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Your data matches 24 different statistics following compositions of up to 3 maps.
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Matching statistic: St000010
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00074: Posets —to graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> ([],1)
=> ([],1)
=> [1]
=> 1
[[2],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 1
[[1,1],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 1
[[2,1],[1]]
=> ([],2)
=> ([],2)
=> [1,1]
=> 2
[[3],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> 1
[[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> 1
[[3,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 2
[[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> 1
[[3,2],[2]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 2
[[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> 1
[[2,2,1],[1,1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 2
[[2,1,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 2
[[3,2,1],[2,1]]
=> ([],3)
=> ([],3)
=> [1,1,1]
=> 3
[[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 1
[[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 1
[[4,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 2
[[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 1
[[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 1
[[4,2],[2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> 2
[[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 1
[[3,2,1],[1,1]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 2
[[3,1,1],[1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> 2
[[4,2,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 3
[[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 1
[[4,3],[3]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 2
[[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 1
[[3,3,1],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 2
[[3,2,1],[2]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 2
[[4,3,1],[3,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 3
[[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 1
[[3,3,2],[2,2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> 2
[[3,2,2],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 2
[[4,3,2],[3,2]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 3
[[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 1
[[2,2,2,1],[1,1,1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 2
[[2,2,1,1],[1,1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> 2
[[3,3,2,1],[2,2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 3
[[2,1,1,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 2
[[3,2,2,1],[2,1,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 3
[[3,2,1,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 3
[[4,3,2,1],[3,2,1]]
=> ([],4)
=> ([],4)
=> [1,1,1,1]
=> 4
[[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> 1
[[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> 1
[[5,1],[1]]
=> ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 2
[[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 1
[[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> 1
[[5,2],[2]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> 2
[[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> 1
[[4,2,1],[1,1]]
=> ([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 2
[[4,1,1],[1]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> 2
Description
The length of the partition.
Matching statistic: St000383
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00074: Posets —to graph⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> ([],1)
=> ([],1)
=> [1] => 1
[[2],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [1,1] => 1
[[1,1],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [1,1] => 1
[[2,1],[1]]
=> ([],2)
=> ([],2)
=> [2] => 2
[[3],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [1,1,1] => 1
[[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [1,1,1] => 1
[[3,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> [1,2] => 2
[[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [1,1,1] => 1
[[3,2],[2]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> [1,2] => 2
[[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [1,1,1] => 1
[[2,2,1],[1,1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> [1,2] => 2
[[2,1,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> [1,2] => 2
[[3,2,1],[2,1]]
=> ([],3)
=> ([],3)
=> [3] => 3
[[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1
[[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1
[[4,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [1,1,2] => 2
[[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => 1
[[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1
[[4,2],[2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2] => 2
[[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1
[[3,2,1],[1,1]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [1,1,2] => 2
[[3,1,1],[1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2] => 2
[[4,2,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> [1,3] => 3
[[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1
[[4,3],[3]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [1,1,2] => 2
[[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1
[[3,3,1],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [1,1,2] => 2
[[3,2,1],[2]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [1,1,2] => 2
[[4,3,1],[3,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> [1,3] => 3
[[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1
[[3,3,2],[2,2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2] => 2
[[3,2,2],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [1,1,2] => 2
[[4,3,2],[3,2]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> [1,3] => 3
[[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1
[[2,2,2,1],[1,1,1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [1,1,2] => 2
[[2,2,1,1],[1,1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2] => 2
[[3,3,2,1],[2,2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> [1,3] => 3
[[2,1,1,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [1,1,2] => 2
[[3,2,2,1],[2,1,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> [1,3] => 3
[[3,2,1,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> [1,3] => 3
[[4,3,2,1],[3,2,1]]
=> ([],4)
=> ([],4)
=> [4] => 4
[[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => 1
[[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => 1
[[5,1],[1]]
=> ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => 2
[[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
[[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => 1
[[5,2],[2]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => 2
[[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => 1
[[4,2,1],[1,1]]
=> ([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => 2
[[4,1,1],[1]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => 2
[[9],[]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> [1,1,1,1,1,1,1,1,1] => ? = 1
[[8,1],[]]
=> ([(0,2),(0,8),(3,5),(4,3),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> [1,1,1,1,1,1,1,1,1] => ? = 1
[[7,1,1],[]]
=> ([(0,7),(0,8),(3,4),(4,6),(5,3),(6,2),(7,5),(8,1)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> [1,1,1,1,1,1,1,1,1] => ? = 1
[[6,1,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,3),(5,2),(6,1),(7,6),(8,4)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> [1,1,1,1,1,1,1,1,1] => ? = 1
[[5,4],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,8),(4,1),(4,7),(5,3),(5,6),(6,8),(8,7)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> [1,1,1,1,1,1,1,1,1] => ? = 1
[[5,1,1,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,6),(5,2),(6,1),(7,3),(8,4)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> [1,1,1,1,1,1,1,1,1] => ? = 1
[[4,4,1],[]]
=> ([(0,4),(0,5),(2,7),(3,2),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(8,7)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> [1,1,1,1,1,1,1,1,1] => ? = 1
[[4,3,2],[]]
=> ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> [1,1,1,1,1,1,1,1,1] => ? = 1
[[4,1,1,1,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,3),(5,2),(6,1),(7,6),(8,4)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> [1,1,1,1,1,1,1,1,1] => ? = 1
[[3,3,2,1],[]]
=> ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> [1,1,1,1,1,1,1,1,1] => ? = 1
[[3,2,2,2],[]]
=> ([(0,4),(0,5),(2,7),(3,2),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(8,7)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> [1,1,1,1,1,1,1,1,1] => ? = 1
[[3,1,1,1,1,1,1],[]]
=> ([(0,7),(0,8),(3,4),(4,6),(5,3),(6,2),(7,5),(8,1)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> [1,1,1,1,1,1,1,1,1] => ? = 1
[[2,2,2,2,1],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,8),(4,1),(4,7),(5,3),(5,6),(6,8),(8,7)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> [1,1,1,1,1,1,1,1,1] => ? = 1
[[2,1,1,1,1,1,1,1],[]]
=> ([(0,2),(0,8),(3,5),(4,3),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> [1,1,1,1,1,1,1,1,1] => ? = 1
[[1,1,1,1,1,1,1,1,1],[]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> [1,1,1,1,1,1,1,1,1] => ? = 1
[[3,3,3,1],[1]]
=> ([(0,3),(0,8),(1,4),(1,8),(3,6),(4,2),(4,7),(6,5),(7,5),(8,6),(8,7)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> [1,1,1,1,1,1,1,1,1] => ? = 1
[[3,3,3,3],[1,1,1]]
=> ([(0,2),(0,3),(1,7),(2,8),(3,4),(3,8),(4,6),(4,7),(6,5),(7,5),(8,6)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> [1,1,1,1,1,1,1,1,1] => ? = 1
[[4,4,3],[1,1]]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,8),(4,7),(4,8),(7,5),(8,5),(8,6)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> [1,1,1,1,1,1,1,1,1] => ? = 1
[[4,4,4],[2,1]]
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,6),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> [1,1,1,1,1,1,1,1,1] => ? = 1
[[5,5],[1]]
=> ([(0,8),(1,4),(1,8),(2,3),(2,7),(3,5),(4,2),(4,6),(6,7),(7,5),(8,6)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> [1,1,1,1,1,1,1,1,1] => ? = 1
[[2,2,2,2,2],[1]]
=> ([(0,8),(1,4),(1,8),(2,3),(2,7),(3,5),(4,2),(4,6),(6,7),(7,5),(8,6)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> [1,1,1,1,1,1,1,1,1] => ? = 1
[[3,3,3,2],[2]]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,8),(4,7),(4,8),(7,5),(8,5),(8,6)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> [1,1,1,1,1,1,1,1,1] => ? = 1
[[3,3,3,3],[2,1]]
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,6),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> [1,1,1,1,1,1,1,1,1] => ? = 1
[[4,3,3],[1]]
=> ([(0,3),(0,8),(1,4),(1,8),(3,6),(4,2),(4,7),(6,5),(7,5),(8,6),(8,7)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> [1,1,1,1,1,1,1,1,1] => ? = 1
[[4,4,4],[3]]
=> ([(0,2),(0,3),(1,7),(2,8),(3,4),(3,8),(4,6),(4,7),(6,5),(7,5),(8,6)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> [1,1,1,1,1,1,1,1,1] => ? = 1
[[5,5,5,5,5],[4,3,2,1]]
=> ([(0,11),(1,10),(2,10),(2,13),(3,11),(3,14),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,6),(13,12),(14,7),(14,12)],15)
=> ([(0,11),(1,10),(2,8),(2,9),(3,10),(3,13),(4,11),(4,14),(5,13),(5,14),(6,8),(6,10),(6,13),(7,9),(7,11),(7,14),(8,12),(9,12),(12,13),(12,14)],15)
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => ? = 1
Description
The last part of an integer composition.
