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Your data matches 70 different statistics following compositions of up to 3 maps.
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Matching statistic: St000383
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,2] => [2] => 2
[2,1] => [1,1] => 1
[1,2,3] => [3] => 3
[1,3,2] => [2,1] => 1
[2,1,3] => [1,2] => 2
[2,3,1] => [2,1] => 1
[3,1,2] => [1,2] => 2
[3,2,1] => [1,1,1] => 1
[1,2,3,4] => [4] => 4
[1,2,4,3] => [3,1] => 1
[1,3,2,4] => [2,2] => 2
[1,3,4,2] => [3,1] => 1
[1,4,2,3] => [2,2] => 2
[1,4,3,2] => [2,1,1] => 1
[2,1,3,4] => [1,3] => 3
[2,1,4,3] => [1,2,1] => 1
[2,3,1,4] => [2,2] => 2
[2,3,4,1] => [3,1] => 1
[2,4,1,3] => [2,2] => 2
[2,4,3,1] => [2,1,1] => 1
[3,1,2,4] => [1,3] => 3
[3,1,4,2] => [1,2,1] => 1
[3,2,1,4] => [1,1,2] => 2
[3,2,4,1] => [1,2,1] => 1
[3,4,1,2] => [2,2] => 2
[3,4,2,1] => [2,1,1] => 1
[4,1,2,3] => [1,3] => 3
[4,1,3,2] => [1,2,1] => 1
[4,2,1,3] => [1,1,2] => 2
[4,2,3,1] => [1,2,1] => 1
[4,3,1,2] => [1,1,2] => 2
[4,3,2,1] => [1,1,1,1] => 1
[1,2,3,4,5] => [5] => 5
[1,2,3,5,4] => [4,1] => 1
[1,2,4,3,5] => [3,2] => 2
[1,2,4,5,3] => [4,1] => 1
[1,2,5,3,4] => [3,2] => 2
[1,2,5,4,3] => [3,1,1] => 1
[1,3,2,4,5] => [2,3] => 3
[1,3,2,5,4] => [2,2,1] => 1
[1,3,4,2,5] => [3,2] => 2
[1,3,4,5,2] => [4,1] => 1
[1,3,5,2,4] => [3,2] => 2
[1,3,5,4,2] => [3,1,1] => 1
[1,4,2,3,5] => [2,3] => 3
[1,4,2,5,3] => [2,2,1] => 1
[1,4,3,2,5] => [2,1,2] => 2
[1,4,3,5,2] => [2,2,1] => 1
[1,4,5,2,3] => [3,2] => 2
Description
The last part of an integer composition.
Matching statistic: St000273
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000273: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000273: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 1
[1,2] => [2] => ([],2)
=> 2
[2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,3] => [3] => ([],3)
=> 3
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[2,1,3] => [1,2] => ([(1,2)],3)
=> 2
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[3,1,2] => [1,2] => ([(1,2)],3)
=> 2
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,3,4] => [4] => ([],4)
=> 4
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> 3
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> 3
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> 3
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,3,4,5] => [5] => ([],5)
=> 5
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 3
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 3
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
Description
The domination number of a graph.
The domination number of a graph is given by the minimum size of a dominating set of vertices. A dominating set of vertices is a subset of the vertex set of such that every vertex is either in this subset or adjacent to an element of this subset.
