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Your data matches 68 different statistics following compositions of up to 3 maps.
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Matching statistic: St000089
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Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000089: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000089: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1] => 0
[[1,2]]
=> [2] => [1,1] => 0
[[1],[2]]
=> [2] => [1,1] => 0
[[1,2,3]]
=> [3] => [1,1,1] => 0
[[1,3],[2]]
=> [2,1] => [2,1] => 1
[[1,2],[3]]
=> [3] => [1,1,1] => 0
[[1],[2],[3]]
=> [3] => [1,1,1] => 0
[[1,2,3,4]]
=> [4] => [1,1,1,1] => 0
[[1,2,4],[3]]
=> [3,1] => [2,1,1] => 1
[[1,2,3],[4]]
=> [4] => [1,1,1,1] => 0
[[1,2],[3,4]]
=> [3,1] => [2,1,1] => 1
[[1,4],[2],[3]]
=> [3,1] => [2,1,1] => 1
[[1,2],[3],[4]]
=> [4] => [1,1,1,1] => 0
[[1],[2],[3],[4]]
=> [4] => [1,1,1,1] => 0
[[1,2,3,4,5]]
=> [5] => [1,1,1,1,1] => 0
[[1,2,3,5],[4]]
=> [4,1] => [2,1,1,1] => 1
[[1,2,3,4],[5]]
=> [5] => [1,1,1,1,1] => 0
[[1,2,3],[4,5]]
=> [4,1] => [2,1,1,1] => 1
[[1,2,5],[3],[4]]
=> [4,1] => [2,1,1,1] => 1
[[1,2,3],[4],[5]]
=> [5] => [1,1,1,1,1] => 0
[[1,2],[3,5],[4]]
=> [4,1] => [2,1,1,1] => 1
[[1,5],[2],[3],[4]]
=> [4,1] => [2,1,1,1] => 1
[[1,2],[3],[4],[5]]
=> [5] => [1,1,1,1,1] => 0
[[1],[2],[3],[4],[5]]
=> [5] => [1,1,1,1,1] => 0
[[1,2,3,4,5,6]]
=> [6] => [1,1,1,1,1,1] => 0
[[1,3,4,5,6],[2]]
=> [2,4] => [1,1,1,2,1] => 2
[[1,2,3,4,6],[5]]
=> [5,1] => [2,1,1,1,1] => 1
[[1,2,3,4,5],[6]]
=> [6] => [1,1,1,1,1,1] => 0
[[1,3,4,6],[2,5]]
=> [2,3,1] => [2,1,2,1] => 3
[[1,3,4,5],[2,6]]
=> [2,4] => [1,1,1,2,1] => 2
[[1,2,3,4],[5,6]]
=> [5,1] => [2,1,1,1,1] => 1
[[1,3,4,6],[2],[5]]
=> [2,3,1] => [2,1,2,1] => 3
[[1,2,3,6],[4],[5]]
=> [5,1] => [2,1,1,1,1] => 1
[[1,3,4,5],[2],[6]]
=> [2,4] => [1,1,1,2,1] => 2
[[1,2,3,4],[5],[6]]
=> [6] => [1,1,1,1,1,1] => 0
[[1,3,4],[2,5,6]]
=> [2,3,1] => [2,1,2,1] => 3
[[1,3,6],[2,4],[5]]
=> [2,3,1] => [2,1,2,1] => 3
[[1,3,4],[2,6],[5]]
=> [2,3,1] => [2,1,2,1] => 3
[[1,2,3],[4,6],[5]]
=> [5,1] => [2,1,1,1,1] => 1
[[1,3,4],[2,5],[6]]
=> [2,4] => [1,1,1,2,1] => 2
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => [2,1,2,1] => 3
[[1,2,6],[3],[4],[5]]
=> [5,1] => [2,1,1,1,1] => 1
[[1,3,4],[2],[5],[6]]
=> [2,4] => [1,1,1,2,1] => 2
[[1,2,3],[4],[5],[6]]
=> [6] => [1,1,1,1,1,1] => 0
[[1,3],[2,4],[5,6]]
=> [2,3,1] => [2,1,2,1] => 3
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => [2,1,2,1] => 3
[[1,2],[3,6],[4],[5]]
=> [5,1] => [2,1,1,1,1] => 1
[[1,3],[2,4],[5],[6]]
=> [2,4] => [1,1,1,2,1] => 2
[[1,6],[2],[3],[4],[5]]
=> [5,1] => [2,1,1,1,1] => 1
[[1,3],[2],[4],[5],[6]]
=> [2,4] => [1,1,1,2,1] => 2
Description
The absolute variation of a composition.
