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Your data matches 15 different statistics following compositions of up to 3 maps.
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Matching statistic: St001588
Mp00047: Ordered trees —to poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001588: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001588: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
=> ([],1)
=> [2]
=> [1,1]
=> 0
[[]]
=> ([(0,1)],2)
=> [3]
=> [1,1,1]
=> 0
[[],[]]
=> ([(0,2),(1,2)],3)
=> [3,2]
=> [2,2,1]
=> 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> 0
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 0
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [8,8,1]
=> 1
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [2,2,2,2,2,2,1]
=> 1
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [2,2,2,2,2,2,1]
=> 1
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [3,3,1,1,1,1,1]
=> 0
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 1
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [2,2,2,2,2,2,1]
=> 1
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [3,3,3,1]
=> 0
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [3,3,1,1,1,1,1]
=> 0
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 0
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5,4,4,4]
=> [4,4,4,4,1]
=> 1
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5,4,4,4]
=> [4,4,4,4,1]
=> 1
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 0
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 0
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> [4,4,2,2,1]
=> 1
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 0
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> [4,4,4,2,1,1]
=> 1
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 0
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5,4,4,4]
=> [4,4,4,4,1]
=> 1
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> [4,4,4,2,1,1]
=> 1
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 0
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 0
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 0
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> [4,4,2,2,1]
=> 1
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 0
[[[],[],[[]]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> [8,6]
=> [2,2,2,2,2,2,1,1]
=> 1
[[[],[[]],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> [8,6]
=> [2,2,2,2,2,2,1,1]
=> 1
[[[],[[],[]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> 0
[[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [6,4]
=> [2,2,2,2,1,1]
=> 1
[[[[]],[],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> [8,6]
=> [2,2,2,2,2,2,1,1]
=> 1
[[[[]],[[]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [5,3,3]
=> [3,3,3,1,1]
=> 0
[[[[],[]],[]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> 0
[[[[[]]],[]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [6,4]
=> [2,2,2,2,1,1]
=> 1
[[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> [5,2,2,2]
=> [4,4,1,1,1]
=> 1
[[[[],[[]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> 0
[[[[[]],[]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> 0
[[[[[],[]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [6,2]
=> [2,2,1,1,1,1]
=> 1
Description
The number of distinct odd parts smaller than the largest even part in an integer partition.
Matching statistic: St000755
Mp00047: Ordered trees —to poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000755: Integer partitions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000755: Integer partitions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%
Values
[]
=> ([],1)
=> [2]
=> [1,1]
=> 1 = 0 + 1
[[]]
=> ([(0,1)],2)
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> [3,2]
=> [2,2,1]
=> 2 = 1 + 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> 2 = 1 + 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 2 = 1 + 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [8,8,1]
=> ? = 1 + 1
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [2,2,2,2,2,2,1]
=> ? = 1 + 1
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [2,2,2,2,2,2,1]
=> ? = 1 + 1
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [3,3,1,1,1,1,1]
=> 1 = 0 + 1
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 2 = 1 + 1
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [2,2,2,2,2,2,1]
=> ? = 1 + 1
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [3,3,3,1]
=> 1 = 0 + 1
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [3,3,1,1,1,1,1]
=> 1 = 0 + 1
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 2 = 1 + 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> 2 = 1 + 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> 2 = 1 + 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 1 = 0 + 1
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5,4,4,4]
=> [4,4,4,4,1]
=> ? = 1 + 1
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5,4,4,4]
=> [4,4,4,4,1]
=> ? = 1 + 1
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> [4,4,2,2,1]
=> 2 = 1 + 1
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> [4,4,4,2,1,1]
=> ? = 1 + 1
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5,4,4,4]
=> [4,4,4,4,1]
=> ? = 1 + 1
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> [4,4,4,2,1,1]
=> ? = 1 + 1
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> [4,4,2,2,1]
=> 2 = 1 + 1
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[[[],[],[[]]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> [8,6]
=> [2,2,2,2,2,2,1,1]
=> ? = 1 + 1
[[[],[[]],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> [8,6]
=> [2,2,2,2,2,2,1,1]
=> ? = 1 + 1
[[[],[[],[]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> 1 = 0 + 1
[[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [6,4]
=> [2,2,2,2,1,1]
=> 2 = 1 + 1
[[[[]],[],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> [8,6]
=> [2,2,2,2,2,2,1,1]
=> ? = 1 + 1
[[[[]],[[]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [5,3,3]
=> [3,3,3,1,1]
=> 1 = 0 + 1
[[[[],[]],[]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> 1 = 0 + 1
[[[[[]]],[]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [6,4]
=> [2,2,2,2,1,1]
=> 2 = 1 + 1
[[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> [5,2,2,2]
=> [4,4,1,1,1]
=> 2 = 1 + 1
[[[[],[[]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[[[[[]],[]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[[[[[],[]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [6,2]
=> [2,2,1,1,1,1]
=> 2 = 1 + 1
[[[[[[]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[[],[[[[],[]]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
=> [11,2,2]
=> [3,3,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> [7,6]
=> [2,2,2,2,2,2,1]
=> ? = 1 + 1
[[[]],[[[[]]]]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[[[]]],[[[]]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> [5,4,4,4]
=> [4,4,4,4,1]
=> ? = 1 + 1
[[[[[]]]],[[]]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[[[[],[]]]],[]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
=> [11,2,2]
=> [3,3,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[[[[[]]]]],[]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> [7,6]
=> [2,2,2,2,2,2,1]
=> ? = 1 + 1
[[[],[[],[[]]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[[],[[[]],[]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[[],[[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> [6,4,2,2]
=> [4,4,2,2,1,1]
=> ? = 1 + 1
[[[],[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[[[[]],[[],[]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> [6,5,3,3]
=> [4,4,4,2,2,1]
=> ? = 1 + 1
