Your data matches 70 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001725
(load all 6 compositions to match this statistic)
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001725: 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)
 => 1
[2,1] => [1,1] => ([(0,1)],2)
 => 2
[1,2,3] => [3] => ([],3)
 => 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[2,1,3] => [1,2] => ([(1,2)],3)
 => 2
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[3,1,2] => [1,2] => ([(1,2)],3)
 => 2
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,2,3,4] => [4] => ([],4)
 => 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 4
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => 3
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 4
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => 3
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,1,3,4] => [1,3] => ([(2,3)],4)
 => 2
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => 3
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 4
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => 3
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,1,2,4] => [1,3] => ([(2,3)],4)
 => 2
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => 3
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,1,2,3] => [1,3] => ([(2,3)],4)
 => 2
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,2,3,4,5] => [5] => ([],5)
 => 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,3,2,4,5] => [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)
 => 5
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,4,2,3,5] => [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)
 => 5
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4
Description
The harmonious chromatic number of a graph. A harmonious colouring is a proper vertex colouring such that any pair of colours appears at most once on adjacent vertices.
Matching statistic: St000171
(load all 6 compositions to match this statistic)
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000171: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
 => 0 = 1 - 1
[1,2] => [2] => ([],2)
 => 0 = 1 - 1
[2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[1,2,3] => [3] => ([],3)
 => 0 = 1 - 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
[2,1,3] => [1,2] => ([(1,2)],3)
 => 1 = 2 - 1
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
[3,1,2] => [1,2] => ([(1,2)],3)
 => 1 = 2 - 1
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[1,2,3,4] => [4] => ([],4)
 => 0 = 1 - 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 4 - 1
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 3 - 1
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 4 - 1
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 3 - 1
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[2,1,3,4] => [1,3] => ([(2,3)],4)
 => 1 = 2 - 1
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 3 - 1
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 4 - 1
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 3 - 1
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[3,1,2,4] => [1,3] => ([(2,3)],4)
 => 1 = 2 - 1
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 3 - 1
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[4,1,2,3] => [1,3] => ([(2,3)],4)
 => 1 = 2 - 1
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[1,2,3,4,5] => [5] => ([],5)
 => 0 = 1 - 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
Description
The degree of the graph. This is the maximal vertex degree of a graph.
Matching statistic: St000987
(load all 6 compositions to match this statistic)
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000987: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
 => 0 = 1 - 1
[1,2] => [2] => ([],2)
 => 0 = 1 - 1
[2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[1,2,3] => [3] => ([],3)
 => 0 = 1 - 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
[2,1,3] => [1,2] => ([(1,2)],3)
 => 1 = 2 - 1
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
[3,1,2] => [1,2] => ([(1,2)],3)
 => 1 = 2 - 1
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[1,2,3,4] => [4] => ([],4)
 => 0 = 1 - 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 4 - 1
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 3 - 1
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 4 - 1
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 3 - 1
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[2,1,3,4] => [1,3] => ([(2,3)],4)
 => 1 = 2 - 1
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 3 - 1
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 4 - 1
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 3 - 1
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[3,1,2,4] => [1,3] => ([(2,3)],4)
 => 1 = 2 - 1
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 3 - 1
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[4,1,2,3] => [1,3] => ([(2,3)],4)
 => 1 = 2 - 1
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[1,2,3,4,5] => [5] => ([],5)
 => 0 = 1 - 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
Description
The number of positive eigenvalues of the Laplacian matrix of the graph. This is the number of vertices minus the number of connected components of the graph.
Matching statistic: St000026
(load all 2 compositions to match this statistic)
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000026: 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] => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 1
[2,1] => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 2
[1,2,3] => [3] => [1,1,1,0,0,0]
 => [1,0,1,1,0,0]
 => 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 3
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 4
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 4
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 4
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 4
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 4
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 4
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 4
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 4
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 4
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 5
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 5
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 5
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 5
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 4
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 5
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4
Description
The position of the first return of a Dyck path.
Matching statistic: St000097
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00147: Graphs —square⟶ Graphs
St000097: Graphs ⟶ 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)
 => ([],2)
 => 1
[2,1] => [1,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,2,3] => [3] => ([],3)
 => ([],3)
 => 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[2,1,3] => [1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,1,2] => [1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,2,3,4] => [4] => ([],4)
 => ([],4)
 => 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,1,3,4] => [1,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,1,2,4] => [1,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,1,2,3] => [1,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,2,3,4,5] => [5] => ([],5)
 => ([],5)
 => 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
Description
The order of the largest clique of the graph. A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000098
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00147: Graphs —square⟶ Graphs
St000098: Graphs ⟶ 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)
 => ([],2)
 => 1
[2,1] => [1,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,2,3] => [3] => ([],3)
 => ([],3)
 => 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[2,1,3] => [1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,1,2] => [1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,2,3,4] => [4] => ([],4)
 => ([],4)
 => 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,1,3,4] => [1,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,1,2,4] => [1,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,1,2,3] => [1,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,2,3,4,5] => [5] => ([],5)
 => ([],5)
 => 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
Description
The chromatic number of a graph. The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.
