6),(5,4),(6,1),(7,5),(8,7)],9)
=> 2
[[[[[[[]]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 1
[[[[[[[[]]]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> 1
[[[[[[[[[]]]]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> 1
Description
The number of maximal chains in a poset.
Matching statistic: St000100
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00047: Ordered trees —to poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
St000100: Posets ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[]
=> ([],1)
=> ? = 1
[[]]
=> ([(0,1)],2)
=> 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> 2
[[[]]]
=> ([(0,2),(2,1)],3)
=> 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> 6
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 24
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 12
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 12
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 8
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 4
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 12
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 6
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 8
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 4
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> 6
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 3
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 3
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 5
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 10
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 10
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 5
[[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 4
[[[[]],[[]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> 6
[[[[[]]],[]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 4
[[[[],[[]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 3
[[[[[]],[]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 3
[[[[[],[]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 2
[[[[[[]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[[]],[[[[]]]]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> 15
[[[[]]],[[[]]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> 20
[[[[[]]]],[[]]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> 15
[[[[[],[[]]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> 3
[[[[[[]],[]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> 3
[[[[[[],[]]]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> 2
[[[[[[[]]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[[[[[[[[]]]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 1
[[[[[[[[[]]]]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> 1
Description
The number of linear extensions of a poset.
Matching statistic: St000110
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[]
=> .
=> ? => ? = 1
[[]]
=> [.,.]
=> [1] => 1
[[],[]]
=> [.,[.,.]]
=> [2,1] => 2
[[[]]]
=> [[.,.],.]
=> [1,2] => 1
[[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 6
[[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 3
[[[]],[]]
=> [[.,.],[.,.]]
=> [3,1,2] => 3
[[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => 2
[[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 24
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 12
[[],[[]],[]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 12
[[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 8
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 4
[[[]],[],[]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 12
[[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 6
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 8
[[[[]]],[]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 4
[[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 6
[[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 3
[[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 3
[[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 2
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 1
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 5
[[[]],[[[]]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 10
[[[[]]],[[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 10
[[[[[]]]],[]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 5
[[[],[[[]]]]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => 4
[[[[]],[[]]]]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => 6
[[[[[]]],[]]]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 4
[[[[],[[]]]]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => 3
[[[[[]],[]]]]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => 3
[[[[[],[]]]]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 2
[[[[[[]]]]]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 1
[[[]],[[[[]]]]]
=> [[.,.],[[[[.,.],.],.],.]]
=> [3,4,5,6,1,2] => 15
[[[[]]],[[[]]]]
=> [[[.,.],.],[[[.,.],.],.]]
=> [4,5,6,1,2,3] => 20
[[[[[]]]],[[]]]
=> [[[[.,.],.],.],[[.,.],.]]
=> [5,6,1,2,3,4] => 15
[[[[[],[[]]]]]]
=> [[[[.,[[.,.],.]],.],.],.]
=> [2,3,1,4,5,6] => 3
[[[[[[]],[]]]]]
=> [[[[[.,.],[.,.]],.],.],.]
=> [3,1,2,4,5,6] => 3
[[[[[[],[]]]]]]
=> [[[[[.,[.,.]],.],.],.],.]
=> [2,1,3,4,5,6] => 2
[[[[[[[]]]]]]]
=> [[[[[[.,.],.],.],.],.],.]
=> [1,2,3,4,5,6] => 1
[[[[[[[[]]]]]]]]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7] => 1
[[[[[[[[[]]]]]]]]]
=> [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7,8] => 1
Description
The number of permutations less than or equal to a permutation in left weak order.
This is the same as the number of permutations less than or equal to the given permutation in right weak order.
