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Your data matches 47 different statistics following compositions of up to 3 maps.
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Matching statistic: St000208
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1] => [1,1]
=> [1]
=> 1
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0]
=> [1,2] => [2,1]
=> [1]
=> 1
[1,1,0,0,1,0]
=> [2,1] => [2,1]
=> [1]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3] => [3,1]
=> [1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> [1]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> [2]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1] => [3,1]
=> [1]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> [1]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> [2,1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [4,1]
=> [1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [4,1]
=> [1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [4,1]
=> [1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [4,1]
=> [1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> [1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> [2,1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> [2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [3,2]
=> [2]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> [2]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> [2]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [3,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [3,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [2,2,1,1]
=> [2,1,1]
=> 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [3,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [4,1,1]
=> [1,1]
=> 1
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has only integer lattice points as vertices.
See also [[St000205]], [[St000206]] and [[St000207]].
Matching statistic: St000455
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 8%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 8%
Values
[1,0,1,0]
=> [1,1] => [2] => ([],2)
=> ? = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1] => [3] => ([],3)
=> ? = 1 - 1
[1,0,1,1,0,0]
=> [1,2] => [1,2] => ([(1,2)],3)
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => ([],4)
=> ? = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,3] => ([(2,3)],4)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[1,1,0,1,0,0,1,0]
=> [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => ([],5)
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,4] => ([(3,4)],5)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [6] => ([],6)
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [1,5] => ([(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [2,4] => ([(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,4] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,2] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,5] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,5] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,5] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,5] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,5] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,1,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,2,1,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,3,1] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,4] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,4] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,1] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,4] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [3,1,2] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,2] => [1,2,1,1,1] => ([(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)
=> ? = 2 - 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,2] => [1,2,1,1,1] => ([(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)
=> ? = 2 - 1
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,2] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,2] => [1,2,1,1,1] => ([(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)
=> ? = 2 - 1
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [4,2] => [1,2,1,1,1] => ([(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)
=> ? = 2 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,2] => [1,2,1,1,1] => ([(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)
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1] => [7] => ([],7)
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,2,2] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,2,1,2] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,2,2,1] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St001060
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 8%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 8%
Values
[1,0,1,0]
=> [1,2] => ([],2)
=> ([],1)
=> ? = 1 + 1
[1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> ([],1)
=> ? = 1 + 1
[1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> ([],1)
=> ? = 1 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 2 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 2 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ? = 2 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,5,2,4,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,6,2,4,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,6,3] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,4,5,2,6,3] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,3,5,1,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,4,1,5,3,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,4,5,1,3,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,1,3,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [3,1,4,2,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [3,1,4,5,2,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [3,1,5,2,4,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [3,4,1,5,2,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,5,1,2,4,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,1,2,5,3,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,1,5,2,3,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,4,6] => ([(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,4,7,5] => ([(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,3,6] => ([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,3,5,7] => ([(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,3,7,5] => ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,6,7,3,5] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,3,5,6] => ([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,3,6,4,7] => ([(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,3,6,7,4] => ([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,3,7,4,6] => ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,3,7,4] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,3,4,6] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,3,4,7,5] => ([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,3,7,4,5] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,4,5,7,2,6] => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,3,4,6,2,5,7] => ([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,4,6,2,7,5] => ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,4,6,7,2,5] => ([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,4,7,2,5,6] => ([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,3,5,2,4,6,7] => ([(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St001487
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 8%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 8%
Values
[1,0,1,0]
=> [1,1,0,0]
=> []
=> [[],[]]
=> ? = 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> [[],[]]
=> ? = 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> [[1],[]]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> [[2,1],[]]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> [[],[]]