Matching statistic: St001363
Values
[[1],[]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
[[2],[]]
=> ([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 1
[[1,1],[]]
=> ([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 1
[[2,1],[1]]
=> ([],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
[[3],[]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[3,1],[1]]
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[3,2],[2]]
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[[2,2,1],[1,1]]
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[2,1,1],[1]]
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[3,2,1],[2,1]]
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 3
[[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[4,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[4,2],[2]]
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 2
[[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[3,2,1],[1,1]]
=> ([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[3,1,1],[1]]
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 2
[[4,2,1],[2,1]]
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 3
[[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[4,3],[3]]
=> ([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[3,3,1],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[3,2,1],[2]]
=> ([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[4,3,1],[3,1]]
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 3
[[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[3,3,2],[2,2]]
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 2
[[3,2,2],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[4,3,2],[3,2]]
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 3
[[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[2,2,2,1],[1,1,1]]
=> ([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[2,2,1,1],[1,1]]
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 2
[[3,3,2,1],[2,2,1]]
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 3
[[2,1,1,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[3,2,2,1],[2,1,1]]
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 3
[[3,2,1,1],[2,1]]
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 3
[[4,3,2,1],[3,2,1]]
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 4
[[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[5,1],[1]]
=> ([(1,4),(3,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[[5,2],[2]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
[[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[[4,2,1],[1,1]]
=> ([(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[4,1,1],[1]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
[[7],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[6,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ? = 1
[[7,1],[1]]
=> ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ? = 2
[[2,1,1,1,1,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ? = 1
[[6,6],[5]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ? = 1
[[7,6],[6]]
=> ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ? = 2
[[2,2,2,2,2,2],[1,1,1,1,1]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ? = 1
[[1,1,1,1,1,1,1],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[2,2,2,2,2,2,1],[1,1,1,1,1,1]]
=> ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ? = 2
[[2,1,1,1,1,1,1],[1]]
=> ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ? = 2
[[5,3],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
=> ([(1,7),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,7),(1,2),(1,4),(1,6),(1,7),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[5,2,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
=> ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7)],8)
=> ([(0,2),(0,7),(1,2),(1,6),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[4,4],[]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[4,2,2],[]]
=> ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ([(1,6),(1,7),(2,4),(2,5),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,1),(0,5),(0,7),(1,5),(1,7),(2,3),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[4,2,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(5,7),(6,4),(6,7)],8)
=> ([(1,2),(1,6),(1,7),(2,4),(2,5),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,1),(0,6),(0,7),(1,6),(1,7),(2,3),(2,5),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
[[3,3,1,1],[]]
=> ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ([(1,6),(1,7),(2,4),(2,5),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,1),(0,5),(0,7),(1,5),(1,7),(2,3),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[3,2,1,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
=> ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7)],8)
=> ([(0,2),(0,7),(1,2),(1,6),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[2,2,2,2],[]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[2,2,2,1,1],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
=> ([(1,7),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,7),(1,2),(1,4),(1,6),(1,7),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[9],[]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([],9)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[[8,1],[]]
=> ([(0,2),(0,8),(3,5),(4,3),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ([(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[[7,1,1],[]]
=> ([(0,7),(0,8),(3,4),(4,6),(5,3),(6,2),(7,5),(8,1)],9)
=> ([(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)
=> ([(0,1),(0,8),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[[6,1,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,3),(5,2),(6,1),(7,6),(8,4)],9)
=> ([(1,6),(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ([(0,1),(0,2),(0,8),(1,2),(1,8),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[[5,4],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,8),(4,1),(4,7),(5,3),(5,6),(6,8),(8,7)],9)
=> ([(1,8),(2,7),(3,6),(3,7),(4,5),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,5),(0,7),(0,8),(1,2),(1,4),(1,6),(1,8),(2,4),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[[5,1,1,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,6),(5,2),(6,1),(7,3),(8,4)],9)
=> ([(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8)],9)
=> ([(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,8),(2,3),(2,4),(2,8),(3,4),(3,8),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[[4,4,1],[]]
=> ([(0,4),(0,5),(2,7),(3,2),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(8,7)],9)
=> ([(1,8),(2,6),(2,8),(3,7),(3,8),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(7,8)],9)
=> ([(0,4),(0,8),(1,3),(1,6),(1,7),(1,8),(2,4),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[[4,3,2],[]]
=> ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8)],9)
=> ([(1,5),(1,8),(2,3),(2,7),(2,8),(3,6),(3,7),(4,6),(4,7),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,5),(0,6),(0,8),(1,2),(1,7),(1,8),(2,4),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[[4,1,1,1,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,3),(5,2),(6,1),(7,6),(8,4)],9)
=> ([(1,6),(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ([(0,1),(0,2),(0,8),(1,2),(1,8),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[[3,3,2,1],[]]
=> ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8)],9)
=> ([(1,5),(1,8),(2,3),(2,7),(2,8),(3,6),(3,7),(4,6),(4,7),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,5),(0,6),(0,8),(1,2),(1,7),(1,8),(2,4),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[[3,2,2,2],[]]
=> ([(0,4),(0,5),(2,7),(3,2),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(8,7)],9)
=> ([(1,8),(2,6),(2,8),(3,7),(3,8),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(7,8)],9)
=> ([(0,4),(0,8),(1,3),(1,6),(1,7),(1,8),(2,4),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[[3,1,1,1,1,1,1],[]]
=> ([(0,7),(0,8),(3,4),(4,6),(5,3),(6,2),(7,5),(8,1)],9)
=> ([(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)
=> ([(0,1),(0,8),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[[2,2,2,2,1],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,8),(4,1),(4,7),(5,3),(5,6),(6,8),(8,7)],9)
=> ([(1,8),(2,7),(3,6),(3,7),(4,5),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,5),(0,7),(0,8),(1,2),(1,4),(1,6),(1,8),(2,4),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[[2,1,1,1,1,1,1,1],[]]
=> ([(0,2),(0,8),(3,5),(4,3),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ([(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[[1,1,1,1,1,1,1,1,1],[]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([],9)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[[5,3,2],[]]
=> ([(0,5),(0,6),(2,9),(3,1),(4,3),(4,8),(5,4),(5,7),(6,2),(6,7),(7,8),(7,9)],10)
=> ([(1,6),(1,9),(2,3),(2,5),(2,6),(2,9),(3,4),(3,7),(3,8),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,6),(0,7),(0,9),(1,2),(1,5),(1,8),(1,9),(2,5),(2,8),(2,9),(3,4),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,8),(4,9),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1
[[4,3,3],[]]
=> ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(1,9),(2,6),(2,8),(3,4),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(5,9),(6,7),(6,9),(7,8),(8,9)],10)
=> ([(0,2),(0,7),(0,9),(1,5),(1,6),(1,8),(1,9),(2,4),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1
[[4,3,2,1],[]]
=> ([(0,5),(0,6),(3,2),(3,8),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,8),(7,9)],10)
=> ([(1,2),(1,7),(1,9),(2,6),(2,8),(3,4),(3,6),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,9),(7,8),(8,9)],10)
=> ([(0,3),(0,8),(0,9),(1,2),(1,7),(1,9),(2,5),(2,7),(2,9),(3,6),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 1
[[3,3,3,1],[]]
=> ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(1,9),(2,6),(2,8),(3,4),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(5,9),(6,7),(6,9),(7,8),(8,9)],10)
=> ([(0,2),(0,7),(0,9),(1,5),(1,6),(1,8),(1,9),(2,4),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1
[[3,3,2,1,1],[]]
=> ([(0,5),(0,6),(2,9),(3,1),(4,3),(4,8),(5,4),(5,7),(6,2),(6,7),(7,8),(7,9)],10)
=> ([(1,6),(1,9),(2,3),(2,5),(2,6),(2,9),(3,4),(3,7),(3,8),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,6),(0,7),(0,9),(1,2),(1,5),(1,8),(1,9),(2,5),(2,8),(2,9),(3,4),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,8),(4,9),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1
[[3,3,3,1],[1,1]]
=> ([(0,4),(0,7),(1,2),(1,3),(2,5),(3,5),(3,7),(5,6),(7,6)],8)
=> ([(0,7),(1,6),(1,7),(2,3),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 1
[[4,4,1],[1]]
=> ([(0,3),(0,7),(1,4),(1,7),(2,6),(4,2),(4,5),(5,6),(7,5)],8)
=> ([(0,7),(1,5),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(6,7)],8)
=> ?
