Your data matches 4 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001876
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1},{2},{3}}
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,3,4}}
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 0
{{1,3},{2,4}}
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
{{1,2,3,4,5}}
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 0
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 0
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 0
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1},{2,4},{3,5}}
 => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 0
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => ([(1,5),(3,4),(4,2),(5,3)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
{{1,2,3,4,5},{6}}
 => [2,3,4,5,1,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
{{1,2,3,4},{5,6}}
 => [2,3,4,1,6,5] => ([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
{{1,2,3,4},{5},{6}}
 => [2,3,4,1,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,3,5},{4,6}}
 => [2,3,5,6,1,4] => ([(0,5),(1,4),(3,2),(4,3),(4,5)],6)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 1
{{1,2,3},{4,5,6}}
 => [2,3,1,5,6,4] => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
{{1,2,3},{4,5},{6}}
 => [2,3,1,5,4,6] => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,3},{4},{5,6}}
 => [2,3,1,4,6,5] => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,4,6},{3,5}}
 => [2,4,5,6,3,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,4},{3,5,6}}
 => [2,4,5,1,6,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(3,5)],6)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 1
{{1,2,4},{3,5},{6}}
 => [2,4,5,1,3,6] => ([(0,4),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
{{1,2},{3,4,5,6}}
 => [2,1,4,5,6,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
{{1,2},{3,4,5},{6}}
 => [2,1,4,5,3,6] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 0
{{1,2},{3,4},{5,6}}
 => [2,1,4,3,6,5] => ([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,5},{3,6},{4}}
 => [2,5,6,4,1,3] => ([(0,5),(1,3),(1,4),(1,5),(4,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1,2},{3,5},{4,6}}
 => [2,1,5,6,3,4] => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
{{1,2},{3},{4,5,6}}
 => [2,1,3,5,6,4] => ([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,6},{3},{4},{5}}
 => [2,6,3,4,5,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 0
{{1,3,4,5},{2,6}}
 => [3,6,4,5,1,2] => ([(0,4),(1,3),(1,5),(5,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1,3,4},{2,6},{5}}
 => [3,6,4,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 0
{{1,3,5,6},{2,4}}
 => [3,4,5,2,6,1] => ([(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 0
{{1,3,5},{2,4},{6}}
 => [3,4,5,2,1,6] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 0
{{1,3},{2,4,5,6}}
 => [3,4,1,5,6,2] => ([(0,4),(1,3),(1,5),(4,5),(5,2)],6)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 1
{{1,3},{2,4,5},{6}}
 => [3,4,1,5,2,6] => ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
{{1,3,6},{2,4},{5}}
 => [3,4,6,2,5,1] => ([(1,5),(2,3),(3,4),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
{{1,3},{2,4},{5,6}}
 => [3,4,1,2,6,5] => ([(0,3),(1,2),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
{{1,3},{2,4},{5},{6}}
 => [3,4,1,2,5,6] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1,3},{2,5},{4,6}}
 => [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,4),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 2
{{1,3,6},{2},{4},{5}}
 => [3,2,6,4,5,1] => ([(1,4),(1,5),(2,4),(2,5),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 0
{{1,4,5},{2,3,6}}
 => [4,3,6,5,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1,4},{2,3,6},{5}}
 => [4,3,6,1,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => 0
{{1,5},{2,3,4,6}}
 => [5,3,4,6,1,2] => ([(0,5),(1,3),(2,4),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1,6},{2,3,4,5}}
