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Your data matches 9 different statistics following compositions of up to 3 maps.
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Matching statistic: St001630
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Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00196: Lattices —The modular quotient of a lattice.⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00196: Lattices —The modular quotient of a lattice.⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> ([(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)
=> 2
[1,1,2,3] => [[4,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,3,2] => [[4,3,1,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
=> ([(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)
=> 2
[1,2,1,3] => [[4,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
=> ([(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)
=> 2
[1,2,2,2] => [[4,3,2,1],[2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[1,2,3,1] => [[4,4,2,1],[3,1]]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,2,4] => [[5,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,1,2] => [[4,3,3,1],[2,2]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,1] => [[4,4,3,1],[3,2]]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,3,3] => [[5,3,1],[2]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[1,4,2] => [[5,4,1],[3]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[2,1,2,2] => [[4,3,2,2],[2,1,1]]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[2,1,3,1] => [[4,4,2,2],[3,1,1]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[2,2,1,2] => [[4,3,3,2],[2,2,1]]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[2,2,3] => [[5,3,2],[2,1]]
=> ([(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)
=> 2
[2,3,1,1] => [[4,4,4,2],[3,3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3,2] => [[5,4,2],[3,1]]
=> ([(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)
=> 2
[2,4,1] => [[5,5,2],[4,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,1,3] => [[5,3,3],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[3,2,2] => [[5,4,3],[3,2]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[3,3,1] => [[5,5,3],[4,2]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[3,4] => [[6,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St000985
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000985: Graphs ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 100%
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000985: Graphs ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 100%
Values
[1,2,2] => [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 1 + 1
[1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 1 + 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,2,3] => [[4,2,1],[1]]
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 1 + 1
[1,3,2] => [[4,3,1],[2]]
=> ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 1 + 1
[2,2,2] => [[4,3,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[2,3,1] => [[4,4,2],[3,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 1 + 1
[3,3] => [[5,3],[2]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 1 + 1
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 2 = 1 + 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> ([(0,5),(0,6),(1,3),(1,6),(4,2),(5,4)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6)],7)
=> 2 = 1 + 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 2 = 1 + 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> ([(0,5),(0,6),(1,3),(1,4),(4,6),(5,2)],7)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> 3 = 2 + 1
[1,1,2,3] => [[4,2,1,1],[1]]
=> ([(0,5),(0,6),(1,4),(1,6),(4,2),(5,3)],7)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 2 = 1 + 1
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> ([(0,4),(1,3),(1,5),(3,6),(4,6),(5,2)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 2 = 1 + 1
[1,1,3,2] => [[4,3,1,1],[2]]
=> ([(0,4),(0,6),(1,3),(1,5),(3,6),(5,2)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 3 = 2 + 1
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
=> ([(0,4),(0,6),(1,2),(1,5),(3,6),(5,3)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
=> ([(0,5),(1,3),(1,6),(2,4),(2,5),(4,6)],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),(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,2,1,3] => [[4,2,2,1],[1,1]]
=> ([(0,4),(0,6),(1,3),(1,5),(3,6),(5,2)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 3 = 2 + 1
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
=> ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> 3 = 2 + 1
[1,2,2,2] => [[4,3,2,1],[2,1]]
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,5)],7)
=> ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,2,3,1] => [[4,4,2,1],[3,1]]
=> ([(0,6),(1,3),(1,5),(2,4),(2,5),(4,6)],7)
=> ([(0,3),(0,4),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
[1,2,4] => [[5,2,1],[1]]
=> ([(0,5),(0,6),(1,3),(1,6),(4,2),(5,4)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6)],7)
=> 2 = 1 + 1
[1,3,1,2] => [[4,3,3,1],[2,2]]
=> ([(0,3),(0,5),(1,2),(1,4),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,3,2,1] => [[4,4,3,1],[3,2]]
=> ([(0,5),(1,5),(1,6),(2,3),(2,4),(4,6)],7)
=> ([(0,3),(0,4),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
[1,3,3] => [[5,3,1],[2]]
=> ([(0,5),(0,6),(1,3),(1,4),(4,6),(5,2)],7)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,4,2] => [[5,4,1],[3]]
=> ([(0,4),(0,6),(1,2),(1,5),(3,6),(5,3)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 3 = 2 + 1
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]]
=> ([(0,5),(1,6),(2,3),(2,5),(3,4),(4,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 3 = 2 + 1
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[2,1,2,2] => [[4,3,2,2],[2,1,1]]
=> ([(0,6),(1,3),(1,5),(2,4),(2,5),(4,6)],7)
=> ([(0,3),(0,4),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
[2,1,3,1] => [[4,4,2,2],[3,1,1]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,5),(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> ([(0,5),(1,5),(1,6),(2,3),(3,4),(4,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6)],7)
=> 2 = 1 + 1
[2,2,1,2] => [[4,3,3,2],[2,2,1]]
=> ([(0,5),(1,5),(1,6),(2,3),(2,4),(4,6)],7)
=> ([(0,3),(0,4),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
[2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,4),(2,4),(2,6),(3,5),(3,6)],7)
=> ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[2,2,3] => [[5,3,2],[2,1]]
=> ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> 3 = 2 + 1
[2,3,1,1] => [[4,4,4,2],[3,3,1]]
=> ([(0,5),(1,3),(2,4),(2,5),(3,6),(4,6)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 3 = 2 + 1
[2,3,2] => [[5,4,2],[3,1]]
=> ([(0,5),(1,3),(1,6),(2,4),(2,5),(4,6)],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),(4,6),(5,6)],7)
=> 3 = 2 + 1
[2,4,1] => [[5,5,2],[4,1]]
=> ([(0,5),(1,6),(2,3),(2,5),(3,4),(4,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 3 = 2 + 1
[3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> ([(0,5),(1,3),(2,4),(2,5),(3,6),(4,6)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 3 = 2 + 1
[3,1,3] => [[5,3,3],[2,2]]
=> ([(0,4),(1,3),(1,5),(3,6),(4,6),(5,2)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 2 = 1 + 1
[3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> ([(0,5),(0,6),(1,4),(2,3),(3,5),(4,6)],7)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 2 = 1 + 1
[3,2,2] => [[5,4,3],[3,2]]
=> ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> 3 = 2 + 1
[3,3,1] => [[5,5,3],[4,2]]
=> ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[3,4] => [[6,3],[2]]
=> ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 2 = 1 + 1
[1,1,1,1,2,1,1] => [[2,2,2,1,1,1,1],[1,1]]
=> ([(0,3),(1,6),(1,7),(3,7),(4,5),(5,2),(6,4)],8)
=> ?