Matching statistic: St000085
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
St000085: Ordered trees ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[]
=> ? = 1
[[]]
=> 1
[[],[]]
=> 2
[[[]]]
=> 1
[[],[],[]]
=> 6
[[],[[]]]
=> 3
[[[]],[]]
=> 3
[[[],[]]]
=> 2
[[[[]]]]
=> 1
[[],[],[],[]]
=> 24
[[],[],[[]]]
=> 12
[[],[[]],[]]
=> 12
[[],[[],[]]]
=> 8
[[],[[[]]]]
=> 4
[[[]],[],[]]
=> 12
[[[]],[[]]]
=> 6
[[[],[]],[]]
=> 8
[[[[]]],[]]
=> 4
[[[],[],[]]]
=> 6
[[[],[[]]]]
=> 3
[[[[]],[]]]
=> 3
[[[[],[]]]]
=> 2
[[[[[]]]]]
=> 1
[[],[[[[]]]]]
=> 5
[[[]],[[[]]]]
=> 10
[[[[]]],[[]]]
=> 10
[[[[[]]]],[]]
=> 5
[[[],[[[]]]]]
=> 4
[[[[]],[[]]]]
=> 6
[[[[[]]],[]]]
=> 4
[[[[],[[]]]]]
=> 3
[[[[[]],[]]]]
=> 3
[[[[[],[]]]]]
=> 2
[[[[[[]]]]]]
=> 1
[[[]],[[[[]]]]]
=> 15
[[[[]]],[[[]]]]
=> 20
[[[[[]]]],[[]]]
=> 15
[[[[[],[[]]]]]]
=> 3
[[[[[[]],[]]]]]
=> 3
[[[[[[],[]]]]]]
=> 2
[[[[[[[]]]]]]]
=> 1
[[[[[[[[]]]]]]]]
=> ? = 1
[[[[[[[[[]]]]]]]]]
=> ? = 1
Description
The number of linear extensions of the tree.
We use Knuth's hook length formula for trees [pg.70, 1]. For an ordered tree $T$ on $n$ vertices, the number of linear extensions is
$$
\frac{n!}{\prod_{v\in T}|T_v|},
$$
where $T_v$ is the number of vertices of the subtree rooted at $v$.
Matching statistic: St000228
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00047: Ordered trees —to poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 79%●distinct values known / distinct values provided: 67%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 79%●distinct values known / distinct values provided: 67%
Values
[]
=> ([],1)
=> [1]
=> 1
[[]]
=> ([(0,1)],2)
=> [1]
=> 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> [2]
=> 2
[[[]]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> [3,3]
=> 6
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 3
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 3
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 2
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,4,4,4,4,4]
=> ? = 24
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> ? = 12
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> ? = 12
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 8
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 4
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> ? = 12
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 6
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 8
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 4
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,3]
=> 6
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 3
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 3
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 2
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 1
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [5]
=> 5
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [5,5]
=> 10
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [5,5]
=> 10
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [5]
=> 5
[[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 4
[[[[]],[[]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 6
[[[[[]]],[]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 4
[[[[],[[]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 3
[[[[[]],[]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 3
[[[[[],[]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 2
[[[[[[]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 1
[[[]],[[[[]]]]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> [6,6,3]
=> ? = 15
[[[[]]],[[[]]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> [6,6,6,2]
=> ? = 20
[[[[[]]]],[[]]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> [6,6,3]
=> ? = 15
[[[[[],[[]]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> [3]
=> 3
[[[[[[]],[]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> [3]
=> 3
[[[[[[],[]]]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> [2]
=> 2
[[[[[[[]]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [1]
=> 1
[[[[[[[[]]]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ?
=> ? = 1
[[[[[[[[[]]]]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ?
=> ? = 1
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St001855
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001855: Signed permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 67%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001855: Signed permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 67%
Values
[]
=> .
=> ? => ? => ? = 1
[[]]
=> [.,.]
=> [1] => [1] => 1
[[],[]]
=> [.,[.,.]]
=> [2,1] => [2,1] => 2
[[[]]]
=> [[.,.],.]
=> [1,2] => [1,2] => 1
[[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => 6
[[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => [2,3,1] => 3
[[[]],[]]
=> [[.,.],[.,.]]