[[[[]],[[[]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[[[],[]],[[]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> [6,5,3,3]
=> [4,4,4,2,2,1]
=> ? = 1 + 1
[[[[[]]],[[]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[[[],[[]]],[]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[[[[]],[]],[]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[[[[],[]]],[]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> [6,4,2,2]
=> [4,4,2,2,1,1]
=> ? = 1 + 1
[[[[[[]]]],[]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[[[[],[],[[]]]]]
=> ([(0,6),(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> [9,6]
=> [2,2,2,2,2,2,1,1,1]
=> ? = 1 + 1
[[[[],[[]],[]]]]
=> ([(0,6),(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> [9,6]
=> [2,2,2,2,2,2,1,1,1]
=> ? = 1 + 1
[[[[],[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> [9,2,2]
=> [3,3,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[[[],[[[]]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> [7,4]
=> [2,2,2,2,1,1,1]
=> 2 = 1 + 1
[[[[[]],[],[]]]]
=> ([(0,6),(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> [9,6]
=> [2,2,2,2,2,2,1,1,1]
=> ? = 1 + 1
[[[[[]],[[]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> [6,3,3]
=> [3,3,3,1,1,1]
=> 1 = 0 + 1
[[[[[],[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> [9,2,2]
=> [3,3,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[[[[[]]],[]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> [7,4]
=> [2,2,2,2,1,1,1]
=> 2 = 1 + 1
[[[[[],[],[]]]]]
=> ([(0,6),(1,6),(2,6),(3,5),(5,4),(6,3)],7)
=> [6,2,2,2]
=> [4,4,1,1,1,1]
=> 2 = 1 + 1
[[[[[],[[]]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[[[[[[]],[]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[[[[[[],[]]]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> [7,2]
=> [2,2,1,1,1,1,1]
=> 2 = 1 + 1
[[[[[[[]]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
Description
The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition.
Consider the recurrence $$f(n)=\sum_{p\in\lambda} f(n-p).$$ This statistic returns the number of distinct real roots of the associated characteristic polynomial.
For example, the partition $(2,1)$ corresponds to the recurrence $f(n)=f(n-1)+f(n-2)$ with associated characteristic polynomial $x^2-x-1$, which has two real roots.
Matching statistic: St000205
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00047: Ordered trees —to poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 100%
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 100%
Values
[]
=> ([],1)
=> [2]
=> 0
[[]]
=> ([(0,1)],2)
=> [3]
=> 0
[[],[]]
=> ([(0,2),(1,2)],3)
=> [3,2]
=> 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> [4]
=> 0
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> ? = 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> 0
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> 0
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 0
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> ? = 1
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> ? = 1
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> ? = 1
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> ? = 0
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> ? = 1
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> ? = 1
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> ? = 0
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> ? = 0
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> ? = 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> ? = 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 0
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 0
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 0
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5,4,4,4]
=> ? = 1
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5,4,4,4]
=> ? = 1
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 0
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 0
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> ? = 1
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [11]
=> ? = 0
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> ? = 1
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 0
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5,4,4,4]
=> ? = 1
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> ? = 1
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 0
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 0
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 0
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> ? = 1
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [11]
=> ? = 0
[[[],[],[[]]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> [8,6]
=> ? = 1
[[[],[[]],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> [8,6]
=> ? = 1
[[[],[[],[]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8,2,2]
=> ? = 0
[[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [6,4]
=> ? = 1
[[[[]],[],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> [8,6]
=> ? = 1
[[[[]],[[]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [5,3,3]
=> ? = 0
[[[[],[]],[]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8,2,2]
=> ? = 0
[[[[[]]],[]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [6,4]
=> ? = 1
[[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> [5,2,2,2]
=> ? = 1
[[[[],[[]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [9]
=> ? = 0
[[[[[]],[]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [9]
=> ? = 0
[[[[[],[]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [6,2]
=> 1
[[[[[[]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 0
[[],[[[[],[]]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
=> [11,2,2]
=> ? = 0
[[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> [7,6]
=> ? = 1
[[[]],[[[[]]]]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> [16]
=> ? = 0
[[[[]]],[[[]]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> [5,4,4,4]
=> ? = 1
[[[[[]]]],[[]]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> [16]
=> ? = 0
[[[[[],[]]]],[]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
=> [11,2,2]
=> ? = 0
[[[[[[]]]]],[]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> [7,6]
=> ? = 1
[[[],[[],[[]]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [16]
=> ? = 0
[[[],[[[]],[]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [16]
=> ? = 0
[[[],[[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> [6,4,2,2]
=> ? = 1
[[[],[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> [12]
=> ? = 0
[[[[]],[[],[]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> [6,5,3,3]
=> ? = 1
[[[[]],[[[]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 0
[[[[[[[]]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [8]
=> 0
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
Matching statistic: St000206
Mp00047: Ordered trees —to poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 100%
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 100%
Values
[]
=> ([],1)
=> [2]
=> [1,1]
=> 0
[[]]
=> ([(0,1)],2)
=> [3]
=> [1,1,1]
=> 0
[[],[]]
=> ([(0,2),(1,2)],3)
=> [3,2]
=> [2,2,1]
=> 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> 0
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> ? = 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 0
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [8,8,1]
=> ? = 1
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [2,2,2,2,2,2,1]
=> ? = 1
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [2,2,2,2,2,2,1]
=> ? = 1
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [3,3,1,1,1,1,1]
=> ? = 0
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> ? = 1
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [2,2,2,2,2,2,1]
=> ? = 1
