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Your data matches 15 different statistics following compositions of up to 3 maps.
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Matching statistic: St001875
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Values
([(0,2),(2,1)],3)
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(0,3),(2,1),(3,2)],4)
=> 4
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 4
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4
([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> 5
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> 4
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> 5
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 5
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 5
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 5
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 5
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 5
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 5
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> 5
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 5
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 5
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> 6
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> 5
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> 5
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> 6
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> 6
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> 6
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> 6
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> 5
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> 5
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> 5
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 6
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> 6
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 5
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> 6
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> 5
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> 6
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 6
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> 6
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 5
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> 6
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> 6
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> 6
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> 6
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 5
([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 6
([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=> 6
Description
The number of simple modules with projective dimension at most 1.
Matching statistic: St001622
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(load all 3 compositions to match this statistic)
Values
([(0,2),(2,1)],3)
=> 2 = 3 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 3 = 4 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3 = 4 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3 = 4 - 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 5 - 1
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> 4 = 5 - 1
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> 4 = 5 - 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> 4 = 5 - 1
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 4 = 5 - 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 4 = 5 - 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4 = 5 - 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 4 = 5 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 3 = 4 - 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 4 = 5 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> 4 = 5 - 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 4 = 5 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3 = 4 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 4 = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5 = 6 - 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> 5 = 6 - 1
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> 5 = 6 - 1
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> 5 = 6 - 1
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> 5 = 6 - 1
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> 5 = 6 - 1
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> 4 = 5 - 1
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> 5 = 6 - 1
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> 5 = 6 - 1
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> 5 = 6 - 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> 4 = 5 - 1
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 5 = 6 - 1
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> 5 = 6 - 1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 4 = 5 - 1
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> 5 = 6 - 1
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> 5 = 6 - 1
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> 5 = 6 - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 4 = 5 - 1
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> 4 = 5 - 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 4 = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> 4 = 5 - 1
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> 5 = 6 - 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> 5 = 6 - 1
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> 4 = 5 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 5 = 6 - 1
([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=> 4 = 5 - 1
Description
The number of join-irreducible elements of a lattice.
An element $j$ of a lattice $L$ is '''join irreducible''' if it is not the least element and if $j=x\vee y$, then $j\in\{x,y\}$ for all $x,y\in L$.
Matching statistic: St000189
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(load all 2 compositions to match this statistic)
Values
([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 2 = 3 - 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 3 = 4 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([],3)
=> 3 = 4 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(1,2)],3)
=> 3 = 4 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,1),(0,2)],3)
=> 3 = 4 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 5 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([],4)
=> 4 = 5 - 1
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(2,3)],4)
=> 4 = 5 - 1
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(1,2),(1,3)],4)
=> 4 = 5 - 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,1),(0,2),(0,3)],4)
=> 4 = 5 - 1
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,2),(0,3),(3,1)],4)
=> 4 = 5 - 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(1,2),(2,3)],4)
=> 4 = 5 - 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(3,1),(3,2)],4)
=> 4 = 5 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([],3)
=> 3 = 4 - 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,3),(1,3),(3,2)],4)
=> 4 = 5 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> 4 = 5 - 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(1,2)],4)
=> 4 = 5 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(1,2)],3)
=> 3 = 4 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 6 - 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 4 = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([],5)
=> 5 = 6 - 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(3,4)],5)
=> 5 = 6 - 1
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ([(2,3),(2,4)],5)
=> 5 = 6 - 1
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(1,2),(1,3),(1,4)],5)
=> 5 = 6 - 1
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> 5 = 6 - 1
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> 5 = 6 - 1
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> ([(0,1),(0,2),(0,3)],4)
=> 4 = 5 - 1
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ([(1,3),(1,4),(4,2)],5)
=> 5 = 6 - 1
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> 5 = 6 - 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(1,2),(1,3)],4)
=> 4 = 5 - 1
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5 = 6 - 1
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 5 = 6 - 1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,2),(0,3),(3,1)],4)
=> 4 = 5 - 1
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(2,3),(3,4)],5)
=> 5 = 6 - 1
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(1,4),(4,2),(4,3)],5)
=> 5 = 6 - 1
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(0,4),(4,1),(4,2),(4,3)],5)
=> 5 = 6 - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([],4)
=> 4 = 5 - 1
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(1,3),(2,3)],4)
=> 4 = 5 - 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([],4)
=> 4 = 5 - 1
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> 5 = 6 - 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ([(2,3)],4)
=> 4 = 5 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([],3)
=> 3 = 4 - 1
([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 5 = 6 - 1
([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 4 = 5 - 1
Description
The number of elements in the poset.
