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Your data matches 10 different statistics following compositions of up to 3 maps.
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Matching statistic: St001621
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(2,5),(3,5),(4,5)],6)
=> [1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(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)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
Description
The number of atoms of a lattice.
An element of a lattice is an '''atom''' if it covers the least element.
Matching statistic: St001878
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(2,5),(3,5),(4,5)],6)
=> [1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(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)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St000455
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 50%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 50%
Values
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
([(2,5),(3,5),(4,5)],6)
=> [1,2,3] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,3,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,2,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,3] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,3,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,2,1,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,1,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,3,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,1,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [2,1,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 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)
=> [3,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 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)],6)
=> [2,3,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 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)
=> [3,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
([(3,6),(4,6),(5,6)],7)
=> [1,2,4] => [2,2,1,1,1] => ([(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 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> [1,3,3] => [2,1,2,1,1] => ([(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)
=> ? = 2 - 1
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,4,2] => [2,1,1,2,1] => ([(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)
=> ? = 2 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> [1,1,2,3] => [3,2,1,1] => ([(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 - 1
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> [1,1,2,1,2] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,3,2] => [3,1,2,1] => ([(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)
=> ? = 2 - 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,2,3] => [3,2,1,1] => ([(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 - 1
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,3,2] => [3,1,2,1] => ([(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)
=> ? = 2 - 1
([(3,5),(3,6),(4,5),(4,6)],7)
=> [1,2,4] => [2,2,1,1,1] => ([(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 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> [2,2,3] => [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> [1,1,3,2] => [3,1,2,1] => ([(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)
=> ? = 2 - 1
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,1,2,1,1,1] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,1,2,1,1,1] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,2,1,1,1] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,6),(1,6),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,6),(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7)
=> [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
=> [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6)],7)
=> [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,4),(0,5),(1,2),(1,3),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,1),(0,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [1,1,2,1,1,1] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5)],7)
=> [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St001545
Values
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([],1)
=> ? = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> ([],1)
=> ? = 1 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? = 1 + 1
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 2 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([],3)
=> ([],1)
=> ? = 1 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ? = 2 + 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],2)
=> ([],1)
=> ? = 1 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ? = 2 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ? = 2 + 1
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],1)
=> ? = 1 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? = 2 + 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 2 + 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([],2)
=> ([],1)
=> ? = 2 + 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],2)
=> ([],1)
=> ? = 2 + 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ? = 2 + 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ? = 1 + 1
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 2 + 1
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> ([],1)
=> ? = 1 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> ([],1)
=> ? = 1 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ? = 2 + 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ? = 2 + 1
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ([],1)
=> ([],1)
=> ? = 1 + 1
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> ([],1)
=> ? = 1 + 1
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ? = 1 + 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ? = 2 + 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ? = 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)
=> ([],1)
=> ([],1)
=> ? = 1 + 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> ([],1)
=> ([],1)
=> ? = 1 + 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ? = 1 + 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ? = 1 + 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([],1)
=> ([],1)
=> ? = 1 + 1
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ? = 1 + 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> ([],1)
=> ? = 1 + 1
([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ? = 1 + 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)],6)
=> ([],1)
=> ([],1)
=> ? = 2 + 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)
=> ([],1)
=> ([],1)
=> ? = 1 + 1
([(3,6),(4,6),(5,6)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ([(1,2)],3)
=> ? = 1 + 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,2)],3)
=> ? = 2 + 1
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,2)],3)
=> ? = 2 + 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1 + 1
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ? = 1 + 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 2 + 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1 + 1
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ? = 1 + 1
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,6),(1,6),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
([(0,6),(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,4),(0,5),(1,2),(1,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,1),(0,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(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)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
Description
The second Elser number of a connected graph.
For a connected graph $G$ the $k$-th Elser number is
$$
els_k(G) = (-1)^{|V(G)|+1} \sum_N (-1)^{|E(N)|} |V(N)|^k
$$
where the sum is over all nuclei of $G$, that is, the connected subgraphs of $G$ whose vertex set is a vertex cover of $G$.
It is clear that this number is even. It was shown in [1] that it is non-negative.