=> [3,1,2] => [3,1,2] => 3
[[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 2
[[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => 24
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,4,2,1] => 12
[[],[[]],[]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [4,2,3,1] => 12
[[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,4,1] => 8
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,4,1] => 4
[[[]],[],[]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [4,3,1,2] => 12
[[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,4,1,2] => 6
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [4,2,1,3] => 8
[[[[]]],[]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => [4,1,2,3] => 4
[[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => 6
[[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1,4] => 3
[[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2,4] => 3
[[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 2
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 1
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,5,1] => ? = 5
[[[]],[[[]]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,4,5,1,2] => ? = 10
[[[[]]],[[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 10
[[[[[]]]],[]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [5,1,2,3,4] => ? = 5
[[[],[[[]]]]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [2,3,4,1,5] => ? = 4
[[[[]],[[]]]]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 6
[[[[[]]],[]]]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => [4,1,2,3,5] => ? = 4
[[[[],[[]]]]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [2,3,1,4,5] => ? = 3
[[[[[]],[]]]]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => [3,1,2,4,5] => ? = 3
[[[[[],[]]]]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? = 2
[[[[[[]]]]]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [1,2,3,4,5] => 1
[[[]],[[[[]]]]]
=> [[.,.],[[[[.,.],.],.],.]]
=> [3,4,5,6,1,2] => [3,4,5,6,1,2] => ? = 15
[[[[]]],[[[]]]]
=> [[[.,.],.],[[[.,.],.],.]]
=> [4,5,6,1,2,3] => [4,5,6,1,2,3] => ? = 20
[[[[[]]]],[[]]]
=> [[[[.,.],.],.],[[.,.],.]]
=> [5,6,1,2,3,4] => [5,6,1,2,3,4] => ? = 15
[[[[[],[[]]]]]]
=> [[[[.,[[.,.],.]],.],.],.]
=> [2,3,1,4,5,6] => [2,3,1,4,5,6] => ? = 3
[[[[[[]],[]]]]]
=> [[[[[.,.],[.,.]],.],.],.]
=> [3,1,2,4,5,6] => [3,1,2,4,5,6] => ? = 3
[[[[[[],[]]]]]]
=> [[[[[.,[.,.]],.],.],.],.]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => ? = 2
[[[[[[[]]]]]]]
=> [[[[[[.,.],.],.],.],.],.]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1
[[[[[[[[]]]]]]]]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 1
[[[[[[[[[]]]]]]]]]
=> [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => ? = 1
Description
The number of signed permutations less than or equal to a signed permutation in left weak order.
Matching statistic: St000909
Values
[]
=> ([],1)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[]]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 6
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 3
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 3
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 24
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? = 12
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? = 12
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 8
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> 4
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? = 12
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(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)
=> ([(0,4),(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)
=> ? = 6
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 8
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> 4
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> ? = 6
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> 3
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> 3
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 5
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(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)
=> ([(0,5),(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)
=> ? = 10
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(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)
=> ([(0,5),(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)
=> ? = 10
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 5
[[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4
[[[[]],[[]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 6
[[[[[]]],[]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4
[[[[],[[]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 3
[[[[[]],[]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 3
[[[[[],[]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 2
[[[[[[]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(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