Matching statistic: St000228
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00263: Lattices —join irreducibles⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> [2]
=> 2 = 3 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> [1,1]
=> 2 = 3 - 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 4 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([],3)
=> [1,1,1]
=> 3 = 4 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(1,2)],3)
=> [2,1]
=> 3 = 4 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,1),(0,2)],3)
=> [2,1]
=> 3 = 4 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 5 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(1,2)],3)
=> [2,1]
=> 3 = 4 - 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([],4)
=> [1,1,1,1]
=> 4 = 5 - 1
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(2,3)],4)
=> [2,1,1]
=> 4 = 5 - 1
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(1,2),(1,3)],4)
=> [2,1,1]
=> 4 = 5 - 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> 4 = 5 - 1
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> 4 = 5 - 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 4 = 5 - 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(1,2),(2,3)],4)
=> [3,1]
=> 4 = 5 - 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> 4 = 5 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([],3)
=> [1,1,1]
=> 3 = 4 - 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 4 = 5 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> 4 = 5 - 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> 4 = 5 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(1,2)],3)
=> [2,1]
=> 3 = 4 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 6 - 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 4 = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([],5)
=> [1,1,1,1,1]
=> 5 = 6 - 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 5 = 6 - 1
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ([(2,3),(2,4)],5)
=> [2,1,1,1]
=> 5 = 6 - 1
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> 5 = 6 - 1
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> 5 = 6 - 1
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> 5 = 6 - 1
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> ([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> 4 = 5 - 1
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> 5 = 6 - 1
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> 5 = 6 - 1
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> 5 = 6 - 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(1,2),(1,3)],4)
=> [2,1,1]
=> 4 = 5 - 1
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 5 = 6 - 1
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> 5 = 6 - 1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> 4 = 5 - 1
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(2,3),(3,4)],5)
=> [3,1,1]
=> 5 = 6 - 1
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> 5 = 6 - 1
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> 5 = 6 - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([],4)
=> [1,1,1,1]
=> 4 = 5 - 1
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(1,3),(2,3)],4)
=> [2,1,1]
=> 4 = 5 - 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> 4 = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([],4)
=> [1,1,1,1]
=> 4 = 5 - 1
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> 5 = 6 - 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> 5 = 6 - 1
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ([(2,3)],4)
=> [2,1,1]
=> 4 = 5 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([],3)
=> [1,1,1]
=> 3 = 4 - 1
([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 5 = 6 - 1
([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> 4 = 5 - 1
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St001746
Values
([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2 = 3 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],2)
=> ([(0,1)],2)
=> 2 = 3 - 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 3 = 4 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 3 = 4 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 4 = 5 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 3 = 4 - 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 4 = 5 - 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 4 = 5 - 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 5 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 4 = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> ([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 4 = 5 - 1
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 4 = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
Description
The coalition number of a graph.
This is the maximal cardinality of a set partition such that each block is either a dominating set of cardinality one, or is not a dominating set but can be joined with a second block to form a dominating set.
Matching statistic: St001615
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3 = 4 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3 = 4 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3 = 4 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3 = 4 - 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? ∊ {5,5,5} - 1
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? ∊ {5,5,5} - 1
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? ∊ {5,5,5} - 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> 4 = 5 - 1
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 4 = 5 - 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 4 = 5 - 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 4 = 5 - 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 4 = 5 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3 = 4 - 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 4 = 5 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(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)
=> 4 = 5 - 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 4 = 5 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3 = 4 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 4 = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([],5)
=> ?
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(3,4)],5)
=> ?
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ([(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(1,11),(1,13),(2,11),(2,12),(3,4),(3,5),(3,12),(3,13),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,15),(6,17),(7,15),(7,18),(8,16),(8,17),(9,16),(9,18),(10,15),(10,16),(11,14),(12,6),(12,7),(12,14),(13,8),(13,9),(13,14),(14,17),(14,18),(15,19),(16,19),(17,19),(18,19)],20)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(1,2),(1,3),(1,4)],5)
=> ([(0,1),(0,2),(1,12),(2,3),(2,4),(2,5),(2,12),(3,8),(3,10),(3,11),(4,7),(4,9),(4,11),(5,6),(5,9),(5,10),(6,13),(6,14),(7,13),(7,15),(8,14),(8,15),(9,13),(9,16),(10,14),(10,16),(11,15),(11,16),(12,6),(12,7),(12,8),(13,17),(14,17),(15,17),(16,17)],18)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1),(1,2),(1,3),(1,4),(1,5),(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)],17)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> 4 = 5 - 1
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ([(1,3),(1,4),(4,2)],5)
=> ([(0,1),(0,2),(1,11),(2,3),(2,4),(2,11),(3,8),(3,10),(4,5),(4,9),(4,10),(5,6),(5,7),(6,13),(7,13),(8,12),(9,7),(9,12),(10,6),(10,12),(11,8),(11,9),(12,13)],14)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 5 = 6 - 1
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 4 = 5 - 1
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(1,9),(2,6),(2,8),(3,7),(4,5),(4,7),(5,1),(5,2),(5,10),(6,11),(7,10),(8,11),(9,11),(10,8),(10,9)],12)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 4 = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(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)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(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)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3 = 4 - 1
([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 5 = 6 - 1
([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=> ([(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)
=> 4 = 5 - 1
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7)
=> ([(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)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7)
=> ([(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(1,10),(2,7),(2,9),(3,11),(3,12),(4,2),(4,11),(4,13),(5,1),(5,12),(5,13),(6,17),(7,15),(8,16),(9,6),(9,15),(10,6),(10,16),(11,7),(11,14),(12,8),(12,14),(13,9),(13,10),(13,14),(14,15),(14,16),(15,17),(16,17)],18)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=> ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 4 = 5 - 1
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 4 = 5 - 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> 4 = 5 - 1
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7)
=> ([(0,4),(1,2),(1,3)],5)
=> ([(0,2),(0,3),(1,11),(2,1),(2,12),(3,4),(3,5),(3,12),(4,8),(4,10),(5,8),(5,9),(6,14),(7,14),(8,13),(9,6),(9,13),(10,7),(10,13),(11,6),(11,7),(12,9),(12,10),(12,11),(13,14)],15)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7)
=> ([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> 5 = 6 - 1
([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
=> ([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> 5 = 6 - 1
([(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2),(6,3)],7)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,3),(0,4),(1,6),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6)],7)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 4 = 5 - 1
([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 4 = 5 - 1
([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> ([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 5 = 6 - 1
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=> ([(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)
=> 5 = 6 - 1
([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7)
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> 5 = 6 - 1
([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
=> ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 4 = 5 - 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=> ([(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)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> ([(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)
=> 5 = 6 - 1
([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 5 = 6 - 1
Description
The number of join prime elements of a lattice.