 => [6,3,4,5,2,1] => ([(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 0
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Matching statistic: St001877
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1},{2},{3}}
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,3,4}}
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 0
{{1,3},{2,4}}
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
{{1,2,3,4,5}}
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 0
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 0
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 0
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1},{2,4},{3,5}}
 => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 0
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => ([(1,5),(3,4),(4,2),(5,3)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
{{1,2,3,4,5},{6}}
 => [2,3,4,5,1,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
{{1,2,3,4},{5,6}}
 => [2,3,4,1,6,5] => ([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
{{1,2,3,4},{5},{6}}
 => [2,3,4,1,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,3,5},{4,6}}
 => [2,3,5,6,1,4] => ([(0,5),(1,4),(3,2),(4,3),(4,5)],6)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 1
{{1,2,3},{4,5,6}}
 => [2,3,1,5,6,4] => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
{{1,2,3},{4,5},{6}}
 => [2,3,1,5,4,6] => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,3},{4},{5,6}}
 => [2,3,1,4,6,5] => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,4,6},{3,5}}
 => [2,4,5,6,3,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,4},{3,5,6}}
 => [2,4,5,1,6,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(3,5)],6)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 1
{{1,2,4},{3,5},{6}}
 => [2,4,5,1,3,6] => ([(0,4),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
{{1,2},{3,4,5,6}}
 => [2,1,4,5,6,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
{{1,2},{3,4,5},{6}}
 => [2,1,4,5,3,6] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 0
{{1,2},{3,4},{5,6}}
 => [2,1,4,3,6,5] => ([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,5},{3,6},{4}}
 => [2,5,6,4,1,3] => ([(0,5),(1,3),(1,4),(1,5),(4,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1,2},{3,5},{4,6}}
 => [2,1,5,6,3,4] => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
{{1,2},{3},{4,5,6}}
 => [2,1,3,5,6,4] => ([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,6},{3},{4},{5}}
 => [2,6,3,4,5,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 0
{{1,3,4,5},{2,6}}
 => [3,6,4,5,1,2] => ([(0,4),(1,3),(1,5),(5,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1,3,4},{2,6},{5}}
 => [3,6,4,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 0
{{1,3,5,6},{2,4}}
 => [3,4,5,2,6,1] => ([(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 0
{{1,3,5},{2,4},{6}}
 => [3,4,5,2,1,6] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 0
{{1,3},{2,4,5,6}}
 => [3,4,1,5,6,2] => ([(0,4),(1,3),(1,5),(4,5),(5,2)],6)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 1
{{1,3},{2,4,5},{6}}
 => [3,4,1,5,2,6] => ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
{{1,3,6},{2,4},{5}}
 => [3,4,6,2,5,1] => ([(1,5),(2,3),(3,4),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
{{1,3},{2,4},{5,6}}
 => [3,4,1,2,6,5] => ([(0,3),(1,2),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
{{1,3},{2,4},{5},{6}}
 => [3,4,1,2,5,6] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1,3},{2,5},{4,6}}
 => [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,4),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 2
{{1,3,6},{2},{4},{5}}
 => [3,2,6,4,5,1] => ([(1,4),(1,5),(2,4),(2,5),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 0
{{1,4,5},{2,3,6}}
 => [4,3,6,5,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1,4},{2,3,6},{5}}
 => [4,3,6,1,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => 0
{{1,5},{2,3,4,6}}
 => [5,3,4,6,1,2] => ([(0,5),(1,3),(2,4),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1,6},{2,3,4,5}}