=> ? = 1 + 1
[1,1,1,1,2,2] => [[3,2,1,1,1,1],[1]]
=> ([(0,6),(0,7),(1,3),(1,7),(4,5),(5,2),(6,4)],8)
=> ?
=> ? = 1 + 1
[1,1,1,2,1,1,1] => [[2,2,2,2,1,1,1],[1,1,1]]
=> ([(0,5),(1,6),(1,7),(3,7),(4,2),(5,3),(6,4)],8)
=> ?
=> ? = 2 + 1
[1,1,1,2,1,2] => [[3,2,2,1,1,1],[1,1]]
=> ([(0,6),(0,7),(1,3),(1,4),(4,7),(5,2),(6,5)],8)
=> ?
=> ? = 2 + 1
[1,1,1,2,2,1] => [[3,3,2,1,1,1],[2,1]]
=> ([(0,6),(1,3),(1,7),(2,6),(2,7),(3,5),(5,4)],8)
=> ?
=> ? = 2 + 1
[1,1,1,3,2] => [[4,3,1,1,1],[2]]
=> ([(0,4),(0,7),(1,3),(1,6),(3,7),(5,2),(6,5)],8)
=> ?
=> ? = 2 + 1
[1,1,2,1,1,1,1] => [[2,2,2,2,2,1,1],[1,1,1,1]]
=> ([(0,5),(1,6),(1,7),(3,7),(4,3),(5,4),(6,2)],8)
=> ?
=> ? = 1 + 1
[1,1,2,1,1,2] => [[3,2,2,2,1,1],[1,1,1]]
=> ([(0,6),(0,7),(1,3),(1,5),(4,7),(5,4),(6,2)],8)
=> ?
=> ? = 2 + 1
[1,1,2,3,1] => [[4,4,2,1,1],[3,1]]
=> ([(0,7),(1,3),(1,6),(2,4),(2,6),(3,7),(4,5)],8)
=> ?
=> ? = 2 + 1
[1,1,3,1,2] => [[4,3,3,1,1],[2,2]]
=> ([(0,3),(0,4),(1,5),(1,6),(4,7),(5,7),(6,2)],8)
=> ?
=> ? = 2 + 1
[1,1,3,2,1] => [[4,4,3,1,1],[3,2]]
=> ([(0,6),(1,6),(1,7),(2,3),(2,4),(3,7),(4,5)],8)
=> ?
=> ? = 2 + 1
[1,1,3,3] => [[5,3,1,1],[2]]
=> ([(0,6),(0,7),(1,4),(1,5),(4,7),(5,3),(6,2)],8)
=> ?
=> ? = 2 + 1
[1,2,1,1,1,2] => [[3,2,2,2,2,1],[1,1,1,1]]
=> ([(0,4),(0,7),(1,2),(1,6),(3,7),(5,3),(6,5)],8)
=> ?
=> ? = 2 + 1
[1,2,1,1,2,1] => [[3,3,2,2,2,1],[2,1,1,1]]
=> ([(0,6),(1,4),(1,6),(2,3),(2,7),(4,5),(5,7)],8)
=> ?
=> ? = 2 + 1
[1,2,1,3,1] => [[4,4,2,2,1],[3,1,1]]
=> ([(0,6),(1,3),(1,7),(2,4),(2,5),(4,6),(5,7)],8)
=> ?
=> ? = 2 + 1
[1,2,2,1,1,1] => [[3,3,3,3,2,1],[2,2,2,1]]
=> ([(0,6),(0,7),(1,4),(2,3),(2,6),(4,5),(5,7)],8)
=> ?
=> ? = 2 + 1
[1,2,3,1,1] => [[4,4,4,2,1],[3,3,1]]
=> ([(0,5),(1,4),(1,6),(2,3),(2,6),(4,7),(5,7)],8)
=> ?
=> ? = 2 + 1
[1,3,1,1,2] => [[4,3,3,3,1],[2,2,2]]
=> ([(0,3),(0,5),(1,4),(1,6),(2,7),(5,7),(6,2)],8)
=> ?
=> ? = 2 + 1
[1,3,1,2,1] => [[4,4,3,3,1],[3,2,2]]
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(4,7),(5,7)],8)
=> ?
=> ? = 2 + 1
[1,3,1,3] => [[5,3,3,1],[2,2]]
=> ([(0,3),(0,4),(1,5),(1,6),(4,7),(5,7),(6,2)],8)
=> ?
=> ? = 2 + 1
[1,3,2,1,1] => [[4,4,4,3,1],[3,3,2]]
=> ([(0,6),(0,7),(1,5),(2,3),(2,4),(4,6),(5,7)],8)
=> ?
=> ? = 2 + 1
[1,4,1,2] => [[5,4,4,1],[3,3]]
=> ([(0,3),(0,5),(1,4),(1,6),(2,7),(5,7),(6,2)],8)
=> ?
=> ? = 2 + 1
[1,4,2,1] => [[5,5,4,1],[4,3]]
=> ([(0,6),(1,6),(1,7),(2,3),(2,4),(4,5),(5,7)],8)
=> ?
=> ? = 2 + 1
[1,4,3] => [[6,4,1],[3]]
=> ([(0,6),(0,7),(1,3),(1,5),(4,7),(5,4),(6,2)],8)
=> ?
=> ? = 2 + 1
[2,1,1,1,2,1] => [[3,3,2,2,2,2],[2,1,1,1,1]]
=> ([(0,7),(1,6),(2,3),(2,6),(3,5),(4,7),(5,4)],8)
=> ?
=> ? = 2 + 1
[2,1,1,2,1,1] => [[3,3,3,2,2,2],[2,2,1,1,1]]
=> ([(0,6),(1,3),(2,4),(2,7),(3,7),(4,5),(5,6)],8)
=> ?
=> ? = 2 + 1
[2,1,1,2,2] => [[4,3,2,2,2],[2,1,1,1]]
=> ([(0,7),(1,3),(1,6),(2,4),(2,6),(4,5),(5,7)],8)
=> ?
=> ? = 2 + 1
[2,1,1,3,1] => [[4,4,2,2,2],[3,1,1,1]]
=> ([(0,7),(1,6),(2,3),(2,4),(3,7),(4,5),(5,6)],8)
=> ?
=> ? = 2 + 1
[2,1,2,1,1,1] => [[3,3,3,3,2,2],[2,2,2,1,1]]
=> ([(0,7),(1,4),(2,3),(2,6),(3,7),(4,5),(5,6)],8)
=> ?
=> ? = 2 + 1
[2,1,2,3] => [[5,3,2,2],[2,1,1]]
=> ([(0,7),(1,3),(1,6),(2,4),(2,6),(3,7),(4,5)],8)
=> ?
=> ? = 2 + 1
[2,1,3,1,1] => [[4,4,4,2,2],[3,3,1,1]]
=> ([(0,6),(1,4),(2,3),(2,5),(3,7),(4,7),(5,6)],8)
=> ?
=> ? = 2 + 1
[2,1,3,2] => [[5,4,2,2],[3,1,1]]
=> ([(0,6),(1,3),(1,7),(2,4),(2,5),(4,6),(5,7)],8)
=> ?
=> ? = 2 + 1
[2,2,1,1,1,1] => [[3,3,3,3,3,2],[2,2,2,2,1]]
=> ([(0,3),(1,6),(2,6),(2,7),(3,5),(4,7),(5,4)],8)
=> ?
=> ? = 1 + 1
[2,2,1,1,2] => [[4,3,3,3,2],[2,2,2,1]]
=> ([(0,6),(1,6),(1,7),(2,3),(2,4),(4,5),(5,7)],8)
=> ?
=> ? = 2 + 1
[2,2,1,3] => [[5,3,3,2],[2,2,1]]
=> ([(0,6),(1,6),(1,7),(2,3),(2,4),(3,7),(4,5)],8)
=> ?
=> ? = 2 + 1
[2,3,1,1,1] => [[4,4,4,4,2],[3,3,3,1]]
=> ([(0,6),(1,4),(2,3),(2,6),(3,7),(4,5),(5,7)],8)
=> ?
=> ? = 2 + 1
[2,3,1,2] => [[5,4,4,2],[3,3,1]]
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(4,7),(5,7)],8)
=> ?
=> ? = 2 + 1
[2,4,1,1] => [[5,5,5,2],[4,4,1]]
=> ([(0,6),(1,3),(2,4),(2,6),(3,7),(4,5),(5,7)],8)
=> ?
=> ? = 2 + 1
[2,4,2] => [[6,5,2],[4,1]]
=> ([(0,6),(1,4),(1,6),(2,3),(2,7),(4,5),(5,7)],8)
=> ?
=> ? = 2 + 1
[3,1,1,2,1] => [[4,4,3,3,3],[3,2,2,2]]
=> ([(0,6),(1,3),(2,4),(2,6),(3,7),(4,5),(5,7)],8)
=> ?
=> ? = 2 + 1
[3,1,1,3] => [[5,3,3,3],[2,2,2]]
=> ([(0,4),(1,5),(1,6),(3,7),(4,7),(5,2),(6,3)],8)
=> ?
=> ? = 1 + 1
[3,1,2,1,1] => [[4,4,4,3,3],[3,3,2,2]]
=> ([(0,3),(1,5),(2,4),(2,6),(3,7),(4,7),(5,6)],8)
=> ?
=> ? = 2 + 1
[3,1,2,2] => [[5,4,3,3],[3,2,2]]
=> ([(0,5),(1,4),(1,6),(2,3),(2,6),(4,7),(5,7)],8)
=> ?
=> ? = 2 + 1
[3,1,3,1] => [[5,5,3,3],[4,2,2]]
=> ([(0,6),(1,4),(2,3),(2,5),(3,7),(4,7),(5,6)],8)
=> ?
=> ? = 2 + 1
[3,2,1,1,1] => [[4,4,4,4,3],[3,3,3,2]]
=> ([(0,6),(0,7),(1,3),(2,4),(3,7),(4,5),(5,6)],8)
=> ?
=> ? = 1 + 1
[3,2,1,2] => [[5,4,4,3],[3,3,2]]
=> ([(0,6),(0,7),(1,5),(2,3),(2,4),(4,6),(5,7)],8)
=> ?
=> ? = 2 + 1
[3,3,1,1] => [[5,5,5,3],[4,4,2]]
=> ([(0,3),(1,5),(2,4),(2,6),(3,7),(4,7),(5,6)],8)
=> ?
=> ? = 2 + 1
[3,4,1] => [[6,6,3],[5,2]]
=> ([(0,6),(1,3),(2,4),(2,7),(3,7),(4,5),(5,6)],8)
=> ?
=> ? = 2 + 1
[4,1,2,1] => [[5,5,4,4],[4,3,3]]
=> ([(0,6),(1,4),(2,3),(2,6),(3,7),(4,5),(5,7)],8)
=> ?
=> ? = 2 + 1
[4,1,3] => [[6,4,4],[3,3]]
=> ([(0,6),(1,4),(1,5),(3,7),(4,7),(5,2),(6,3)],8)
=> ?
=> ? = 1 + 1
Description
The number of positive eigenvalues of the adjacency matrix of the graph.
Matching statistic: St000273
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000273: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000273: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 100%
Values
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 1 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 1 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2 = 1 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 1 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 3 = 2 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 1 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 1 + 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 3 = 2 + 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 1 + 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2 = 1 + 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 1
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,1,2,3,7,4,8,6] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [4,1,2,8,3,5,6,7] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 1
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,1,2,7,3,5,8,6] => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
=> ? = 2 + 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [4,1,2,6,3,8,5,7] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 1
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,1,2,6,3,7,8,5] => ([(0,6),(1,6),(2,7),(3,7),(4,5),(4,7),(5,6)],8)
=> ? = 1 + 1
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [4,1,2,5,8,3,6,7] => ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,1,2,5,7,3,8,6] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [3,1,7,2,4,5,8,6] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 1
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,6,2,4,8,5,7] => ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 2 + 1
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,1,6,2,4,7,8,5] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [3,1,5,2,8,4,6,7] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 1
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,7,4,8,6] => ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 1
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,5,2,6,8,4,7] => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,1,5,2,6,7,8,4] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 1
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,4,7,2,5,8,6] => ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,1,4,6,2,8,5,7] => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,4,6,2,7,8,5] => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
=> ? = 2 + 1
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [3,1,4,5,7,2,8,6] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 1
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,6,1,3,4,8,5,7] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 1
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,5,1,3,8,4,6,7] => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
=> ? = 2 + 1
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,7,4,8,6] => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,5,1,3,6,8,4,7] => ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,4,1,8,3,5,6,7] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 1
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,4,1,7,3,5,8,6] => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,8,5,7] => ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 1
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,6,3,7,8,5] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 1
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,4,1,5,8,3,6,7] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,4,1,5,7,3,8,6] => ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 2 + 1
[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,4,1,5,6,8,3,7] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 1
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [2,3,6,1,4,8,5,7] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [2,3,6,1,4,7,8,5] => ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 1
[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [2,3,5,1,8,4,6,7] => ([(0,6),(1,6),(2,7),(3,7),(4,5),(4,7),(5,6)],8)
=> ? = 1 + 1
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [2,3,5,1,7,4,8,6] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 1
[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [2,3,5,1,6,8,4,7] => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
=> ? = 2 + 1
[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [2,3,5,1,6,7,8,4] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 1
[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [2,3,4,6,1,8,5,7] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 1
[4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [2,3,4,6,1,7,8,5] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 1
[1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,1,2,3,4,9,5,7,8] => ([(0,8),(1,8),(2,8),(3,8),(4,7),(5,7),(6,7),(6,8)],9)
=> ? = 1 + 1
[1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,1,2,3,4,8,5,9,7] => ([(0,8),(1,8),(2,8),(3,8),(4,5),(5,7),(6,7),(6,8)],9)
=> ? = 1 + 1
[1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,1,2,3,9,4,6,7,8] => ([(0,8),(1,8),(2,8),(3,7),(4,7),(5,7),(6,7),(6,8)],9)
=> ? = 2 + 1
[1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,1,2,3,8,4,6,9,7] => ([(0,8),(1,8),(2,8),(3,7),(4,5),(5,7),(6,7),(6,8)],9)
=> ? = 2 + 1
[1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,9,6,8] => ?
=> ? = 2 + 1
[1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,1,2,3,6,8,4,9,7] => ?
=> ? = 2 + 1
[1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [4,1,2,9,3,5,6,7,8] => ?
=> ? = 1 + 1
[1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,1,2,8,3,5,6,9,7] => ([(0,8),(1,8),(2,7),(3,7),(4,5),(5,8),(6,7),(6,8)],9)
=> ? = 2 + 1
[1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [4,1,2,6,3,7,9,5,8] => ?
=> ? = 2 + 1
[1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,1,2,5,8,3,6,9,7] => ?
=> ? = 2 + 1
[1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [4,1,2,5,7,3,9,6,8] => ?
=> ? = 2 + 1
[1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,1,2,5,7,3,8,9,6] => ?
=> ? = 2 + 1
Description
The domination number of a graph.
The domination number of a graph is given by the minimum size of a dominating set of vertices. A dominating set of vertices is a subset of the vertex set of such that every vertex is either in this subset or adjacent to an element of this subset.