[[[]],[[[[]]]]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> ([(0,6),(0,7),(2,5),(2,15),(3,12),(4,11),(5,4),(5,13),(6,2),(6,14),(7,3),(7,14),(8,1),(9,10),(10,8),(11,8),(12,9),(13,10),(13,11),(14,12),(14,15),(15,9),(15,13)],16)
=> ([(0,6),(0,7),(2,5),(2,15),(3,12),(4,11),(5,4),(5,13),(6,2),(6,14),(7,3),(7,14),(8,1),(9,10),(10,8),(11,8),(12,9),(13,10),(13,11),(14,12),(14,15),(15,9),(15,13)],16)
=> ? = 15
[[[[]]],[[[]]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(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)
=> ([(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)
=> ? = 20
[[[[[]]]],[[]]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> ([(0,6),(0,7),(2,5),(2,15),(3,12),(4,11),(5,4),(5,13),(6,2),(6,14),(7,3),(7,14),(8,1),(9,10),(10,8),(11,8),(12,9),(13,10),(13,11),(14,12),(14,15),(15,9),(15,13)],16)
=> ([(0,6),(0,7),(2,5),(2,15),(3,12),(4,11),(5,4),(5,13),(6,2),(6,14),(7,3),(7,14),(8,1),(9,10),(10,8),(11,8),(12,9),(13,10),(13,11),(14,12),(14,15),(15,9),(15,13)],16)
=> ? = 15
[[[[[],[[]]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,7),(2,9),(3,8),(4,6),(5,4),(6,1),(7,2),(7,8),(8,9),(9,5)],10)
=> ([(0,3),(0,7),(2,9),(3,8),(4,6),(5,4),(6,1),(7,2),(7,8),(8,9),(9,5)],10)
=> ? = 3
[[[[[[]],[]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,7),(2,9),(3,8),(4,6),(5,4),(6,1),(7,2),(7,8),(8,9),(9,5)],10)
=> ([(0,3),(0,7),(2,9),(3,8),(4,6),(5,4),(6,1),(7,2),(7,8),(8,9),(9,5)],10)
=> ? = 3
[[[[[[],[]]]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(0,2),(0,3),(2,8),(3,8),(4,6),(5,4),(6,1),(7,5),(8,7)],9)
=> ([(0,2),(0,3),(2,8),(3,8),(4,6),(5,4),(6,1),(7,5),(8,7)],9)
=> ? = 2
[[[[[[[]]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 1
[[[[[[[[]]]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> 1
[[[[[[[[[]]]]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ? = 1
Description
The number of maximal chains of maximal size in a poset.
Matching statistic: St001633
Values
[]
=> ([],1)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
[[]]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 2 - 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ? = 6 - 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 3 - 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 3 - 1
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1 = 2 - 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 24 - 1
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? = 12 - 1
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? = 12 - 1
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 8 - 1
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 4 - 1
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? = 12 - 1
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(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)
=> ([(0,4),(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)
=> ? = 6 - 1
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 8 - 1
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 4 - 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> ? = 6 - 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 3 - 1
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 3 - 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 1 = 2 - 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 5 - 1
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(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)
=> ([(0,5),(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)
=> ? = 10 - 1
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(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)
=> ([(0,5),(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)
=> ? = 10 - 1
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 5 - 1
[[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 - 1
[[[[]],[[]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 6 - 1
[[[[[]]],[]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 - 1
[[[[],[[]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 3 - 1
[[[[[]],[]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 3 - 1
[[[[[],[]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 2 - 1
[[[[[[]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(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)
=> 0 = 1 - 1
[[[]],[[[[]]]]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> ([(0,6),(0,7),(2,5),(2,15),(3,12),(4,11),(5,4),(5,13),(6,2),(6,14),(7,3),(7,14),(8,1),(9,10),(10,8),(11,8),(12,9),(13,10),(13,11),(14,12),(14,15),(15,9),(15,13)],16)
=> ([(0,6),(0,7),(2,5),(2,15),(3,12),(4,11),(5,4),(5,13),(6,2),(6,14),(7,3),(7,14),(8,1),(9,10),(10,8),(11,8),(12,9),(13,10),(13,11),(14,12),(14,15),(15,9),(15,13)],16)
=> ? = 15 - 1
[[[[]]],[[[]]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(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)
=> ([(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)
=> ? = 20 - 1
[[[[[]]]],[[]]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> ([(0,6),(0,7),(2,5),(2,15),(3,12),(4,11),(5,4),(5,13),(6,2),(6,14),(7,3),(7,14),(8,1),(9,10),(10,8),(11,8),(12,9),(13,10),(13,11),(14,12),(14,15),(15,9),(15,13)],16)