=> ? = 1
[[3,3,3,1],[1]]
=> ([(0,3),(0,8),(1,4),(1,8),(3,6),(4,2),(4,7),(6,5),(7,5),(8,6),(8,7)],9)
=> ([(0,8),(1,5),(1,7),(2,3),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> ?
=> ? = 1
[[3,3,3,2],[1,1,1]]
=> ([(0,7),(1,3),(1,4),(2,6),(2,7),(3,5),(4,2),(4,5),(5,6)],8)
=> ([(0,7),(1,5),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(6,7)],8)
=> ?
=> ? = 1
[[4,4,2],[1,1]]
=> ([(0,7),(1,3),(1,4),(2,6),(3,5),(3,7),(4,2),(4,5),(5,6)],8)
=> ([(0,7),(1,6),(1,7),(2,3),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 1
[[4,4,4],[2,2]]
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ([(1,6),(1,7),(2,4),(2,5),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,1),(0,5),(0,7),(1,5),(1,7),(2,3),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[5,5],[2]]
=> ([(0,3),(1,4),(1,7),(2,6),(3,7),(4,2),(4,5),(5,6),(7,5)],8)
=> ([(1,7),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,7),(1,2),(1,4),(1,6),(1,7),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[3,3,3,3],[1,1,1]]
=> ([(0,2),(0,3),(1,7),(2,8),(3,4),(3,8),(4,6),(4,7),(6,5),(7,5),(8,6)],9)
=> ([(1,8),(2,6),(2,8),(3,7),(3,8),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(7,8)],9)
=> ([(0,4),(0,8),(1,3),(1,6),(1,7),(1,8),(2,4),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[[4,4,3],[1,1]]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,8),(4,7),(4,8),(7,5),(8,5),(8,6)],9)
=> ([(0,8),(1,5),(1,7),(2,3),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> ?
=> ? = 1
[[4,4,4],[2,1]]
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,6),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(1,5),(1,8),(2,3),(2,7),(2,8),(3,6),(3,7),(4,6),(4,7),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,5),(0,6),(0,8),(1,2),(1,7),(1,8),(2,4),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[[5,5],[1]]
=> ([(0,8),(1,4),(1,8),(2,3),(2,7),(3,5),(4,2),(4,6),(6,7),(7,5),(8,6)],9)
=> ([(1,8),(2,7),(3,6),(3,7),(4,5),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,5),(0,7),(0,8),(1,2),(1,4),(1,6),(1,8),(2,4),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
Description
The Euler characteristic of a graph according to Knill.
This is $$\sum_{k\geq 1} (-1)^{k-1} c_k,$$
where $c_k$ is the number of cliques with $k$ vertices.
Matching statistic: St000544
Values
[[1],[]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
[[2],[]]
=> ([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 1
[[1,1],[]]
=> ([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 1
[[2,1],[1]]
=> ([],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
[[3],[]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[3,1],[1]]
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[3,2],[2]]
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[[2,2,1],[1,1]]
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[2,1,1],[1]]
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[3,2,1],[2,1]]
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 3
[[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[4,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[4,2],[2]]
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 2
[[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[3,2,1],[1,1]]
=> ([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[3,1,1],[1]]
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 2
[[4,2,1],[2,1]]
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 3
[[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[4,3],[3]]
=> ([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[3,3,1],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[3,2,1],[2]]
=> ([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[4,3,1],[3,1]]
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 3
[[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[3,3,2],[2,2]]
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 2
[[3,2,2],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[4,3,2],[3,2]]
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 3
[[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[2,2,2,1],[1,1,1]]
=> ([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[2,2,1,1],[1,1]]
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 2
[[3,3,2,1],[2,2,1]]
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 3
[[2,1,1,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[3,2,2,1],[2,1,1]]
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 3
[[3,2,1,1],[2,1]]
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 3
[[4,3,2,1],[3,2,1]]
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 4
[[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[5,1],[1]]
=> ([(1,4),(3,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[[5,2],[2]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
[[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[[4,2,1],[1,1]]
=> ([(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[4,1,1],[1]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
[[5,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[6,3],[2]]
=> ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 1
[[7,3],[3]]
=> ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[5,1,1,1],[1]]
=> ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[6,4],[3]]
=> ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 1
[[7,4],[4]]
=> ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[4,2,2],[1]]
=> ([(0,3),(0,6),(1,4),(1,6),(3,5),(4,2),(6,5)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(1,5),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[5,2,2],[2]]
=> ([(0,5),(1,3),(1,4),(3,6),(4,6),(5,2)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[4,2,1,1],[1]]
=> ([(0,5),(0,6),(1,4),(1,6),(4,2),(5,3)],7)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,6),(1,6),(2,3),(2,5),(3,5),(4,5),(4,6)],7)
=> ? = 1
[[5,3,3],[2,2]]
=> ([(0,4),(1,3),(1,5),(3,6),(4,6),(5,2)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,6),(1,6),(2,3),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[3,1,1,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[4,1,1,1,1],[1]]
=> ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[3,3,1,1],[1]]
=> ([(0,3),(0,6),(1,4),(1,6),(3,5),(4,2),(6,5)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(1,5),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[4,4,3],[2,2]]
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(3,6),(4,6)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(1,5),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[5,5,3],[3,3]]
=> ([(0,5),(1,3),(1,4),(3,6),(4,6),(5,2)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[3,3,1,1,1],[1,1]]
=> ([(0,5),(1,3),(1,4),(3,6),(4,6),(5,2)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[5,5,5],[4,4]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[3,3,3,2],[2,2]]
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(3,6),(4,6)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(1,5),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[3,3,3,1,1],[2,2]]
=> ([(0,4),(1,3),(1,5),(3,6),(4,6),(5,2)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,6),(1,6),(2,3),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[4,4,4,3],[3,3,2]]
=> ([(0,5),(0,6),(1,4),(2,3),(3,5),(4,6)],7)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,6),(1,6),(2,3),(2,5),(3,5),(4,5),(4,6)],7)
=> ? = 1
[[2,2,2,1,1,1],[1,1]]
=> ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 1
[[4,4,4,4],[3,3,3]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[5,5,5,4],[4,4,4]]
=> ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[3,3,3,2,2],[2,2,2]]
=> ([(0,5),(1,3),(1,4),(3,6),(4,6),(5,2)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[2,2,2,2,1,1],[1,1,1]]
=> ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 1
[[3,3,3,3,3],[2,2,2,2]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[4,4,4,4,3],[3,3,3,3]]
=> ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[2,2,2,2,1,1,1],[1,1,1,1]]
=> ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[2,2,2,1,1,1,1],[1,1,1]]
=> ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[5,3],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
=> ([(1,7),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,7),(1,2),(1,4),(1,6),(1,7),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[5,2,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
=> ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7)],8)
=> ([(0,2),(0,7),(1,2),(1,6),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[4,4],[]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[4,2,2],[]]
=> ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ([(1,6),(1,7),(2,4),(2,5),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,1),(0,5),(0,7),(1,5),(1,7),(2,3),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[4,2,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(5,7),(6,4),(6,7)],8)
=> ([(1,2),(1,6),(1,7),(2,4),(2,5),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,1),(0,6),(0,7),(1,6),(1,7),(2,3),(2,5),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
[[3,3,1,1],[]]
=> ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ([(1,6),(1,7),(2,4),(2,5),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,1),(0,5),(0,7),(1,5),(1,7),(2,3),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[3,2,1,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
=> ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7)],8)
=> ([(0,2),(0,7),(1,2),(1,6),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[2,2,2,2],[]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[2,2,2,1,1],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
=> ([(1,7),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,7),(1,2),(1,4),(1,6),(1,7),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[9],[]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([],9)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[[8,1],[]]
=> ([(0,2),(0,8),(3,5),(4,3),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ([(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[[7,1,1],[]]
=> ([(0,7),(0,8),(3,4),(4,6),(5,3),(6,2),(7,5),(8,1)],9)
=> ([(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)
=> ([(0,1),(0,8),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[[6,1,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,3),(5,2),(6,1),(7,6),(8,4)],9)
=> ([(1,6),(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ([(0,1),(0,2),(0,8),(1,2),(1,8),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[[5,4],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,8),(4,1),(4,7),(5,3),(5,6),(6,8),(8,7)],9)
=> ([(1,8),(2,7),(3,6),(3,7),(4,5),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,5),(0,7),(0,8),(1,2),(1,4),(1,6),(1,8),(2,4),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[[5,1,1,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,6),(5,2),(6,1),(7,3),(8,4)],9)
=> ([(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8)],9)
=> ([(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,8),(2,3),(2,4),(2,8),(3,4),(3,8),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[[4,4,1],[]]
=> ([(0,4),(0,5),(2,7),(3,2),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(8,7)],9)
=> ([(1,8),(2,6),(2,8),(3,7),(3,8),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(7,8)],9)
=> ([(0,4),(0,8),(1,3),(1,6),(1,7),(1,8),(2,4),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[[4,3,2],[]]
=> ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8)],9)
=> ([(1,5),(1,8),(2,3),(2,7),(2,8),(3,6),(3,7),(4,6),(4,7),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,5),(0,6),(0,8),(1,2),(1,7),(1,8),(2,4),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[[4,1,1,1,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,3),(5,2),(6,1),(7,6),(8,4)],9)
=> ([(1,6),(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ([(0,1),(0,2),(0,8),(1,2),(1,8),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[[3,3,2,1],[]]
=> ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8)],9)
=> ([(1,5),(1,8),(2,3),(2,7),(2,8),(3,6),(3,7),(4,6),(4,7),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,5),(0,6),(0,8),(1,2),(1,7),(1,8),(2,4),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[[3,2,2,2],[]]
=> ([(0,4),(0,5),(2,7),(3,2),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(8,7)],9)
=> ([(1,8),(2,6),(2,8),(3,7),(3,8),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(7,8)],9)
=> ([(0,4),(0,8),(1,3),(1,6),(1,7),(1,8),(2,4),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
Description
The cop number of a graph.