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [3,3,3,1]
=> ? = 0
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [3,3,1,1,1,1,1]
=> ? = 0
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> ? = 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> ? = 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 0
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5,4,4,4]
=> [4,4,4,4,1]
=> ? = 1
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5,4,4,4]
=> [4,4,4,4,1]
=> ? = 1
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> [4,4,2,2,1]
=> ? = 1
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> [4,4,4,2,1,1]
=> ? = 1
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5,4,4,4]
=> [4,4,4,4,1]
=> ? = 1
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> [4,4,4,2,1,1]
=> ? = 1
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> [4,4,2,2,1]
=> ? = 1
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[[[],[],[[]]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> [8,6]
=> [2,2,2,2,2,2,1,1]
=> ? = 1
[[[],[[]],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> [8,6]
=> [2,2,2,2,2,2,1,1]
=> ? = 1
[[[],[[],[]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> ? = 0
[[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [6,4]
=> [2,2,2,2,1,1]
=> ? = 1
[[[[]],[],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> [8,6]
=> [2,2,2,2,2,2,1,1]
=> ? = 1
[[[[]],[[]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [5,3,3]
=> [3,3,3,1,1]
=> ? = 0
[[[[],[]],[]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> ? = 0
[[[[[]]],[]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [6,4]
=> [2,2,2,2,1,1]
=> ? = 1
[[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> [5,2,2,2]
=> [4,4,1,1,1]
=> ? = 1
[[[[],[[]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 0
[[[[[]],[]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 0
[[[[[],[]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [6,2]
=> [2,2,1,1,1,1]
=> 1
[[[[[[]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0
[[],[[[[],[]]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
=> [11,2,2]
=> [3,3,1,1,1,1,1,1,1,1,1]
=> ? = 0
[[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> [7,6]
=> [2,2,2,2,2,2,1]
=> ? = 1
[[[]],[[[[]]]]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[[[[]]],[[[]]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> [5,4,4,4]
=> [4,4,4,4,1]
=> ? = 1
[[[[[]]]],[[]]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[[[[[],[]]]],[]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
=> [11,2,2]
=> [3,3,1,1,1,1,1,1,1,1,1]
=> ? = 0
[[[[[[]]]]],[]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> [7,6]
=> [2,2,2,2,2,2,1]
=> ? = 1
[[[],[[],[[]]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[[[],[[[]],[]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[[[],[[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> [6,4,2,2]
=> [4,4,2,2,1,1]
=> ? = 1
[[[],[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[[[[]],[[],[]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> [6,5,3,3]
=> [4,4,4,2,2,1]
=> ? = 1
[[[[]],[[[]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[[[[[[[]]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
See also [[St000205]].
Each value in this statistic is greater than or equal to corresponding value in [[St000205]].
Matching statistic: St000208
Mp00047: Ordered trees —to poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 100%
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 100%
Values
[]
=> ([],1)
=> [2]
=> [1,1]
=> 1 = 0 + 1
[[]]
=> ([(0,1)],2)
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> [3,2]
=> [2,2,1]
=> 2 = 1 + 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> ? = 1 + 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 2 = 1 + 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [8,8,1]
=> ? = 1 + 1
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [2,2,2,2,2,2,1]
=> ? = 1 + 1
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [2,2,2,2,2,2,1]
=> ? = 1 + 1
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [3,3,1,1,1,1,1]
=> ? = 0 + 1
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> ? = 1 + 1
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [2,2,2,2,2,2,1]
=> ? = 1 + 1
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [3,3,3,1]
=> ? = 0 + 1
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [3,3,1,1,1,1,1]
=> ? = 0 + 1
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> ? = 1 + 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> ? = 1 + 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> 2 = 1 + 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 1 = 0 + 1
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5,4,4,4]
=> [4,4,4,4,1]
=> ? = 1 + 1
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5,4,4,4]
=> [4,4,4,4,1]
=> ? = 1 + 1
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> [4,4,2,2,1]
=> ? = 1 + 1
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> [4,4,4,2,1,1]
=> ? = 1 + 1
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5,4,4,4]
=> [4,4,4,4,1]
=> ? = 1 + 1
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> [4,4,4,2,1,1]
=> ? = 1 + 1
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> [4,4,2,2,1]
=> ? = 1 + 1
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[[],[],[[]]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> [8,6]
=> [2,2,2,2,2,2,1,1]
=> ? = 1 + 1
[[[],[[]],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> [8,6]
=> [2,2,2,2,2,2,1,1]
=> ? = 1 + 1
[[[],[[],[]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> ? = 0 + 1
[[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [6,4]
=> [2,2,2,2,1,1]
=> ? = 1 + 1
[[[[]],[],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> [8,6]
=> [2,2,2,2,2,2,1,1]
=> ? = 1 + 1
[[[[]],[[]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [5,3,3]
=> [3,3,3,1,1]
=> ? = 0 + 1
[[[[],[]],[]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> ? = 0 + 1
[[[[[]]],[]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [6,4]
=> [2,2,2,2,1,1]
=> ? = 1 + 1
[[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> [5,2,2,2]
=> [4,4,1,1,1]
=> ? = 1 + 1
[[[[],[[]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[[[[]],[]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[[[[],[]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [6,2]
=> [2,2,1,1,1,1]
=> 2 = 1 + 1
[[[[[[]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[[],[[[[],[]]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
=> [11,2,2]
=> [3,3,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> [7,6]
=> [2,2,2,2,2,2,1]
=> ? = 1 + 1
[[[]],[[[[]]]]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[[[]]],[[[]]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> [5,4,4,4]
=> [4,4,4,4,1]
=> ? = 1 + 1
[[[[[]]]],[[]]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[[[[],[]]]],[]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
=> [11,2,2]
=> [3,3,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[[[[[]]]]],[]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> [7,6]
=> [2,2,2,2,2,2,1]
=> ? = 1 + 1
[[[],[[],[[]]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[[],[[[]],[]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[[],[[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> [6,4,2,2]
=> [4,4,2,2,1,1]
=> ? = 1 + 1
[[[],[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[[[]],[[],[]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> [6,5,3,3]
=> [4,4,4,2,2,1]
=> ? = 1 + 1
[[[[]],[[[]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[[[[[[[]]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has only integer lattice points as vertices.