Matching statistic: St000456
Values
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(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),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> ? = 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 1
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ? = 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> ? = 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> ? = 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> ? = 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> ? = 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 2
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ? = 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? = 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ? = 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([],2)
=> ? = 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> ? = 2
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> ? = 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> ? = 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> ? = 1
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ? = 2
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 2
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> ? = 2
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([],1)
=> ? = 1
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 1
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 2
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 2
([(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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 1
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 1
([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 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)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 1
([(3,6),(4,6),(5,6)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(2,5),(3,5),(4,5)],6)
=> ? = 1
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(2,3)],4)
=> ? = 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 1
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 1
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 1
([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 1
([(0,6),(1,6),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 1
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
=> ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(1,2)],3)
=> 1
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 1
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(1,2)],3)
=> 1
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 1
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 1
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 1
([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(0,1),(0,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 1
([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 1
([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(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)
=> ([(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)
=> ([(0,2),(1,2)],3)
=> 1
([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ([(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)
=> ([(0,2),(1,2)],3)
=> 1
([(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(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)
=> ([(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)
=> ([(0,2),(1,2)],3)
=> 1
([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ([(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)
=> ([(0,2),(1,2)],3)
=> 1
([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5)],7)
=> ([(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)
=> ([(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)
=> ([(0,2),(1,2)],3)
=> 1
([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> ([(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)
=> ([(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)
=> ([(0,2),(1,2)],3)
=> 1
([(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)
=> ([(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)
=> ([(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)
=> ([(0,2),(1,2)],3)
=> 1
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 1
([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 1
([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 1
([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 1
([(0,1),(0,6),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 1
([(0,1),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(1,2)],3)
=> 1
([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 1
([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 1
Description
The monochromatic index of a connected graph.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St001645
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 50%
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 50%
Values
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
([(2,5),(3,5),(4,5)],6)
=> [1,2,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,3,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,2,2] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,2] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
([(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,2] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,3,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1 + 6
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,2,1,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,1,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,2,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,3,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1 + 6
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,1,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,2,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,2,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> [1,2,2,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,2,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,1,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,2,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,2] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,2] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(0,6),(1,6),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,6),(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,4),(0,5),(1,2),(1,3),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,1),(0,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(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)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 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)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,1),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,1,1,2,2] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(0,4),(0,6),(1,2),(1,3),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(4,5),(5,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,6),(1,2),(1,3),(1,6),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,1),(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)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
([(0,5),(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)],7)
=> [1,1,1,2,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
Description
The pebbling number of a connected graph.
Matching statistic: St000788
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000788: Perfect matchings ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000788: Perfect matchings ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Values
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
=> ? = 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> ? = 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
=> ? = 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
=> ? = 2
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
=> ? = 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? = 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> ? = 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> ? = 2
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
=> ? = 1
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 2
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? = 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? = 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> ? = 2
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 2
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? = 1
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> ? = 2
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> ? = 2
([(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)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
=> ? = 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> ? = 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? = 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> ? = 1
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> ? = 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? = 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)],6)
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
=> ? = 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)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
=> ? = 1
([(3,6),(4,6),(5,6)],7)
=> [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,6),(4,5),(7,14),(8,13),(9,12),(10,11)]
=> ? = 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,14),(10,13),(11,12)]
=> ? = 2
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7),(11,14),(12,13)]
=> ? = 2
([(1,2),(3,6),(4,6),(5,6)],7)
=> [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,14),(10,13),(11,12)]
=> ? = 1
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,14),(12,13)]
=> ? = 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,14),(12,13)]
=> ? = 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,14),(10,13),(11,12)]
=> ? = 1
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12),(13,14)]
=> ? = 1
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,14),(12,13)]
=> ? = 2
([(3,5),(3,6),(4,5),(4,6)],7)
=> [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,6),(4,5),(7,14),(8,13),(9,12),(10,11)]
=> ? = 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,14),(10,13),(11,12)]
=> ? = 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,14),(12,13)]
=> ? = 2
([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,14),(12,13)]
=> ? = 1
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12),(13,14)]
=> ? = 1
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12),(13,14)]
=> ? = 1
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12),(13,14)]
=> ? = 1
([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,14),(12,13)]
=> ? = 1
([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12),(13,14)]
=> ? = 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,14),(10,13),(11,12)]
=> ? = 1
Description
The number of nesting-similar perfect matchings of a perfect matching.