=> ([(0,6),(0,7),(2,5),(2,15),(3,12),(4,11),(5,4),(5,13),(6,2),(6,14),(7,3),(7,14),(8,1),(9,10),(10,8),(11,8),(12,9),(13,10),(13,11),(14,12),(14,15),(15,9),(15,13)],16)
=> ? = 15 - 1
[[[[[],[[]]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,7),(2,9),(3,8),(4,6),(5,4),(6,1),(7,2),(7,8),(8,9),(9,5)],10)
=> ([(0,3),(0,7),(2,9),(3,8),(4,6),(5,4),(6,1),(7,2),(7,8),(8,9),(9,5)],10)
=> ? = 3 - 1
[[[[[[]],[]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,7),(2,9),(3,8),(4,6),(5,4),(6,1),(7,2),(7,8),(8,9),(9,5)],10)
=> ([(0,3),(0,7),(2,9),(3,8),(4,6),(5,4),(6,1),(7,2),(7,8),(8,9),(9,5)],10)
=> ? = 3 - 1
[[[[[[],[]]]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(0,2),(0,3),(2,8),(3,8),(4,6),(5,4),(6,1),(7,5),(8,7)],9)
=> ([(0,2),(0,3),(2,8),(3,8),(4,6),(5,4),(6,1),(7,5),(8,7)],9)
=> ? = 2 - 1
[[[[[[[]]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 1 - 1
[[[[[[[[]]]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 1 - 1
[[[[[[[[[]]]]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ? = 1 - 1
Description
The number of simple modules with projective dimension two in the incidence algebra of the poset.
Matching statistic: St001876
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[]
=> ([],1)
=> ([],1)
=> ? = 1 - 1
[[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1 - 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ? = 6 - 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ? = 3 - 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ? = 3 - 1
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 2 - 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 24 - 1
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ? = 12 - 1
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ? = 12 - 1
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 8 - 1
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ? = 4 - 1
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ? = 12 - 1
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ? = 6 - 1
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 8 - 1
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ? = 4 - 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ? = 6 - 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ? = 3 - 1
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ? = 3 - 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1 = 2 - 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> ? = 5 - 1
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7)
=> ? = 10 - 1
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7)
=> ? = 10 - 1
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> ? = 5 - 1
[[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> ? = 4 - 1
[[[[]],[[]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=> ? = 6 - 1
[[[[[]]],[]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> ? = 4 - 1
[[[[],[[]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=> ? = 3 - 1
[[[[[]],[]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=> ? = 3 - 1
[[[[[],[]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 1 = 2 - 1
[[[[[[]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[[[]],[[[[]]]]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> ([(0,5),(0,6),(1,7),(2,7),(3,4),(4,2),(5,3),(6,1)],8)
=> ? = 15 - 1
[[[[]]],[[[]]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 20 - 1
[[[[[]]]],[[]]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> ([(0,5),(0,6),(1,7),(2,7),(3,4),(4,2),(5,3),(6,1)],8)
=> ? = 15 - 1
[[[[[],[[]]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(7,5)],8)
=> ? = 3 - 1
[[[[[[]],[]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,3),(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(7,5)],8)
=> ? = 3 - 1
[[[[[[],[]]]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> 1 = 2 - 1
[[[[[[[]]]]]]]
=> ([(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)
=> 0 = 1 - 1
[[[[[[[[]]]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ?
=> ? = 1 - 1
[[[[[[[[[]]]]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ?
=> ? = 1 - 1
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
The following 32 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001877Number of indecomposable injective modules with projective dimension 2. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000632The jump number of the poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001645The pebbling number of a connected graph. St001779The order of promotion on the set of linear extensions of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001330The hat guessing number of a graph. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001964The interval resolution global dimension of a poset. St001722The number of minimal chains with small intervals between a binary word and the top element.
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