=> ? = 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [[1],[]]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [[2,1],[]]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [[2],[]]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [[3,1],[]]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1],[]]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[2,1],[]]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[2],[]]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[3,1],[]]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[3,2],[]]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[3],[]]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> ? = 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[4,2,1,1],[]]
=> ? = 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[4,3,2,1],[]]
=> ? = 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[4,2,2,1],[]]
=> ? = 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[4,3,1,1],[]]
=> ? = 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ? = 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[3,3,2,1],[]]
=> ? = 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[4,2,1],[]]
=> ? = 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[4,3,2],[]]
=> ? = 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[3,3,2],[]]
=> ? = 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[4,1],[]]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> [[1],[]]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [[2,1],[]]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> [[2],[]]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [[3,1],[]]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [[3,2],[]]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> [[3],[]]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1]
=> [[4,2,1,1],[]]
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [[4,3,2,1],[]]
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [[4,2,2,1],[]]
=> ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1]
=> [[4,3,1,1],[]]
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> [[3,3,2,1],[]]
=> ? = 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1]
=> [[3,2,2,1],[]]
=> ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [[3,3,1,1],[]]
=> ? = 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,2,1]
=> [[4,2,1],[]]
=> ? = 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> [[4,3,2],[]]
=> ? = 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [[4,2,2],[]]
=> ? = 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,3,1]
=> [[4,3,1],[]]
=> ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [[3,3,2],[]]
=> ? = 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [[3,2,2],[]]
=> ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,3,1]
=> [[3,3,1],[]]
=> ? = 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [[4,1],[]]
=> 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,2]
=> [[4,2],[]]
=> ? = 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> [[4,3],[]]
=> ? = 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [[3,3],[]]
=> ? = 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> [[4],[]]
=> 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [[5,1,1,1,1],[]]
=> ? = 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[5,2,1,1,1],[]]
=> ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[5,3,2,1,1],[]]
=> ? = 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[5,2,2,1,1],[]]
=> ? = 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[5,3,1,1,1],[]]
=> ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[5,4,2,2,1],[]]
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1]
=> [[1],[]]
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,1]
=> [[2,1],[]]
=> 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2]
=> [[2],[]]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [3,1]
=> [[3,1],[]]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [3,2]
=> [[3,2],[]]
=> 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1
Description
The number of inner corners of a skew partition.
Matching statistic: St001490
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 8%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 8%
Values
[1,0,1,0]
=> [1,1,0,0]
=> []
=> [[],[]]
=> ? = 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> [[],[]]
=> ? = 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> [[1],[]]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> [[2,1],[]]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> [[],[]]
=> ? = 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [[1],[]]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [[2,1],[]]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [[2],[]]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [[3,1],[]]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1],[]]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[2,1],[]]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[2],[]]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[3,1],[]]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[3,2],[]]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[3],[]]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> ? = 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[4,2,1,1],[]]
=> ? = 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[4,3,2,1],[]]
=> ? = 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[4,2,2,1],[]]
=> ? = 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[4,3,1,1],[]]
=> ? = 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ? = 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[3,3,2,1],[]]
=> ? = 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[4,2,1],[]]
=> ? = 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[4,3,2],[]]
=> ? = 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[3,3,2],[]]
=> ? = 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[4,1],[]]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> [[1],[]]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [[2,1],[]]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> [[2],[]]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [[3,1],[]]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [[3,2],[]]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> [[3],[]]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1]
=> [[4,2,1,1],[]]
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [[4,3,2,1],[]]
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [[4,2,2,1],[]]
=> ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1]
=> [[4,3,1,1],[]]
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> [[3,3,2,1],[]]
=> ? = 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1]
=> [[3,2,2,1],[]]
=> ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [[3,3,1,1],[]]
=> ? = 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,2,1]
=> [[4,2,1],[]]
=> ? = 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> [[4,3,2],[]]
=> ? = 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [[4,2,2],[]]
=> ? = 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,3,1]
=> [[4,3,1],[]]
=> ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [[3,3,2],[]]
=> ? = 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [[3,2,2],[]]
=> ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,3,1]
=> [[3,3,1],[]]
=> ? = 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [[4,1],[]]
=> 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,2]
=> [[4,2],[]]
=> ? = 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> [[4,3],[]]
=> ? = 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [[3,3],[]]
=> ? = 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> [[4],[]]
=> 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [[5,1,1,1,1],[]]
=> ? = 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[5,2,1,1,1],[]]
=> ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[5,3,2,1,1],[]]
=> ? = 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[5,2,2,1,1],[]]
=> ? = 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[5,3,1,1,1],[]]
=> ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[5,4,2,2,1],[]]
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1]
=> [[1],[]]
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,1]
=> [[2,1],[]]
=> 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2]
=> [[2],[]]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [3,1]
=> [[3,1],[]]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [3,2]
=> [[3,2],[]]
=> 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1
Description
The number of connected components of a skew partition.