An element $x$ of a lattice $L$ is join-prime (or coprime) if $x \leq a \vee b$ implies $x \leq a$ or $x \leq b$ for every $a, b \in L$.
Matching statistic: St001617
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3 = 4 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3 = 4 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3 = 4 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3 = 4 - 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? ∊ {5,5,5} - 1
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? ∊ {5,5,5} - 1
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? ∊ {5,5,5} - 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> 4 = 5 - 1
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 4 = 5 - 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 4 = 5 - 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 4 = 5 - 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 4 = 5 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3 = 4 - 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 4 = 5 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(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)
=> 4 = 5 - 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 4 = 5 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3 = 4 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 4 = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([],5)
=> ?
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(3,4)],5)
=> ?
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ([(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(1,11),(1,13),(2,11),(2,12),(3,4),(3,5),(3,12),(3,13),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,15),(6,17),(7,15),(7,18),(8,16),(8,17),(9,16),(9,18),(10,15),(10,16),(11,14),(12,6),(12,7),(12,14),(13,8),(13,9),(13,14),(14,17),(14,18),(15,19),(16,19),(17,19),(18,19)],20)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(1,2),(1,3),(1,4)],5)
=> ([(0,1),(0,2),(1,12),(2,3),(2,4),(2,5),(2,12),(3,8),(3,10),(3,11),(4,7),(4,9),(4,11),(5,6),(5,9),(5,10),(6,13),(6,14),(7,13),(7,15),(8,14),(8,15),(9,13),(9,16),(10,14),(10,16),(11,15),(11,16),(12,6),(12,7),(12,8),(13,17),(14,17),(15,17),(16,17)],18)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1),(1,2),(1,3),(1,4),(1,5),(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)],17)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> 4 = 5 - 1
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ([(1,3),(1,4),(4,2)],5)
=> ([(0,1),(0,2),(1,11),(2,3),(2,4),(2,11),(3,8),(3,10),(4,5),(4,9),(4,10),(5,6),(5,7),(6,13),(7,13),(8,12),(9,7),(9,12),(10,6),(10,12),(11,8),(11,9),(12,13)],14)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 5 = 6 - 1
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 4 = 5 - 1
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(1,9),(2,6),(2,8),(3,7),(4,5),(4,7),(5,1),(5,2),(5,10),(6,11),(7,10),(8,11),(9,11),(10,8),(10,9)],12)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 4 = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(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)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(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)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3 = 4 - 1
([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 5 = 6 - 1
([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=> ([(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)
=> 4 = 5 - 1
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7)
=> ([(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)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7)
=> ([(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(1,10),(2,7),(2,9),(3,11),(3,12),(4,2),(4,11),(4,13),(5,1),(5,12),(5,13),(6,17),(7,15),(8,16),(9,6),(9,15),(10,6),(10,16),(11,7),(11,14),(12,8),(12,14),(13,9),(13,10),(13,14),(14,15),(14,16),(15,17),(16,17)],18)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=> ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 4 = 5 - 1
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 4 = 5 - 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> 4 = 5 - 1
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7)
=> ([(0,4),(1,2),(1,3)],5)
=> ([(0,2),(0,3),(1,11),(2,1),(2,12),(3,4),(3,5),(3,12),(4,8),(4,10),(5,8),(5,9),(6,14),(7,14),(8,13),(9,6),(9,13),(10,7),(10,13),(11,6),(11,7),(12,9),(12,10),(12,11),(13,14)],15)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7)
=> ([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> 5 = 6 - 1
([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
=> ([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> 5 = 6 - 1
([(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2),(6,3)],7)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,3),(0,4),(1,6),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6)],7)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 4 = 5 - 1
([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 4 = 5 - 1
([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> ([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 5 = 6 - 1
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=> ([(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)
=> 5 = 6 - 1
([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7)
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> 5 = 6 - 1
([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
=> ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 4 = 5 - 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=> ([(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)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> ([(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)
=> 5 = 6 - 1
([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 5 = 6 - 1
Description
The dimension of the space of valuations of a lattice.
A valuation, or modular function, on a lattice $L$ is a function $v:L\mapsto\mathbb R$ satisfying
$$
v(a\vee b) + v(a\wedge b) = v(a) + v(b).
$$
It was shown by Birkhoff [1, thm. X.2], that a lattice with a positive valuation must be modular. This was sharpened by Fleischer and Traynor [2, thm. 1], which states that the modular functions on an arbitrary lattice are in bijection with the modular functions on its modular quotient [[Mp00196]].
Moreover, Birkhoff [1, thm. X.2] showed that the dimension of the space of modular functions equals the number of subsets of projective prime intervals.
Matching statistic: St001820
Values
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 3
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 4
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 4
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 4
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,6),(2,10),(2,11),(2,12),(3,8),(3,9),(3,12),(4,7),(4,9),(4,11),(5,7),(5,8),(5,10),(6,2),(6,3),(6,4),(6,5),(7,13),(7,16),(8,13),(8,14),(9,13),(9,15),(10,14),(10,16),(11,15),(11,16),(12,14),(12,15),(13,17),(14,17),(15,17),(16,17),(17,1)],18)
=> ? ∊ {4,4,5,5,5,5,5,5,5}
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,6),(2,10),(2,11),(3,7),(3,9),(4,7),(4,8),(5,2),(5,8),(5,9),(6,3),(6,4),(6,5),(7,12),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13),(13,1)],14)
=> ? ∊ {4,4,5,5,5,5,5,5,5}
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,6),(2,7),(3,8),(3,10),(4,8),(4,9),(5,3),(5,4),(5,7),(6,2),(6,5),(7,9),(7,10),(8,11),(9,11),(10,11),(11,1)],12)
=> ? ∊ {4,4,5,5,5,5,5,5,5}
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,5),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,6),(6,2),(6,3),(6,4),(7,10),(8,10),(9,10),(10,1)],11)
=> ? ∊ {4,4,5,5,5,5,5,5,5}
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> 5
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> 5
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? ∊ {4,4,5,5,5,5,5,5,5}
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> 5
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,6),(2,10),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(6,3),(6,4),(6,5),(7,11),(8,11),(9,2),(9,11),(10,1),(11,10)],12)
=> ? ∊ {4,4,5,5,5,5,5,5,5}
([(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)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> 5
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,6),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,2),(6,3),(6,4),(7,10),(8,10),(9,10),(10,5)],11)
=> ? ∊ {4,4,5,5,5,5,5,5,5}
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,6),(2,9),(3,8),(4,3),(4,7),(5,2),(5,7),(6,4),(6,5),(7,8),(7,9),(8,10),(9,10),(10,1)],11)
=> ? ∊ {4,4,5,5,5,5,5,5,5}
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? ∊ {4,4,5,5,5,5,5,5,5}
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> 5
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ?
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ?
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,6),(2,8),(3,10),(3,11),(3,14),(4,9),(4,11),(4,13),(5,9),(5,10),(5,12),(6,2),(6,7),(7,3),(7,4),(7,5),(7,8),(8,12),(8,13),(8,14),(9,17),(9,18),(10,15),(10,18),(11,16),(11,18),(12,15),(12,17),(13,16),(13,17),(14,15),(14,16),(15,19),(16,19),(17,19),(18,19),(19,1)],20)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,6),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,7),(7,2),(7,3),(7,4),(7,5),(8,14),(8,17),(9,14),(9,15),(10,14),(10,16),(11,15),(11,17),(12,16),(12,17),(13,15),(13,16),(14,18),(15,18),(16,18),(17,18),(18,1)],19)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ([(0,6),(2,11),(2,12),(3,8),(3,10),(4,8),(4,9),(5,2),(5,9),(5,10),(6,7),(7,3),(7,4),(7,5),(8,13),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14),(14,1)],15)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(2,11),(3,9),(3,10),(4,8),(4,10),(5,8),(5,9),(6,7),(7,3),(7,4),(7,5),(8,12),(9,12),(10,2),(10,12),(11,1),(12,11)],13)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(2,9),(2,10),(3,8),(3,10),(4,8),(4,9),(5,1),(6,7),(7,2),(7,3),(7,4),(8,11),(9,11),(10,11),(11,5)],12)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ([(0,7),(1,9),(3,8),(3,12),(4,10),(4,11),(5,4),(5,8),(5,13),(6,3),(6,5),(6,9),(7,1),(7,6),(8,11),(8,14),(9,12),(9,13),(10,15),(11,15),(12,14),(13,10),(13,14),(14,15),(15,2)],16)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ([(0,5),(2,9),(2,11),(3,9),(3,10),(4,8),(5,6),(6,4),(6,7),(7,2),(7,3),(7,8),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(0,7),(2,8),(3,12),(4,10),(4,11),(5,9),(5,11),(6,4),(6,5),(6,8),(7,2),(7,6),(8,9),(8,10),(9,13),(10,13),(11,3),(11,13),(12,1),(13,12)],14)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ([(0,6),(2,10),(3,9),(4,3),(4,8),(5,2),(5,8),(6,7),(7,4),(7,5),(8,9),(8,10),(9,11),(10,11),(11,1)],12)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(0,7),(2,8),(2,10),(3,8),(3,9),(4,11),(4,12),(5,6),(5,9),(5,10),(6,4),(6,13),(6,14),(7,2),(7,3),(7,5),(8,15),(9,13),(9,15),(10,14),(10,15),(11,17),(12,17),(13,11),(13,16),(14,12),(14,16),(15,16),(16,17),(17,1)],18)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(0,7),(1,9),(3,8),(3,11),(4,8),(4,10),(5,6),(5,9),(6,3),(6,4),(6,12),(7,1),(7,5),(8,13),(9,12),(10,13),(11,13),(12,10),(12,11),(13,2)],14)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(0,6),(2,9),(2,10),(3,8),(3,10),(4,8),(4,9),(5,7),(6,5),(7,2),(7,3),(7,4),(8,11),(9,11),(10,11),(11,1)],12)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ?