See also [[St000205]], [[St000206]] and [[St000207]].
Matching statistic: St001876
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[]
=> ([],1)
=> ([],1)
=> ? = 0
[[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 0
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ? = 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ? = 0
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ? = 0
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[],[],[],[]]
=> ([(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)
=> ? = 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)
=> ? = 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)
=> ? = 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)
=> ? = 0
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ? = 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)
=> ? = 1
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ? = 0
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 0
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ? = 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)
=> ? = 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ? = 0
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ? = 0
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ? = 1
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ? = 1
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7)
=> ? = 0
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7)
=> ? = 0
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ? = 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)
=> ? = 0
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ? = 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)
=> ? = 0
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ? = 1
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ? = 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)
=> ? = 0
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7)
=> ? = 0
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7)
=> ? = 0
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ? = 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)
=> ? = 0
[[[],[],[[]]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> ([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7)
=> ? = 1
[[[],[[]],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> ([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7)
=> ? = 1
[[[],[[],[]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=> ? = 0
[[[],[[[]]]]]
=> ([(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)
=> ? = 1
[[[[]],[],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> ([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7)
=> ? = 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)
=> ? = 0
[[[[],[]],[]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=> ? = 0
[[[[[]]],[]]]
=> ([(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)
=> ? = 1
[[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ? = 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)
=> ? = 0
[[[[[]],[]]]]
=> ([(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)
=> ? = 0
[[[[[],[]]]]]
=> ([(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
[[[[[[]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[[],[[[[],[]]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
=> ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(7,5)],8)
=> ? = 0
[[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ([(0,2),(0,6),(1,7),(2,7),(3,5),(4,3),(5,1),(6,4)],8)
=> ? = 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)
=> ? = 0
[[[[]]],[[[]]]]
=> ([(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)
=> ? = 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)
=> ? = 0
[[[[[],[]]]],[]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
=> ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(7,5)],8)
=> ? = 0
[[[[[[]]]]],[]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ([(0,2),(0,6),(1,7),(2,7),(3,5),(4,3),(5,1),(6,4)],8)
=> ? = 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
[[[[[[[]]]]]]]
=> ([(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
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Matching statistic: St000689
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000689: Dyck paths ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000689: Dyck paths ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 100%
Values
[]
=> []
=> [1,0]
=> [1,1,0,0]
=> 0
[[]]
=> [1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 0
[[],[]]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[[[]]]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 0
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 0
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 0
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 0
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 0
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> ? = 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 0
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 0
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> ? = 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> ? = 0
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> ? = 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 0
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 0
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> ? = 0
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 0
[[[],[],[[]]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 1
[[[],[[]],[]]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> ? = 1
[[[],[[],[]]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 0
[[[],[[[]]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[[[]],[],[]]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 1
[[[[]],[[]]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> ? = 0
[[[[],[]],[]]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 0
[[[[[]]],[]]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 1
[[[[],[],[]]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [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]
=> ? = 1
[[[[],[[]]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 0
[[[[[]],[]]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 0
[[[[[],[]]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1
[[[[[[]]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[[],[[[[],[]]]]]
=> [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]
=> [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 0
[[],[[[[[]]]]]]
=> [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]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[[[]],[[[[]]]]]
=> [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]
=> [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[[[[]]],[[[]]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> ? = 1
[[[[[]]]],[[]]]
=> [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]
=> [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> ? = 0
[[[[[],[]]]],[]]
=> [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]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
=> ? = 0
[[[[[[]]]]],[]]
=> [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]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 1
[[[],[[],[[]]]]]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 0
Description
The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid.
The correspondence between LNakayama algebras and Dyck paths is explained in [[St000684]]. A module $M$ is $n$-rigid, if $\operatorname{Ext}^i(M,M)=0$ for $1\leq i\leq n$.
This statistic gives the maximal $n$ such that the minimal generator-cogenerator module $A \oplus D(A)$ of the LNakayama algebra $A$ corresponding to a Dyck path is $n$-rigid.
An application is to check for maximal $n$-orthogonal objects in the module category in the sense of [2].