 => [6,3,4,5,2,1] => ([(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 0
Description
Number of indecomposable injective modules with projective dimension 2.
Matching statistic: St001597
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001597: Skew partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 67%
Values
{{1},{2},{3}}
 => [1,2,3] => [1,0,1,0,1,0]
 => [[1,1,1],[]]
 => 1 = 0 + 1
{{1,2,3,4}}
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [[4],[]]
 => 1 = 0 + 1
{{1,3},{2,4}}
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => 2 = 1 + 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => 1 = 0 + 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 1 = 0 + 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => 1 = 0 + 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => [[4,3],[2]]
 => 1 = 0 + 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => 2 = 1 + 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => [[4,2],[1]]
 => 1 = 0 + 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [[2,2,2,2],[]]
 => ? = 0 + 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2],[]]
 => 2 = 1 + 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => [[2,2,2,2],[1]]
 => 2 = 1 + 1
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [[4,1],[]]
 => 1 = 0 + 1
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => [[4,4],[]]
 => ? = 1 + 1
{{1},{2,4},{3,5}}
 => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [[2,2,2,1],[]]
 => 2 = 1 + 1
{{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3],[]]
 => ? = 0 + 1
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [[1,1,1,1,1],[]]
 => 1 = 0 + 1
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [[6],[]]
 => 1 = 0 + 1
{{1,2,3,4,5},{6}}
 => [2,3,4,5,1,6] => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [[5,5],[4]]
 => 1 = 0 + 1
{{1,2,3,4},{5,6}}
 => [2,3,4,1,6,5] => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [[5,4],[3]]
 => 1 = 0 + 1
{{1,2,3,4},{5},{6}}
 => [2,3,4,1,5,6] => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [[4,4,4],[3,3]]
 => 1 = 0 + 1
{{1,2,3,5},{4,6}}
 => [2,3,5,6,1,4] => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [[4,4,4],[2,2]]
 => ? = 1 + 1
{{1,2,3},{4,5,6}}
 => [2,3,1,5,6,4] => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [[5,3],[2]]
 => 1 = 0 + 1
{{1,2,3},{4,5},{6}}
 => [2,3,1,5,4,6] => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [[4,4,3],[3,2]]
 => 1 = 0 + 1
{{1,2,3},{4},{5,6}}
 => [2,3,1,4,6,5] => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [[4,3,3],[2,2]]
 => 1 = 0 + 1
{{1,2,4,6},{3,5}}
 => [2,4,5,6,3,1] => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [[3,3,3,3],[1,1,1]]
 => ? = 0 + 1
{{1,2,4},{3,5,6}}
 => [2,4,5,1,6,3] => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [[4,3,3],[1,1]]
 => ? = 1 + 1
{{1,2,4},{3,5},{6}}
 => [2,4,5,1,3,6] => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [[3,3,3,3],[2,1,1]]
 => ? = 1 + 1
{{1,2},{3,4,5,6}}
 => [2,1,4,5,6,3] => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [[5,2],[1]]
 => 1 = 0 + 1
{{1,2},{3,4,5},{6}}
 => [2,1,4,5,3,6] => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [[4,4,2],[3,1]]
 => 1 = 0 + 1
{{1,2},{3,4},{5,6}}
 => [2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [[4,3,2],[2,1]]
 => 1 = 0 + 1
{{1,2,5},{3,6},{4}}
 => [2,5,6,4,1,3] => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [[5,5],[1]]
 => ? = 1 + 1
{{1,2},{3,5},{4,6}}
 => [2,1,5,6,3,4] => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [[3,3,3,2],[1,1,1]]
 => ? = 1 + 1
{{1,2},{3},{4,5,6}}
 => [2,1,3,5,6,4] => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [[4,2,2],[1,1]]
 => 1 = 0 + 1
{{1,2,6},{3},{4},{5}}
 => [2,6,3,4,5,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [[4,4,4],[1,1]]
 => ? = 0 + 1
{{1,3,4,5},{2,6}}
 => [3,6,4,5,1,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [[3,3,3,2],[]]
 => ? = 1 + 1
{{1,3,4},{2,6},{5}}