Matching statistic: St001435
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 8%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 8%
Values
[1,0,1,0]
=> [1,1,0,0]
=> []
=> [[],[]]
=> ? = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> [[],[]]
=> ? = 1 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> [[1],[]]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> [[2,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> [[],[]]
=> ? = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [[1],[]]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [[2,1],[]]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [[2],[]]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 2 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [[3,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[2,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[2],[]]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[3,1],[]]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[3,2],[]]
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[3],[]]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[4,2,1,1],[]]
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[4,3,2,1],[]]
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[4,2,2,1],[]]
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[4,3,1,1],[]]
=> ? = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ? = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[3,3,2,1],[]]
=> ? = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[4,2,1],[]]
=> ? = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[4,3,2],[]]
=> ? = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[3,3,2],[]]
=> ? = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[4,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> [[1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [[2,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> [[2],[]]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [[3,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [[3,2],[]]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> [[3],[]]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1]
=> [[4,2,1,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [[4,3,2,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [[4,2,2,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1]
=> [[4,3,1,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 1 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> [[3,3,2,1],[]]
=> ? = 1 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1]
=> [[3,2,2,1],[]]
=> ? = 1 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 1 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [[3,3,1,1],[]]
=> ? = 1 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 1 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,2,1]
=> [[4,2,1],[]]
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> [[4,3,2],[]]
=> ? = 1 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [[4,2,2],[]]
=> ? = 1 - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,3,1]
=> [[4,3,1],[]]
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [[3,3,2],[]]
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [[3,2,2],[]]
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 1 - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,3,1]
=> [[3,3,1],[]]
=> ? = 1 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [[4,1],[]]
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,2]
=> [[4,2],[]]
=> ? = 1 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> [[4,3],[]]
=> ? = 1 - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [[3,3],[]]
=> ? = 1 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> [[4],[]]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [[5,1,1,1,1],[]]
=> ? = 1 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[5,2,1,1,1],[]]
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[5,3,2,1,1],[]]
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[5,2,2,1,1],[]]
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[5,3,1,1,1],[]]
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[5,4,2,2,1],[]]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1]
=> [[1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,1]
=> [[2,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2]
=> [[2],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [3,1]
=> [[3,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [3,2]
=> [[3,2],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 0 = 1 - 1
Description
The number of missing boxes in the first row.
Matching statistic: St001438
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 8%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 8%
Values
[1,0,1,0]
=> [1,1,0,0]
=> []
=> [[],[]]
=> ? = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> [[],[]]
=> ? = 1 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> [[1],[]]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> [[2,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> [[],[]]
=> ? = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [[1],[]]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [[2,1],[]]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [[2],[]]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 2 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [[3,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[2,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[2],[]]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[3,1],[]]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[3,2],[]]
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[3],[]]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[4,2,1,1],[]]
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[4,3,2,1],[]]
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[4,2,2,1],[]]
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[4,3,1,1],[]]
=> ? = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ? = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[3,3,2,1],[]]
=> ? = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[4,2,1],[]]
=> ? = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[4,3,2],[]]
=> ? = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[3,3,2],[]]
=> ? = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[4,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> [[1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [[2,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> [[2],[]]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [[3,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [[3,2],[]]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> [[3],[]]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1]
=> [[4,2,1,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [[4,3,2,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [[4,2,2,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1]
=> [[4,3,1,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 1 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> [[3,3,2,1],[]]
=> ? = 1 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1]
=> [[3,2,2,1],[]]
=> ? = 1 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 1 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [[3,3,1,1],[]]
=> ? = 1 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 1 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,2,1]
=> [[4,2,1],[]]
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> [[4,3,2],[]]
=> ? = 1 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> [[4,2,2],[]]
=> ? = 1 - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,3,1]
=> [[4,3,1],[]]
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [[3,3,2],[]]
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [[3,2,2],[]]
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 1 - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,3,1]
=> [[3,3,1],[]]
=> ? = 1 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [[4,1],[]]
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,2]
=> [[4,2],[]]
=> ? = 1 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> [[4,3],[]]
=> ? = 1 - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [[3,3],[]]
=> ? = 1 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> [[4],[]]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [[5,1,1,1,1],[]]
=> ? = 1 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[5,2,1,1,1],[]]
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[5,3,2,1,1],[]]
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[5,2,2,1,1],[]]
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[5,3,1,1,1],[]]
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[5,4,2,2,1],[]]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1]
=> [[1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,1]
=> [[2,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2]
=> [[2],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [3,1]
=> [[3,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [3,2]
=> [[3,2],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 0 = 1 - 1
Description
The number of missing boxes of a skew partition.