Matching statistic: St001036
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(load all 4 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1,0]
=> 0
[[1,2]]
=> [2] => [1,1,0,0]
=> 0
[[1],[2]]
=> [2] => [1,1,0,0]
=> 0
[[1,2,3]]
=> [3] => [1,1,1,0,0,0]
=> 0
[[1,3],[2]]
=> [2,1] => [1,1,0,0,1,0]
=> 1
[[1,2],[3]]
=> [3] => [1,1,1,0,0,0]
=> 0
[[1],[2],[3]]
=> [3] => [1,1,1,0,0,0]
=> 0
[[1,2,3,4]]
=> [4] => [1,1,1,1,0,0,0,0]
=> 0
[[1,2,4],[3]]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[[1,2,3],[4]]
=> [4] => [1,1,1,1,0,0,0,0]
=> 0
[[1,2],[3,4]]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[[1,4],[2],[3]]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[[1,2],[3],[4]]
=> [4] => [1,1,1,1,0,0,0,0]
=> 0
[[1],[2],[3],[4]]
=> [4] => [1,1,1,1,0,0,0,0]
=> 0
[[1,2,3,4,5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[1,2,3,5],[4]]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,2,3,4],[5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[1,2,3],[4,5]]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,2,5],[3],[4]]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,2,3],[4],[5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[1,2],[3,5],[4]]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,5],[2],[3],[4]]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,2],[3],[4],[5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[1],[2],[3],[4],[5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[1,2,3,4,5,6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[[1,3,4,5,6],[2]]
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[[1,2,3,4,6],[5]]
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1,2,3,4,5],[6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[[1,3,4,6],[2,5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 3
[[1,3,4,5],[2,6]]
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[[1,2,3,4],[5,6]]
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1,3,4,6],[2],[5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 3
[[1,2,3,6],[4],[5]]
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1,3,4,5],[2],[6]]
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[[1,2,3,4],[5],[6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[[1,3,4],[2,5,6]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 3
[[1,3,6],[2,4],[5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 3
[[1,3,4],[2,6],[5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 3
[[1,2,3],[4,6],[5]]
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1,3,4],[2,5],[6]]
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 3
[[1,2,6],[3],[4],[5]]
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1,3,4],[2],[5],[6]]
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[[1,2,3],[4],[5],[6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[[1,3],[2,4],[5,6]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 3
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 3
[[1,2],[3,6],[4],[5]]
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1,3],[2,4],[5],[6]]
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[[1,6],[2],[3],[4],[5]]
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1,3],[2],[4],[5],[6]]
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
Description
The number of inner corners of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000388
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Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000388: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000388: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => ([],1)
=> 1 = 0 + 1
[[1,2]]
=> [2] => ([],2)
=> 1 = 0 + 1
[[1],[2]]
=> [2] => ([],2)
=> 1 = 0 + 1
[[1,2,3]]
=> [3] => ([],3)
=> 1 = 0 + 1
[[1,3],[2]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1,2],[3]]
=> [3] => ([],3)
=> 1 = 0 + 1
[[1],[2],[3]]
=> [3] => ([],3)
=> 1 = 0 + 1
[[1,2,3,4]]
=> [4] => ([],4)
=> 1 = 0 + 1
[[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,2,3],[4]]
=> [4] => ([],4)
=> 1 = 0 + 1
[[1,2],[3,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,4],[2],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,2],[3],[4]]
=> [4] => ([],4)
=> 1 = 0 + 1
[[1],[2],[3],[4]]
=> [4] => ([],4)
=> 1 = 0 + 1
[[1,2,3,4,5]]
=> [5] => ([],5)
=> 1 = 0 + 1
[[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2,3,4],[5]]
=> [5] => ([],5)
=> 1 = 0 + 1
[[1,2,3],[4,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2,3],[4],[5]]
=> [5] => ([],5)
=> 1 = 0 + 1
[[1,2],[3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,5],[2],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2],[3],[4],[5]]
=> [5] => ([],5)
=> 1 = 0 + 1
[[1],[2],[3],[4],[5]]
=> [5] => ([],5)
=> 1 = 0 + 1
[[1,2,3,4,5,6]]
=> [6] => ([],6)
=> 1 = 0 + 1
[[1,3,4,5,6],[2]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,2,3,4,5],[6]]
=> [6] => ([],6)
=> 1 = 0 + 1
[[1,3,4,6],[2,5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,3,4,5],[2,6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3,4,6],[2],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3,4,5],[2],[6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,2,3,4],[5],[6]]
=> [6] => ([],6)
=> 1 = 0 + 1
[[1,3,4],[2,5,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,3,6],[2,4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,3,4],[2,6],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,2,3],[4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3,4],[2,5],[6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,2,6],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3,4],[2],[5],[6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,2,3],[4],[5],[6]]
=> [6] => ([],6)
=> 1 = 0 + 1
[[1,3],[2,4],[5,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,2],[3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3],[2,4],[5],[6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,6],[2],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3],[2],[4],[5],[6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
Description
The number of orbits of vertices of a graph under automorphisms.