 => [3,6,4,1,5,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [[3,3,3,2],[]]
 => ? = 0 + 1
{{1,3,5,6},{2,4}}
 => [3,4,5,2,6,1] => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [[3,2,2,2],[]]
 => ? = 0 + 1
{{1,3,5},{2,4},{6}}
 => [3,4,5,2,1,6] => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [[2,2,2,2,2],[1]]
 => ? = 0 + 1
{{1,3},{2,4,5,6}}
 => [3,4,1,5,6,2] => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [[4,2,2],[]]
 => ? = 1 + 1
{{1,3},{2,4,5},{6}}
 => [3,4,1,5,2,6] => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [[3,3,2,2],[2]]
 => ? = 1 + 1
{{1,3,6},{2,4},{5}}
 => [3,4,6,2,5,1] => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [[3,3,2,2],[]]
 => ? = 0 + 1
{{1,3},{2,4},{5,6}}
 => [3,4,1,2,6,5] => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [[3,2,2,2],[1]]
 => ? = 1 + 1
{{1,3},{2,4},{5},{6}}
 => [3,4,1,2,5,6] => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [[2,2,2,2,2],[1,1]]
 => ? = 1 + 1
{{1,3},{2,5},{4,6}}
 => [3,5,1,6,2,4] => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [[3,3,3,2],[1]]
 => ? = 2 + 1
{{1,3,6},{2},{4},{5}}
 => [3,2,6,4,5,1] => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [[4,4,2],[1]]
 => ? = 0 + 1
{{1,4,5},{2,3,6}}
 => [4,3,6,5,1,2] => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [[4,4,3],[1]]
 => ? = 1 + 1
{{1,4},{2,3,6},{5}}
 => [4,3,6,1,5,2] => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [[4,4,3],[1]]
 => ? = 0 + 1
{{1,5},{2,3,4,6}}
 => [5,3,4,6,1,2] => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [[3,3,3,3],[1]]
 => ? = 1 + 1
{{1,6},{2,3,4,5}}
 => [6,3,4,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [[4,4,4],[]]
 => ? = 0 + 1
{{1},{2,3,4,5,6}}
 => [1,3,4,5,6,2] => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [[5,1],[]]
 => 1 = 0 + 1
{{1},{2,3,4,5},{6}}
 => [1,3,4,5,2,6] => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [[4,4,1],[3]]
 => 1 = 0 + 1
{{1},{2,3,4},{5,6}}
 => [1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [[4,3,1],[2]]
 => 1 = 0 + 1
{{1,5},{2,3,6},{4}}
 => [5,3,6,4,1,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [[4,4,3],[]]
 => ? = 2 + 1
{{1},{2,3,5},{4,6}}
 => [1,3,5,6,2,4] => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3,1],[1,1]]
 => ? = 1 + 1
{{1},{2,3},{4,5,6}}
 => [1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [[4,2,1],[1]]
 => 1 = 0 + 1
{{1,4,5},{2,6},{3}}
 => [4,6,3,5,1,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [[4,4,4],[1]]
 => ? = 2 + 1
{{1,4,5},{2},{3,6}}
 => [4,2,6,5,1,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [[4,4,3],[1]]
 => ? = 0 + 1
{{1,4},{2,5,6},{3}}
 => [4,5,3,1,6,2] => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [[5,4],[]]
 => ? = 1 + 1
{{1,4},{2,5},{3},{6}}
 => [4,5,3,1,2,6] => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[4,4,4],[3]]
 => ? = 1 + 1
{{1,4,6},{2},{3,5}}
 => [4,2,5,6,3,1] => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [[3,3,3,3],[1,1]]
 => ? = 0 + 1
{{1,4,6},{2},{3},{5}}
 => [4,2,3,6,5,1] => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [[4,4,3],[2]]
 => ? = 0 + 1
{{1,4},{2,6},{3},{5}}
 => [4,6,3,1,5,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [[4,4,4],[1]]
 => ? = 1 + 1
{{1,6},{2,4},{3,5}}
 => [6,4,5,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [[4,4,4],[]]
 => ? = 1 + 1
{{1},{2,4,6},{3,5}}
 => [1,4,5,6,3,2] => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [[2,2,2,2,1],[]]
 => ? = 0 + 1
{{1},{2,4},{3,5,6}}
 => [1,4,5,2,6,3] => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2,1],[]]
 => ? = 1 + 1
{{1},{2,4},{3,5},{6}}
 => [1,4,5,2,3,6] => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [[2,2,2,2,1],[1]]
 => ? = 1 + 1
{{1,5},{2,6},{3,4}}
 => [5,6,4,3,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [[3,3,3,3],[]]
 => ? = 1 + 1
{{1,5},{2},{3,4,6}}
 => [5,2,4,6,1,3] => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [[3,3,3,3],[1]]
 => ? = 0 + 1
{{1},{2},{3,4,5,6}}