Matching statistic: St001271
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00047: Ordered trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001271: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 17%
Mp00047: Ordered trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001271: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 17%
Values
[1,0,1,0]
=> [[],[]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [[[],[]],[],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [[[],[],[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[]]
=> ([(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)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[],[[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(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 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[],[],[],[[]],[]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(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 = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[],[],[],[[],[]]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(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 = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[],[],[],[[[]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(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 = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[],[],[[]],[],[]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(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 = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[],[],[[]],[[]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(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)
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[],[],[[],[]],[]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(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 = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[],[],[[],[],[]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
=> ([(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 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[],[],[[],[[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7)
=> ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[],[],[[[]]],[]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(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 = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[],[],[[[]],[]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7)
=> ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[],[[]],[],[[]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(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)
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[],[[]],[[]],[]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(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)
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[],[[]],[[],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[],[[]],[[[]]]]
=> ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[],[[],[]],[[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [[],[[[]]],[[]]]
=> ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 1 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [[[]],[],[],[[]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(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)
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [[[]],[],[[]],[]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(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)
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [[[]],[],[[],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [[[]],[],[[[]]]]
=> ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [[[]],[[]],[],[]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(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)
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [[[]],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [[[]],[[],[]],[]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [[[]],[[],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(6,5)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [[[]],[[],[[]]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [[[]],[[[]]],[]]
=> ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [[[]],[[[]],[]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [[[]],[[[],[]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
=> ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [[[]],[[[[]]]]]
=> ([(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)
=> ? = 2 - 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [[[],[]],[],[[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [[[],[]],[[]],[]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [[[],[]],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,6),(5,6)],7)
=> ([(1,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [[[],[]],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [[[],[],[]],[[]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(6,5)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 - 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [[[],[[]]],[[]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 - 1
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [[[[]]],[],[[]]]
=> ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [[[[]]],[[]],[]]
=> ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [[[[]]],[[],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [[[[]]],[[[]]]]
=> ([(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)
=> ? = 3 - 1
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [[[[]],[]],[[]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 - 1
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [[[[]],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 1 - 1
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [[[[],[]]],[[]]]
=> ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
=> ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [[[[[]]]],[[]]]
=> ([(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)
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [[],[],[],[],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[],[],[],[],[[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[],[],[],[],[[[]]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [[],[],[],[[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [[],[],[],[[]],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [[],[],[],[[],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [[],[],[],[[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [[],[],[],[[],[[]]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [[],[],[],[[[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [[],[],[],[[[]],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [[],[],[],[[[],[]]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ? = 1 - 1
Description
The competition number of a graph.
The competition graph of a digraph $D$ is a (simple undirected) graph which has the same vertex set as $D$ and has an edge between $x$ and $y$ if and only if there exists a vertex $v$ in $D$ such that $(x, v)$ and $(y, v)$ are arcs of $D$. For any graph, $G$ together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number $k(G)$ is the smallest number of such isolated vertices.
