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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 $S_0$ 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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