 => [1,2,4,5,6,3] => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [[4,1,1],[]]
 => 1 = 0 + 1
{{1,5,6},{2},{3},{4}}
 => [5,2,3,4,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [[4,3,3],[]]
 => ? = 0 + 1
{{1,5},{2},{3,6},{4}}
 => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [[4,4,3],[]]
 => ? = 1 + 1
{{1,5},{2},{3},{4},{6}}
 => [5,2,3,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[3,3,3,3],[2]]
 => ? = 0 + 1
{{1},{2,5},{3,6},{4}}
 => [1,5,6,4,2,3] => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[4,4,1],[]]
 => ? = 1 + 1
{{1},{2},{3,5},{4,6}}
 => [1,2,5,6,3,4] => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [[2,2,2,1,1],[]]
 => ? = 1 + 1
{{1,6},{2},{3},{4},{5}}
 => [6,2,3,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [[4,4,4],[]]
 => ? = 0 + 1
{{1},{2,6},{3},{4},{5}}
 => [1,6,3,4,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3,1],[]]
 => ? = 0 + 1
{{1},{2},{3},{4},{5},{6}}
 => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [[1,1,1,1,1,1],[]]
 => 1 = 0 + 1
{{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [[7],[]]
 => 1 = 0 + 1
{{1,2,3,4,5,6},{7}}
 => [2,3,4,5,6,1,7] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [[6,6],[5]]
 => 1 = 0 + 1
{{1,2,3,4,5},{6,7}}
 => [2,3,4,5,1,7,6] => [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => [[6,5],[4]]
 => 1 = 0 + 1
{{1,2,3,4,5},{6},{7}}
 => [2,3,4,5,1,6,7] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [[5,5,5],[4,4]]
 => 1 = 0 + 1
{{1,2,3,4,6},{5,7}}
 => [2,3,4,6,7,1,5] => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [[5,5,5],[3,3]]
 => ? = 1 + 1
{{1,2,3,4},{5,6,7}}
 => [2,3,4,1,6,7,5] => [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => [[6,4],[3]]
 => 1 = 0 + 1
{{1,2,3,4},{5,6},{7}}
 => [2,3,4,1,6,5,7] => [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
 => [[5,5,4],[4,3]]
 => 1 = 0 + 1
{{1,2,3,4},{5},{6,7}}
 => [2,3,4,1,5,7,6] => [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
 => [[5,4,4],[3,3]]
 => 1 = 0 + 1
{{1,2,3,4},{5},{6},{7}}
 => [2,3,4,1,5,6,7] => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [[4,4,4,4],[3,3,3]]
 => 1 = 0 + 1
{{1,2,3,5,7},{4,6}}
 => [2,3,5,6,7,4,1] => [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [[4,4,4,4],[2,2,2]]
 => ? = 0 + 1
{{1,2,3,5},{4,6,7}}
 => [2,3,5,6,1,7,4] => [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [[5,4,4],[2,2]]
 => ? = 1 + 1
{{1,2,3},{4,5,6,7}}
 => [2,3,1,5,6,7,4] => [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [[6,3],[2]]
 => 1 = 0 + 1
{{1,2,3},{4,5,6},{7}}
 => [2,3,1,5,6,4,7] => [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => [[5,5,3],[4,2]]
 => 1 = 0 + 1
{{1,2,3},{4,5},{6,7}}
 => [2,3,1,5,4,7,6] => [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => [[5,4,3],[3,2]]
 => 1 = 0 + 1
{{1,2,3},{4,5},{6},{7}}
 => [2,3,1,5,4,6,7] => [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => [[4,4,4,3],[3,3,2]]
 => 1 = 0 + 1
{{1,2,3},{4},{5,6,7}}
 => [2,3,1,4,6,7,5] => [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => [[5,3,3],[2,2]]
 => 1 = 0 + 1
{{1,2,3},{4},{5,6},{7}}
 => [2,3,1,4,6,5,7] => [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
 => [[4,4,3,3],[3,2,2]]
 => 1 = 0 + 1
{{1,2,3},{4},{5},{6,7}}
 => [2,3,1,4,5,7,6] => [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => [[4,3,3,3],[2,2,2]]
 => 1 = 0 + 1
{{1,2},{3,4,5,6,7}}
 => [2,1,4,5,6,7,3] => [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => [[6,2],[1]]
 => 1 = 0 + 1
{{1,2},{3,4,5,6},{7}}
 => [2,1,4,5,6,3,7] => [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => [[5,5,2],[4,1]]
 => 1 = 0 + 1
{{1,2},{3,4,5},{6,7}}
 => [2,1,4,5,3,7,6] => [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => [[5,4,2],[3,1]]
 => 1 = 0 + 1
{{1,2},{3,4,5},{6},{7}}
 => [2,1,4,5,3,6,7] => [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => [[4,4,4,2],[3,3,1]]
 => 1 = 0 + 1
Description
The Frobenius rank of a skew partition. This is the minimal number of border strips in a border strip decomposition of the skew partition.
