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Your data matches 40 different statistics following compositions of up to 3 maps.
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Mp00037: Graphs to partition of connected componentsInteger partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 1 = 2 - 1
([],2)
=> [1,1]
=> 2 = 3 - 1
([(0,1)],2)
=> [2]
=> 0 = 1 - 1
([],3)
=> [1,1,1]
=> 3 = 4 - 1
([(1,2)],3)
=> [2,1]
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> [3]
=> 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0 = 1 - 1
([],4)
=> [1,1,1,1]
=> 4 = 5 - 1
([(2,3)],4)
=> [2,1,1]
=> 2 = 3 - 1
([(1,3),(2,3)],4)
=> [3,1]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [4]
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> [4]
=> 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0 = 1 - 1
([],5)
=> [1,1,1,1,1]
=> 5 = 6 - 1
([(3,4)],5)
=> [2,1,1,1]
=> 3 = 4 - 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 0 = 1 - 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> 1 = 2 - 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> 0 = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0 = 1 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> 0 = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 0 = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0 = 1 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> 0 = 1 - 1
Description
The number of parts equal to 1 in a partition.
Matching statistic: St000160
Mp00259: Graphs vertex additionGraphs
Mp00037: Graphs to partition of connected componentsInteger partitions
St000160: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],2)
=> [1,1]
=> 2
([],2)
=> ([],3)
=> [1,1,1]
=> 3
([(0,1)],2)
=> ([(1,2)],3)
=> [2,1]
=> 1
([],3)
=> ([],4)
=> [1,1,1,1]
=> 4
([(1,2)],3)
=> ([(2,3)],4)
=> [2,1,1]
=> 2
([(0,2),(1,2)],3)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1
([],4)
=> ([],5)
=> [1,1,1,1,1]
=> 5
([(2,3)],4)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 3
([(1,3),(2,3)],4)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 1
([(0,3),(1,2)],4)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 1
([(1,2),(1,3),(2,3)],4)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
([],5)
=> ([],6)
=> [1,1,1,1,1,1]
=> 6
([(3,4)],5)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> 4
([(2,4),(3,4)],5)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> 3
([(1,4),(2,4),(3,4)],5)
=> ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1
([(1,4),(2,3)],5)
=> ([(2,5),(3,4)],6)
=> [2,2,1,1]
=> 2
([(1,4),(2,3),(3,4)],5)
=> ([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> 1
([(2,3),(2,4),(3,4)],5)
=> ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1
Description
The multiplicity of the smallest part of a partition. This counts the number of occurrences of the smallest part spt(λ) of a partition λ. The sum spt(n)=λnspt(λ) satisfies the congruences \begin{align*} spt(5n+4) &\equiv 0\quad \pmod{5}\\\ spt(7n+5) &\equiv 0\quad \pmod{7}\\\ spt(13n+6) &\equiv 0\quad \pmod{13}, \end{align*} analogous to those of the counting function of partitions, see [1] and [2].
Mp00037: Graphs to partition of connected componentsInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00284: Standard tableaux rowsSet partitions
St000247: Set partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [[1]]
=> {{1}}
=> ? = 2 - 1
([],2)
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 2 = 3 - 1
([(0,1)],2)
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0 = 1 - 1
([],3)
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 3 = 4 - 1
([(1,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0 = 1 - 1
([],4)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 4 = 5 - 1
([(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 2 = 3 - 1
([(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 0 = 1 - 1
([],5)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 5 = 6 - 1
([(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 3 = 4 - 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0 = 1 - 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 1 = 2 - 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0 = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0 = 1 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0 = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0 = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0 = 1 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0 = 1 - 1
Description
The number of singleton blocks of a set partition.
Mp00037: Graphs to partition of connected componentsInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00227: Dyck paths Delest-Viennot-inverseDyck paths
St000674: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1,0]
=> [1,0]
=> ? = 2 - 1
([],2)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 3 - 1
([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 4 - 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
Description
The number of hills of a Dyck path. A hill is a peak with up step starting and down step ending at height zero.
Mp00147: Graphs squareGraphs
Mp00117: Graphs Ore closureGraphs
St000986: Graphs ⟶ ℤResult quality: 99% values known / values provided: 99%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> ([],1)
=> 1 = 2 - 1
([],2)
=> ([],2)
=> ([],2)
=> 2 = 3 - 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
([],3)
=> ([],3)
=> ([],3)
=> 3 = 4 - 1
([(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([],4)
=> ([],4)
=> ([],4)
=> 4 = 5 - 1
([(2,3)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 2 = 3 - 1
([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],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)
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],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)
=> 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],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)
=> 0 = 1 - 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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
([],5)
=> ([],5)
=> ([],5)
=> 5 = 6 - 1
([(3,4)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> 3 = 4 - 1
([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 1 = 2 - 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
Description
The multiplicity of the eigenvalue zero of the adjacency matrix of the graph.