Matching statistic: St000100
Values
[]
=> ([],1)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
[[]]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 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 = 1 + 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 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)
=> ? = 1 + 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)
=> ? = 0 + 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)
=> ? = 0 + 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)
=> 2 = 1 + 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)
=> 1 = 0 + 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)
=> ? = 1 + 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)
=> ? = 1 + 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)
=> ? = 1 + 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)
=> ? = 0 + 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)
=> ? = 1 + 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)
=> ? = 1 + 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)
=> ? = 0 + 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)
=> ? = 0 + 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)
=> ? = 1 + 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)
=> ? = 1 + 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)
=> ? = 0 + 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)
=> ? = 0 + 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 + 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)
=> 1 = 0 + 1
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,7),(2,8),(3,2),(3,10),(3,11),(4,9),(4,13),(5,9),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,14),(10,7),(10,16),(11,8),(11,16),(12,10),(12,14),(13,11),(13,14),(14,16),(15,1),(16,15)],17)
=> ([(0,4),(0,5),(0,6),(2,7),(2,8),(3,2),(3,10),(3,11),(4,9),(4,13),(5,9),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,14),(10,7),(10,16),(11,8),(11,16),(12,10),(12,14),(13,11),(13,14),(14,16),(15,1),(16,15)],17)
=> ? = 1 + 1
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,7),(2,8),(3,2),(3,10),(3,11),(4,9),(4,13),(5,9),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,14),(10,7),(10,16),(11,8),(11,16),(12,10),(12,14),(13,11),(13,14),(14,16),(15,1),(16,15)],17)
=> ([(0,4),(0,5),(0,6),(2,7),(2,8),(3,2),(3,10),(3,11),(4,9),(4,13),(5,9),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,14),(10,7),(10,16),(11,8),(11,16),(12,10),(12,14),(13,11),(13,14),(14,16),(15,1),(16,15)],17)
=> ? = 1 + 1
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ? = 0 + 1
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ? = 0 + 1
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ? = 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)
=> ? = 0 + 1
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
=> ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
=> ? = 1 + 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)
=> ? = 0 + 1
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,7),(2,8),(3,2),(3,10),(3,11),(4,9),(4,13),(5,9),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,14),(10,7),(10,16),(11,8),(11,16),(12,10),(12,14),(13,11),(13,14),(14,16),(15,1),(16,15)],17)
=> ([(0,4),(0,5),(0,6),(2,7),(2,8),(3,2),(3,10),(3,11),(4,9),(4,13),(5,9),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,14),(10,7),(10,16),(11,8),(11,16),(12,10),(12,14),(13,11),(13,14),(14,16),(15,1),(16,15)],17)
=> ? = 1 + 1
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
=> ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
=> ? = 1 + 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)
=> ? = 0 + 1
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ? = 0 + 1
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ? = 0 + 1
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ? = 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)
=> ? = 0 + 1
[[[],[],[[]]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> ([(0,3),(0,4),(0,6),(1,10),(1,11),(3,8),(3,9),(4,7),(4,9),(5,2),(6,1),(6,7),(6,8),(7,10),(7,13),(8,11),(8,13),(9,13),(10,12),(11,12),(12,5),(13,12)],14)
=> ([(0,3),(0,4),(0,6),(1,10),(1,11),(3,8),(3,9),(4,7),(4,9),(5,2),(6,1),(6,7),(6,8),(7,10),(7,13),(8,11),(8,13),(9,13),(10,12),(11,12),(12,5),(13,12)],14)
=> ? = 1 + 1
[[[],[[]],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> ([(0,3),(0,4),(0,6),(1,10),(1,11),(3,8),(3,9),(4,7),(4,9),(5,2),(6,1),(6,7),(6,8),(7,10),(7,13),(8,11),(8,13),(9,13),(10,12),(11,12),(12,5),(13,12)],14)
=> ([(0,3),(0,4),(0,6),(1,10),(1,11),(3,8),(3,9),(4,7),(4,9),(5,2),(6,1),(6,7),(6,8),(7,10),(7,13),(8,11),(8,13),(9,13),(10,12),(11,12),(12,5),(13,12)],14)
=> ? = 1 + 1
[[[],[[],[]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ? = 0 + 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)
=> ? = 1 + 1
[[[[]],[],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> ([(0,3),(0,4),(0,6),(1,10),(1,11),(3,8),(3,9),(4,7),(4,9),(5,2),(6,1),(6,7),(6,8),(7,10),(7,13),(8,11),(8,13),(9,13),(10,12),(11,12),(12,5),(13,12)],14)
=> ([(0,3),(0,4),(0,6),(1,10),(1,11),(3,8),(3,9),(4,7),(4,9),(5,2),(6,1),(6,7),(6,8),(7,10),(7,13),(8,11),(8,13),(9,13),(10,12),(11,12),(12,5),(13,12)],14)
=> ? = 1 + 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)
=> ? = 0 + 1
[[[[],[]],[]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ? = 0 + 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)
=> ? = 1 + 1
[[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> ([(0,2),(0,3),(0,4),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,5),(7,10),(8,10),(9,10),(10,6)],11)
=> ([(0,2),(0,3),(0,4),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,5),(7,10),(8,10),(9,10),(10,6)],11)
=> ? = 1 + 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)
=> ? = 0 + 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)
=> ? = 0 + 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)
=> ? = 1 + 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)
=> 1 = 0 + 1
[[],[[[[],[]]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
=> ([(0,3),(0,4),(0,5),(2,13),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(6,12),(7,6),(7,11),(8,14),(9,14),(10,7),(10,14),(11,12),(12,13),(13,1),(14,11)],15)
=> ([(0,3),(0,4),(0,5),(2,13),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(6,12),(7,6),(7,11),(8,14),(9,14),(10,7),(10,14),(11,12),(12,13),(13,1),(14,11)],15)
=> ? = 0 + 1
[[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ([(0,3),(0,7),(1,9),(3,8),(4,6),(4,10),(5,4),(5,12),(6,1),(6,11),(7,5),(7,8),(8,12),(9,2),(10,11),(11,9),(12,10)],13)
=> ([(0,3),(0,7),(1,9),(3,8),(4,6),(4,10),(5,4),(5,12),(6,1),(6,11),(7,5),(7,8),(8,12),(9,2),(10,11),(11,9),(12,10)],13)
=> ? = 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)
=> ? = 0 + 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)
=> ? = 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)
=> ? = 0 + 1
[[[[[],[]]]],[]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
=> ([(0,3),(0,4),(0,5),(2,13),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(6,12),(7,6),(7,11),(8,14),(9,14),(10,7),(10,14),(11,12),(12,13),(13,1),(14,11)],15)
=> ([(0,3),(0,4),(0,5),(2,13),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(6,12),(7,6),(7,11),(8,14),(9,14),(10,7),(10,14),(11,12),(12,13),(13,1),(14,11)],15)
=> ? = 0 + 1
[[[[[[]]]]],[]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ([(0,3),(0,7),(1,9),(3,8),(4,6),(4,10),(5,4),(5,12),(6,1),(6,11),(7,5),(7,8),(8,12),(9,2),(10,11),(11,9),(12,10)],13)
=> ([(0,3),(0,7),(1,9),(3,8),(4,6),(4,10),(5,4),(5,12),(6,1),(6,11),(7,5),(7,8),(8,12),(9,2),(10,11),(11,9),(12,10)],13)
=> ? = 1 + 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 = 0 + 1
Description
The number of linear extensions of a poset.
Matching statistic: St000632
Values
[]
=> ([],1)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[[]]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
[[],[]]
=> ([(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
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[],[],[]]
=> ([(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)
=> ? = 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)
=> ? = 0
[[[]],[]]
=> ([(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)
=> ? = 0
[[[],[]]]
=> ([(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
[[[[]]]]
=> ([(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
[[],[],[],[]]
=> ([(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)
=> ? = 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)
=> ? = 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)
=> ? = 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)
=> ? = 0
[[],[[[]]]]
=> ([(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)
=> ? = 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)
=> ? = 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)
=> ? = 0
[[[],[]],[]]
=> ([(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)
=> ? = 0
[[[[]]],[]]
=> ([(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)
=> ? = 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)
=> ? = 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)
=> ? = 0
[[[[]],[]]]
=> ([(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)
=> ? = 0
[[[[],[]]]]
=> ([(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
[[[[[]]]]]
=> ([(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
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,7),(2,8),(3,2),(3,10),(3,11),(4,9),(4,13),(5,9),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,14),(10,7),(10,16),(11,8),(11,16),(12,10),(12,14),(13,11),(13,14),(14,16),(15,1),(16,15)],17)
=> ([(0,4),(0,5),(0,6),(2,7),(2,8),(3,2),(3,10),(3,11),(4,9),(4,13),(5,9),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,14),(10,7),(10,16),(11,8),(11,16),(12,10),(12,14),(13,11),(13,14),(14,16),(15,1),(16,15)],17)
=> ? = 1
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,7),(2,8),(3,2),(3,10),(3,11),(4,9),(4,13),(5,9),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,14),(10,7),(10,16),(11,8),(11,16),(12,10),(12,14),(13,11),(13,14),(14,16),(15,1),(16,15)],17)
=> ([(0,4),(0,5),(0,6),(2,7),(2,8),(3,2),(3,10),(3,11),(4,9),(4,13),(5,9),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,14),(10,7),(10,16),(11,8),(11,16),(12,10),(12,14),(13,11),(13,14),(14,16),(15,1),(16,15)],17)
=> ? = 1
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ? = 0
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ? = 0
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ? = 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)
=> ? = 0
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
=> ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
=> ? = 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)
=> ? = 0
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,7),(2,8),(3,2),(3,10),(3,11),(4,9),(4,13),(5,9),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,14),(10,7),(10,16),(11,8),(11,16),(12,10),(12,14),(13,11),(13,14),(14,16),(15,1),(16,15)],17)
=> ([(0,4),(0,5),(0,6),(2,7),(2,8),(3,2),(3,10),(3,11),(4,9),(4,13),(5,9),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,14),(10,7),(10,16),(11,8),(11,16),(12,10),(12,14),(13,11),(13,14),(14,16),(15,1),(16,15)],17)
=> ? = 1
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
=> ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
=> ? = 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)
=> ? = 0
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ? = 0
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ? = 0
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ? = 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)
=> ? = 0
[[[],[],[[]]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> ([(0,3),(0,4),(0,6),(1,10),(1,11),(3,8),(3,9),(4,7),(4,9),(5,2),(6,1),(6,7),(6,8),(7,10),(7,13),(8,11),(8,13),(9,13),(10,12),(11,12),(12,5),(13,12)],14)
=> ([(0,3),(0,4),(0,6),(1,10),(1,11),(3,8),(3,9),(4,7),(4,9),(5,2),(6,1),(6,7),(6,8),(7,10),(7,13),(8,11),(8,13),(9,13),(10,12),(11,12),(12,5),(13,12)],14)
=> ? = 1
[[[],[[]],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> ([(0,3),(0,4),(0,6),(1,10),(1,11),(3,8),(3,9),(4,7),(4,9),(5,2),(6,1),(6,7),(6,8),(7,10),(7,13),(8,11),(8,13),(9,13),(10,12),(11,12),(12,5),(13,12)],14)
=> ([(0,3),(0,4),(0,6),(1,10),(1,11),(3,8),(3,9),(4,7),(4,9),(5,2),(6,1),(6,7),(6,8),(7,10),(7,13),(8,11),(8,13),(9,13),(10,12),(11,12),(12,5),(13,12)],14)
=> ? = 1
[[[],[[],[]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ? = 0
[[[],[[[]]]]]
=> ([(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)
=> ? = 1
[[[[]],[],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> ([(0,3),(0,4),(0,6),(1,10),(1,11),(3,8),(3,9),(4,7),(4,9),(5,2),(6,1),(6,7),(6,8),(7,10),(7,13),(8,11),(8,13),(9,13),(10,12),(11,12),(12,5),(13,12)],14)
=> ([(0,3),(0,4),(0,6),(1,10),(1,11),(3,8),(3,9),(4,7),(4,9),(5,2),(6,1),(6,7),(6,8),(7,10),(7,13),(8,11),(8,13),(9,13),(10,12),(11,12),(12,5),(13,12)],14)
=> ? = 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)
=> ? = 0
[[[[],[]],[]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ? = 0
[[[[[]]],[]]]
=> ([(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)
=> ? = 1
[[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> ([(0,2),(0,3),(0,4),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,5),(7,10),(8,10),(9,10),(10,6)],11)
=> ([(0,2),(0,3),(0,4),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,5),(7,10),(8,10),(9,10),(10,6)],11)
=> ? = 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)
=> ? = 0
[[[[[]],[]]]]
=> ([(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)
=> ? = 0
[[[[[],[]]]]]
=> ([(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)
=> ? = 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
[[],[[[[],[]]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
=> ([(0,3),(0,4),(0,5),(2,13),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(6,12),(7,6),(7,11),(8,14),(9,14),(10,7),(10,14),(11,12),(12,13),(13,1),(14,11)],15)
=> ([(0,3),(0,4),(0,5),(2,13),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(6,12),(7,6),(7,11),(8,14),(9,14),(10,7),(10,14),(11,12),(12,13),(13,1),(14,11)],15)
=> ? = 0
[[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ([(0,3),(0,7),(1,9),(3,8),(4,6),(4,10),(5,4),(5,12),(6,1),(6,11),(7,5),(7,8),(8,12),(9,2),(10,11),(11,9),(12,10)],13)
=> ([(0,3),(0,7),(1,9),(3,8),(4,6),(4,10),(5,4),(5,12),(6,1),(6,11),(7,5),(7,8),(8,12),(9,2),(10,11),(11,9),(12,10)],13)
=> ? = 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)
=> ? = 0
[[[[]]],[[[]]]]
=> ([(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)
=> ? = 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)
=> ? = 0
[[[[[],[]]]],[]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
=> ([(0,3),(0,4),(0,5),(2,13),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(6,12),(7,6),(7,11),(8,14),(9,14),(10,7),(10,14),(11,12),(12,13),(13,1),(14,11)],15)
=> ([(0,3),(0,4),(0,5),(2,13),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(6,12),(7,6),(7,11),(8,14),(9,14),(10,7),(10,14),(11,12),(12,13),(13,1),(14,11)],15)
=> ? = 0
[[[[[[]]]]],[]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ([(0,3),(0,7),(1,9),(3,8),(4,6),(4,10),(5,4),(5,12),(6,1),(6,11),(7,5),(7,8),(8,12),(9,2),(10,11),(11,9),(12,10)],13)
=> ([(0,3),(0,7),(1,9),(3,8),(4,6),(4,10),(5,4),(5,12),(6,1),(6,11),(7,5),(7,8),(8,12),(9,2),(10,11),(11,9),(12,10)],13)
=> ? = 1
Description
The jump number of the poset.
A jump in a linear extension $e_1, \dots, e_n$ of a poset $P$ is a pair $(e_i, e_{i+1})$ so that $e_{i+1}$ does not cover $e_i$ in $P$. The jump number of a poset is the minimal number of jumps in linear extensions of a poset.
Matching statistic: St000298
Values
[]
=> ([],1)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
[[]]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 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 = 1 + 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 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)
=> ? = 1 + 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)
=> ? = 0 + 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)
=> ? = 0 + 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)
=> 2 = 1 + 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)
=> 1 = 0 + 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)
=> ? = 1 + 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)
=> ? = 1 + 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)
=> ? = 1 + 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)
=> ? = 0 + 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)
=> ? = 1 + 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)
=> ? = 1 + 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)
=> ? = 0 + 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)
=> ? = 0 + 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)
=> ? = 1 + 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)
=> ? = 1 + 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)
=> ? = 0 + 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)
=> ? = 0 + 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 + 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)
=> 1 = 0 + 1
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,7),(2,8),(3,2),(3,10),(3,11),(4,9),(4,13),(5,9),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,14),(10,7),(10,16),(11,8),(11,16),(12,10),(12,14),(13,11),(13,14),(14,16),(15,1),(16,15)],17)
=> ([(0,4),(0,5),(0,6),(2,7),(2,8),(3,2),(3,10),(3,11),(4,9),(4,13),(5,9),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,14),(10,7),(10,16),(11,8),(11,16),(12,10),(12,14),(13,11),(13,14),(14,16),(15,1),(16,15)],17)
=> ? = 1 + 1
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,7),(2,8),(3,2),(3,10),(3,11),(4,9),(4,13),(5,9),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,14),(10,7),(10,16),(11,8),(11,16),(12,10),(12,14),(13,11),(13,14),(14,16),(15,1),(16,15)],17)
=> ([(0,4),(0,5),(0,6),(2,7),(2,8),(3,2),(3,10),(3,11),(4,9),(4,13),(5,9),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,14),(10,7),(10,16),(11,8),(11,16),(12,10),(12,14),(13,11),(13,14),(14,16),(15,1),(16,15)],17)
=> ? = 1 + 1
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ? = 0 + 1
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ? = 0 + 1
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ? = 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)
=> ? = 0 + 1
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
=> ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
=> ? = 1 + 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)
=> ? = 0 + 1
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,7),(2,8),(3,2),(3,10),(3,11),(4,9),(4,13),(5,9),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,14),(10,7),(10,16),(11,8),(11,16),(12,10),(12,14),(13,11),(13,14),(14,16),(15,1),(16,15)],17)
=> ([(0,4),(0,5),(0,6),(2,7),(2,8),(3,2),(3,10),(3,11),(4,9),(4,13),(5,9),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,14),(10,7),(10,16),(11,8),(11,16),(12,10),(12,14),(13,11),(13,14),(14,16),(15,1),(16,15)],17)
=> ? = 1 + 1
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
=> ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
=> ? = 1 + 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)
=> ? = 0 + 1
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ? = 0 + 1
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ? = 0 + 1
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ? = 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)
=> ? = 0 + 1
[[[],[],[[]]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> ([(0,3),(0,4),(0,6),(1,10),(1,11),(3,8),(3,9),(4,7),(4,9),(5,2),(6,1),(6,7),(6,8),(7,10),(7,13),(8,11),(8,13),(9,13),(10,12),(11,12),(12,5),(13,12)],14)
=> ([(0,3),(0,4),(0,6),(1,10),(1,11),(3,8),(3,9),(4,7),(4,9),(5,2),(6,1),(6,7),(6,8),(7,10),(7,13),(8,11),(8,13),(9,13),(10,12),(11,12),(12,5),(13,12)],14)
=> ? = 1 + 1
[[[],[[]],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> ([(0,3),(0,4),(0,6),(1,10),(1,11),(3,8),(3,9),(4,7),(4,9),(5,2),(6,1),(6,7),(6,8),(7,10),(7,13),(8,11),(8,13),(9,13),(10,12),(11,12),(12,5),(13,12)],14)
=> ([(0,3),(0,4),(0,6),(1,10),(1,11),(3,8),(3,9),(4,7),(4,9),(5,2),(6,1),(6,7),(6,8),(7,10),(7,13),(8,11),(8,13),(9,13),(10,12),(11,12),(12,5),(13,12)],14)
=> ? = 1 + 1
[[[],[[],[]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ? = 0 + 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)
=> ? = 1 + 1
[[[[]],[],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> ([(0,3),(0,4),(0,6),(1,10),(1,11),(3,8),(3,9),(4,7),(4,9),(5,2),(6,1),(6,7),(6,8),(7,10),(7,13),(8,11),(8,13),(9,13),(10,12),(11,12),(12,5),(13,12)],14)
=> ([(0,3),(0,4),(0,6),(1,10),(1,11),(3,8),(3,9),(4,7),(4,9),(5,2),(6,1),(6,7),(6,8),(7,10),(7,13),(8,11),(8,13),(9,13),(10,12),(11,12),(12,5),(13,12)],14)
=> ? = 1 + 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)
=> ? = 0 + 1
[[[[],[]],[]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ? = 0 + 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)
=> ? = 1 + 1
[[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> ([(0,2),(0,3),(0,4),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,5),(7,10),(8,10),(9,10),(10,6)],11)
=> ([(0,2),(0,3),(0,4),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,5),(7,10),(8,10),(9,10),(10,6)],11)
=> ? = 1 + 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)
=> ? = 0 + 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)
=> ? = 0 + 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)
=> ? = 1 + 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)
=> 1 = 0 + 1
[[],[[[[],[]]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
=> ([(0,3),(0,4),(0,5),(2,13),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(6,12),(7,6),(7,11),(8,14),(9,14),(10,7),(10,14),(11,12),(12,13),(13,1),(14,11)],15)
=> ([(0,3),(0,4),(0,5),(2,13),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(6,12),(7,6),(7,11),(8,14),(9,14),(10,7),(10,14),(11,12),(12,13),(13,1),(14,11)],15)
=> ? = 0 + 1
[[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ([(0,3),(0,7),(1,9),(3,8),(4,6),(4,10),(5,4),(5,12),(6,1),(6,11),(7,5),(7,8),(8,12),(9,2),(10,11),(11,9),(12,10)],13)
=> ([(0,3),(0,7),(1,9),(3,8),(4,6),(4,10),(5,4),(5,12),(6,1),(6,11),(7,5),(7,8),(8,12),(9,2),(10,11),(11,9),(12,10)],13)
=> ? = 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)
=> ? = 0 + 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)
=> ? = 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)
=> ? = 0 + 1
[[[[[],[]]]],[]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
=> ([(0,3),(0,4),(0,5),(2,13),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(6,12),(7,6),(7,11),(8,14),(9,14),(10,7),(10,14),(11,12),(12,13),(13,1),(14,11)],15)
=> ([(0,3),(0,4),(0,5),(2,13),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(6,12),(7,6),(7,11),(8,14),(9,14),(10,7),(10,14),(11,12),(12,13),(13,1),(14,11)],15)
=> ? = 0 + 1
[[[[[[]]]]],[]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ([(0,3),(0,7),(1,9),(3,8),(4,6),(4,10),(5,4),(5,12),(6,1),(6,11),(7,5),(7,8),(8,12),(9,2),(10,11),(11,9),(12,10)],13)
=> ([(0,3),(0,7),(1,9),(3,8),(4,6),(4,10),(5,4),(5,12),(6,1),(6,11),(7,5),(7,8),(8,12),(9,2),(10,11),(11,9),(12,10)],13)
=> ? = 1 + 1
Description
The order dimension or Dushnik-Miller dimension of a poset.
This is the minimal number of linear orderings whose intersection is the given poset.
The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St000264The girth of a graph, which is not a tree.
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