Matching statistic: St001271
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001271: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 67%
Values
{{1},{2},{3}}
 => [1,2,3] => ([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
{{1,2,3,4}}
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,3},{2,4}}
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => ([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => 1 = 0 + 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 1 + 1
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
{{1},{2,4},{3,5}}
 => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 1 + 1
{{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => 1 = 0 + 1
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1 = 0 + 1
{{1,2,3,4,5},{6}}
 => [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1 = 0 + 1
{{1,2,3,4},{5,6}}
 => [2,3,4,1,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1 = 0 + 1
{{1,2,3,4},{5},{6}}
 => [2,3,4,1,5,6] => ([(2,5),(3,5),(4,5)],6)
 => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1 = 0 + 1
{{1,2,3,5},{4,6}}
 => [2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
{{1,2,3},{4,5,6}}
 => [2,3,1,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => 1 = 0 + 1
{{1,2,3},{4,5},{6}}
 => [2,3,1,5,4,6] => ([(1,2),(3,5),(4,5)],6)
 => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1 = 0 + 1
{{1,2,3},{4},{5,6}}
 => [2,3,1,4,6,5] => ([(1,2),(3,5),(4,5)],6)
 => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1 = 0 + 1
{{1,2,4,6},{3,5}}
 => [2,4,5,6,3,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => 1 = 0 + 1
{{1,2,4},{3,5,6}}
 => [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
{{1,2,4},{3,5},{6}}
 => [2,4,5,1,3,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
{{1,2},{3,4,5,6}}
 => [2,1,4,5,6,3] => ([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1 = 0 + 1
{{1,2},{3,4,5},{6}}
 => [2,1,4,5,3,6] => ([(1,2),(3,5),(4,5)],6)
 => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1 = 0 + 1
{{1,2},{3,4},{5,6}}
 => [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6)
 => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
 => 1 = 0 + 1
{{1,2,5},{3,6},{4}}
 => [2,5,6,4,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
{{1,2},{3,5},{4,6}}
 => [2,1,5,6,3,4] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
{{1,2},{3},{4,5,6}}
 => [2,1,3,5,6,4] => ([(1,2),(3,5),(4,5)],6)
 => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1 = 0 + 1
{{1,2,6},{3},{4},{5}}
 => [2,6,3,4,5,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => 1 = 0 + 1
{{1,3,4,5},{2,6}}
 => [3,6,4,5,1,2] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(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),(4,6),(5,6)],7)
 => ? = 1 + 1
{{1,3,4},{2,6},{5}}
 => [3,6,4,1,5,2] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 1
{{1,3,5,6},{2,4}}
 => [3,4,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => 1 = 0 + 1
{{1,3,5},{2,4},{6}}
 => [3,4,5,2,1,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => 1 = 0 + 1
{{1,3},{2,4,5,6}}
 => [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
{{1,3},{2,4,5},{6}}
 => [3,4,1,5,2,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
{{1,3,6},{2,4},{5}}
 => [3,4,6,2,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => 1 = 0 + 1
{{1,3},{2,4},{5,6}}
 => [3,4,1,2,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
{{1,3},{2,4},{5},{6}}
 => [3,4,1,2,5,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
{{1,3},{2,5},{4,6}}
 => [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
{{1,3,6},{2},{4},{5}}
 => [3,2,6,4,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => 1 = 0 + 1
{{1,4,5},{2,3,6}}
 => [4,3,6,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(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),(4,6),(5,6)],7)
 => ? = 1 + 1
{{1,4},{2,3,6},{5}}
 => [4,3,6,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 1
{{1,5},{2,3,4,6}}
 => [5,3,4,6,1,2] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(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),(4,6),(5,6)],7)
 => ? = 1 + 1
{{1,6},{2,3,4,5}}
 => [6,3,4,5,2,1] => ([(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)
 => ([(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)
 => 1 = 0 + 1
{{1},{2,3,4,5,6}}
 => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1 = 0 + 1
{{1},{2,3,4,5},{6}}
 => [1,3,4,5,2,6] => ([(2,5),(3,5),(4,5)],6)
 => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1 = 0 + 1
{{1},{2,3,4},{5,6}}
 => [1,3,4,2,6,5] => ([(1,2),(3,5),(4,5)],6)
 => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1 = 0 + 1
{{1,5},{2,3,6},{4}}