Mp00147: Graphs squareGraphs
Mp00117: Graphs Ore closureGraphs
St001691: Graphs ⟶ ℤResult quality: 99% values known / values provided: 99%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> ([],1)
=> 1 = 2 - 1
([],2)
=> ([],2)
=> ([],2)
=> 2 = 3 - 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
([],3)
=> ([],3)
=> ([],3)
=> 3 = 4 - 1
([(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([],4)
=> ([],4)
=> ([],4)
=> 4 = 5 - 1
([(2,3)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 2 = 3 - 1
([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],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)
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],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)
=> 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],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)
=> 0 = 1 - 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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
([],5)
=> ([],5)
=> ([],5)
=> 5 = 6 - 1
([(3,4)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> 3 = 4 - 1
([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 1 = 2 - 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
Description
The number of kings in a graph. A vertex of a graph is a king, if all its neighbours have smaller degree. In particular, an isolated vertex is a king.
Matching statistic: St000241
Mp00037: Graphs to partition of connected componentsInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00119: Dyck paths to 321-avoiding permutation (Krattenthaler)Permutations
St000241: Permutations ⟶ ℤResult quality: 68% values known / values provided: 68%distinct values known / distinct values provided: 88%
Values
([],1)
=> [1]
=> [1,0]
=> [1] => 1 = 2 - 1
([],2)
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 2 = 3 - 1
([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> [1,2] => 0 = 1 - 1
([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 3 = 4 - 1
([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1 = 2 - 1
([(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4 = 5 - 1
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2 = 3 - 1
([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = 1 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = 1 - 1
([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 5 = 6 - 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 3 = 4 - 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 1 - 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1 = 2 - 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 1 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 0 = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 1 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1 = 2 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 1 - 1
([],7)
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => ? = 8 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? = 3 - 1
([(3,6),(4,5),(5,6)],7)
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? = 4 - 1
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ? = 2 - 1
([(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? = 4 - 1
([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ? = 2 - 1
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 1 - 1
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ? = 2 - 1
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ? = 2 - 1
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 1 - 1
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ? = 2 - 1
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 1 - 1
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ? = 2 - 1
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 1 - 1
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? = 3 - 1
([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 1 - 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? = 4 - 1
([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ? = 2 - 1
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 1 - 1
([(0,5),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 1 - 1
([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ? = 2 - 1
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 1 - 1
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6)],7)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 1 - 1
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 1 - 1
([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ? = 2 - 1
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 1 - 1
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 1 - 1
([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 1 - 1
([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ? = 2 - 1
([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 1 - 1
([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 1 - 1
([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ? = 2 - 1
([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 1 - 1
([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 1 - 1
([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ? = 2 - 1
([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 1 - 1
([(0,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 1 - 1
([(0,4),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 1 - 1
([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 1 - 1
([(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)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ? = 2 - 1
([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 1 - 1
([(0,5),(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 1 - 1
([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 1 - 1
([(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)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 1 - 1
([(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)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 1 - 1
([(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)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 1 - 1
([(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)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 1 - 1
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 1 - 1
([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 1 - 1
([(1,5),(1,6),(2,3),(2,4),(3,4),(3,6),(4,5),(5,6)],7)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ? = 2 - 1
Description
The number of cyclical small excedances. A cyclical small excedance is an index i such that \pi_i = i+1 considered cyclically.
Matching statistic: St000312
Mp00259: Graphs vertex additionGraphs
Mp00203: Graphs coneGraphs
St000312: Graphs ⟶ ℤResult quality: 66% values known / values provided: 66%distinct values known / distinct values provided: 88%
Values
([],1)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 2
([],2)
=> ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
([(0,1)],2)
=> ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([],3)
=> ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(1,2)],3)
=> ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,2),(1,2)],3)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([],4)
=> ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
([(2,3)],4)
=> ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(1,3),(2,3)],4)
=> ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,3),(1,2)],4)
=> ([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(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
([(1,2),(1,3),(2,3)],4)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(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
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([],5)
=> ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6
([(3,4)],5)
=> ([(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
([(2,4),(3,4)],5)
=> ([(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(1,4),(2,4),(3,4)],5)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(1,4),(2,3)],5)
=> ([(2,5),(3,4)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> 2
([(1,4),(2,3),(3,4)],5)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(1,2),(3,5),(4,5)],6)
=> ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(2,3),(2,4),(3,4)],5)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([],7)
=> ([],8)
=> ?
=> ? = 8
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 3
([(3,6),(4,5)],7)
=> ([(4,7),(5,6)],8)
=> ?
=> ? = 4
([(3,6),(4,5),(5,6)],7)
=> ([(4,7),(5,6),(6,7)],8)
=> ?
=> ? = 4
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 4
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(2,7),(3,7),(4,6),(5,6)],8)
=> ?
=> ? = 2
([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 2
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 2
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 1
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ?