Consider the infinite tree $T$ defined in [1] as follows. $T$ has the perfect matchings on $\{1,\dots,2n\}$ on level $n$, with children obtained by inserting an arc with opener $1$. For example, the matching $[(1,2)]$ has the three children $[(1,2),(3,4)]$, $[(1,3),(2,4)]$ and $[(1,4),(2,3)]$.
Two perfect matchings $M$ and $N$ on $\{1,\dots,2n\}$ are nesting-similar, if the distribution of the number of nestings agrees on all levels of the subtrees of $T$ rooted at $M$ and $N$.
[thm 1.2, 1] shows that to find out whether $M$ and $N$ are nesting-similar, it is enough to check that $M$ and $N$ have the same number of nestings, and that the distribution of nestings agrees for their direct children.
[thm 3.5, 1], see also [2], gives the number of equivalence classes of nesting-similar matchings with $n$ arcs as $$2\cdot 4^{n-1} - \frac{3n-1}{2n+2}\binom{2n}{n}.$$ [prop 3.6, 1] has further interpretations of this number.
Matching statistic: St001132
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001132: Perfect matchings ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001132: Perfect matchings ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Values
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
=> ? = 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> ? = 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
=> ? = 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
=> ? = 2
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
=> ? = 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? = 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> ? = 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> ? = 2
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
=> ? = 1
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 2
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? = 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? = 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> ? = 2
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 2
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? = 1
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> ? = 2
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> ? = 2
([(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)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
=> ? = 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> ? = 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? = 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> ? = 1
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> ? = 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? = 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)],6)
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
=> ? = 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)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
=> ? = 1
([(3,6),(4,6),(5,6)],7)
=> [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,6),(4,5),(7,14),(8,13),(9,12),(10,11)]
=> ? = 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,14),(10,13),(11,12)]
=> ? = 2
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7),(11,14),(12,13)]
=> ? = 2
([(1,2),(3,6),(4,6),(5,6)],7)
=> [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,14),(10,13),(11,12)]
=> ? = 1
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,14),(12,13)]
=> ? = 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,14),(12,13)]
=> ? = 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,14),(10,13),(11,12)]
=> ? = 1
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12),(13,14)]
=> ? = 1
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,14),(12,13)]
=> ? = 2
([(3,5),(3,6),(4,5),(4,6)],7)
=> [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,6),(4,5),(7,14),(8,13),(9,12),(10,11)]
=> ? = 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,14),(10,13),(11,12)]
=> ? = 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,14),(12,13)]
=> ? = 2
([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,14),(12,13)]
=> ? = 1
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12),(13,14)]
=> ? = 1
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12),(13,14)]
=> ? = 1
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12),(13,14)]
=> ? = 1
([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,14),(12,13)]
=> ? = 1
([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12),(13,14)]
=> ? = 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,14),(10,13),(11,12)]
=> ? = 1
Description
The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching.
The bijection between perfect matchings of $\{1,\dots,2n\}$ and trees with $n+1$ leaves is described in Example 5.2.6 of [1].
Matching statistic: St000787
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000787: Perfect matchings ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000787: Perfect matchings ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Values
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
=> [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
=> ? = 1 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> ? = 2 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0 = 1 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
=> ? = 1 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 2 - 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0 = 1 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 0 = 1 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0 = 1 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 2 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0 = 1 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
=> ? = 2 - 1
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
=> ? = 1 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 2 - 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 2 - 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 2 - 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> ? = 2 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> ? = 2 - 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0 = 1 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 2 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
=> ? = 1 - 1
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? = 1 - 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0 = 1 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? = 1 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> ? = 2 - 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 2 - 1
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0 = 1 - 1
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? = 1 - 1
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0 = 1 - 1
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0 = 1 - 1
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0 = 1 - 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> ? = 2 - 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> ? = 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)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
=> ? = 1 - 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0 = 1 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0 = 1 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> ? = 1 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? = 1 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> ? = 1 - 1
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> ? = 1 - 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0 = 1 - 1
([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? = 1 - 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)],6)
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
=> ? = 2 - 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)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
=> ? = 1 - 1
([(3,6),(4,6),(5,6)],7)
=> [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,6),(4,5),(7,14),(8,13),(9,12),(10,11)]
=> ? = 1 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,14),(10,13),(11,12)]
=> ? = 2 - 1
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7),(11,14),(12,13)]
=> ? = 2 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,14),(10,13),(11,12)]
=> ? = 1 - 1
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,14),(12,13)]
=> ? = 1 - 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,14),(12,13)]
=> ? = 2 - 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,14),(10,13),(11,12)]
=> ? = 1 - 1
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12),(13,14)]
=> ? = 1 - 1
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,14),(12,13)]
=> ? = 2 - 1
([(3,5),(3,6),(4,5),(4,6)],7)
=> [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,6),(4,5),(7,14),(8,13),(9,12),(10,11)]
=> ? = 1 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,14),(10,13),(11,12)]
=> ? = 2 - 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,14),(12,13)]
=> ? = 2 - 1
([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,14),(12,13)]
=> ? = 1 - 1
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12),(13,14)]
=> ? = 1 - 1
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12),(13,14)]
=> ? = 1 - 1
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12),(13,14)]
=> ? = 1 - 1
([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,14),(12,13)]
=> ? = 1 - 1
([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12),(13,14)]
=> ? = 1 - 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,14),(10,13),(11,12)]
=> ? = 1 - 1
Description
The number of flips required to make a perfect matching noncrossing.