 => [5,3,6,4,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(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),(4,6),(5,6)],7)
 => ? = 2 + 1
{{1},{2,3,5},{4,6}}
 => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
{{1},{2,3},{4,5,6}}
 => [1,3,2,5,6,4] => ([(1,2),(3,5),(4,5)],6)
 => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1 = 0 + 1
{{1,4,5},{2,6},{3}}
 => [4,6,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(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),(4,6),(5,6)],7)
 => ? = 2 + 1
{{1,4,5},{2},{3,6}}
 => [4,2,6,5,1,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 1
{{1,4},{2,5,6},{3}}
 => [4,5,3,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
{{1,4},{2,5},{3},{6}}
 => [4,5,3,1,2,6] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
{{1,4,6},{2},{3,5}}
 => [4,2,5,6,3,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => 1 = 0 + 1
{{1,4,6},{2},{3},{5}}
 => [4,2,3,6,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => 1 = 0 + 1
{{1,4},{2,6},{3},{5}}
 => [4,6,3,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
{{1,6},{2,4},{3,5}}
 => [6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => ? = 1 + 1
{{1},{2,4,6},{3,5}}
 => [1,4,5,6,3,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => 1 = 0 + 1
{{1},{2,4},{3,5,6}}
 => [1,4,5,2,6,3] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
{{1},{2,4},{3,5},{6}}
 => [1,4,5,2,3,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
{{1,5},{2,6},{3,4}}
 => [5,6,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => ? = 1 + 1
{{1,5},{2},{3,4,6}}
 => [5,2,4,6,1,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 1
{{1},{2},{3,4,5,6}}
 => [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1 = 0 + 1
{{1,5,6},{2},{3},{4}}
 => [5,2,3,4,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => 1 = 0 + 1
{{1,5},{2},{3,6},{4}}
 => [5,2,6,4,1,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
{{1,5},{2},{3},{4},{6}}
 => [5,2,3,4,1,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => 1 = 0 + 1
{{1},{2,5},{3,6},{4}}
 => [1,5,6,4,2,3] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
{{1},{2},{3,5},{4,6}}
 => [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
{{1,6},{2},{3},{4},{5}}
 => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => 1 = 0 + 1
{{1},{2,6},{3},{4},{5}}
 => [1,6,3,4,5,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => 1 = 0 + 1
{{1},{2},{3},{4},{5},{6}}
 => [1,2,3,4,5,6] => ([],6)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1 = 0 + 1
{{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0 + 1
{{1,2,3,4,5,6},{7}}
 => [2,3,4,5,6,1,7] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0 + 1
{{1,2,3,4,5},{6,7}}
 => [2,3,4,5,1,7,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0 + 1
{{1,2,3,4,5},{6},{7}}
 => [2,3,4,5,1,6,7] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0 + 1
{{1,2,3,4,6},{5,7}}
 => [2,3,4,6,7,1,5] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 1 + 1
{{1,2,3,4},{5,6,7}}
 => [2,3,4,1,6,7,5] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
 => ? = 0 + 1
{{1,2,3,4},{5,6},{7}}
 => [2,3,4,1,6,5,7] => ([(1,2),(3,6),(4,6),(5,6)],7)
 => ([(0,7),(1,2),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0 + 1
{{1,2,3,4},{5},{6,7}}
 => [2,3,4,1,5,7,6] => ([(1,2),(3,6),(4,6),(5,6)],7)
 => ([(0,7),(1,2),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0 + 1
{{1,2,3,4},{5},{6},{7}}
 => [2,3,4,1,5,6,7] => ([(3,6),(4,6),(5,6)],7)
 => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0 + 1
{{1,2,3,5,7},{4,6}}
 => [2,3,5,6,7,4,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0 + 1
{{1,2,3,5},{4,6,7}}
 => [2,3,5,6,1,7,4] => ([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 1 + 1
{{1,2,3,5},{4,6},{7}}
 => [2,3,5,6,1,4,7] => ([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1 + 1
{{1,2,3},{4,5,6,7}}
 => [2,3,1,5,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
 => ? = 0 + 1
{{1,2,3},{4,5,6},{7}}
 => [2,3,1,5,6,4,7] => ([(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
 => ? = 0 + 1
{{1,2,3},{4,5},{6,7}}
 => [2,3,1,5,4,7,6] => ([(0,3),(1,2),(4,6),(5,6)],7)
 => ([(0,3),(0,7),(1,2),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0 + 1
Description
The competition number of a graph. The competition graph of a digraph $D$ is a (simple undirected) graph which has the same vertex set as $D$ and has an edge between $x$ and $y$ if and only if there exists a vertex $v$ in $D$ such that $(x, v)$ and $(y, v)$ are arcs of $D$. For any graph, $G$ together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number $k(G)$ is the smallest number of such isolated vertices.