This is the minimal number of cops needed to catch the robber. The algorithm is from [2].
Matching statistic: St000286
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
[[1],[]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
[[2],[]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> 1
[[1,1],[]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> 1
[[2,1],[1]]
=> ([],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[[3],[]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 1
[[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[[3,1],[1]]
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[[3,2],[2]]
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 1
[[2,2,1],[1,1]]
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[[2,1,1],[1]]
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[[3,2,1],[2,1]]
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 1
[[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1
[[4,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> 1
[[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[4,2],[2]]
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2
[[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1
[[3,2,1],[1,1]]
=> ([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[3,1,1],[1]]
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2
[[4,2,1],[2,1]]
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1
[[4,3],[3]]
=> ([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[3,3,1],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[3,2,1],[2]]
=> ([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[4,3,1],[3,1]]
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1
[[3,3,2],[2,2]]
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2
[[3,2,2],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[4,3,2],[3,2]]
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 1
[[2,2,2,1],[1,1,1]]
=> ([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[[2,2,1,1],[1,1]]
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2
[[3,3,2,1],[2,2,1]]
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[2,1,1,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[[3,2,2,1],[2,1,1]]
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[3,2,1,1],[2,1]]
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[4,3,2,1],[3,2,1]]
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1
[[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1
[[5,1],[1]]
=> ([(1,4),(3,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 1
[[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[5,2],[2]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
[[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> 1
[[4,2,1],[1,1]]
=> ([(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[4,1,1],[1]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
[[4,3],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1
[[4,4],[1]]
=> ([(0,6),(1,3),(1,6),(2,4),(3,2),(3,5),(5,4),(6,5)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1
[[3,3,1],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ([(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[4,3,1],[1]]
=> ([(0,3),(0,6),(1,4),(1,6),(4,2),(4,5),(6,5)],7)
=> ([(0,4),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 1
[[5,4,1],[2,1]]
=> ([(1,5),(2,3),(2,5),(3,4),(3,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ([(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[4,3,2],[1,1]]
=> ([(0,6),(1,3),(1,4),(3,5),(3,6),(4,2),(4,5)],7)
=> ([(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[4,3,2,1],[1,1,1]]
=> ([(1,4),(1,5),(4,3),(4,6),(5,2),(5,6)],7)
=> ([(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[5,4,2,1],[2,2,1]]
=> ([(2,3),(2,4),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[4,4,1],[2]]
=> ([(0,4),(0,6),(1,2),(1,3),(3,6),(4,5),(6,5)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[3,3,2],[1]]
=> ([(0,3),(0,6),(1,2),(1,6),(2,4),(3,5),(6,4),(6,5)],7)
=> ([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[4,4,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5),(3,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 1
[[4,3,2],[2]]
=> ([(0,4),(0,6),(1,2),(1,3),(2,5),(3,5),(3,6)],7)
=> ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 1
[[5,4,2],[3,1]]
=> ([(0,5),(1,3),(1,6),(2,4),(2,5),(4,6)],7)
=> ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1
[[4,4,2,1],[2,1,1]]
=> ([(1,3),(1,5),(2,4),(2,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[5,5,2,1],[3,2,1]]
=> ([(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[5,4,2,1],[3,1,1]]
=> ([(1,4),(1,6),(2,3),(2,5),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[3,3,3],[1,1]]
=> ([(0,2),(0,3),(1,5),(2,6),(3,5),(3,6),(5,4),(6,4)],7)
=> ([(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[4,3,3],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[2,2,2,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1
[[3,3,2,1],[1,1]]
=> ([(0,4),(0,6),(1,2),(1,3),(2,5),(3,5),(3,6)],7)
=> ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 1
[[4,4,3,1],[2,2,1]]
=> ([(1,5),(2,3),(2,4),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[3,2,2,1],[1]]
=> ([(0,3),(0,6),(1,4),(1,6),(4,2),(4,5),(6,5)],7)
=> ([(0,4),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 1
[[4,3,3,1],[2,1,1]]
=> ([(1,3),(1,5),(2,4),(2,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[4,3,2,1],[2,1]]
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,5)],7)
=> ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[5,4,3,1],[3,2,1]]
=> ([(1,5),(2,5),(2,6),(3,4),(3,6)],7)
=> ([(0,1),(0,3),(0,5),(0,6),(1,2),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,3),(0,5),(0,6),(1,2),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[5,4,2,1],[3,2]]
=> ([(0,6),(1,5),(1,6),(2,3),(2,4)],7)
=> ([(0,2),(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[3,3,2,2],[1,1,1]]
=> ([(0,6),(1,3),(1,4),(2,6),(3,5),(4,2),(4,5)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[4,3,2,2],[2,1,1]]
=> ([(0,6),(1,3),(1,5),(2,4),(2,5),(4,6)],7)
=> ([(0,3),(0,4),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 1
[[5,4,2,2],[3,2,1]]
=> ([(0,5),(1,5),(2,6),(3,4),(3,6)],7)
=> ([(0,2),(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[4,4,3,2,1],[2,2,2,1]]
=> ([(2,3),(2,4),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[5,4,3,2,1],[3,2,2,1]]
=> ([(2,5),(2,6),(3,4),(3,6)],7)
=> ([(0,1),(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[6,5,3,2,1],[4,3,2,1]]
=> ([(3,6),(4,5),(4,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[4,3,3,1],[2,2]]
=> ([(0,3),(0,5),(1,2),(1,4),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[3,2,2,2],[1,1]]
=> ([(0,4),(0,6),(1,2),(1,3),(3,6),(4,5),(6,5)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[4,3,3,2],[2,2,1]]
=> ([(0,5),(1,5),(1,6),(2,3),(2,4),(4,6)],7)
=> ([(0,3),(0,4),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 1
[[5,4,3,2],[3,3,1]]
=> ([(0,6),(1,5),(1,6),(2,3),(2,4)],7)
=> ([(0,2),(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[4,3,3,2,1],[2,2,1,1]]
=> ([(1,4),(1,6),(2,3),(2,5),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[5,4,4,2,1],[3,3,2,1]]
=> ([(2,6),(3,4),(3,5),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[5,4,3,2,1],[3,3,1,1]]
=> ([(1,5),(1,6),(2,3),(2,4)],7)
=> ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[4,3,2,2,1],[2,2,1]]
=> ([(0,6),(1,5),(1,6),(2,3),(2,4)],7)
=> ([(0,2),(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[5,4,3,3,1],[3,3,2,1]]
=> ([(1,6),(2,6),(3,4),(3,5)],7)
=> ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[5,4,3,2,1],[3,3,2]]
=> ([(1,5),(1,6),(2,3),(2,4)],7)
=> ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[5,4,3,2,2],[3,3,2,1]]
=> ([(1,6),(2,6),(3,4),(3,5)],7)
=> ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[6,5,4,3,2,1],[4,4,3,2,1]]
=> ([(4,5),(4,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[[4,4,2],[3]]
=> ([(0,6),(1,3),(1,4),(2,6),(3,5),(4,2),(4,5)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[3,3,3],[2]]
=> ([(0,2),(0,3),(1,5),(2,6),(3,5),(3,6),(5,4),(6,4)],7)
=> ([(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[4,4,3],[3,1]]
=> ([(0,4),(1,5),(2,3),(2,4),(3,5),(3,6),(4,6)],7)
=> ([(0,4),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 1
[[5,4,3],[4,1]]
=> ([(1,5),(2,3),(2,5),(3,4),(3,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[[3,3,2,1],[2]]
=> ([(0,6),(1,3),(1,4),(3,5),(3,6),(4,2),(4,5)],7)
=> ([(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
Description
The number of connected components of the complement of a graph.