=> ? = 3
([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(2,7),(3,6),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 4
([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
=> ([(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,7)],8)
=> ?
=> ? = 1
([(0,5),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,6),(2,3),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 1
([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6)],7)
=> ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 1
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,6),(2,7),(3,6),(3,7),(4,5),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 1
([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 2
([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 1
([(0,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> ([(1,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(6,7)],8)
=> ?
=> ? = 1
([(0,4),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> ([(1,5),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(6,7)],8)
=> ?
=> ? = 1
([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(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
([(0,5),(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(1,7),(2,5),(2,6),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(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)
=> ?
=> ? = 1
([(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)
=> ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 1
([(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)
=> ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(6,7)],8)
=> ?
=> ? = 1
([(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)
=> ([(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
([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(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)
=> ?
=> ? = 1
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(1,7),(2,5),(2,7),(3,4),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1
Description
The number of leaves in a graph. That is, the number of vertices of a graph that have degree 1.
St000315: Graphs ⟶ ℤResult quality: 66% values known / values provided: 66%distinct values known / distinct values provided: 88%
Values
([],1)
=> 1 = 2 - 1
([],2)
=> 2 = 3 - 1
([(0,1)],2)
=> 0 = 1 - 1
([],3)
=> 3 = 4 - 1
([(1,2)],3)
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([],4)
=> 4 = 5 - 1
([(2,3)],4)
=> 2 = 3 - 1
([(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
([],5)
=> 5 = 6 - 1
([(3,4)],5)
=> 3 = 4 - 1
([(2,4),(3,4)],5)
=> 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(1,4),(2,3)],5)
=> 1 = 2 - 1
([(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> 0 = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 0 = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
([],7)
=> ? = 8 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
([(3,6),(4,5)],7)
=> ? = 4 - 1
([(3,6),(4,5),(5,6)],7)
=> ? = 4 - 1
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
([(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 4 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ? = 2 - 1
([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 - 1
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 - 1
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 1 - 1
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 1 - 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 1
([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
=> ? = 1 - 1
([(0,5),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 1 - 1
([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6)],7)
=> ? = 1 - 1
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 1 - 1
([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 2 - 1
([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 1 - 1
([(0,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> ? = 1 - 1
([(0,4),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> ? = 1 - 1
([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 1 - 1
([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
([(0,5),(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 1 - 1
([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 1 - 1
([(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)
=> ? = 1 - 1
([(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)
=> ? = 1 - 1
([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1 - 1
Description
The number of isolated vertices of a graph.
Mp00203: Graphs coneGraphs
St001672: Graphs ⟶ ℤResult quality: 63% values known / values provided: 63%distinct values known / distinct values provided: 88%
Values
([],1)
=> ([(0,1)],2)
=> 2
([],2)
=> ([(0,2),(1,2)],3)
=> 3
([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 4
([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5
([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(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)
=> 1
([(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
([(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)
=> 1
([(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)
=> 1
([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 6
([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
([(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
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(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)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(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)
=> 1
([(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)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,4),(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
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,6),(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(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
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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
([],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 8
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(3,6),(4,5)],7)
=> ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 4
([(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
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 4
([(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)
=> ? = 2
([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(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
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],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),(5,6),(5,7),(6,7)],8)
=> ? = 2
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(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)
=> ? = 2
([(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
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],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),(5,6),(5,7),(6,7)],8)
=> ? = 2
([(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)
=> ? = 1
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(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)
=> ? = 3
([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(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)
=> ? = 4
([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(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)
=> ? = 2
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
=> ([(0,4),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
([(0,5),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,7),(1,2),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(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)
=> ? = 2
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(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),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(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)
=> ? = 1
([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,5),(1,6),(1,7),(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)
=> ? = 2
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,5),(1,6),(1,7),(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
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(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
([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(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)
=> ? = 2
([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(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
([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(0,7),(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)
=> ? = 1
([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(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)
=> ? = 2
([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(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)
=> ? = 1
([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(0,7),(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)
=> ? = 1
([(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)
=> ? = 2
([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
([(0,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> ([(0,4),(0,7),(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)
=> ? = 1
([(0,4),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> ([(0,4),(0,6),(0,7),(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)
=> ? = 1
Description
The restrained domination number of a graph. This is the minimal size of a set of vertices D such that every vertex not in D is adjacent to a vertex in D and to a vertex not in D.
The following 30 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001826The maximal number of leaves on a vertex of a graph. St001342The number of vertices in the center of a graph. St001368The number of vertices of maximal degree in a graph. St001479The number of bridges of a graph. St001657The number of twos in an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000699The toughness times the least common multiple of 1,. St000456The monochromatic index of a connected graph. St001545The second Elser number of a connected graph. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000264The girth of a graph, which is not a tree. St000618The number of self-evacuating tableaux of given shape. St001432The order dimension of the partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition.