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Your data matches 5 different statistics following compositions of up to 3 maps.
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Matching statistic: St000171
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00181: Skew partitions —row lengths⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000171: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00181: Skew partitions —row lengths⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000171: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[1],[]]
=> [1] => ([],1)
=> 0
[1,0,1,0]
=> [[1,1],[]]
=> [1,1] => ([(0,1)],2)
=> 1
[1,1,0,0]
=> [[2],[]]
=> [2] => ([],2)
=> 0
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,0,1,1,0,0]
=> [[2,1],[]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1,2] => ([(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [[3],[]]
=> [3] => ([],3)
=> 0
[1,1,1,0,0,0]
=> [[2,2],[]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [1,3] => ([(2,3)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> [4] => ([],4)
=> 0
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> [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)
=> 5
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [1,4] => ([(3,4)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> [5] => ([],5)
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 3
Description
The degree of the graph.
This is the maximal vertex degree of a graph.
Matching statistic: St000987
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00181: Skew partitions —row lengths⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000987: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00181: Skew partitions —row lengths⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000987: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[1],[]]
=> [1] => ([],1)
=> 0
[1,0,1,0]
=> [[1,1],[]]
=> [1,1] => ([(0,1)],2)
=> 1
[1,1,0,0]
=> [[2],[]]
=> [2] => ([],2)
=> 0
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,0,1,1,0,0]
=> [[2,1],[]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1,2] => ([(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [[3],[]]
=> [3] => ([],3)
=> 0
[1,1,1,0,0,0]
=> [[2,2],[]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [1,3] => ([(2,3)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> [4] => ([],4)
=> 0
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> [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)
=> 5
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [1,4] => ([(3,4)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> [5] => ([],5)
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 3
Description
The number of positive eigenvalues of the Laplacian matrix of the graph.
This is the number of vertices minus the number of connected components of the graph.
Matching statistic: St001725
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00181: Skew partitions —row lengths⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001725: Graphs ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Mp00181: Skew partitions —row lengths⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001725: Graphs ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[1],[]]
=> [1] => ([],1)
=> 1 = 0 + 1
[1,0,1,0]
=> [[1,1],[]]
=> [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,0,0]
=> [[2],[]]
=> [2] => ([],2)
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,0,1,1,0,0]
=> [[2,1],[]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [[3],[]]
=> [3] => ([],3)
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [[2,2],[]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> [4] => ([],4)
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> [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)
=> 6 = 5 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> [5] => ([],5)
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 4 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1]]
=> [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1],[1]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1,1]]
=> [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 + 1
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,2],[1,1,1]]
=> [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 + 1
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [[3,2,2,2],[1,1]]
=> [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1,1],[]]
=> [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1,1],[]]
=> [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1,1],[1]]
=> [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1,1],[]]
=> [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1,1],[1,1]]
=> [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1,1],[1]]
=> [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1,1],[2]]
=> [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1,1],[]]
=> [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1,1],[1,1]]
=> [2,1,2,1,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1,1],[2,1]]
=> [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1,1],[1]]
=> [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1,1],[2]]
=> [2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1,1],[3]]
=> [1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [[2,2,2,2,2,1],[1,1,1,1]]
=> [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [[3,2,2,2,1],[1,1,1]]
=> [2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [[3,3,2,2,1],[2,1,1]]
=> [1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [[4,2,2,1],[1,1]]
=> [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2,1],[2,2,1]]
=> [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [[4,3,2,1],[2,1]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [[4,4,2,1],[3,1]]
=> [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [[3,3,3,3,1],[2,2,2]]
=> [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [[4,3,3,1],[2,2]]
=> [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[2,2,2,2,2,2],[1,1,1,1,1]]
=> [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 + 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [[3,2,2,2,2],[1,1,1,1]]
=> [2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 + 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [[3,3,2,2,2],[2,1,1,1]]
=> [1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 + 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [[4,2,2,2],[1,1,1]]
=> [3,1,1,2] => ([(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)
=> ? = 5 + 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2,2],[2,2,1,1]]
=> [1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 + 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [[4,3,2,2],[2,1,1]]
=> [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 + 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [[4,4,2,2],[3,1,1]]
=> [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 + 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [[3,3,3,3,2],[2,2,2,1]]
=> [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 + 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [[4,3,3,2],[2,2,1]]
=> [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 + 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[3,3,3,3,3],[2,2,2,2]]
=> [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [[4,3,3,3],[2,2,2]]
=> [2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
Description
The harmonious chromatic number of a graph.
A harmonious colouring is a proper vertex colouring such that any pair of colours appears at most once on adjacent vertices.
Matching statistic: St000029
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000029: Permutations ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000029: Permutations ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,2] => [2,1] => 1
[1,1,0,0]
=> [1,0,1,0]
=> [2,1] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [2,3,1] => 2
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => [3,2,1] => 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => [1,3,2] => 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => [1,2,3] => 0
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [2,3,4,1] => 3
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [2,4,3,1] => 3
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [3,2,4,1] => 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [4,2,3,1] => 3
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,4,2,1] => 4
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [1,3,4,2] => 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [1,4,3,2] => 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [1,2,4,3] => 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [1,2,3,4] => 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [1,4,2,3] => 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,4,2] => 3
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => 3
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [3,4,1,2] => 4
[1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,1,2,3] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [2,3,4,5,1] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [2,3,5,4,1] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [2,4,3,5,1] => 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [2,5,3,4,1] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [2,4,5,3,1] => 5
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [3,2,4,5,1] => 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [3,2,5,4,1] => 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [4,2,3,5,1] => 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [5,2,3,4,1] => 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [4,2,5,3,1] => 5
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [3,4,2,5,1] => 5
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [3,5,2,4,1] => 5
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [3,4,5,2,1] => 6
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [4,5,2,3,1] => 6
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [1,3,4,5,2] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [1,3,5,4,2] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [1,4,3,5,2] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [1,5,3,4,2] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [1,4,5,3,2] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [1,2,4,5,3] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [1,2,5,4,3] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [1,2,3,5,4] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [1,2,5,3,4] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [1,4,2,5,3] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [1,5,2,4,3] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [1,4,5,2,3] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [1,5,2,3,4] => 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 6
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 6
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 6
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,2,3,4,6,7,5] => [2,3,4,7,5,6,1] => ? = 6
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,2,3,5,4,6,7] => [2,3,5,4,6,7,1] => ? = 6
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,2,3,5,4,7,6] => [2,3,5,4,7,6,1] => ? = 6
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,2,3,5,6,4,7] => [2,3,6,4,5,7,1] => ? = 6
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,2,3,5,6,7,4] => [2,3,7,4,5,6,1] => ? = 6
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,2,4,3,5,6,7] => [2,4,3,5,6,7,1] => ? = 6
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,2,4,3,5,7,6] => [2,4,3,5,7,6,1] => ? = 6
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5,7] => [2,4,3,6,5,7,1] => ? = 6
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,6,7,5] => [2,4,3,7,5,6,1] => ? = 6
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,2,4,5,3,7,6] => [2,5,3,4,7,6,1] => ? = 6
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,2,4,5,6,3,7] => [2,6,3,4,5,7,1] => ? = 6
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,2,4,5,6,7,3] => [2,7,3,4,5,6,1] => ? = 6
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,3,2,4,5,6,7] => [3,2,4,5,6,7,1] => ? = 6
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,3,2,4,5,7,6] => [3,2,4,5,7,6,1] => ? = 6
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2,4,6,5,7] => [3,2,4,6,5,7,1] => ? = 6
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,2,4,6,7,5] => [3,2,4,7,5,6,1] => ? = 6
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,5,4,6,7] => [3,2,5,4,6,7,1] => ? = 6
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,7,6] => [3,2,5,4,7,6,1] => ? = 6
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,5,6,4,7] => [3,2,6,4,5,7,1] => ? = 6
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,5,6,7,4] => [3,2,7,4,5,6,1] => ? = 6
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,3,4,2,5,6,7] => [4,2,3,5,6,7,1] => ? = 6
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,3,4,2,5,7,6] => [4,2,3,5,7,6,1] => ? = 6
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,3,4,2,6,5,7] => [4,2,3,6,5,7,1] => ? = 6
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,3,4,2,6,7,5] => [4,2,3,7,5,6,1] => ? = 6
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,3,4,5,2,6,7] => [5,2,3,4,6,7,1] => ? = 6
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,3,4,5,2,7,6] => [5,2,3,4,7,6,1] => ? = 6
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,6,2,7] => [6,2,3,4,5,7,1] => ? = 6
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,7,2] => [7,2,3,4,5,6,1] => ? = 6
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => [1,3,4,5,6,7,2] => ? = 5
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,3,4,5,7,6] => [1,3,4,5,7,6,2] => ? = 5
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => [1,3,4,6,5,7,2] => ? = 5
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => [1,3,4,7,5,6,2] => ? = 5
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => [1,3,5,4,6,7,2] => ? = 5
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => [1,3,5,4,7,6,2] => ? = 5
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,1,3,5,6,4,7] => [1,3,6,4,5,7,2] => ? = 5
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,1,3,5,6,7,4] => [1,3,7,4,5,6,2] => ? = 5
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => [1,4,3,5,6,7,2] => ? = 5
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,5,7,6] => [1,4,3,5,7,6,2] => ? = 5
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,7] => [1,4,3,6,5,7,2] => ? = 5
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5] => [1,4,3,7,5,6,2] => ? = 5
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,4,5,3,6,7] => [1,5,3,4,6,7,2] => ? = 5
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => [1,5,3,4,7,6,2] => ? = 5
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,4,5,6,3,7] => [1,6,3,4,5,7,2] => ? = 5
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => [1,7,3,4,5,6,2] => ? = 5
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6,7] => [1,2,4,5,6,7,3] => ? = 4
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,3,1,4,5,7,6] => [1,2,4,5,7,6,3] => ? = 4
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,3,1,4,6,5,7] => [1,2,4,6,5,7,3] => ? = 4
Description
The depth of a permutation.
This is given by
$$\operatorname{dp}(\sigma) = \sum_{\sigma_i>i} (\sigma_i-i) = |\{ i \leq j : \sigma_i > j\}|.$$
The depth is half of the total displacement [4], Problem 5.1.1.28, or Spearman’s disarray [3] $\sum_i |\sigma_i-i|$.
Permutations with depth at most $1$ are called ''almost-increasing'' in [5].
Matching statistic: St000030
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000030: Permutations ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000030: Permutations ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [.,.]
=> [1] => [1] => 0
[1,0,1,0]
=> [.,[.,.]]
=> [2,1] => [2,1] => 1
[1,1,0,0]
=> [[.,.],.]
=> [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => 2
[1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> [2,3,1] => [3,1,2] => 2
[1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 0
[1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> [3,1,2] => [2,3,1] => 2
[1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => 3
[1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,3,1,2] => 3
[1,0,1,1,0,0,1,0]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,2,1,3] => 3
[1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,1,2,3] => 3
[1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [4,2,3,1] => 4
[1,1,0,0,1,0,1,0]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => 2
[1,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1,2,4] => 2
[1,1,0,1,0,0,1,0]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,1,0,1,1,0,0,0]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => [2,3,1,4] => 2
[1,1,1,0,0,0,1,0]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,4,2,1] => 3
[1,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,4,1,2] => 3
[1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,2,4,1] => 4
[1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => [2,3,4,1] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,4,3,1,2] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,4,2,1,3] => 4
[1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,4,1,2,3] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [5,4,2,3,1] => 5
[1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,3,2,1,4] => 4
[1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,3,1,2,4] => 4
[1,0,1,1,0,1,0,0,1,0]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,2,1,3,4] => 4
[1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,1,2,3,4] => 4
[1,0,1,1,0,1,1,0,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [5,2,3,1,4] => 5
[1,0,1,1,1,0,0,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [5,3,4,2,1] => 5
[1,0,1,1,1,0,0,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [5,3,4,1,2] => 5
[1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [5,3,2,4,1] => 6
[1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [5,2,3,4,1] => 6
[1,1,0,0,1,0,1,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1,5] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [4,3,1,2,5] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [4,2,1,3,5] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [4,1,2,3,5] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [4,2,3,1,5] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [3,2,1,4,5] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [3,1,2,4,5] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => [2,3,1,4,5] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => [3,4,2,1,5] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => [3,4,1,2,5] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => [3,2,4,1,5] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => [2,3,4,1,5] => 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 6
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [6,7,5,4,3,2,1] => [7,6,5,4,3,1,2] => ? = 6
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [6,5,7,4,3,2,1] => [7,6,5,4,2,1,3] => ? = 6
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [5,6,7,4,3,2,1] => [7,6,5,4,1,2,3] => ? = 6
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [6,5,4,7,3,2,1] => [7,6,5,3,2,1,4] => ? = 6
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [5,6,4,7,3,2,1] => [7,6,5,3,1,2,4] => ? = 6
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [5,4,6,7,3,2,1] => [7,6,5,2,1,3,4] => ? = 6
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[.,[[[[.,.],.],.],.]]]]
=> [4,5,6,7,3,2,1] => [7,6,5,1,2,3,4] => ? = 6
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [6,5,4,3,7,2,1] => [7,6,4,3,2,1,5] => ? = 6
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [5,6,4,3,7,2,1] => [7,6,4,3,1,2,5] => ? = 6
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [5,4,6,3,7,2,1] => [7,6,4,2,1,3,5] => ? = 6
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [.,[.,[[.,[[[.,.],.],.]],.]]]
=> [4,5,6,3,7,2,1] => [7,6,4,1,2,3,5] => ? = 6
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [.,[.,[[[.,[[.,.],.]],.],.]]]
=> [4,5,3,6,7,2,1] => [7,6,3,1,2,4,5] => ? = 6
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [.,[.,[[[[.,[.,.]],.],.],.]]]
=> [4,3,5,6,7,2,1] => [7,6,2,1,3,4,5] => ? = 6
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[.,[[[[[.,.],.],.],.],.]]]
=> [3,4,5,6,7,2,1] => [7,6,1,2,3,4,5] => ? = 6
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [6,5,4,3,2,7,1] => [7,5,4,3,2,1,6] => ? = 6
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [.,[[.,[.,[.,[[.,.],.]]]],.]]
=> [5,6,4,3,2,7,1] => [7,5,4,3,1,2,6] => ? = 6
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [.,[[.,[.,[[.,[.,.]],.]]],.]]
=> [5,4,6,3,2,7,1] => [7,5,4,2,1,3,6] => ? = 6
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [.,[[.,[.,[[[.,.],.],.]]],.]]
=> [4,5,6,3,2,7,1] => [7,5,4,1,2,3,6] => ? = 6
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [.,[[.,[[.,[.,[.,.]]],.]],.]]
=> [5,4,3,6,2,7,1] => [7,5,3,2,1,4,6] => ? = 6
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [.,[[.,[[.,[[.,.],.]],.]],.]]
=> [4,5,3,6,2,7,1] => [7,5,3,1,2,4,6] => ? = 6
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [.,[[.,[[[.,[.,.]],.],.]],.]]
=> [4,3,5,6,2,7,1] => [7,5,2,1,3,4,6] => ? = 6
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [.,[[.,[[[[.,.],.],.],.]],.]]
=> [3,4,5,6,2,7,1] => [7,5,1,2,3,4,6] => ? = 6
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [.,[[[.,[.,[.,[.,.]]]],.],.]]
=> [5,4,3,2,6,7,1] => [7,4,3,2,1,5,6] => ? = 6
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [.,[[[.,[.,[[.,.],.]]],.],.]]
=> [4,5,3,2,6,7,1] => [7,4,3,1,2,5,6] => ? = 6
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [.,[[[.,[[.,[.,.]],.]],.],.]]
=> [4,3,5,2,6,7,1] => [7,4,2,1,3,5,6] => ? = 6
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [.,[[[.,[[[.,.],.],.]],.],.]]
=> [3,4,5,2,6,7,1] => [7,4,1,2,3,5,6] => ? = 6
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [.,[[[[.,[.,[.,.]]],.],.],.]]
=> [4,3,2,5,6,7,1] => [7,3,2,1,4,5,6] => ? = 6
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [.,[[[[.,[[.,.],.]],.],.],.]]
=> [3,4,2,5,6,7,1] => [7,3,1,2,4,5,6] => ? = 6
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [.,[[[[[.,[.,.]],.],.],.],.]]
=> [3,2,4,5,6,7,1] => [7,2,1,3,4,5,6] => ? = 6
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => [7,1,2,3,4,5,6] => ? = 6
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => ? = 5
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [[.,[.,[.,[.,[[.,.],.]]]]],.]
=> [5,6,4,3,2,1,7] => [6,5,4,3,1,2,7] => ? = 5
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [[.,[.,[.,[[.,[.,.]],.]]]],.]
=> [5,4,6,3,2,1,7] => [6,5,4,2,1,3,7] => ? = 5
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [[.,[.,[.,[[[.,.],.],.]]]],.]
=> [4,5,6,3,2,1,7] => [6,5,4,1,2,3,7] => ? = 5
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [[.,[.,[[.,[.,[.,.]]],.]]],.]
=> [5,4,3,6,2,1,7] => [6,5,3,2,1,4,7] => ? = 5
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [[.,[.,[[.,[[.,.],.]],.]]],.]
=> [4,5,3,6,2,1,7] => [6,5,3,1,2,4,7] => ? = 5
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [[.,[.,[[[.,[.,.]],.],.]]],.]
=> [4,3,5,6,2,1,7] => [6,5,2,1,3,4,7] => ? = 5
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [[.,[.,[[[[.,.],.],.],.]]],.]
=> [3,4,5,6,2,1,7] => [6,5,1,2,3,4,7] => ? = 5
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [[.,[[.,[.,[.,[.,.]]]],.]],.]
=> [5,4,3,2,6,1,7] => [6,4,3,2,1,5,7] => ? = 5
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [[.,[[.,[.,[[.,.],.]]],.]],.]
=> [4,5,3,2,6,1,7] => [6,4,3,1,2,5,7] => ? = 5
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [[.,[[.,[[.,[.,.]],.]],.]],.]
=> [4,3,5,2,6,1,7] => [6,4,2,1,3,5,7] => ? = 5
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [[.,[[.,[[[.,.],.],.]],.]],.]
=> [3,4,5,2,6,1,7] => [6,4,1,2,3,5,7] => ? = 5
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [[.,[[[.,[.,[.,.]]],.],.]],.]
=> [4,3,2,5,6,1,7] => [6,3,2,1,4,5,7] => ? = 5
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [[.,[[[.,[[.,.],.]],.],.]],.]
=> [3,4,2,5,6,1,7] => [6,3,1,2,4,5,7] => ? = 5
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [[.,[[[[.,[.,.]],.],.],.]],.]
=> [3,2,4,5,6,1,7] => [6,2,1,3,4,5,7] => ? = 5
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [[.,[[[[[.,.],.],.],.],.]],.]
=> [2,3,4,5,6,1,7] => [6,1,2,3,4,5,7] => ? = 5
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[[.,[.,[.,[.,[.,.]]]]],.],.]
=> [5,4,3,2,1,6,7] => [5,4,3,2,1,6,7] => ? = 4
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [[[.,[.,[.,[[.,.],.]]]],.],.]
=> [4,5,3,2,1,6,7] => [5,4,3,1,2,6,7] => ? = 4
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [[[.,[.,[[.,[.,.]],.]]],.],.]
=> [4,3,5,2,1,6,7] => [5,4,2,1,3,6,7] => ? = 4
Description
The sum of the descent differences of a permutations.
This statistic is given by
$$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} (\pi_i-\pi_{i+1}).$$
See [[St000111]] and [[St000154]] for the sum of the descent tops and the descent bottoms, respectively. This statistic was studied in [1] and [2] where is was called the ''drop'' of a permutation.
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