Matching statistic: St000722
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000722: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000722: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => ([],1)
=> 1 = 0 + 1
[[1,2]]
=> [2] => ([],2)
=> 1 = 0 + 1
[[1],[2]]
=> [2] => ([],2)
=> 1 = 0 + 1
[[1,2,3]]
=> [3] => ([],3)
=> 1 = 0 + 1
[[1,3],[2]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1,2],[3]]
=> [3] => ([],3)
=> 1 = 0 + 1
[[1],[2],[3]]
=> [3] => ([],3)
=> 1 = 0 + 1
[[1,2,3,4]]
=> [4] => ([],4)
=> 1 = 0 + 1
[[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,2,3],[4]]
=> [4] => ([],4)
=> 1 = 0 + 1
[[1,2],[3,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,4],[2],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,2],[3],[4]]
=> [4] => ([],4)
=> 1 = 0 + 1
[[1],[2],[3],[4]]
=> [4] => ([],4)
=> 1 = 0 + 1
[[1,2,3,4,5]]
=> [5] => ([],5)
=> 1 = 0 + 1
[[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2,3,4],[5]]
=> [5] => ([],5)
=> 1 = 0 + 1
[[1,2,3],[4,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2,3],[4],[5]]
=> [5] => ([],5)
=> 1 = 0 + 1
[[1,2],[3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,5],[2],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2],[3],[4],[5]]
=> [5] => ([],5)
=> 1 = 0 + 1
[[1],[2],[3],[4],[5]]
=> [5] => ([],5)
=> 1 = 0 + 1
[[1,2,3,4,5,6]]
=> [6] => ([],6)
=> 1 = 0 + 1
[[1,3,4,5,6],[2]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,2,3,4,5],[6]]
=> [6] => ([],6)
=> 1 = 0 + 1
[[1,3,4,6],[2,5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,3,4,5],[2,6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3,4,6],[2],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3,4,5],[2],[6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,2,3,4],[5],[6]]
=> [6] => ([],6)
=> 1 = 0 + 1
[[1,3,4],[2,5,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,3,6],[2,4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,3,4],[2,6],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,2,3],[4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3,4],[2,5],[6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,2,6],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3,4],[2],[5],[6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,2,3],[4],[5],[6]]
=> [6] => ([],6)
=> 1 = 0 + 1
[[1,3],[2,4],[5,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,2],[3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3],[2,4],[5],[6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,6],[2],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3],[2],[4],[5],[6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
Description
The number of different neighbourhoods in a graph.
Matching statistic: St000767
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000767: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000767: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1] => 1 = 0 + 1
[[1,2]]
=> [2] => [1,1] => 1 = 0 + 1
[[1],[2]]
=> [2] => [1,1] => 1 = 0 + 1
[[1,2,3]]
=> [3] => [1,1,1] => 1 = 0 + 1
[[1,3],[2]]
=> [2,1] => [2,1] => 2 = 1 + 1
[[1,2],[3]]
=> [3] => [1,1,1] => 1 = 0 + 1
[[1],[2],[3]]
=> [3] => [1,1,1] => 1 = 0 + 1
[[1,2,3,4]]
=> [4] => [1,1,1,1] => 1 = 0 + 1
[[1,2,4],[3]]
=> [3,1] => [2,1,1] => 2 = 1 + 1
[[1,2,3],[4]]
=> [4] => [1,1,1,1] => 1 = 0 + 1
[[1,2],[3,4]]
=> [3,1] => [2,1,1] => 2 = 1 + 1
[[1,4],[2],[3]]
=> [3,1] => [2,1,1] => 2 = 1 + 1
[[1,2],[3],[4]]
=> [4] => [1,1,1,1] => 1 = 0 + 1
[[1],[2],[3],[4]]
=> [4] => [1,1,1,1] => 1 = 0 + 1
[[1,2,3,4,5]]
=> [5] => [1,1,1,1,1] => 1 = 0 + 1
[[1,2,3,5],[4]]
=> [4,1] => [2,1,1,1] => 2 = 1 + 1
[[1,2,3,4],[5]]
=> [5] => [1,1,1,1,1] => 1 = 0 + 1
[[1,2,3],[4,5]]
=> [4,1] => [2,1,1,1] => 2 = 1 + 1
[[1,2,5],[3],[4]]
=> [4,1] => [2,1,1,1] => 2 = 1 + 1
[[1,2,3],[4],[5]]
=> [5] => [1,1,1,1,1] => 1 = 0 + 1
[[1,2],[3,5],[4]]
=> [4,1] => [2,1,1,1] => 2 = 1 + 1
[[1,5],[2],[3],[4]]
=> [4,1] => [2,1,1,1] => 2 = 1 + 1
[[1,2],[3],[4],[5]]
=> [5] => [1,1,1,1,1] => 1 = 0 + 1
[[1],[2],[3],[4],[5]]
=> [5] => [1,1,1,1,1] => 1 = 0 + 1
[[1,2,3,4,5,6]]
=> [6] => [1,1,1,1,1,1] => 1 = 0 + 1
[[1,3,4,5,6],[2]]
=> [2,4] => [1,1,1,2,1] => 3 = 2 + 1
[[1,2,3,4,6],[5]]
=> [5,1] => [2,1,1,1,1] => 2 = 1 + 1
[[1,2,3,4,5],[6]]
=> [6] => [1,1,1,1,1,1] => 1 = 0 + 1
[[1,3,4,6],[2,5]]
=> [2,3,1] => [2,1,2,1] => 4 = 3 + 1
[[1,3,4,5],[2,6]]
=> [2,4] => [1,1,1,2,1] => 3 = 2 + 1
[[1,2,3,4],[5,6]]
=> [5,1] => [2,1,1,1,1] => 2 = 1 + 1
[[1,3,4,6],[2],[5]]
=> [2,3,1] => [2,1,2,1] => 4 = 3 + 1
[[1,2,3,6],[4],[5]]
=> [5,1] => [2,1,1,1,1] => 2 = 1 + 1
[[1,3,4,5],[2],[6]]
=> [2,4] => [1,1,1,2,1] => 3 = 2 + 1
[[1,2,3,4],[5],[6]]
=> [6] => [1,1,1,1,1,1] => 1 = 0 + 1
[[1,3,4],[2,5,6]]
=> [2,3,1] => [2,1,2,1] => 4 = 3 + 1
[[1,3,6],[2,4],[5]]
=> [2,3,1] => [2,1,2,1] => 4 = 3 + 1
[[1,3,4],[2,6],[5]]
=> [2,3,1] => [2,1,2,1] => 4 = 3 + 1
[[1,2,3],[4,6],[5]]
=> [5,1] => [2,1,1,1,1] => 2 = 1 + 1
[[1,3,4],[2,5],[6]]
=> [2,4] => [1,1,1,2,1] => 3 = 2 + 1
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => [2,1,2,1] => 4 = 3 + 1
[[1,2,6],[3],[4],[5]]
=> [5,1] => [2,1,1,1,1] => 2 = 1 + 1
[[1,3,4],[2],[5],[6]]
=> [2,4] => [1,1,1,2,1] => 3 = 2 + 1
[[1,2,3],[4],[5],[6]]
=> [6] => [1,1,1,1,1,1] => 1 = 0 + 1
[[1,3],[2,4],[5,6]]
=> [2,3,1] => [2,1,2,1] => 4 = 3 + 1
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => [2,1,2,1] => 4 = 3 + 1
[[1,2],[3,6],[4],[5]]
=> [5,1] => [2,1,1,1,1] => 2 = 1 + 1
[[1,3],[2,4],[5],[6]]
=> [2,4] => [1,1,1,2,1] => 3 = 2 + 1
[[1,6],[2],[3],[4],[5]]
=> [5,1] => [2,1,1,1,1] => 2 = 1 + 1
[[1,3],[2],[4],[5],[6]]
=> [2,4] => [1,1,1,2,1] => 3 = 2 + 1
Description
The number of runs in an integer composition.
Writing the composition as $c_1^{e_1} \dots c_\ell^{e_\ell}$, where $c_i \neq c_{i+1}$ for all $i$, the number of runs is $\ell$, see [def.2.8, 1].
It turns out that the total number of runs in all compositions of $n$ equals the total number of odd parts in all these compositions, see [1].