Matching statistic: St000916
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000916: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000916: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 100%
Values
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 1 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 1 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2 = 1 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 1 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 3 = 2 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 1 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 1 + 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 3 = 2 + 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 1 + 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2 = 1 + 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 1
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,1,2,3,7,4,8,6] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [4,1,2,8,3,5,6,7] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 1
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,1,2,7,3,5,8,6] => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
=> ? = 2 + 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [4,1,2,6,3,8,5,7] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 1
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,1,2,6,3,7,8,5] => ([(0,6),(1,6),(2,7),(3,7),(4,5),(4,7),(5,6)],8)
=> ? = 1 + 1
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [4,1,2,5,8,3,6,7] => ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,1,2,5,7,3,8,6] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [3,1,7,2,4,5,8,6] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 1
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,6,2,4,8,5,7] => ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 2 + 1
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,1,6,2,4,7,8,5] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [3,1,5,2,8,4,6,7] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 1
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,7,4,8,6] => ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 1
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,5,2,6,8,4,7] => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,1,5,2,6,7,8,4] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 1
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,4,7,2,5,8,6] => ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,1,4,6,2,8,5,7] => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,4,6,2,7,8,5] => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
=> ? = 2 + 1
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [3,1,4,5,7,2,8,6] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 1
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,6,1,3,4,8,5,7] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 1
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,5,1,3,8,4,6,7] => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
=> ? = 2 + 1
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,7,4,8,6] => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,5,1,3,6,8,4,7] => ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,4,1,8,3,5,6,7] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 1
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,4,1,7,3,5,8,6] => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,8,5,7] => ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 1
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,6,3,7,8,5] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 1
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,4,1,5,8,3,6,7] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,4,1,5,7,3,8,6] => ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 2 + 1
[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,4,1,5,6,8,3,7] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 1
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [2,3,6,1,4,8,5,7] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [2,3,6,1,4,7,8,5] => ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 1
[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [2,3,5,1,8,4,6,7] => ([(0,6),(1,6),(2,7),(3,7),(4,5),(4,7),(5,6)],8)
=> ? = 1 + 1
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [2,3,5,1,7,4,8,6] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 1
[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [2,3,5,1,6,8,4,7] => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
=> ? = 2 + 1
[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [2,3,5,1,6,7,8,4] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 1
[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [2,3,4,6,1,8,5,7] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 1
[4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [2,3,4,6,1,7,8,5] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 1
[1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,1,2,3,4,9,5,7,8] => ([(0,8),(1,8),(2,8),(3,8),(4,7),(5,7),(6,7),(6,8)],9)
=> ? = 1 + 1
[1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,1,2,3,4,8,5,9,7] => ([(0,8),(1,8),(2,8),(3,8),(4,5),(5,7),(6,7),(6,8)],9)
=> ? = 1 + 1
[1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,1,2,3,9,4,6,7,8] => ([(0,8),(1,8),(2,8),(3,7),(4,7),(5,7),(6,7),(6,8)],9)
=> ? = 2 + 1
[1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,1,2,3,8,4,6,9,7] => ([(0,8),(1,8),(2,8),(3,7),(4,5),(5,7),(6,7),(6,8)],9)
=> ? = 2 + 1
[1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,9,6,8] => ?
=> ? = 2 + 1
[1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,1,2,3,6,8,4,9,7] => ?
=> ? = 2 + 1
[1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [4,1,2,9,3,5,6,7,8] => ?
=> ? = 1 + 1
[1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,1,2,8,3,5,6,9,7] => ([(0,8),(1,8),(2,7),(3,7),(4,5),(5,8),(6,7),(6,8)],9)
=> ? = 2 + 1
[1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [4,1,2,6,3,7,9,5,8] => ?
=> ? = 2 + 1
[1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,1,2,5,8,3,6,9,7] => ?
=> ? = 2 + 1
[1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [4,1,2,5,7,3,9,6,8] => ?
=> ? = 2 + 1
[1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,1,2,5,7,3,8,9,6] => ?
=> ? = 2 + 1
Description
The packing number of a graph.
This is the size of a largest subset of vertices of a graph, such that any two distinct vertices in the subset have disjoint closed neighbourhoods, or, equivalently, have distance greater than two.
Matching statistic: St001322
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001322: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001322: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 100%
Values
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 1 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 1 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2 = 1 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 1 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 3 = 2 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 1 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 1 + 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 3 = 2 + 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 1 + 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2 = 1 + 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 1
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,1,2,3,7,4,8,6] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [4,1,2,8,3,5,6,7] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 1
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,1,2,7,3,5,8,6] => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
=> ? = 2 + 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [4,1,2,6,3,8,5,7] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 1
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,1,2,6,3,7,8,5] => ([(0,6),(1,6),(2,7),(3,7),(4,5),(4,7),(5,6)],8)
=> ? = 1 + 1
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [4,1,2,5,8,3,6,7] => ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,1,2,5,7,3,8,6] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [3,1,7,2,4,5,8,6] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 1
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,6,2,4,8,5,7] => ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 2 + 1
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,1,6,2,4,7,8,5] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [3,1,5,2,8,4,6,7] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 1
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,7,4,8,6] => ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 1
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,5,2,6,8,4,7] => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,1,5,2,6,7,8,4] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 1
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,4,7,2,5,8,6] => ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,1,4,6,2,8,5,7] => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,4,6,2,7,8,5] => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
=> ? = 2 + 1
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [3,1,4,5,7,2,8,6] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 1
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,6,1,3,4,8,5,7] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 1
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,5,1,3,8,4,6,7] => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
=> ? = 2 + 1
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,7,4,8,6] => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,5,1,3,6,8,4,7] => ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,4,1,8,3,5,6,7] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 1
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,4,1,7,3,5,8,6] => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,8,5,7] => ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 1
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,6,3,7,8,5] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 1
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,4,1,5,8,3,6,7] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,4,1,5,7,3,8,6] => ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 2 + 1
[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,4,1,5,6,8,3,7] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 1
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [2,3,6,1,4,8,5,7] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [2,3,6,1,4,7,8,5] => ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 1
[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [2,3,5,1,8,4,6,7] => ([(0,6),(1,6),(2,7),(3,7),(4,5),(4,7),(5,6)],8)
=> ? = 1 + 1
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [2,3,5,1,7,4,8,6] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 1
[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [2,3,5,1,6,8,4,7] => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
=> ? = 2 + 1
[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [2,3,5,1,6,7,8,4] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 1
[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [2,3,4,6,1,8,5,7] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 1
[4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [2,3,4,6,1,7,8,5] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 1
[1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,1,2,3,4,9,5,7,8] => ([(0,8),(1,8),(2,8),(3,8),(4,7),(5,7),(6,7),(6,8)],9)
=> ? = 1 + 1
[1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,1,2,3,4,8,5,9,7] => ([(0,8),(1,8),(2,8),(3,8),(4,5),(5,7),(6,7),(6,8)],9)
=> ? = 1 + 1
[1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,1,2,3,9,4,6,7,8] => ([(0,8),(1,8),(2,8),(3,7),(4,7),(5,7),(6,7),(6,8)],9)
=> ? = 2 + 1
[1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,1,2,3,8,4,6,9,7] => ([(0,8),(1,8),(2,8),(3,7),(4,5),(5,7),(6,7),(6,8)],9)
=> ? = 2 + 1
[1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,9,6,8] => ?