Matching statistic: St000382
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1
[1,2] => [2] => [2] => 2
[2,1] => [1,1] => [1,1] => 1
[1,2,3] => [3] => [3] => 3
[1,3,2] => [2,1] => [1,2] => 1
[2,1,3] => [1,2] => [2,1] => 2
[2,3,1] => [2,1] => [1,2] => 1
[3,1,2] => [1,2] => [2,1] => 2
[3,2,1] => [1,1,1] => [1,1,1] => 1
[1,2,3,4] => [4] => [4] => 4
[1,2,4,3] => [3,1] => [1,3] => 1
[1,3,2,4] => [2,2] => [2,2] => 2
[1,3,4,2] => [3,1] => [1,3] => 1
[1,4,2,3] => [2,2] => [2,2] => 2
[1,4,3,2] => [2,1,1] => [1,1,2] => 1
[2,1,3,4] => [1,3] => [3,1] => 3
[2,1,4,3] => [1,2,1] => [1,2,1] => 1
[2,3,1,4] => [2,2] => [2,2] => 2
[2,3,4,1] => [3,1] => [1,3] => 1
[2,4,1,3] => [2,2] => [2,2] => 2
[2,4,3,1] => [2,1,1] => [1,1,2] => 1
[3,1,2,4] => [1,3] => [3,1] => 3
[3,1,4,2] => [1,2,1] => [1,2,1] => 1
[3,2,1,4] => [1,1,2] => [2,1,1] => 2
[3,2,4,1] => [1,2,1] => [1,2,1] => 1
[3,4,1,2] => [2,2] => [2,2] => 2
[3,4,2,1] => [2,1,1] => [1,1,2] => 1
[4,1,2,3] => [1,3] => [3,1] => 3
[4,1,3,2] => [1,2,1] => [1,2,1] => 1
[4,2,1,3] => [1,1,2] => [2,1,1] => 2
[4,2,3,1] => [1,2,1] => [1,2,1] => 1
[4,3,1,2] => [1,1,2] => [2,1,1] => 2
[4,3,2,1] => [1,1,1,1] => [1,1,1,1] => 1
[1,2,3,4,5] => [5] => [5] => 5
[1,2,3,5,4] => [4,1] => [1,4] => 1
[1,2,4,3,5] => [3,2] => [2,3] => 2
[1,2,4,5,3] => [4,1] => [1,4] => 1
[1,2,5,3,4] => [3,2] => [2,3] => 2
[1,2,5,4,3] => [3,1,1] => [1,1,3] => 1
[1,3,2,4,5] => [2,3] => [3,2] => 3
[1,3,2,5,4] => [2,2,1] => [1,2,2] => 1
[1,3,4,2,5] => [3,2] => [2,3] => 2
[1,3,4,5,2] => [4,1] => [1,4] => 1
[1,3,5,2,4] => [3,2] => [2,3] => 2
[1,3,5,4,2] => [3,1,1] => [1,1,3] => 1
[1,4,2,3,5] => [2,3] => [3,2] => 3
[1,4,2,5,3] => [2,2,1] => [1,2,2] => 1
[1,4,3,2,5] => [2,1,2] => [2,1,2] => 2
[1,4,3,5,2] => [2,2,1] => [1,2,2] => 1
[1,4,5,2,3] => [3,2] => [2,3] => 2
Description
The first part of an integer composition.
Matching statistic: St000544
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000544: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000544: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 1
[1,2] => [2] => ([],2)
=> 2
[2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,3] => [3] => ([],3)
=> 3
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[2,1,3] => [1,2] => ([(1,2)],3)
=> 2
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[3,1,2] => [1,2] => ([(1,2)],3)
=> 2
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,3,4] => [4] => ([],4)
=> 4
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> 3
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> 3
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> 3
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,3,4,5] => [5] => ([],5)
=> 5
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 3
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 3
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
Description
The cop number of a graph.
This is the minimal number of cops needed to catch the robber. The algorithm is from [2].
Matching statistic: St000745
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [[1]]
=> 1
[1,2] => [2,1] => [[1],[2]]
=> 2
[2,1] => [1,2] => [[1,2]]
=> 1
[1,2,3] => [3,2,1] => [[1],[2],[3]]
=> 3
[1,3,2] => [2,3,1] => [[1,2],[3]]
=> 1
[2,1,3] => [3,1,2] => [[1,3],[2]]
=> 2
[2,3,1] => [1,3,2] => [[1,2],[3]]
=> 1
[3,1,2] => [2,1,3] => [[1,3],[2]]
=> 2
[3,2,1] => [1,2,3] => [[1,2,3]]
=> 1
[1,2,3,4] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 4
[1,2,4,3] => [3,4,2,1] => [[1,2],[3],[4]]
=> 1
[1,3,2,4] => [4,2,3,1] => [[1,3],[2],[4]]
=> 2
[1,3,4,2] => [2,4,3,1] => [[1,2],[3],[4]]
=> 1
[1,4,2,3] => [3,2,4,1] => [[1,3],[2],[4]]
=> 2
[1,4,3,2] => [2,3,4,1] => [[1,2,3],[4]]
=> 1
[2,1,3,4] => [4,3,1,2] => [[1,4],[2],[3]]
=> 3
[2,1,4,3] => [3,4,1,2] => [[1,2],[3,4]]
=> 1
[2,3,1,4] => [4,1,3,2] => [[1,3],[2],[4]]
=> 2
[2,3,4,1] => [1,4,3,2] => [[1,2],[3],[4]]
=> 1
[2,4,1,3] => [3,1,4,2] => [[1,3],[2,4]]
=> 2
[2,4,3,1] => [1,3,4,2] => [[1,2,3],[4]]
=> 1
[3,1,2,4] => [4,2,1,3] => [[1,4],[2],[3]]
=> 3
[3,1,4,2] => [2,4,1,3] => [[1,2],[3,4]]
=> 1
[3,2,1,4] => [4,1,2,3] => [[1,3,4],[2]]
=> 2
[3,2,4,1] => [1,4,2,3] => [[1,2,4],[3]]
=> 1
[3,4,1,2] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[3,4,2,1] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[4,1,2,3] => [3,2,1,4] => [[1,4],[2],[3]]
=> 3
[4,1,3,2] => [2,3,1,4] => [[1,2,4],[3]]
=> 1
[4,2,1,3] => [3,1,2,4] => [[1,3,4],[2]]
=> 2
[4,2,3,1] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[4,3,1,2] => [2,1,3,4] => [[1,3,4],[2]]
=> 2
[4,3,2,1] => [1,2,3,4] => [[1,2,3,4]]
=> 1
[1,2,3,4,5] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 5
[1,2,3,5,4] => [4,5,3,2,1] => [[1,2],[3],[4],[5]]
=> 1
[1,2,4,3,5] => [5,3,4,2,1] => [[1,3],[2],[4],[5]]
=> 2
[1,2,4,5,3] => [3,5,4,2,1] => [[1,2],[3],[4],[5]]
=> 1
[1,2,5,3,4] => [4,3,5,2,1] => [[1,3],[2],[4],[5]]
=> 2
[1,2,5,4,3] => [3,4,5,2,1] => [[1,2,3],[4],[5]]
=> 1
[1,3,2,4,5] => [5,4,2,3,1] => [[1,4],[2],[3],[5]]
=> 3
[1,3,2,5,4] => [4,5,2,3,1] => [[1,2],[3,4],[5]]
=> 1
[1,3,4,2,5] => [5,2,4,3,1] => [[1,3],[2],[4],[5]]
=> 2
[1,3,4,5,2] => [2,5,4,3,1] => [[1,2],[3],[4],[5]]
=> 1
[1,3,5,2,4] => [4,2,5,3,1] => [[1,3],[2,4],[5]]
=> 2
[1,3,5,4,2] => [2,4,5,3,1] => [[1,2,3],[4],[5]]
=> 1
[1,4,2,3,5] => [5,3,2,4,1] => [[1,4],[2],[3],[5]]
=> 3
[1,4,2,5,3] => [3,5,2,4,1] => [[1,2],[3,4],[5]]
=> 1
[1,4,3,2,5] => [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> 2
[1,4,3,5,2] => [2,5,3,4,1] => [[1,2,4],[3],[5]]
=> 1
[1,4,5,2,3] => [3,2,5,4,1] => [[1,3],[2,4],[5]]
=> 2
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000916
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000916: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000916: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 1
[1,2] => [2] => ([],2)
=> 2
[2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,3] => [3] => ([],3)
=> 3
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[2,1,3] => [1,2] => ([(1,2)],3)
=> 2
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[3,1,2] => [1,2] => ([(1,2)],3)
=> 2
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,3,4] => [4] => ([],4)
=> 4
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> 3
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> 3
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> 3
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,3,4,5] => [5] => ([],5)
=> 5
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 3
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 3
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
Description
The packing number of a graph.
This is the size of a largest subset of vertices of a graph, such that any two distinct vertices in the subset have disjoint closed neighbourhoods, or, equivalently, have distance greater than two.
Matching statistic: St001829
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001829: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001829: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 1
[1,2] => [2] => ([],2)
=> 2
[2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,3] => [3] => ([],3)
=> 3
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[2,1,3] => [1,2] => ([(1,2)],3)
=> 2
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[3,1,2] => [1,2] => ([(1,2)],3)
=> 2
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,3,4] => [4] => ([],4)
=> 4
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> 3
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> 3
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> 3
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,3,4,5] => [5] => ([],5)
=> 5
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 3
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 3
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
Description
The common independence number of a graph.
The common independence number of a graph G is the greatest integer r such that every vertex of G belongs to some independent set X of vertices of cardinality at least r.