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(0,7),(2,12),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(6,11),(7,3),(7,4),(7,5),(8,13),(9,13),(10,6),(10,13),(11,12),(12,1),(13,11)],14)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,7),(2,9),(3,9),(4,8),(5,8),(6,2),(6,3),(7,4),(7,5),(8,6),(9,1)],10)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,7),(1,14),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,11),(6,12),(6,13),(7,3),(7,4),(7,5),(7,6),(8,17),(8,18),(9,15),(9,18),(10,16),(10,18),(11,15),(11,17),(12,16),(12,17),(13,15),(13,16),(14,2),(15,19),(16,19),(17,19),(18,1),(18,19),(19,14)],20)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(0,7),(2,9),(2,10),(3,8),(3,10),(4,8),(4,9),(5,1),(6,5),(7,2),(7,3),(7,4),(8,11),(9,11),(10,11),(11,6)],12)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,1),(7,2),(7,3),(7,4),(7,5),(8,14),(8,17),(9,14),(9,15),(10,14),(10,16),(11,15),(11,17),(12,16),(12,17),(13,15),(13,16),(14,18),(15,18),(16,18),(17,18),(18,6)],19)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ([(0,7),(2,11),(2,12),(3,8),(4,10),(4,13),(5,9),(5,13),(6,2),(6,9),(6,10),(7,4),(7,5),(7,6),(8,14),(9,11),(9,16),(10,12),(10,16),(11,15),(12,15),(13,3),(13,16),(14,1),(15,14),(16,8),(16,15)],17)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,7),(2,11),(2,12),(3,10),(4,9),(5,8),(5,11),(6,8),(6,12),(7,2),(7,5),(7,6),(8,14),(9,13),(10,13),(11,4),(11,14),(12,3),(12,14),(13,1),(14,9),(14,10)],15)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(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)
=> ([(0,7),(2,8),(3,8),(4,5),(5,1),(6,4),(7,2),(7,3),(8,6)],9)
=> 6
([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=> ([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,10),(3,9),(3,11),(4,8),(4,11),(5,8),(5,9),(6,1),(7,3),(7,4),(7,5),(8,12),(9,12),(10,6),(11,2),(11,12),(12,10)],13)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
=> ([(0,7),(1,13),(3,11),(3,12),(4,8),(4,10),(5,8),(5,9),(6,3),(6,9),(6,10),(7,4),(7,5),(7,6),(8,15),(9,11),(9,15),(10,12),(10,15),(11,14),(12,14),(13,2),(14,13),(15,1),(15,14)],16)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7)
=> ([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7)
=> ([(0,7),(1,11),(1,12),(3,8),(3,10),(4,8),(4,9),(5,2),(6,1),(6,9),(6,10),(7,3),(7,4),(7,6),(8,14),(9,11),(9,14),(10,12),(10,14),(11,13),(12,13),(13,5),(14,13)],15)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7)
=> ([(0,2),(2,3),(2,4),(2,5),(3,13),(3,14),(4,7),(4,14),(4,15),(5,6),(5,13),(5,15),(6,9),(6,11),(7,10),(7,12),(8,19),(9,17),(10,18),(11,8),(11,17),(12,8),(12,18),(13,9),(13,16),(14,10),(14,16),(15,11),(15,12),(15,16),(16,17),(16,18),(17,19),(18,19),(19,1)],20)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
=> ([(0,7),(2,9),(2,12),(3,8),(3,14),(4,11),(4,13),(5,8),(5,10),(6,4),(6,10),(6,14),(7,3),(7,5),(7,6),(8,16),(9,17),(10,13),(10,16),(11,12),(11,15),(12,17),(13,15),(14,2),(14,11),(14,16),(15,17),(16,9),(16,15),(17,1)],18)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=> ([(0,7),(2,11),(3,10),(4,9),(5,6),(5,11),(6,4),(6,8),(7,2),(7,5),(8,9),(8,10),(9,12),(10,12),(11,3),(11,8),(12,1)],13)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(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)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> ([(0,7),(1,11),(3,8),(4,10),(5,4),(5,8),(6,1),(6,9),(7,3),(7,5),(8,6),(8,10),(9,11),(10,9),(11,2)],12)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7)
=> ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7)
=> ([(0,7),(2,8),(2,12),(3,8),(3,11),(4,10),(5,4),(5,9),(6,2),(6,3),(6,9),(7,5),(7,6),(8,15),(9,10),(9,11),(9,12),(10,13),(10,14),(11,13),(11,15),(12,14),(12,15),(13,16),(14,16),(15,16),(16,1)],17)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7)
=> ([(0,7),(2,13),(3,11),(3,12),(4,8),(4,10),(5,8),(5,9),(6,4),(6,5),(6,13),(7,2),(7,6),(8,14),(9,11),(9,14),(10,12),(10,14),(11,15),(12,15),(13,3),(13,9),(13,10),(14,15),(15,1)],16)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> ([(0,6),(2,10),(3,8),(4,2),(4,9),(5,4),(5,8),(6,7),(7,3),(7,5),(8,9),(9,10),(10,1)],11)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