A crossing in a perfect matching is a pair of arcs $\{a,b\}$ and $\{c,d\}$ such that $a < c < b < d$. Replacing any such pair by either $\{a,c\}$ and $\{b,d\}$ or by $\{a,d\}$, $\{b,c\}$ produces a perfect matching with fewer crossings.
This statistic is the minimal number of such flips required to turn a given matching into a noncrossing matching.
Matching statistic: St001043
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001043: Perfect matchings ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001043: Perfect matchings ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Values
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 2 = 1 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 2 = 1 + 1
([(2,5),(3,5),(4,5)],6)
=> [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
=> ? = 1 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> ? = 2 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 2 = 1 + 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2 = 1 + 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 2 = 1 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
=> ? = 1 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 2 + 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2 = 1 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 2 = 1 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2 = 1 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 2 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2 = 1 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
=> ? = 2 + 1
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
=> ? = 1 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 2 + 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 2 + 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? = 1 + 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 2 + 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> ? = 2 + 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> ? = 2 + 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2 = 1 + 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 2 + 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
=> ? = 1 + 1
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 2 + 1
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? = 1 + 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2 = 1 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? = 1 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> ? = 2 + 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 2 + 1
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2 = 1 + 1
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? = 1 + 1
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2 = 1 + 1
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2 = 1 + 1
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2 = 1 + 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> ? = 2 + 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> ? = 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)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
=> ? = 1 + 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2 = 1 + 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2 = 1 + 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> ? = 1 + 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? = 1 + 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> ? = 1 + 1
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> ? = 1 + 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2 = 1 + 1
([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? = 1 + 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)],6)
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
=> ? = 2 + 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)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
=> ? = 1 + 1
([(3,6),(4,6),(5,6)],7)
=> [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,6),(4,5),(7,14),(8,13),(9,12),(10,11)]
=> ? = 1 + 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,14),(10,13),(11,12)]
=> ? = 2 + 1
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7),(11,14),(12,13)]
=> ? = 2 + 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,14),(10,13),(11,12)]
=> ? = 1 + 1
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,14),(12,13)]
=> ? = 1 + 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,14),(12,13)]
=> ? = 2 + 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,14),(10,13),(11,12)]
=> ? = 1 + 1
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12),(13,14)]
=> ? = 1 + 1
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,14),(12,13)]
=> ? = 2 + 1
([(3,5),(3,6),(4,5),(4,6)],7)
=> [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,6),(4,5),(7,14),(8,13),(9,12),(10,11)]
=> ? = 1 + 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,14),(10,13),(11,12)]
=> ? = 2 + 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,14),(12,13)]
=> ? = 2 + 1
([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,14),(12,13)]
=> ? = 1 + 1
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12),(13,14)]
=> ? = 1 + 1
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12),(13,14)]
=> ? = 1 + 1
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12),(13,14)]
=> ? = 1 + 1
([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,14),(12,13)]
=> ? = 1 + 1
([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12),(13,14)]
=> ? = 1 + 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,14),(10,13),(11,12)]
=> ? = 1 + 1
Description
The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching.
The bijection between perfect matchings of $\{1,\dots,2n\}$ and trees with $n+1$ leaves is described in Example 5.2.6 of [1].
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