Matching statistic: St001352
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001352: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001352: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => ([],1)
=> 1 = 0 + 1
[[1,2]]
=> [2] => ([],2)
=> 1 = 0 + 1
[[1],[2]]
=> [2] => ([],2)
=> 1 = 0 + 1
[[1,2,3]]
=> [3] => ([],3)
=> 1 = 0 + 1
[[1,3],[2]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1,2],[3]]
=> [3] => ([],3)
=> 1 = 0 + 1
[[1],[2],[3]]
=> [3] => ([],3)
=> 1 = 0 + 1
[[1,2,3,4]]
=> [4] => ([],4)
=> 1 = 0 + 1
[[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,2,3],[4]]
=> [4] => ([],4)
=> 1 = 0 + 1
[[1,2],[3,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,4],[2],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,2],[3],[4]]
=> [4] => ([],4)
=> 1 = 0 + 1
[[1],[2],[3],[4]]
=> [4] => ([],4)
=> 1 = 0 + 1
[[1,2,3,4,5]]
=> [5] => ([],5)
=> 1 = 0 + 1
[[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2,3,4],[5]]
=> [5] => ([],5)
=> 1 = 0 + 1
[[1,2,3],[4,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2,3],[4],[5]]
=> [5] => ([],5)
=> 1 = 0 + 1
[[1,2],[3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,5],[2],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2],[3],[4],[5]]
=> [5] => ([],5)
=> 1 = 0 + 1
[[1],[2],[3],[4],[5]]
=> [5] => ([],5)
=> 1 = 0 + 1
[[1,2,3,4,5,6]]
=> [6] => ([],6)
=> 1 = 0 + 1
[[1,3,4,5,6],[2]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,2,3,4,5],[6]]
=> [6] => ([],6)
=> 1 = 0 + 1
[[1,3,4,6],[2,5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,3,4,5],[2,6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3,4,6],[2],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3,4,5],[2],[6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,2,3,4],[5],[6]]
=> [6] => ([],6)
=> 1 = 0 + 1
[[1,3,4],[2,5,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,3,6],[2,4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,3,4],[2,6],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,2,3],[4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3,4],[2,5],[6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,2,6],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3,4],[2],[5],[6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,2,3],[4],[5],[6]]
=> [6] => ([],6)
=> 1 = 0 + 1
[[1,3],[2,4],[5,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,2],[3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3],[2,4],[5],[6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,6],[2],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3],[2],[4],[5],[6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
Description
The number of internal nodes in the modular decomposition of a graph.
Matching statistic: St001951
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001951: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001951: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => ([],1)
=> 1 = 0 + 1
[[1,2]]
=> [2] => ([],2)
=> 1 = 0 + 1
[[1],[2]]
=> [2] => ([],2)
=> 1 = 0 + 1
[[1,2,3]]
=> [3] => ([],3)
=> 1 = 0 + 1
[[1,3],[2]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1,2],[3]]
=> [3] => ([],3)
=> 1 = 0 + 1
[[1],[2],[3]]
=> [3] => ([],3)
=> 1 = 0 + 1
[[1,2,3,4]]
=> [4] => ([],4)
=> 1 = 0 + 1
[[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,2,3],[4]]
=> [4] => ([],4)
=> 1 = 0 + 1
[[1,2],[3,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,4],[2],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,2],[3],[4]]
=> [4] => ([],4)
=> 1 = 0 + 1
[[1],[2],[3],[4]]
=> [4] => ([],4)
=> 1 = 0 + 1
[[1,2,3,4,5]]
=> [5] => ([],5)
=> 1 = 0 + 1
[[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2,3,4],[5]]
=> [5] => ([],5)
=> 1 = 0 + 1
[[1,2,3],[4,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2,3],[4],[5]]
=> [5] => ([],5)
=> 1 = 0 + 1
[[1,2],[3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,5],[2],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2],[3],[4],[5]]
=> [5] => ([],5)
=> 1 = 0 + 1
[[1],[2],[3],[4],[5]]
=> [5] => ([],5)
=> 1 = 0 + 1
[[1,2,3,4,5,6]]
=> [6] => ([],6)
=> 1 = 0 + 1
[[1,3,4,5,6],[2]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,2,3,4,5],[6]]
=> [6] => ([],6)
=> 1 = 0 + 1
[[1,3,4,6],[2,5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,3,4,5],[2,6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3,4,6],[2],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3,4,5],[2],[6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,2,3,4],[5],[6]]
=> [6] => ([],6)
=> 1 = 0 + 1
[[1,3,4],[2,5,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,3,6],[2,4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,3,4],[2,6],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,2,3],[4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3,4],[2,5],[6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,2,6],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3,4],[2],[5],[6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,2,3],[4],[5],[6]]
=> [6] => ([],6)
=> 1 = 0 + 1
[[1,3],[2,4],[5,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,2],[3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3],[2,4],[5],[6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,6],[2],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3],[2],[4],[5],[6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
Description
The number of factors in the disjoint direct product decomposition of the automorphism group of a graph.
The disjoint direct product decomposition of a permutation group factors the group corresponding to the product $(G, X) \ast (H, Y) = (G\times H, Z)$, where $Z$ is the disjoint union of $X$ and $Y$.
In particular, for an asymmetric graph, i.e., with trivial automorphism group, this statistic equals the number of vertices, because the trivial action factors completely.
Matching statistic: St000340
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1] => [1,0]
=> 0
[[1,2]]
=> [2] => [1,1] => [1,0,1,0]
=> 0
[[1],[2]]
=> [2] => [1,1] => [1,0,1,0]
=> 0
[[1,2,3]]
=> [3] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[[1,3],[2]]
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[[1,2],[3]]
=> [3] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[[1],[2],[3]]
=> [3] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[[1,2,3,4]]
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[[1,2,4],[3]]
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[[1,2,3],[4]]
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[[1,2],[3,4]]
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[[1,4],[2],[3]]
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[[1,2],[3],[4]]
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[[1],[2],[3],[4]]
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[[1,2,3,4,5]]
=> [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[[1,2,3,5],[4]]
=> [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[[1,2,3,4],[5]]
=> [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[[1,2,3],[4,5]]
=> [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[[1,2,5],[3],[4]]
=> [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[[1,2,3],[4],[5]]
=> [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[[1,2],[3,5],[4]]
=> [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[[1,5],[2],[3],[4]]
=> [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[[1,2],[3],[4],[5]]
=> [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[[1],[2],[3],[4],[5]]
=> [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[[1,2,3,4,5,6]]
=> [6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[[1,3,4,5,6],[2]]
=> [2,4] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 2
[[1,2,3,4,6],[5]]
=> [5,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[[1,2,3,4,5],[6]]
=> [6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[[1,3,4,6],[2,5]]
=> [2,3,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 3
[[1,3,4,5],[2,6]]
=> [2,4] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 2
[[1,2,3,4],[5,6]]
=> [5,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[[1,3,4,6],[2],[5]]
=> [2,3,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 3
[[1,2,3,6],[4],[5]]
=> [5,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[[1,3,4,5],[2],[6]]
=> [2,4] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 2
[[1,2,3,4],[5],[6]]
=> [6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[[1,3,4],[2,5,6]]
=> [2,3,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 3
[[1,3,6],[2,4],[5]]
=> [2,3,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 3
[[1,3,4],[2,6],[5]]
=> [2,3,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 3
[[1,2,3],[4,6],[5]]
=> [5,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[[1,3,4],[2,5],[6]]
=> [2,4] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 2
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 3
[[1,2,6],[3],[4],[5]]
=> [5,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[[1,3,4],[2],[5],[6]]
=> [2,4] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 2
[[1,2,3],[4],[5],[6]]
=> [6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[[1,3],[2,4],[5,6]]
=> [2,3,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 3
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 3
[[1,2],[3,6],[4],[5]]
=> [5,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[[1,3],[2,4],[5],[6]]
=> [2,4] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 2
[[1,6],[2],[3],[4],[5]]
=> [5,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[[1,3],[2],[4],[5],[6]]
=> [2,4] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 2
Description
The number of non-final maximal constant sub-paths of length greater than one.
This is the total number of occurrences of the patterns $110$ and $001$.