=> ([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
=> ([(0,7),(1,11),(3,8),(4,9),(4,12),(5,9),(5,10),(6,4),(6,5),(6,8),(7,3),(7,6),(8,10),(8,12),(9,13),(10,13),(11,2),(12,1),(12,13),(13,11)],14)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,6),(2,9),(3,8),(4,2),(4,8),(5,1),(6,7),(7,3),(7,4),(8,9),(9,5)],10)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2),(6,3)],7)
=> ([(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2),(6,3)],7)
=> ([(0,7),(2,8),(3,9),(3,11),(4,9),(4,10),(5,1),(6,3),(6,4),(6,8),(7,2),(7,6),(8,10),(8,11),(9,12),(10,12),(11,12),(12,5)],13)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(0,3),(0,4),(1,6),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(1,6),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6)],7)
=> ([(0,7),(1,10),(1,11),(3,9),(4,8),(5,4),(5,13),(6,3),(6,13),(7,5),(7,6),(8,10),(8,12),(9,11),(9,12),(10,14),(11,14),(12,14),(13,1),(13,8),(13,9),(14,2)],15)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
([(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)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 7
([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,6),(2,8),(3,8),(4,7),(5,1),(6,4),(7,2),(7,3),(8,5)],9)
=> 6
Description
The size of the image of the pop stack sorting operator.
The pop stack sorting operator is defined by $Pop_L^\downarrow(x) = x\wedge\bigwedge\{y\in L\mid y\lessdot x\}$. This statistic returns the size of $Pop_L^\downarrow(L)\}$.
Matching statistic: St001720
Values
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 4 = 3 + 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 4 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> 5 = 4 + 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 5 = 4 + 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 5 = 4 + 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,6),(2,10),(2,11),(2,12),(3,8),(3,9),(3,12),(4,7),(4,9),(4,11),(5,7),(5,8),(5,10),(6,2),(6,3),(6,4),(6,5),(7,13),(7,16),(8,13),(8,14),(9,13),(9,15),(10,14),(10,16),(11,15),(11,16),(12,14),(12,15),(13,17),(14,17),(15,17),(16,17),(17,1)],18)
=> ? ∊ {4,4,5,5,5,5,5,5,5} + 1
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,6),(2,10),(2,11),(3,7),(3,9),(4,7),(4,8),(5,2),(5,8),(5,9),(6,3),(6,4),(6,5),(7,12),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13),(13,1)],14)
=> ? ∊ {4,4,5,5,5,5,5,5,5} + 1
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,6),(2,7),(3,8),(3,10),(4,8),(4,9),(5,3),(5,4),(5,7),(6,2),(6,5),(7,9),(7,10),(8,11),(9,11),(10,11),(11,1)],12)
=> ? ∊ {4,4,5,5,5,5,5,5,5} + 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,5),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,6),(6,2),(6,3),(6,4),(7,10),(8,10),(9,10),(10,1)],11)
=> ? ∊ {4,4,5,5,5,5,5,5,5} + 1
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> 6 = 5 + 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> 6 = 5 + 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? ∊ {4,4,5,5,5,5,5,5,5} + 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> 6 = 5 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,6),(2,10),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(6,3),(6,4),(6,5),(7,11),(8,11),(9,2),(9,11),(10,1),(11,10)],12)
=> ? ∊ {4,4,5,5,5,5,5,5,5} + 1
([(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)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> 6 = 5 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,6),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,2),(6,3),(6,4),(7,10),(8,10),(9,10),(10,5)],11)
=> ? ∊ {4,4,5,5,5,5,5,5,5} + 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,6),(2,9),(3,8),(4,3),(4,7),(5,2),(5,7),(6,4),(6,5),(7,8),(7,9),(8,10),(9,10),(10,1)],11)
=> ? ∊ {4,4,5,5,5,5,5,5,5} + 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? ∊ {4,4,5,5,5,5,5,5,5} + 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> 6 = 5 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ?
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ?
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,6),(2,8),(3,10),(3,11),(3,14),(4,9),(4,11),(4,13),(5,9),(5,10),(5,12),(6,2),(6,7),(7,3),(7,4),(7,5),(7,8),(8,12),(8,13),(8,14),(9,17),(9,18),(10,15),(10,18),(11,16),(11,18),(12,15),(12,17),(13,16),(13,17),(14,15),(14,16),(15,19),(16,19),(17,19),(18,19),(19,1)],20)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,6),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,7),(7,2),(7,3),(7,4),(7,5),(8,14),(8,17),(9,14),(9,15),(10,14),(10,16),(11,15),(11,17),(12,16),(12,17),(13,15),(13,16),(14,18),(15,18),(16,18),(17,18),(18,1)],19)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ([(0,6),(2,11),(2,12),(3,8),(3,10),(4,8),(4,9),(5,2),(5,9),(5,10),(6,7),(7,3),(7,4),(7,5),(8,13),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14),(14,1)],15)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(2,11),(3,9),(3,10),(4,8),(4,10),(5,8),(5,9),(6,7),(7,3),(7,4),(7,5),(8,12),(9,12),(10,2),(10,12),(11,1),(12,11)],13)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(2,9),(2,10),(3,8),(3,10),(4,8),(4,9),(5,1),(6,7),(7,2),(7,3),(7,4),(8,11),(9,11),(10,11),(11,5)],12)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ([(0,7),(1,9),(3,8),(3,12),(4,10),(4,11),(5,4),(5,8),(5,13),(6,3),(6,5),(6,9),(7,1),(7,6),(8,11),(8,14),(9,12),(9,13),(10,15),(11,15),(12,14),(13,10),(13,14),(14,15),(15,2)],16)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ([(0,5),(2,9),(2,11),(3,9),(3,10),(4,8),(5,6),(6,4),(6,7),(7,2),(7,3),(7,8),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(0,7),(2,8),(3,12),(4,10),(4,11),(5,9),(5,11),(6,4),(6,5),(6,8),(7,2),(7,6),(8,9),(8,10),(9,13),(10,13),(11,3),(11,13),(12,1),(13,12)],14)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ([(0,6),(2,10),(3,9),(4,3),(4,8),(5,2),(5,8),(6,7),(7,4),(7,5),(8,9),(8,10),(9,11),(10,11),(11,1)],12)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(0,7),(2,8),(2,10),(3,8),(3,9),(4,11),(4,12),(5,6),(5,9),(5,10),(6,4),(6,13),(6,14),(7,2),(7,3),(7,5),(8,15),(9,13),(9,15),(10,14),(10,15),(11,17),(12,17),(13,11),(13,16),(14,12),(14,16),(15,16),(16,17),(17,1)],18)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(0,7),(1,9),(3,8),(3,11),(4,8),(4,10),(5,6),(5,9),(6,3),(6,4),(6,12),(7,1),(7,5),(8,13),(9,12),(10,13),(11,13),(12,10),(12,11),(13,2)],14)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(0,6),(2,9),(2,10),(3,8),(3,10),(4,8),(4,9),(5,7),(6,5),(7,2),(7,3),(7,4),(8,11),(9,11),(10,11),(11,1)],12)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ?
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(0,7),(2,12),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(6,11),(7,3),(7,4),(7,5),(8,13),(9,13),(10,6),(10,13),(11,12),(12,1),(13,11)],14)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,7),(2,9),(3,9),(4,8),(5,8),(6,2),(6,3),(7,4),(7,5),(8,6),(9,1)],10)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,7),(1,14),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,11),(6,12),(6,13),(7,3),(7,4),(7,5),(7,6),(8,17),(8,18),(9,15),(9,18),(10,16),(10,18),(11,15),(11,17),(12,16),(12,17),(13,15),(13,16),(14,2),(15,19),(16,19),(17,19),(18,1),(18,19),(19,14)],20)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(0,7),(2,9),(2,10),(3,8),(3,10),(4,8),(4,9),(5,1),(6,5),(7,2),(7,3),(7,4),(8,11),(9,11),(10,11),(11,6)],12)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,1),(7,2),(7,3),(7,4),(7,5),(8,14),(8,17),(9,14),(9,15),(10,14),(10,16),(11,15),(11,17),(12,16),(12,17),(13,15),(13,16),(14,18),(15,18),(16,18),(17,18),(18,6)],19)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ([(0,7),(2,11),(2,12),(3,8),(4,10),(4,13),(5,9),(5,13),(6,2),(6,9),(6,10),(7,4),(7,5),(7,6),(8,14),(9,11),(9,16),(10,12),(10,16),(11,15),(12,15),(13,3),(13,16),(14,1),(15,14),(16,8),(16,15)],17)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,7),(2,11),(2,12),(3,10),(4,9),(5,8),(5,11),(6,8),(6,12),(7,2),(7,5),(7,6),(8,14),(9,13),(10,13),(11,4),(11,14),(12,3),(12,14),(13,1),(14,9),(14,10)],15)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(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)
=> ([(0,7),(2,8),(3,8),(4,5),(5,1),(6,4),(7,2),(7,3),(8,6)],9)
=> 7 = 6 + 1
([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=> ([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,10),(3,9),(3,11),(4,8),(4,11),(5,8),(5,9),(6,1),(7,3),(7,4),(7,5),(8,12),(9,12),(10,6),(11,2),(11,12),(12,10)],13)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
=> ([(0,7),(1,13),(3,11),(3,12),(4,8),(4,10),(5,8),(5,9),(6,3),(6,9),(6,10),(7,4),(7,5),(7,6),(8,15),(9,11),(9,15),(10,12),(10,15),(11,14),(12,14),(13,2),(14,13),(15,1),(15,14)],16)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7)
=> ([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7)
=> ([(0,7),(1,11),(1,12),(3,8),(3,10),(4,8),(4,9),(5,2),(6,1),(6,9),(6,10),(7,3),(7,4),(7,6),(8,14),(9,11),(9,14),(10,12),(10,14),(11,13),(12,13),(13,5),(14,13)],15)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7)
=> ([(0,2),(2,3),(2,4),(2,5),(3,13),(3,14),(4,7),(4,14),(4,15),(5,6),(5,13),(5,15),(6,9),(6,11),(7,10),(7,12),(8,19),(9,17),(10,18),(11,8),(11,17),(12,8),(12,18),(13,9),(13,16),(14,10),(14,16),(15,11),(15,12),(15,16),(16,17),(16,18),(17,19),(18,19),(19,1)],20)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