The complement of a graph is the graph on the same vertex set with complementary edges.
Matching statistic: St000914
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00185: Skew partitions —cell poset⟶ Posets
St000914: Posets ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 100%
St000914: Posets ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> ([],1)
=> ? = 1
[[2],[]]
=> ([(0,1)],2)
=> 1
[[1,1],[]]
=> ([(0,1)],2)
=> 1
[[2,1],[1]]
=> ([],2)
=> 2
[[3],[]]
=> ([(0,2),(2,1)],3)
=> 1
[[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 1
[[3,1],[1]]
=> ([(1,2)],3)
=> 2
[[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> 1
[[3,2],[2]]
=> ([(1,2)],3)
=> 2
[[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 1
[[2,2,1],[1,1]]
=> ([(1,2)],3)
=> 2
[[2,1,1],[1]]
=> ([(1,2)],3)
=> 2
[[3,2,1],[2,1]]
=> ([],3)
=> 3
[[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
[[4,1],[1]]
=> ([(1,2),(2,3)],4)
=> 2
[[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 1
[[4,2],[2]]
=> ([(0,3),(1,2)],4)
=> 2
[[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
[[3,2,1],[1,1]]
=> ([(1,2),(1,3)],4)
=> 2
[[3,1,1],[1]]
=> ([(0,3),(1,2)],4)
=> 2
[[4,2,1],[2,1]]
=> ([(2,3)],4)
=> 3
[[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[4,3],[3]]
=> ([(1,2),(2,3)],4)
=> 2
[[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 1
[[3,3,1],[2,1]]
=> ([(1,3),(2,3)],4)
=> 2
[[3,2,1],[2]]
=> ([(1,2),(1,3)],4)
=> 2
[[4,3,1],[3,1]]
=> ([(2,3)],4)
=> 3
[[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[3,3,2],[2,2]]
=> ([(0,3),(1,2)],4)
=> 2
[[3,2,2],[2,1]]
=> ([(1,3),(2,3)],4)
=> 2
[[4,3,2],[3,2]]
=> ([(2,3)],4)
=> 3
[[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,2,2,1],[1,1,1]]
=> ([(1,2),(2,3)],4)
=> 2
[[2,2,1,1],[1,1]]
=> ([(0,3),(1,2)],4)
=> 2
[[3,3,2,1],[2,2,1]]
=> ([(2,3)],4)
=> 3
[[2,1,1,1],[1]]
=> ([(1,2),(2,3)],4)
=> 2
[[3,2,2,1],[2,1,1]]
=> ([(2,3)],4)
=> 3
[[3,2,1,1],[2,1]]
=> ([(2,3)],4)
=> 3
[[4,3,2,1],[3,2,1]]
=> ([],4)
=> 4
[[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[[5,1],[1]]
=> ([(1,4),(3,2),(4,3)],5)
=> 2
[[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 1
[[5,2],[2]]
=> ([(0,3),(1,4),(4,2)],5)
=> 2
[[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 1
[[4,2,1],[1,1]]
=> ([(1,3),(1,4),(4,2)],5)
=> 2
[[4,1,1],[1]]
=> ([(0,3),(1,4),(4,2)],5)
=> 2
[[5,2,1],[2,1]]
=> ([(2,3),(3,4)],5)
=> 3
[[7],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 1
[[6,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> ? = 1
[[7,1],[1]]
=> ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 2
[[5,2],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> ? = 1
[[6,2],[1]]
=> ([(0,6),(1,5),(1,6),(3,4),(4,2),(5,3)],7)
=> ? = 1
[[7,2],[2]]
=> ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 2
[[5,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> ? = 1
[[6,2,1],[1,1]]
=> ([(1,3),(1,6),(4,5),(5,2),(6,4)],7)
=> ? = 2
[[6,1,1],[1]]
=> ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 2
[[7,2,1],[2,1]]
=> ([(2,6),(4,5),(5,3),(6,4)],7)
=> ? = 3
[[4,3],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ? = 1
[[5,3],[1]]
=> ([(0,6),(1,4),(1,6),(3,2),(4,3),(4,5),(6,5)],7)
=> ? = 1
[[6,3],[2]]
=> ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
=> ? = 1
[[7,3],[3]]
=> ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 2
[[4,2,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1
[[5,3,1],[1,1]]
=> ([(1,3),(1,5),(3,6),(4,2),(5,4),(5,6)],7)
=> ? = 2
[[5,2,1],[1]]
=> ([(0,5),(0,6),(1,3),(1,6),(4,2),(5,4)],7)
=> ? = 1
[[6,2,1],[2]]
=> ([(0,6),(1,3),(1,4),(5,2),(6,5)],7)
=> ? = 2
[[7,3,1],[3,1]]
=> ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 3
[[5,2,2],[1,1]]
=> ([(0,6),(1,3),(1,5),(3,6),(4,2),(5,4)],7)
=> ? = 1
[[6,3,2],[2,2]]
=> ([(0,4),(1,3),(1,6),(5,2),(6,5)],7)
=> ? = 2
[[7,3,2],[3,2]]
=> ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 3
[[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> ? = 1
[[5,2,2,1],[1,1,1]]
=> ([(1,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 2
[[5,2,1,1],[1,1]]
=> ([(0,4),(1,3),(1,6),(5,2),(6,5)],7)
=> ? = 2
[[6,3,2,1],[2,2,1]]
=> ([(2,4),(2,6),(5,3),(6,5)],7)
=> ? = 3
[[5,1,1,1],[1]]
=> ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 2
[[6,2,2,1],[2,1,1]]
=> ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 3
[[6,2,1,1],[2,1]]
=> ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 3
[[4,4],[1]]
=> ([(0,6),(1,3),(1,6),(2,4),(3,2),(3,5),(5,4),(6,5)],7)
=> ? = 1
[[5,4],[2]]
=> ([(0,3),(1,4),(1,6),(3,6),(4,2),(4,5),(6,5)],7)
=> ? = 1
[[6,4],[3]]
=> ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
=> ? = 1
[[7,4],[4]]
=> ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 2
[[3,3,1],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1
[[4,4,1],[1,1]]
=> ([(1,3),(1,4),(2,6),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 2
[[4,3,1],[1]]
=> ([(0,3),(0,6),(1,4),(1,6),(4,2),(4,5),(6,5)],7)
=> ? = 1
[[5,3,1],[2]]
=> ([(0,5),(0,6),(1,3),(1,4),(4,6),(5,2)],7)
=> ? = 1
[[6,4,1],[3,1]]
=> ([(1,4),(2,3),(2,6),(3,5),(4,6)],7)
=> ? = 2
[[6,3,1],[3]]
=> ([(0,6),(1,4),(1,5),(5,3),(6,2)],7)
=> ? = 2
[[7,4,1],[4,1]]
=> ([(1,6),(2,5),(5,3),(6,4)],7)
=> ? = 3
[[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1
[[4,3,2],[1,1]]
=> ([(0,6),(1,3),(1,4),(3,5),(3,6),(4,2),(4,5)],7)
=> ? = 1
[[5,4,2],[2,2]]
=> ([(0,4),(1,3),(1,5),(3,6),(5,2),(5,6)],7)
=> ? = 2
[[4,2,2],[1]]
=> ([(0,3),(0,6),(1,4),(1,6),(3,5),(4,2),(6,5)],7)
=> ? = 1
[[5,3,2],[2,1]]
=> ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7)
=> ? = 1
[[6,4,2],[3,2]]
=> ([(0,6),(1,3),(2,4),(2,6),(4,5)],7)
=> ? = 2
[[5,2,2],[2]]
=> ([(0,5),(1,3),(1,4),(3,6),(4,6),(5,2)],7)
=> ? = 2
[[6,3,2],[3,1]]
=> ([(0,6),(1,4),(2,3),(2,6),(4,5)],7)
=> ? = 2
[[7,4,2],[4,2]]
=> ([(0,5),(1,4),(2,6),(6,3)],7)
=> ? = 3
Description
The sum of the values of the Möbius function of a poset.
The Möbius function $\mu$ of a finite poset is defined as
$$\mu (x,y)=\begin{cases} 1& \text{if }x = y\\
-\sum _{z: x\leq z < y}\mu (x,z)& \text{for }x < y\\
0&\text{otherwise}.
\end{cases}
$$
Since $\mu(x,y)=0$ whenever $x\not\leq y$, this statistic is
$$
\sum_{x\leq y} \mu(x,y).
$$
If the poset has a minimal or a maximal element, then the definition implies immediately that the statistic equals $1$. Moreover, the statistic equals the sum of the statistics of the connected components.
This statistic is also called the magnitude of a poset.
Matching statistic: St000287
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
[[1],[]]
=> ([],1)
=> ([],1)
=> 1
[[2],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[2,1],[1]]
=> ([],2)
=> ([],2)
=> 2
[[3],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[3,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[3,2],[2]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[2,2,1],[1,1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[[2,1,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[[3,2,1],[2,1]]
=> ([],3)
=> ([],3)
=> 3
[[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[4,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
[[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[4,2],[2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
[[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[3,2,1],[1,1]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[3,1,1],[1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
[[4,2,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 3
[[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[4,3],[3]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[3,3,1],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[3,2,1],[2]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[4,3,1],[3,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 3
[[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[3,3,2],[2,2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
[[3,2,2],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[4,3,2],[3,2]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 3
[[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[2,2,2,1],[1,1,1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[2,2,1,1],[1,1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
[[3,3,2,1],[2,2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 3
[[2,1,1,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[3,2,2,1],[2,1,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 3
[[3,2,1,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 3
[[4,3,2,1],[3,2,1]]
=> ([],4)
=> ([],4)
=> 4
[[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[5,1],[1]]
=> ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[5,2],[2]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 2
[[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[4,2,1],[1,1]]
=> ([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[[4,1,1],[1]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 2
[[7],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 1
[[6,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 1
[[7,1],[1]]
=> ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ? = 2
[[5,2],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ? = 1
[[6,2],[1]]
=> ([(0,6),(1,5),(1,6),(3,4),(4,2),(5,3)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 1
[[7,2],[2]]
=> ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ? = 2
[[5,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 1
[[6,2,1],[1,1]]
=> ([(1,3),(1,6),(4,5),(5,2),(6,4)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ? = 2
[[6,1,1],[1]]
=> ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ? = 2
[[7,2,1],[2,1]]
=> ([(2,6),(4,5),(5,3),(6,4)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 3
[[4,3],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 1
[[5,3],[1]]
=> ([(0,6),(1,4),(1,6),(3,2),(4,3),(4,5),(6,5)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ? = 1
[[6,3],[2]]
=> ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 1
[[7,3],[3]]
=> ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ? = 2
[[4,2,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ? = 1
[[5,3,1],[1,1]]
=> ([(1,3),(1,5),(3,6),(4,2),(5,4),(5,6)],7)
=> ([(1,5),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> ? = 2
[[5,2,1],[1]]
=> ([(0,5),(0,6),(1,3),(1,6),(4,2),(5,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 1
[[6,3,1],[2,1]]
=> ([(1,6),(2,3),(2,6),(3,5),(5,4)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ? = 2
[[6,2,1],[2]]
=> ([(0,6),(1,3),(1,4),(5,2),(6,5)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ? = 2
[[7,3,1],[3,1]]
=> ([(1,6),(2,4),(5,3),(6,5)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ? = 3
[[5,2,2],[1,1]]
=> ([(0,6),(1,3),(1,5),(3,6),(4,2),(5,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 1
[[6,3,2],[2,2]]
=> ([(0,4),(1,3),(1,6),(5,2),(6,5)],7)
=> ([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ? = 2
[[6,2,2],[2,1]]
=> ([(0,3),(1,6),(2,6),(3,5),(5,4)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ? = 2
[[7,3,2],[3,2]]
=> ([(1,6),(2,4),(5,3),(6,5)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ? = 3
[[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 1
[[5,2,2,1],[1,1,1]]
=> ([(1,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ? = 2
[[5,2,1,1],[1,1]]
=> ([(0,4),(1,3),(1,6),(5,2),(6,5)],7)
=> ([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ? = 2
[[6,3,2,1],[2,2,1]]
=> ([(2,4),(2,6),(5,3),(6,5)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 3
[[5,1,1,1],[1]]
=> ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ? = 2
[[6,2,2,1],[2,1,1]]
=> ([(1,6),(2,4),(5,3),(6,5)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ? = 3
[[6,2,1,1],[2,1]]
=> ([(1,6),(2,4),(5,3),(6,5)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ? = 3
[[7,3,2,1],[3,2,1]]
=> ([(3,4),(4,6),(6,5)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 4
[[4,4],[1]]
=> ([(0,6),(1,3),(1,6),(2,4),(3,2),(3,5),(5,4),(6,5)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 1
[[5,4],[2]]
=> ([(0,3),(1,4),(1,6),(3,6),(4,2),(4,5),(6,5)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ? = 1
[[6,4],[3]]
=> ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 1
[[7,4],[4]]
=> ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ? = 2
[[3,3,1],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 1
[[4,4,1],[1,1]]
=> ([(1,3),(1,4),(2,6),(3,5),(4,2),(4,5),(5,6)],7)
=> ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ? = 2
[[4,3,1],[1]]
=> ([(0,3),(0,6),(1,4),(1,6),(4,2),(4,5),(6,5)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ? = 1
[[5,4,1],[2,1]]
=> ([(1,5),(2,3),(2,5),(3,4),(3,6),(5,6)],7)
=> ([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2
[[5,3,1],[2]]
=> ([(0,5),(0,6),(1,3),(1,4),(4,6),(5,2)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 1
[[6,4,1],[3,1]]
=> ([(1,4),(2,3),(2,6),(3,5),(4,6)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ? = 2
[[6,3,1],[3]]
=> ([(0,6),(1,4),(1,5),(5,3),(6,2)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ? = 2
[[7,4,1],[4,1]]
=> ([(1,6),(2,5),(5,3),(6,4)],7)
=> ([(1,6),(2,6),(3,5),(4,5)],7)
=> ? = 3
[[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 1
[[4,3,2],[1,1]]
=> ([(0,6),(1,3),(1,4),(3,5),(3,6),(4,2),(4,5)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ? = 1
[[5,4,2],[2,2]]
=> ([(0,4),(1,3),(1,5),(3,6),(5,2),(5,6)],7)
=> ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2
[[4,2,2],[1]]
=> ([(0,3),(0,6),(1,4),(1,6),(3,5),(4,2),(6,5)],7)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ? = 1
[[5,3,2],[2,1]]
=> ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 1
[[6,4,2],[3,2]]
=> ([(0,6),(1,3),(2,4),(2,6),(4,5)],7)
=> ([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ? = 2
Description
The number of connected components of a graph.