Matching statistic: St000451
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%
Values
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,3,2,4] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,3,1,4] => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,4,3,2,5] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,4,3,1,5] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [2,1,4,3,5] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,2,4,3,5] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [3,4,2,1,5] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,1,4,2,5] => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [3,2,4,1,5] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [2,3,4,1,5] => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,6] => [5,4,3,2,1,6] => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [4,5,3,2,1,6] => [1,5,4,3,2,6] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [5,3,4,2,1,6] => [2,5,4,3,1,6] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,1,6] => [2,1,5,4,3,6] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,1,6] => [1,2,5,4,3,6] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [5,4,2,3,1,6] => [3,5,4,2,1,6] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [4,5,2,3,1,6] => [3,1,5,4,2,6] => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [5,3,2,4,1,6] => [3,2,5,4,1,6] => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,1,6] => [3,2,1,5,4,6] => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,1,6] => [1,3,2,5,4,6] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,1,6] => [2,3,5,4,1,6] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,1,6] => [2,3,1,5,4,6] => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,1,6] => [2,1,3,5,4,6] => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,1,6] => [1,2,3,5,4,6] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [5,4,3,1,2,6] => [4,5,3,2,1,6] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [4,5,3,1,2,6] => [4,1,5,3,2,6] => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [5,3,4,1,2,6] => [4,2,5,3,1,6] => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [4,3,5,1,2,6] => [4,2,1,5,3,6] => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,4,5,1,2,6] => [1,4,2,5,3,6] => 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [5,4,2,1,3,6] => [4,3,5,2,1,6] => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,5,2,1,3,6] => [4,3,1,5,2,6] => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1,4,6] => [4,3,2,5,1,6] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,3,1,4,6] => [2,4,3,5,1,6] => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [5,4,1,2,3,6] => [3,4,5,2,1,6] => 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,5,1,2,3,6] => [3,4,1,5,2,6] => 3 = 2 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [5,3,1,2,4,6] => [3,4,2,5,1,6] => 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,2,1,3,4,6] => [3,2,4,5,1,6] => 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,4,6] => [2,3,4,5,1,6] => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,2,1,7] => [1,6,5,4,3,2,7] => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,2,1,7] => [2,6,5,4,3,1,7] => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,2,1,7] => [2,1,6,5,4,3,7] => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,2,1,7] => [1,2,6,5,4,3,7] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,2,1,7] => [3,6,5,4,2,1,7] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,2,1,7] => [3,1,6,5,4,2,7] => ? = 2 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,2,1,7] => [3,2,6,5,4,1,7] => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,2,1,7] => [3,2,1,6,5,4,7] => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,2,1,7] => [1,3,2,6,5,4,7] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,2,1,7] => [2,3,6,5,4,1,7] => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,2,1,7] => [2,3,1,6,5,4,7] => ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,2,1,7] => [2,1,3,6,5,4,7] => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,1,7] => [1,2,3,6,5,4,7] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,3,1,7] => [4,6,5,3,2,1,7] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [5,6,4,2,3,1,7] => [4,1,6,5,3,2,7] => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [6,4,5,2,3,1,7] => [4,2,6,5,3,1,7] => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [5,4,6,2,3,1,7] => [4,2,1,6,5,3,7] => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [4,5,6,2,3,1,7] => [1,4,2,6,5,3,7] => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [6,5,3,2,4,1,7] => [4,3,6,5,2,1,7] => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [5,6,3,2,4,1,7] => [4,3,1,6,5,2,7] => ? = 2 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [6,4,3,2,5,1,7] => [4,3,2,6,5,1,7] => ? = 1 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,6,1,7] => [4,3,2,1,6,5,7] => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,1,7] => [2,4,3,6,5,1,7] => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [5,3,4,2,6,1,7] => [2,4,3,1,6,5,7] => ? = 1 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [4,3,5,2,6,1,7] => [2,1,4,3,6,5,7] => ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [6,5,2,3,4,1,7] => [3,4,6,5,2,1,7] => ? = 1 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [5,6,2,3,4,1,7] => [3,4,1,6,5,2,7] => ? = 2 + 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [6,4,2,3,5,1,7] => [3,4,2,6,5,1,7] => ? = 1 + 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [5,4,2,3,6,1,7] => [3,4,2,1,6,5,7] => ? = 1 + 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [4,5,2,3,6,1,7] => [3,1,4,2,6,5,7] => ? = 1 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [6,3,2,4,5,1,7] => [3,2,4,6,5,1,7] => ? = 1 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [5,3,2,4,6,1,7] => [3,2,4,1,6,5,7] => ? = 1 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,5,6,1,7] => [3,2,1,4,6,5,7] => ? = 1 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [6,2,3,4,5,1,7] => [2,3,4,6,5,1,7] => ? = 1 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [5,2,3,4,6,1,7] => [2,3,4,1,6,5,7] => ? = 1 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,1,7] => [2,3,1,4,6,5,7] => ? = 1 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,1,7] => [2,1,3,4,6,5,7] => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,1,2,7] => [5,1,6,4,3,2,7] => ? = 2 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,1,2,7] => [5,2,6,4,3,1,7] => ? = 2 + 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,1,2,7] => [5,2,1,6,4,3,7] => ? = 2 + 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,1,2,7] => [1,5,2,6,4,3,7] => ? = 2 + 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,1,2,7] => [5,3,6,4,2,1,7] => ? = 2 + 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,1,2,7] => [5,3,1,6,4,2,7] => ? = 3 + 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,1,2,7] => [5,3,2,6,4,1,7] => ? = 2 + 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,1,2,7] => [5,3,2,1,6,4,7] => ? = 2 + 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,1,2,7] => [1,5,3,2,6,4,7] => ? = 2 + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,1,2,7] => [2,5,3,6,4,1,7] => ? = 2 + 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,1,2,7] => [2,5,3,1,6,4,7] => ? = 2 + 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,1,2,7] => [2,1,5,3,6,4,7] => ? = 2 + 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [5,6,4,2,1,3,7] => [5,4,1,6,3,2,7] => ? = 2 + 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [6,4,5,2,1,3,7] => [5,4,2,6,3,1,7] => ? = 2 + 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [5,4,6,2,1,3,7] => [5,4,2,1,6,3,7] => ? = 3 + 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [4,5,6,2,1,3,7] => [1,5,4,2,6,3,7] => ? = 3 + 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [5,6,3,2,1,4,7] => [5,4,3,1,6,2,7] => ? = 2 + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,1,5,7] => [2,5,4,3,6,1,7] => ? = 1 + 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [6,5,2,3,1,4,7] => [3,5,4,6,2,1,7] => ? = 1 + 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [5,6,2,3,1,4,7] => [3,5,4,1,6,2,7] => ? = 2 + 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [6,4,2,3,1,5,7] => [3,5,4,2,6,1,7] => ? = 1 + 1
Description
The length of the longest pattern of the form k 1 2...(k-1).
Matching statistic: St001545
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001545: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 8%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001545: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 8%
Values
[1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,0,0,1,0]
=> [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,6] => ([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ? = 2 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,5,6,3] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,5,3,6] => ([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ? = 2 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ? = 2 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,6,3,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ? = 2 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,4,1,5,3,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? = 2 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 1 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ? = 1 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 1 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,3,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ? = 1 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ? = 2 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [3,5,1,2,6,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [4,5,1,2,6,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,7,1,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,4,6,1,7,5] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,4,6,7,1,5] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,7,1,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,3,5,1,6,7,4] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,3,5,6,1,7,4] => ([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,1,4] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,6,1,7,4,5] => ([(0,6),(1,6),(2,3),(2,4),(3,5),(4,5),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,6,7,1,4,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,7,1,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,4,1,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,4,5,1,6,7,3] => ([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,4,5,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,7,1,3] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [2,5,1,6,7,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [2,5,6,1,7,3,4] => ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [2,5,6,7,1,3,4] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [2,6,1,7,3,4,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
Description
The second Elser number of a connected graph.
For a connected graph $G$ the $k$-th Elser number is
$$
els_k(G) = (-1)^{|V(G)|+1} \sum_N (-1)^{|E(N)|} |V(N)|^k
$$
where the sum is over all nuclei of $G$, that is, the connected subgraphs of $G$ whose vertex set is a vertex cover of $G$.
It is clear that this number is even. It was shown in [1] that it is non-negative.
The following 37 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. St001330The hat guessing number of a graph. St000360The number of occurrences of the pattern 32-1. St000527The width of the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000632The jump number of the poset. St000640The rank of the largest boolean interval in a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000314The number of left-to-right-maxima of a permutation. St000570The Edelman-Greene number of a permutation. St001737The number of descents of type 2 in a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000906The length of the shortest maximal chain in a poset. St000068The number of minimal elements in a poset. St001868The number of alignments of type NE of a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001890The maximum magnitude of the Möbius function of a poset. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001846The number of elements which do not have a complement in the lattice. St001513The number of nested exceedences of a permutation. St001866The nesting alignments of a signed permutation. St000031The number of cycles in the cycle decomposition of a permutation. St001533The largest coefficient of the Poincare polynomial of the poset cone.
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