=> ([(0,7),(2,9),(2,12),(3,8),(3,14),(4,11),(4,13),(5,8),(5,10),(6,4),(6,10),(6,14),(7,3),(7,5),(7,6),(8,16),(9,17),(10,13),(10,16),(11,12),(11,15),(12,17),(13,15),(14,2),(14,11),(14,16),(15,17),(16,9),(16,15),(17,1)],18)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=> ([(0,7),(2,11),(3,10),(4,9),(5,6),(5,11),(6,4),(6,8),(7,2),(7,5),(8,9),(8,10),(9,12),(10,12),(11,3),(11,8),(12,1)],13)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(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)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> ([(0,7),(1,11),(3,8),(4,10),(5,4),(5,8),(6,1),(6,9),(7,3),(7,5),(8,6),(8,10),(9,11),(10,9),(11,2)],12)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7)
=> ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7)
=> ([(0,7),(2,8),(2,12),(3,8),(3,11),(4,10),(5,4),(5,9),(6,2),(6,3),(6,9),(7,5),(7,6),(8,15),(9,10),(9,11),(9,12),(10,13),(10,14),(11,13),(11,15),(12,14),(12,15),(13,16),(14,16),(15,16),(16,1)],17)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7)
=> ([(0,7),(2,13),(3,11),(3,12),(4,8),(4,10),(5,8),(5,9),(6,4),(6,5),(6,13),(7,2),(7,6),(8,14),(9,11),(9,14),(10,12),(10,14),(11,15),(12,15),(13,3),(13,9),(13,10),(14,15),(15,1)],16)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> ([(0,6),(2,10),(3,8),(4,2),(4,9),(5,4),(5,8),(6,7),(7,3),(7,5),(8,9),(9,10),(10,1)],11)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
=> ([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
=> ([(0,7),(1,11),(3,8),(4,9),(4,12),(5,9),(5,10),(6,4),(6,5),(6,8),(7,3),(7,6),(8,10),(8,12),(9,13),(10,13),(11,2),(12,1),(12,13),(13,11)],14)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,6),(2,9),(3,8),(4,2),(4,8),(5,1),(6,7),(7,3),(7,4),(8,9),(9,5)],10)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2),(6,3)],7)
=> ([(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2),(6,3)],7)
=> ([(0,7),(2,8),(3,9),(3,11),(4,9),(4,10),(5,1),(6,3),(6,4),(6,8),(7,2),(7,6),(8,10),(8,11),(9,12),(10,12),(11,12),(12,5)],13)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
([(0,3),(0,4),(1,6),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(1,6),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6)],7)
=> ([(0,7),(1,10),(1,11),(3,9),(4,8),(5,4),(5,13),(6,3),(6,13),(7,5),(7,6),(8,10),(8,12),(9,11),(9,12),(10,14),(11,14),(12,14),(13,1),(13,8),(13,9),(14,2)],15)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 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)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 8 = 7 + 1
([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,6),(2,8),(3,8),(4,7),(5,1),(6,4),(7,2),(7,3),(8,5)],9)
=> 7 = 6 + 1
Description
The minimal length of a chain of small intervals in a lattice.
An interval $[a, b]$ is small if $b$ is a join of elements covering $a$.
Matching statistic: St001644
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,4} - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,4} - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,5,5,5,5,5,5,5,5,5} - 1
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,5,5,5,5,5,5,5,5,5} - 1
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,5,5,5,5,5,5,5,5,5} - 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,5,5,5,5,5,5,5,5,5} - 1
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,5,5,5,5,5,5,5,5,5} - 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,5,5,5,5,5,5,5,5,5} - 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,5,5,5,5,5,5,5,5,5} - 1
([(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)
=> ([(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,5,5,5,5,5,5,5,5,5} - 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,5,5,5,5,5,5,5,5,5} - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,5,5,5,5,5,5,5,5,5} - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,5,5,5,5,5,5,5,5,5} - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 6 - 1
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(3,6),(4,5)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(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)
=> ([(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 6 - 1
([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=> ([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
=> ([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7)
=> ([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7)
=> ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7)
=> ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,3),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> ([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7)
=> ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
=> ([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 6 - 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)
=> ([],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 7 - 1
([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 6 - 1
Description
The dimension of a graph.
The dimension of a graph is the least integer $n$ such that there exists a representation of the graph in the Euclidean space of dimension $n$ with all vertices distinct and all edges having unit length. Edges are allowed to intersect, however.
The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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