Matching statistic: St000454
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1] => ([],1)
=> 0
[[1,2]]
=> [2] => [2] => ([],2)
=> 0
[[1],[2]]
=> [2] => [2] => ([],2)
=> 0
[[1,2,3]]
=> [3] => [3] => ([],3)
=> 0
[[1,3],[2]]
=> [2,1] => [1,2] => ([(1,2)],3)
=> 1
[[1,2],[3]]
=> [3] => [3] => ([],3)
=> 0
[[1],[2],[3]]
=> [3] => [3] => ([],3)
=> 0
[[1,2,3,4]]
=> [4] => [4] => ([],4)
=> 0
[[1,2,4],[3]]
=> [3,1] => [1,3] => ([(2,3)],4)
=> 1
[[1,2,3],[4]]
=> [4] => [4] => ([],4)
=> 0
[[1,2],[3,4]]
=> [3,1] => [1,3] => ([(2,3)],4)
=> 1
[[1,4],[2],[3]]
=> [3,1] => [1,3] => ([(2,3)],4)
=> 1
[[1,2],[3],[4]]
=> [4] => [4] => ([],4)
=> 0
[[1],[2],[3],[4]]
=> [4] => [4] => ([],4)
=> 0
[[1,2,3,4,5]]
=> [5] => [5] => ([],5)
=> 0
[[1,2,3,5],[4]]
=> [4,1] => [1,4] => ([(3,4)],5)
=> 1
[[1,2,3,4],[5]]
=> [5] => [5] => ([],5)
=> 0
[[1,2,3],[4,5]]
=> [4,1] => [1,4] => ([(3,4)],5)
=> 1
[[1,2,5],[3],[4]]
=> [4,1] => [1,4] => ([(3,4)],5)
=> 1
[[1,2,3],[4],[5]]
=> [5] => [5] => ([],5)
=> 0
[[1,2],[3,5],[4]]
=> [4,1] => [1,4] => ([(3,4)],5)
=> 1
[[1,5],[2],[3],[4]]
=> [4,1] => [1,4] => ([(3,4)],5)
=> 1
[[1,2],[3],[4],[5]]
=> [5] => [5] => ([],5)
=> 0
[[1],[2],[3],[4],[5]]
=> [5] => [5] => ([],5)
=> 0
[[1,2,3,4,5,6]]
=> [6] => [6] => ([],6)
=> 0
[[1,3,4,5,6],[2]]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[1,2,3,4,6],[5]]
=> [5,1] => [1,5] => ([(4,5)],6)
=> 1
[[1,2,3,4,5],[6]]
=> [6] => [6] => ([],6)
=> 0
[[1,3,4,6],[2,5]]
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,3,4,5],[2,6]]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[1,2,3,4],[5,6]]
=> [5,1] => [1,5] => ([(4,5)],6)
=> 1
[[1,3,4,6],[2],[5]]
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2,3,6],[4],[5]]
=> [5,1] => [1,5] => ([(4,5)],6)
=> 1
[[1,3,4,5],[2],[6]]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[1,2,3,4],[5],[6]]
=> [6] => [6] => ([],6)
=> 0
[[1,3,4],[2,5,6]]
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,3,6],[2,4],[5]]
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,3,4],[2,6],[5]]
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2,3],[4,6],[5]]
=> [5,1] => [1,5] => ([(4,5)],6)
=> 1
[[1,3,4],[2,5],[6]]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2,6],[3],[4],[5]]
=> [5,1] => [1,5] => ([(4,5)],6)
=> 1
[[1,3,4],[2],[5],[6]]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[1,2,3],[4],[5],[6]]
=> [6] => [6] => ([],6)
=> 0
[[1,3],[2,4],[5,6]]
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2],[3,6],[4],[5]]
=> [5,1] => [1,5] => ([(4,5)],6)
=> 1
[[1,3],[2,4],[5],[6]]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[1,6],[2],[3],[4],[5]]
=> [5,1] => [1,5] => ([(4,5)],6)
=> 1
[[1,3],[2],[4],[5],[6]]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St000691
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1] => 1 => 0
[[1,2]]
=> [2] => [1,1] => 11 => 0
[[1],[2]]
=> [2] => [1,1] => 11 => 0
[[1,2,3]]
=> [3] => [1,1,1] => 111 => 0
[[1,3],[2]]
=> [2,1] => [1,2] => 110 => 1
[[1,2],[3]]
=> [3] => [1,1,1] => 111 => 0
[[1],[2],[3]]
=> [3] => [1,1,1] => 111 => 0
[[1,2,3,4]]
=> [4] => [1,1,1,1] => 1111 => 0
[[1,2,4],[3]]
=> [3,1] => [1,1,2] => 1110 => 1
[[1,2,3],[4]]
=> [4] => [1,1,1,1] => 1111 => 0
[[1,2],[3,4]]
=> [3,1] => [1,1,2] => 1110 => 1
[[1,4],[2],[3]]
=> [3,1] => [1,1,2] => 1110 => 1
[[1,2],[3],[4]]
=> [4] => [1,1,1,1] => 1111 => 0
[[1],[2],[3],[4]]
=> [4] => [1,1,1,1] => 1111 => 0
[[1,2,3,4,5]]
=> [5] => [1,1,1,1,1] => 11111 => 0
[[1,2,3,5],[4]]
=> [4,1] => [1,1,1,2] => 11110 => 1
[[1,2,3,4],[5]]
=> [5] => [1,1,1,1,1] => 11111 => 0
[[1,2,3],[4,5]]
=> [4,1] => [1,1,1,2] => 11110 => 1
[[1,2,5],[3],[4]]
=> [4,1] => [1,1,1,2] => 11110 => 1
[[1,2,3],[4],[5]]
=> [5] => [1,1,1,1,1] => 11111 => 0