=> ? = 2 + 1
[1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,1,2,3,6,8,4,9,7] => ?
=> ? = 2 + 1
[1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [4,1,2,9,3,5,6,7,8] => ?
=> ? = 1 + 1
[1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,1,2,8,3,5,6,9,7] => ([(0,8),(1,8),(2,7),(3,7),(4,5),(5,8),(6,7),(6,8)],9)
=> ? = 2 + 1
[1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [4,1,2,6,3,7,9,5,8] => ?
=> ? = 2 + 1
[1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,1,2,5,8,3,6,9,7] => ?
=> ? = 2 + 1
[1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [4,1,2,5,7,3,9,6,8] => ?
=> ? = 2 + 1
[1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,1,2,5,7,3,8,9,6] => ?
=> ? = 2 + 1
Description
The size of a minimal independent dominating set in a graph.
Matching statistic: St001339
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001339: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001339: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 100%
Values
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 1 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 1 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2 = 1 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 1 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 3 = 2 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 1 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 1 + 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 3 = 2 + 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 3 = 2 + 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 1 + 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2 = 1 + 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 1
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,1,2,3,7,4,8,6] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [4,1,2,8,3,5,6,7] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 1
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,1,2,7,3,5,8,6] => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
=> ? = 2 + 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [4,1,2,6,3,8,5,7] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 1
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,1,2,6,3,7,8,5] => ([(0,6),(1,6),(2,7),(3,7),(4,5),(4,7),(5,6)],8)
=> ? = 1 + 1
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [4,1,2,5,8,3,6,7] => ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,1,2,5,7,3,8,6] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [3,1,7,2,4,5,8,6] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 1
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,6,2,4,8,5,7] => ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 2 + 1
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,1,6,2,4,7,8,5] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [3,1,5,2,8,4,6,7] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 1
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,7,4,8,6] => ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 1
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,5,2,6,8,4,7] => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,1,5,2,6,7,8,4] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 1
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,4,7,2,5,8,6] => ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,1,4,6,2,8,5,7] => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,4,6,2,7,8,5] => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
=> ? = 2 + 1
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [3,1,4,5,7,2,8,6] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 1
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,6,1,3,4,8,5,7] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 1
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,5,1,3,8,4,6,7] => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
=> ? = 2 + 1
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,7,4,8,6] => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,5,1,3,6,8,4,7] => ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,4,1,8,3,5,6,7] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 1
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,4,1,7,3,5,8,6] => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,8,5,7] => ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 1
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,6,3,7,8,5] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 1
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,4,1,5,8,3,6,7] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,4,1,5,7,3,8,6] => ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 2 + 1
[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,4,1,5,6,8,3,7] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 1
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [2,3,6,1,4,8,5,7] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 2 + 1
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [2,3,6,1,4,7,8,5] => ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 1
[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [2,3,5,1,8,4,6,7] => ([(0,6),(1,6),(2,7),(3,7),(4,5),(4,7),(5,6)],8)
=> ? = 1 + 1
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [2,3,5,1,7,4,8,6] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 1
[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [2,3,5,1,6,8,4,7] => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
=> ? = 2 + 1
[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [2,3,5,1,6,7,8,4] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 1
[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [2,3,4,6,1,8,5,7] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 1
[4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [2,3,4,6,1,7,8,5] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 1
[1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,1,2,3,4,9,5,7,8] => ([(0,8),(1,8),(2,8),(3,8),(4,7),(5,7),(6,7),(6,8)],9)
=> ? = 1 + 1
[1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,1,2,3,4,8,5,9,7] => ([(0,8),(1,8),(2,8),(3,8),(4,5),(5,7),(6,7),(6,8)],9)
=> ? = 1 + 1
[1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,1,2,3,9,4,6,7,8] => ([(0,8),(1,8),(2,8),(3,7),(4,7),(5,7),(6,7),(6,8)],9)
=> ? = 2 + 1
[1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,1,2,3,8,4,6,9,7] => ([(0,8),(1,8),(2,8),(3,7),(4,5),(5,7),(6,7),(6,8)],9)
=> ? = 2 + 1
[1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,9,6,8] => ?
=> ? = 2 + 1
[1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,1,2,3,6,8,4,9,7] => ?
=> ? = 2 + 1
[1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [4,1,2,9,3,5,6,7,8] => ?
=> ? = 1 + 1
[1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,1,2,8,3,5,6,9,7] => ([(0,8),(1,8),(2,7),(3,7),(4,5),(5,8),(6,7),(6,8)],9)
=> ? = 2 + 1
[1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [4,1,2,6,3,7,9,5,8] => ?
=> ? = 2 + 1
[1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,1,2,5,8,3,6,9,7] => ?
=> ? = 2 + 1
[1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [4,1,2,5,7,3,9,6,8] => ?
=> ? = 2 + 1
[1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,1,2,5,7,3,8,9,6] => ?
=> ? = 2 + 1
Description
The irredundance number of a graph.
A set S of vertices is irredundant, if there is no vertex in S, whose closed neighbourhood is contained in the union of the closed neighbourhoods of the other vertices of S.
The irredundance number is the smallest size of a maximal irredundant set.
Matching statistic: St000918
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000918: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000918: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 100%
Values
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 4 = 1 + 3
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 4 = 1 + 3
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 4 = 1 + 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 4 = 1 + 3
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 5 = 2 + 3
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 5 = 2 + 3
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 4 = 1 + 3
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 5 = 2 + 3
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 5 = 2 + 3
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 4 = 1 + 3
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 5 = 2 + 3
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 5 = 2 + 3
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 4 = 1 + 3
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 4 = 1 + 3
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 3
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,1,2,3,7,4,8,6] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 3
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [4,1,2,8,3,5,6,7] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 3
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,1,2,7,3,5,8,6] => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
=> ? = 2 + 3
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [4,1,2,6,3,8,5,7] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 3
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,1,2,6,3,7,8,5] => ([(0,6),(1,6),(2,7),(3,7),(4,5),(4,7),(5,6)],8)
=> ? = 1 + 3
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [4,1,2,5,8,3,6,7] => ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 3
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,1,2,5,7,3,8,6] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 2 + 3
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [3,1,7,2,4,5,8,6] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 3
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,6,2,4,8,5,7] => ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 2 + 3
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,1,6,2,4,7,8,5] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 2 + 3
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [3,1,5,2,8,4,6,7] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 3
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,7,4,8,6] => ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 3
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,5,2,6,8,4,7] => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 3
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,1,5,2,6,7,8,4] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 3
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,4,7,2,5,8,6] => ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 2 + 3
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,1,4,6,2,8,5,7] => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 3
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,4,6,2,7,8,5] => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
=> ? = 2 + 3
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [3,1,4,5,7,2,8,6] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 3
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,6,1,3,4,8,5,7] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 3
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,5,1,3,8,4,6,7] => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
=> ? = 2 + 3
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,7,4,8,6] => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 3
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,5,1,3,6,8,4,7] => ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 2 + 3
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,4,1,8,3,5,6,7] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 3
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,4,1,7,3,5,8,6] => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 3
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,8,5,7] => ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 3
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,6,3,7,8,5] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 3
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,4,1,5,8,3,6,7] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 2 + 3
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,4,1,5,7,3,8,6] => ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 2 + 3
[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,4,1,5,6,8,3,7] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 3
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [2,3,6,1,4,8,5,7] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 2 + 3
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [2,3,6,1,4,7,8,5] => ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 3
[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [2,3,5,1,8,4,6,7] => ([(0,6),(1,6),(2,7),(3,7),(4,5),(4,7),(5,6)],8)
=> ? = 1 + 3
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [2,3,5,1,7,4,8,6] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 3
[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [2,3,5,1,6,8,4,7] => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
=> ? = 2 + 3
[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [2,3,5,1,6,7,8,4] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 3
[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [2,3,4,6,1,8,5,7] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 3
[4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [2,3,4,6,1,7,8,5] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 3
[1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,1,2,3,4,9,5,7,8] => ([(0,8),(1,8),(2,8),(3,8),(4,7),(5,7),(6,7),(6,8)],9)
=> ? = 1 + 3
[1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,1,2,3,4,8,5,9,7] => ([(0,8),(1,8),(2,8),(3,8),(4,5),(5,7),(6,7),(6,8)],9)
=> ? = 1 + 3
[1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,1,2,3,9,4,6,7,8] => ([(0,8),(1,8),(2,8),(3,7),(4,7),(5,7),(6,7),(6,8)],9)
=> ? = 2 + 3
[1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,1,2,3,8,4,6,9,7] => ([(0,8),(1,8),(2,8),(3,7),(4,5),(5,7),(6,7),(6,8)],9)
=> ? = 2 + 3
[1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,9,6,8] => ?
=> ? = 2 + 3
[1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,1,2,3,6,8,4,9,7] => ?
=> ? = 2 + 3
[1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [4,1,2,9,3,5,6,7,8] => ?
=> ? = 1 + 3
[1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,1,2,8,3,5,6,9,7] => ([(0,8),(1,8),(2,7),(3,7),(4,5),(5,8),(6,7),(6,8)],9)
=> ? = 2 + 3
[1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [4,1,2,6,3,7,9,5,8] => ?
=> ? = 2 + 3
[1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,1,2,5,8,3,6,9,7] => ?
=> ? = 2 + 3
[1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [4,1,2,5,7,3,9,6,8] => ?
=> ? = 2 + 3
[1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,1,2,5,7,3,8,9,6] => ?
=> ? = 2 + 3
Description
The 2-limited packing number of a graph.
A subset B of the set of vertices of a graph is a k-limited packing set if its intersection with the (closed) neighbourhood of any vertex is at most k. The k-limited packing number is the largest number of vertices in a k-limited packing set.
Matching statistic: St000453
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000453: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000453: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 100%
Values
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 6 = 1 + 5
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 6 = 1 + 5
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 6 = 1 + 5
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 6 = 1 + 5
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 7 = 2 + 5
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 7 = 2 + 5
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 6 = 1 + 5
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 7 = 2 + 5
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 7 = 2 + 5
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 6 = 1 + 5
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 7 = 2 + 5
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 7 = 2 + 5
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 6 = 1 + 5
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 6 = 1 + 5
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,1,2,3,8,4,6,7] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 5
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,1,2,3,7,4,8,6] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 5
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [4,1,2,8,3,5,6,7] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 5
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,1,2,7,3,5,8,6] => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
=> ? = 2 + 5
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [4,1,2,6,3,8,5,7] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 5
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,1,2,6,3,7,8,5] => ([(0,6),(1,6),(2,7),(3,7),(4,5),(4,7),(5,6)],8)
=> ? = 1 + 5
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [4,1,2,5,8,3,6,7] => ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 5
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,1,2,5,7,3,8,6] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 2 + 5
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [3,1,7,2,4,5,8,6] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 5
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,6,2,4,8,5,7] => ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 2 + 5
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,1,6,2,4,7,8,5] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 2 + 5
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [3,1,5,2,8,4,6,7] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 5
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,7,4,8,6] => ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 5
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,5,2,6,8,4,7] => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 5
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,1,5,2,6,7,8,4] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 5
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,4,7,2,5,8,6] => ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 2 + 5
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,1,4,6,2,8,5,7] => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 5
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,4,6,2,7,8,5] => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
=> ? = 2 + 5
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [3,1,4,5,7,2,8,6] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 5
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,6,1,3,4,8,5,7] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 5
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,5,1,3,8,4,6,7] => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
=> ? = 2 + 5
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,7,4,8,6] => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 5
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,5,1,3,6,8,4,7] => ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 2 + 5
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,4,1,8,3,5,6,7] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 5
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,4,1,7,3,5,8,6] => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 2 + 5
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,8,5,7] => ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 5
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,6,3,7,8,5] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 5
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,4,1,5,8,3,6,7] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 2 + 5
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,4,1,5,7,3,8,6] => ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 2 + 5
[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,4,1,5,6,8,3,7] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 5
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [2,3,6,1,4,8,5,7] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 2 + 5
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [2,3,6,1,4,7,8,5] => ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 5
[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [2,3,5,1,8,4,6,7] => ([(0,6),(1,6),(2,7),(3,7),(4,5),(4,7),(5,6)],8)
=> ? = 1 + 5
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [2,3,5,1,7,4,8,6] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 2 + 5
[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [2,3,5,1,6,8,4,7] => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
=> ? = 2 + 5
[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [2,3,5,1,6,7,8,4] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 5
[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [2,3,4,6,1,8,5,7] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 5
[4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [2,3,4,6,1,7,8,5] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ? = 1 + 5
[1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,1,2,3,4,9,5,7,8] => ([(0,8),(1,8),(2,8),(3,8),(4,7),(5,7),(6,7),(6,8)],9)
=> ? = 1 + 5
[1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,1,2,3,4,8,5,9,7] => ([(0,8),(1,8),(2,8),(3,8),(4,5),(5,7),(6,7),(6,8)],9)
=> ? = 1 + 5
[1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,1,2,3,9,4,6,7,8] => ([(0,8),(1,8),(2,8),(3,7),(4,7),(5,7),(6,7),(6,8)],9)
=> ? = 2 + 5
[1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,1,2,3,8,4,6,9,7] => ([(0,8),(1,8),(2,8),(3,7),(4,5),(5,7),(6,7),(6,8)],9)
=> ? = 2 + 5
[1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,9,6,8] => ?
=> ? = 2 + 5
[1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,1,2,3,6,8,4,9,7] => ?
=> ? = 2 + 5
[1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [4,1,2,9,3,5,6,7,8] => ?
=> ? = 1 + 5
[1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,1,2,8,3,5,6,9,7] => ([(0,8),(1,8),(2,7),(3,7),(4,5),(5,8),(6,7),(6,8)],9)
=> ? = 2 + 5
[1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [4,1,2,6,3,7,9,5,8] => ?
=> ? = 2 + 5
[1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,1,2,5,8,3,6,9,7] => ?
=> ? = 2 + 5
[1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [4,1,2,5,7,3,9,6,8] => ?
=> ? = 2 + 5
[1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,1,2,5,7,3,8,9,6] => ?
=> ? = 2 + 5
Description
The number of distinct Laplacian eigenvalues of a graph.
Matching statistic: St001875
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 50%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 50%
Values
[1,2,2] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? = 1 + 1
[2,2,1] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? = 1 + 1
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ? = 1 + 1
[1,1,2,2] => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? = 1 + 1
[1,2,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? = 2 + 1
[1,2,2,1] => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? = 2 + 1
[1,2,3] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 1 + 1
[1,3,2] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 2 + 1
[2,1,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? = 2 + 1
[2,2,1,1] => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? = 1 + 1
[2,2,2] => [3] => [[3],[]]
=> ([],1)
=> ? = 2 + 1
[2,3,1] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 2 + 1
[3,2,1] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 1 + 1
[3,3] => [2] => [[2],[]]
=> ([],1)
=> ? = 1 + 1
[1,1,1,2,1,1] => [3,1,2] => [[4,3,3],[2,2]]
=> ([(0,1)],2)
=> ? = 1 + 1
[1,1,1,2,2] => [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? = 1 + 1
[1,1,2,1,1,1] => [2,1,3] => [[4,2,2],[1,1]]
=> ([(0,1)],2)
=> ? = 1 + 1
[1,1,2,1,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ? = 2 + 1
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[1,1,2,3] => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? = 1 + 1
[1,1,3,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ? = 1 + 1
[1,1,3,2] => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? = 2 + 1
[1,2,1,1,2] => [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ? = 2 + 1
[1,2,1,2,1] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ? = 2 + 1
[1,2,1,3] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? = 2 + 1
[1,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[1,2,2,2] => [1,3] => [[3,1],[]]
=> ([],1)
=> ? = 2 + 1
[1,2,3,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? = 2 + 1
[1,2,4] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 1 + 1
[1,3,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? = 2 + 1
[1,3,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? = 2 + 1
[1,3,3] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? = 2 + 1
[1,4,2] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 2 + 1
[2,1,1,2,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ? = 2 + 1
[2,1,2,1,1] => [1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ? = 2 + 1
[2,1,2,2] => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? = 2 + 1
[2,1,3,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? = 2 + 1
[2,2,1,1,1] => [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? = 1 + 1
[2,2,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? = 2 + 1
[2,2,2,1] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? = 2 + 1
[2,2,3] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? = 2 + 1
[2,3,1,1] => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? = 2 + 1
[2,3,2] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 2 + 1
[2,4,1] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 2 + 1
[3,1,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? = 2 + 1
[3,1,3] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 1 + 1
[3,2,1,1] => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? = 1 + 1
[3,2,2] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? = 2 + 1
[3,3,1] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? = 2 + 1
[3,4] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? = 1 + 1
[4,2,1] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 1 + 1
[4,3] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? = 1 + 1
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[1,1,1,2,2,1] => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[1,1,2,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,2,1,1,2,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[1,2,2,1,1,1] => [1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,1,1,2,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,1,1,2,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,2,1,1,2] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
Description
The number of simple modules with projective dimension at most 1.
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