Matching statistic: St000010
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> [1]
=> 1
[1,2] => [2] => ([],2)
=> [1,1]
=> 2
[2,1] => [1,1] => ([(0,1)],2)
=> [2]
=> 1
[1,2,3] => [3] => ([],3)
=> [1,1,1]
=> 3
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> [3]
=> 1
[2,1,3] => [1,2] => ([(1,2)],3)
=> [2,1]
=> 2
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> [3]
=> 1
[3,1,2] => [1,2] => ([(1,2)],3)
=> [2,1]
=> 2
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,2,3,4] => [4] => ([],4)
=> [1,1,1,1]
=> 4
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 2
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 2
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> [2,1,1]
=> 3
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 2
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 2
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> [2,1,1]
=> 3
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 2
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 2
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> [2,1,1]
=> 3
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 2
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 2
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[1,2,3,4,5] => [5] => ([],5)
=> [1,1,1,1,1]
=> 5
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 1
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 2
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 1
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 2
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 2
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 1
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 2
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 2
Description
The length of the partition.
Matching statistic: St000011
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 1
[1,2] => [1,2] => [1,0,1,0]
=> [1,0,1,0]
=> 2
[2,1] => [2,1] => [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,3,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,4,2,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,4,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[2,1,4,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,3,1,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,3,4,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,4,1,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[3,1,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[3,2,4,1] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,4,2,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,2,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,2,4,5,3] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,2,5,3,4] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,3,2,5,4] => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,3,4,2,5] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,3,4,5,2] => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,3,5,2,4] => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,3,5,4,2] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,4,2,3,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,4,2,5,3] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,4,3,2,5] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,4,3,5,2] => [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,4,5,2,3] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000025
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [.,.]
=> [1,0]
=> 1
[1,2] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 2
[2,1] => [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 3
[1,3,2] => [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 2
[2,3,1] => [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 2
[3,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,2,4,3] => [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,3,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,3,4,2] => [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,4,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,4,3,2] => [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,1,3,4] => [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 3
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,3,1,4] => [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,3,4,1] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,4,1,3] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,4,3,1] => [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,1,2,4] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 3
[3,1,4,2] => [3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,2,1,4] => [3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 2
[3,2,4,1] => [3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,4,1,2] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,4,2,1] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[4,1,2,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 3
[4,1,3,2] => [4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[4,2,1,3] => [2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 2
[4,2,3,1] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,3,1,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 2
[4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,2,3,5,4] => [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,2,4,3,5] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,2,4,5,3] => [4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,2,5,3,4] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,2,5,4,3] => [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,3,2,4,5] => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[1,3,2,5,4] => [3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,3,4,2,5] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,3,4,5,2] => [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,2,4] => [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,3,5,4,2] => [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,4,2,5,3] => [4,1,5,2,3] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,4,3,2,5] => [1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,4,3,5,2] => [4,1,3,2,5] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,4,5,2,3] => [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of D.
The following 60 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000026The position of the first return of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000297The number of leading ones in a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000917The open packing number of a graph. St001050The number of terminal closers of a set partition. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001463The number of distinct columns in the nullspace of a graph. St001672The restrained domination number of a graph. St001691The number of kings in a graph. St001733The number of weak left to right maxima of a Dyck path. St000439The position of the first down step of a Dyck path. St001176The size of a partition minus its first part. St000678The number of up steps after the last double rise of a Dyck path. St000287The number of connected components of a graph. St000553The number of blocks of a graph. St000363The number of minimal vertex covers of a graph. St001828The Euler characteristic of a graph. St001316The domatic number of a graph. St000759The smallest missing part in an integer partition. St000617The number of global maxima of a Dyck path. St000989The number of final rises of a permutation. St000234The number of global ascents of a permutation. St000542The number of left-to-right-minima of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000993The multiplicity of the largest part of an integer partition. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St000654The first descent of a permutation. St000990The first ascent of a permutation. St000007The number of saliances of the permutation. St000546The number of global descents of a permutation. St000054The first entry of the permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000056The decomposition (or block) number of a permutation. St000286The number of connected components of the complement of a graph. St000314The number of left-to-right-maxima of a permutation. St001201The grade of the simple module S0 in the special CNakayama algebra corresponding to the Dyck path. St001481The minimal height of a peak of a Dyck path. St000090The variation of a composition. St000258The burning number of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001340The cardinality of a minimal non-edge isolating set of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001904The length of the initial strictly increasing segment of a parking function. St000942The number of critical left to right maxima of the parking functions. St001937The size of the center of a parking function.
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