Matching statistic: St000553
Values
[[1],[]]
=> ([],1)
=> ([],1)
=> ([(0,1)],2)
=> 1
[[2],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,1],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[[2,1],[1]]
=> ([],2)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 2
[[3],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[3,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[3,2],[2]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[2,2,1],[1,1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[2,1,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[3,2,1],[2,1]]
=> ([],3)
=> ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[4,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
[[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[4,2],[2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[3,2,1],[1,1]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[3,1,1],[1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[[4,2,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[4,3],[3]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[3,3,1],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[3,2,1],[2]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[4,3,1],[3,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[3,3,2],[2,2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[[3,2,2],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[4,3,2],[3,2]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[2,2,2,1],[1,1,1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[2,2,1,1],[1,1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[[3,3,2,1],[2,2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[2,1,1,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[3,2,2,1],[2,1,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[3,2,1,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[4,3,2,1],[3,2,1]]
=> ([],4)
=> ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1
[[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1
[[5,1],[1]]
=> ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1
[[5,2],[2]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1
[[4,2,1],[1,1]]
=> ([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[4,1,1],[1]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[7],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
[[6,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
[[7,1],[1]]
=> ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ([(0,7),(1,6),(1,7),(2,5),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 2
[[5,2],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,7),(1,2),(1,3),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
[[6,2],[1]]
=> ([(0,6),(1,5),(1,6),(3,4),(4,2),(5,3)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
[[7,2],[2]]
=> ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,1),(0,7),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[[5,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
[[6,2,1],[1,1]]
=> ([(1,3),(1,6),(4,5),(5,2),(6,4)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ([(0,7),(1,6),(1,7),(2,5),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 2
[[6,1,1],[1]]
=> ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,1),(0,7),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[[7,2,1],[2,1]]
=> ([(2,6),(4,5),(5,3),(6,4)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 3
[[4,3],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,2),(1,4),(1,7),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[5,3],[1]]
=> ([(0,6),(1,4),(1,6),(3,2),(4,3),(4,5),(6,5)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 1
[[6,3],[2]]
=> ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
[[7,3],[3]]
=> ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,4),(2,7),(3,4),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[[4,2,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 1
[[5,3,1],[1,1]]
=> ([(1,3),(1,5),(3,6),(4,2),(5,4),(5,6)],7)
=> ([(1,5),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> ([(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[[5,2,1],[1]]
=> ([(0,5),(0,6),(1,3),(1,6),(4,2),(5,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
[[6,3,1],[2,1]]
=> ([(1,6),(2,3),(2,6),(3,5),(5,4)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ([(0,7),(1,6),(1,7),(2,5),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 2
[[6,2,1],[2]]
=> ([(0,6),(1,3),(1,4),(5,2),(6,5)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,4),(2,7),(3,4),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[[7,3,1],[3,1]]
=> ([(1,6),(2,4),(5,3),(6,5)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(0,7),(1,2),(1,7),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[[5,2,2],[1,1]]
=> ([(0,6),(1,3),(1,5),(3,6),(4,2),(5,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
[[6,3,2],[2,2]]
=> ([(0,4),(1,3),(1,6),(5,2),(6,5)],7)
=> ([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,1),(0,7),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[[6,2,2],[2,1]]
=> ([(0,3),(1,6),(2,6),(3,5),(5,4)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,4),(2,7),(3,4),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[[7,3,2],[3,2]]
=> ([(1,6),(2,4),(5,3),(6,5)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(0,7),(1,2),(1,7),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
[[5,2,2,1],[1,1,1]]
=> ([(1,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ([(0,7),(1,6),(1,7),(2,5),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 2
[[5,2,1,1],[1,1]]
=> ([(0,4),(1,3),(1,6),(5,2),(6,5)],7)
=> ([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,1),(0,7),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[[6,3,2,1],[2,2,1]]
=> ([(2,4),(2,6),(5,3),(6,5)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 3
[[5,1,1,1],[1]]
=> ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,4),(2,7),(3,4),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[[6,2,2,1],[2,1,1]]
=> ([(1,6),(2,4),(5,3),(6,5)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(0,7),(1,2),(1,7),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[[6,2,1,1],[2,1]]
=> ([(1,6),(2,4),(5,3),(6,5)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(0,7),(1,2),(1,7),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[[7,3,2,1],[3,2,1]]
=> ([(3,4),(4,6),(6,5)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[[4,4],[1]]
=> ([(0,6),(1,3),(1,6),(2,4),(3,2),(3,5),(5,4),(6,5)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,2),(1,4),(1,7),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[5,4],[2]]
=> ([(0,3),(1,4),(1,6),(3,6),(4,2),(4,5),(6,5)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 1
[[6,4],[3]]
=> ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
[[7,4],[4]]
=> ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,4),(2,7),(3,4),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[[3,3,1],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,2),(1,4),(1,7),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[4,4,1],[1,1]]
=> ([(1,3),(1,4),(2,6),(3,5),(4,2),(4,5),(5,6)],7)
=> ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,7),(1,4),(1,6),(1,7),(2,3),(2,6),(2,7),(3,5),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[[4,3,1],[1]]
=> ([(0,3),(0,6),(1,4),(1,6),(4,2),(4,5),(6,5)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 1
[[5,4,1],[2,1]]
=> ([(1,5),(2,3),(2,5),(3,4),(3,6),(5,6)],7)
=> ([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,7),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 2
[[5,3,1],[2]]
=> ([(0,5),(0,6),(1,3),(1,4),(4,6),(5,2)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
[[6,4,1],[3,1]]
=> ([(1,4),(2,3),(2,6),(3,5),(4,6)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ([(0,7),(1,6),(1,7),(2,5),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 2
[[6,3,1],[3]]
=> ([(0,6),(1,4),(1,5),(5,3),(6,2)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,4),(2,7),(3,4),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[[7,4,1],[4,1]]
=> ([(1,6),(2,5),(5,3),(6,4)],7)
=> ([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
[[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,2),(1,4),(1,7),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[4,3,2],[1,1]]
=> ([(0,6),(1,3),(1,4),(3,5),(3,6),(4,2),(4,5)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 1
[[5,4,2],[2,2]]
=> ([(0,4),(1,3),(1,5),(3,6),(5,2),(5,6)],7)
=> ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,7),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 2
[[4,2,2],[1]]
=> ([(0,3),(0,6),(1,4),(1,6),(3,5),(4,2),(6,5)],7)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,7),(1,2),(1,3),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
[[5,3,2],[2,1]]
=> ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
[[6,4,2],[3,2]]
=> ([(0,6),(1,3),(2,4),(2,6),(4,5)],7)
=> ([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,1),(0,7),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
Description
The number of blocks of a graph.
A cut vertex is a vertex whose deletion increases the number of connected components. A block is a maximal connected subgraph which itself has no cut vertices. Two distinct blocks cannot overlap in more than a single cut vertex.
Matching statistic: St000773
Values
[[1],[]]
=> ([],1)
=> ([],1)
=> ([(0,1)],2)
=> 1
[[2],[]]
=> ([(0,1)],2)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1],[]]
=> ([(0,1)],2)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 1
[[2,1],[1]]
=> ([],2)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[[3],[]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[3,1],[1]]
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[3,2],[2]]
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[[2,2,1],[1,1]]
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[2,1,1],[1]]
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[3,2,1],[2,1]]
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[4,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[4,2],[2]]
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[3,2,1],[1,1]]
=> ([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[3,1,1],[1]]
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[[4,2,1],[2,1]]
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[4,3],[3]]
=> ([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[3,3,1],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[3,2,1],[2]]
=> ([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[4,3,1],[3,1]]
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[3,3,2],[2,2]]
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[[3,2,2],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[4,3,2],[3,2]]
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[2,2,2,1],[1,1,1]]
=> ([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[2,2,1,1],[1,1]]
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[[3,3,2,1],[2,2,1]]
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[2,1,1,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[3,2,2,1],[2,1,1]]
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[3,2,1,1],[2,1]]
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[4,3,2,1],[3,2,1]]
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[5,1],[1]]
=> ([(1,4),(3,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[5,2],[2]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 2
[[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1
[[4,2,1],[1,1]]
=> ([(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[4,1,1],[1]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 2
[[7],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[[6,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[7,1],[1]]
=> ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[[5,2],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
[[6,2],[1]]
=> ([(0,6),(1,5),(1,6),(3,4),(4,2),(5,3)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
[[7,2],[2]]
=> ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 2
[[5,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
[[6,2,1],[1,1]]
=> ([(1,3),(1,6),(4,5),(5,2),(6,4)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[[6,1,1],[1]]
=> ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 2
[[7,2,1],[2,1]]
=> ([(2,6),(4,5),(5,3),(6,4)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[[4,3],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,5),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[5,3],[1]]
=> ([(0,6),(1,4),(1,6),(3,2),(4,3),(4,5),(6,5)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,5),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[6,3],[2]]
=> ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
[[7,3],[3]]
=> ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 2
[[4,2,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
[[5,3,1],[1,1]]
=> ([(1,3),(1,5),(3,6),(4,2),(5,4),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[[5,2,1],[1]]
=> ([(0,5),(0,6),(1,3),(1,6),(4,2),(5,4)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6)],7)
=> ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
[[6,3,1],[2,1]]
=> ([(1,6),(2,3),(2,6),(3,5),(5,4)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[[6,2,1],[2]]
=> ([(0,6),(1,3),(1,4),(5,2),(6,5)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[[7,3,1],[3,1]]
=> ([(1,6),(2,4),(5,3),(6,5)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[[5,2,2],[1,1]]
=> ([(0,6),(1,3),(1,5),(3,6),(4,2),(5,4)],7)
=> ([(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),(0,7),(1,2),(1,3),(1,4),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[6,3,2],[2,2]]
=> ([(0,4),(1,3),(1,6),(5,2),(6,5)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 2
[[6,2,2],[2,1]]
=> ([(0,3),(1,6),(2,6),(3,5),(5,4)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[[7,3,2],[3,2]]
=> ([(1,6),(2,4),(5,3),(6,5)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[[5,2,2,1],[1,1,1]]
=> ([(1,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(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),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[[5,2,1,1],[1,1]]
=> ([(0,4),(1,3),(1,6),(5,2),(6,5)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 2
[[6,3,2,1],[2,2,1]]
=> ([(2,4),(2,6),(5,3),(6,5)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[[5,1,1,1],[1]]
=> ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 2
[[6,2,2,1],[2,1,1]]
=> ([(1,6),(2,4),(5,3),(6,5)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[[6,2,1,1],[2,1]]
=> ([(1,6),(2,4),(5,3),(6,5)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[[7,3,2,1],[3,2,1]]
=> ([(3,4),(4,6),(6,5)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[[4,4],[1]]
=> ([(0,6),(1,3),(1,6),(2,4),(3,2),(3,5),(5,4),(6,5)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,5),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[5,4],[2]]
=> ([(0,3),(1,4),(1,6),(3,6),(4,2),(4,5),(6,5)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,5),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[6,4],[3]]
=> ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
[[7,4],[4]]
=> ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 2
[[3,3,1],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ([(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,3),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[4,4,1],[1,1]]
=> ([(1,3),(1,4),(2,6),(3,5),(4,2),(4,5),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[[4,3,1],[1]]
=> ([(0,3),(0,6),(1,4),(1,6),(4,2),(4,5),(6,5)],7)
=> ([(0,4),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[5,4,1],[2,1]]
=> ([(1,5),(2,3),(2,5),(3,4),(3,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[[5,3,1],[2]]
=> ([(0,5),(0,6),(1,3),(1,4),(4,6),(5,2)],7)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,5),(0,6),(0,7),(1,3),(1,4),(1,7),(2,3),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[6,4,1],[3,1]]
=> ([(1,4),(2,3),(2,6),(3,5),(4,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[[6,3,1],[3]]
=> ([(0,6),(1,4),(1,5),(5,3),(6,2)],7)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[[7,4,1],[4,1]]
=> ([(1,6),(2,5),(5,3),(6,4)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ([(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,3),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[4,3,2],[1,1]]
=> ([(0,6),(1,3),(1,4),(3,5),(3,6),(4,2),(4,5)],7)
=> ([(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,2),(1,4),(1,6),(1,7),(2,3),(2,5),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[5,4,2],[2,2]]
=> ([(0,4),(1,3),(1,5),(3,6),(5,2),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 2
[[4,2,2],[1]]
=> ([(0,3),(0,6),(1,4),(1,6),(3,5),(4,2),(6,5)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(0,7),(1,3),(1,4),(1,7),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
[[5,3,2],[2,1]]
=> ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[[6,4,2],[3,2]]
=> ([(0,6),(1,3),(2,4),(2,6),(4,5)],7)
=> ([(0,4),(0,5),(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)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
Description
The multiplicity of the largest Laplacian eigenvalue in a graph.
Matching statistic: St001490
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
St001490: Skew partitions ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 71%
Values
[[1],[]]
=> 1
[[2],[]]
=> 1
[[1,1],[]]
=> 1
[[2,1],[1]]
=> 2
[[3],[]]
=> 1
[[2,1],[]]
=> 1
[[3,1],[1]]
=> 2
[[2,2],[1]]
=> 1
[[3,2],[2]]
=> 2
[[1,1,1],[]]
=> 1
[[2,2,1],[1,1]]
=> 2
[[2,1,1],[1]]
=> 2
[[3,2,1],[2,1]]
=> 3
[[4],[]]
=> 1
[[3,1],[]]
=> 1
[[4,1],[1]]
=> 2
[[2,2],[]]
=> 1
[[3,2],[1]]
=> 1
[[4,2],[2]]
=> 2
[[2,1,1],[]]
=> 1
[[3,2,1],[1,1]]
=> 2
[[3,1,1],[1]]
=> 2
[[4,2,1],[2,1]]
=> 3
[[3,3],[2]]
=> 1
[[4,3],[3]]
=> 2
[[2,2,1],[1]]
=> 1
[[3,3,1],[2,1]]
=> 2
[[3,2,1],[2]]
=> 2
[[4,3,1],[3,1]]
=> 3
[[2,2,2],[1,1]]
=> 1
[[3,3,2],[2,2]]
=> 2
[[3,2,2],[2,1]]
=> 2
[[4,3,2],[3,2]]
=> 3
[[1,1,1,1],[]]
=> 1
[[2,2,2,1],[1,1,1]]
=> 2
[[2,2,1,1],[1,1]]
=> 2
[[3,3,2,1],[2,2,1]]
=> 3
[[2,1,1,1],[1]]
=> 2
[[3,2,2,1],[2,1,1]]
=> 3
[[3,2,1,1],[2,1]]
=> 3
[[4,3,2,1],[3,2,1]]
=> 4
[[5],[]]
=> 1
[[4,1],[]]
=> 1
[[5,1],[1]]
=> 2
[[3,2],[]]
=> 1
[[4,2],[1]]
=> 1
[[5,2],[2]]
=> 2
[[3,1,1],[]]
=> 1
[[4,2,1],[1,1]]
=> 2
[[4,1,1],[1]]
=> 2
[[6],[]]
=> ? = 1
[[5,1],[]]
=> ? = 1
[[6,1],[1]]
=> ? = 2
[[4,2],[]]
=> ? = 1
[[5,2],[1]]
=> ? = 1
[[6,2],[2]]
=> ? = 2
[[4,1,1],[]]
=> ? = 1
[[5,2,1],[1,1]]
=> ? = 2
[[5,1,1],[1]]
=> ? = 2
[[6,2,1],[2,1]]
=> ? = 3
[[3,3],[]]
=> ? = 1
[[4,3],[1]]
=> ? = 1
[[5,3],[2]]
=> ? = 1
[[6,3],[3]]
=> ? = 2
[[3,2,1],[]]
=> ? = 1
[[4,3,1],[1,1]]
=> ? = 2
[[4,2,1],[1]]
=> ? = 1
[[5,3,1],[2,1]]
=> ? = 2
[[5,2,1],[2]]
=> ? = 2
[[6,3,1],[3,1]]
=> ? = 3
[[4,2,2],[1,1]]
=> ? = 1
[[5,3,2],[2,2]]
=> ? = 2
[[5,2,2],[2,1]]
=> ? = 2
[[6,3,2],[3,2]]
=> ? = 3
[[3,1,1,1],[]]
=> ? = 1
[[4,2,2,1],[1,1,1]]
=> ? = 2
[[4,2,1,1],[1,1]]
=> ? = 2
[[5,3,2,1],[2,2,1]]
=> ? = 3
[[4,1,1,1],[1]]
=> ? = 2
[[5,2,2,1],[2,1,1]]
=> ? = 3
[[5,2,1,1],[2,1]]
=> ? = 3
[[6,3,2,1],[3,2,1]]
=> ? = 4
[[4,4],[2]]
=> ? = 1
[[5,4],[3]]
=> ? = 1
[[6,4],[4]]
=> ? = 2
[[3,3,1],[1]]
=> ? = 1
[[4,4,1],[2,1]]
=> ? = 2
[[4,3,1],[2]]
=> ? = 1
[[5,4,1],[3,1]]
=> ? = 2
[[5,3,1],[3]]
=> ? = 2
[[6,4,1],[4,1]]
=> ? = 3
[[2,2,2],[]]
=> ? = 1
[[3,3,2],[1,1]]
=> ? = 1
[[4,4,2],[2,2]]
=> ? = 2
[[3,2,2],[1]]
=> ? = 1
[[4,3,2],[2,1]]
=> ? = 1
[[5,4,2],[3,2]]
=> ? = 2
[[4,2,2],[2]]
=> ? = 2
[[5,3,2],[3,1]]
=> ? = 2
[[6,4,2],[4,2]]
=> ? = 3
Description
The number of connected components of a skew partition.
The following 14 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000181The number of connected components of the Hasse diagram for the poset. St001570The minimal number of edges to add to make a graph Hamiltonian. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001545The second Elser number of a connected graph. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices.
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