[[1,2],[3,5],[4]]
=> [4,1] => [1,1,1,2] => 11110 => 1
[[1,5],[2],[3],[4]]
=> [4,1] => [1,1,1,2] => 11110 => 1
[[1,2],[3],[4],[5]]
=> [5] => [1,1,1,1,1] => 11111 => 0
[[1],[2],[3],[4],[5]]
=> [5] => [1,1,1,1,1] => 11111 => 0
[[1,2,3,4,5,6]]
=> [6] => [1,1,1,1,1,1] => 111111 => 0
[[1,3,4,5,6],[2]]
=> [2,4] => [1,2,1,1,1] => 110111 => 2
[[1,2,3,4,6],[5]]
=> [5,1] => [1,1,1,1,2] => 111110 => 1
[[1,2,3,4,5],[6]]
=> [6] => [1,1,1,1,1,1] => 111111 => 0
[[1,3,4,6],[2,5]]
=> [2,3,1] => [1,2,1,2] => 110110 => 3
[[1,3,4,5],[2,6]]
=> [2,4] => [1,2,1,1,1] => 110111 => 2
[[1,2,3,4],[5,6]]
=> [5,1] => [1,1,1,1,2] => 111110 => 1
[[1,3,4,6],[2],[5]]
=> [2,3,1] => [1,2,1,2] => 110110 => 3
[[1,2,3,6],[4],[5]]
=> [5,1] => [1,1,1,1,2] => 111110 => 1
[[1,3,4,5],[2],[6]]
=> [2,4] => [1,2,1,1,1] => 110111 => 2
[[1,2,3,4],[5],[6]]
=> [6] => [1,1,1,1,1,1] => 111111 => 0
[[1,3,4],[2,5,6]]
=> [2,3,1] => [1,2,1,2] => 110110 => 3
[[1,3,6],[2,4],[5]]
=> [2,3,1] => [1,2,1,2] => 110110 => 3
[[1,3,4],[2,6],[5]]
=> [2,3,1] => [1,2,1,2] => 110110 => 3
[[1,2,3],[4,6],[5]]
=> [5,1] => [1,1,1,1,2] => 111110 => 1
[[1,3,4],[2,5],[6]]
=> [2,4] => [1,2,1,1,1] => 110111 => 2
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => [1,2,1,2] => 110110 => 3
[[1,2,6],[3],[4],[5]]
=> [5,1] => [1,1,1,1,2] => 111110 => 1
[[1,3,4],[2],[5],[6]]
=> [2,4] => [1,2,1,1,1] => 110111 => 2
[[1,2,3],[4],[5],[6]]
=> [6] => [1,1,1,1,1,1] => 111111 => 0
[[1,3],[2,4],[5,6]]
=> [2,3,1] => [1,2,1,2] => 110110 => 3
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => [1,2,1,2] => 110110 => 3
[[1,2],[3,6],[4],[5]]
=> [5,1] => [1,1,1,1,2] => 111110 => 1
[[1,3],[2,4],[5],[6]]
=> [2,4] => [1,2,1,1,1] => 110111 => 2
[[1,6],[2],[3],[4],[5]]
=> [5,1] => [1,1,1,1,2] => 111110 => 1
[[1,3],[2],[4],[5],[6]]
=> [2,4] => [1,2,1,1,1] => 110111 => 2
Description
The number of changes of a binary word.
This is the number of indices $i$ such that $w_i \neq w_{i+1}$.
The following 58 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001120The length of a longest path in a graph. St001270The bandwidth of a graph. St001512The minimum rank of a graph. St001962The proper pathwidth of a graph. St000071The number of maximal chains in a poset. St000453The number of distinct Laplacian eigenvalues of a graph. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001644The dimension of a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000938The number of zeros of the symmetric group character corresponding to the partition. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000331The number of upper interactions of a Dyck path. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000455The second largest eigenvalue of a graph if it is integral. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000035The number of left outer peaks of a permutation. St000662The staircase size of the code of a permutation. St000884The number of isolated descents of a permutation. St000264The girth of a graph, which is not a tree. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000422The energy of a graph, if it is integral. St001487The number of inner corners of a skew partition. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000355The number of occurrences of the pattern 21-3. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000021The number of descents of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000699The toughness times the least common multiple of 1,. St001811The Castelnuovo-Mumford regularity of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001569The maximal modular displacement of a permutation.
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