Matching statistic: St000147
(load all 2 compositions to match this statistic)
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000147: 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]
 => 1
[2,1] => [1,1] => ([(0,1)],2)
 => [2]
 => 2
[1,2,3] => [3] => ([],3)
 => [1,1,1]
 => 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => [3]
 => 3
[2,1,3] => [1,2] => ([(1,2)],3)
 => [2,1]
 => 2
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => [3]
 => 3
[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]
 => 3
[1,2,3,4] => [4] => ([],4)
 => [1,1,1,1]
 => 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => 4
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => [3,1]
 => 3
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => 4
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => [3,1]
 => 3
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
[2,1,3,4] => [1,3] => ([(2,3)],4)
 => [2,1,1]
 => 2
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => [3,1]
 => 3
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => 4
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => [3,1]
 => 3
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
[3,1,2,4] => [1,3] => ([(2,3)],4)
 => [2,1,1]
 => 2
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 3
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => [3,1]
 => 3
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
[4,1,2,3] => [1,3] => ([(2,3)],4)
 => [2,1,1]
 => 2
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 3
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 3
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
[1,2,3,4,5] => [5] => ([],5)
 => [1,1,1,1,1]
 => 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => 5
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 4
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => 5
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 4
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
[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]
 => 5
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 4
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => 5
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 4
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
[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]
 => 5
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 4
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 4
Description
The largest part of an integer partition.
Matching statistic: St000172
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00147: Graphs —square⟶ Graphs
St000172: Graphs ⟶ 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)
 => ([],2)
 => 1
[2,1] => [1,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,2,3] => [3] => ([],3)
 => ([],3)
 => 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[2,1,3] => [1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,1,2] => [1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,2,3,4] => [4] => ([],4)
 => ([],4)
 => 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,1,3,4] => [1,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,1,2,4] => [1,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,1,2,3] => [1,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,2,3,4,5] => [5] => ([],5)
 => ([],5)
 => 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
Description
The Grundy number of a graph. The Grundy number $\Gamma(G)$ is defined to be the largest $k$ such that $G$ admits a greedy $k$-coloring. Any order of the vertices of $G$ induces a greedy coloring by assigning to the $i$-th vertex in this order the smallest positive integer such that the partial coloring remains a proper coloring. In particular, we have that $\chi(G) \leq \Gamma(G) \leq \Delta(G) + 1$, where $\chi(G)$ is the chromatic number of $G$ ([[St000098]]), and where $\Delta(G)$ is the maximal degree of a vertex of $G$ ([[St000171]]).
Matching statistic: St000363
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00147: Graphs —square⟶ Graphs
St000363: Graphs ⟶ 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)
 => ([],2)
 => 1
[2,1] => [1,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,2,3] => [3] => ([],3)
 => ([],3)
 => 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[2,1,3] => [1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,1,2] => [1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,2,3,4] => [4] => ([],4)
 => ([],4)
 => 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,1,3,4] => [1,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,1,2,4] => [1,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,1,2,3] => [1,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 2
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,2,3,4,5] => [5] => ([],5)
 => ([],5)
 => 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
Description
The number of minimal vertex covers of a graph. A '''vertex cover''' of a graph $G$ is a subset $S$ of the vertices of $G$ such that each edge of $G$ contains at least one vertex of $S$. A vertex cover is minimal if it contains the least possible number of vertices. This is also the leading coefficient of the clique polynomial of the complement of $G$. This is also the number of independent sets of maximal cardinality of $G$.
Matching statistic: St000734
Mapping
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000734: 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]]
 => 1
[2,1] => [[.,.],.]
 => [1,2] => [[1,2]]
 => 2
[1,2,3] => [.,[.,[.,.]]]
 => [3,2,1] => [[1],[2],[3]]
 => 1
[1,3,2] => [.,[[.,.],.]]
 => [2,3,1] => [[1,3],[2]]
 => 3
[2,1,3] => [[.,.],[.,.]]
 => [1,3,2] => [[1,2],[3]]
 => 2
[2,3,1] => [[.,[.,.]],.]
 => [2,1,3] => [[1,3],[2]]
 => 3
[3,1,2] => [[.,.],[.,.]]
 => [1,3,2] => [[1,2],[3]]
 => 2
[3,2,1] => [[[.,.],.],.]
 => [1,2,3] => [[1,2,3]]
 => 3
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [[1],[2],[3],[4]]
 => 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => [[1,4],[2],[3]]
 => 4
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => [2,4,3,1] => [[1,3],[2],[4]]
 => 3
[1,3,4,2] => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => [[1,4],[2],[3]]
 => 4
[1,4,2,3] => [.,[[.,.],[.,.]]]
 => [2,4,3,1] => [[1,3],[2],[4]]
 => 3
[1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => [[1,3,4],[2]]
 => 4
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => [[1,2],[3],[4]]
 => 2
[2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => [[1,2,4],[3]]
 => 4
[2,3,1,4] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => [[1,3],[2,4]]
 => 3
[2,3,4,1] => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => [[1,4],[2],[3]]
 => 4
[2,4,1,3] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => [[1,3],[2,4]]
 => 3
[2,4,3,1] => [[.,[[.,.],.]],.]
 => [2,3,1,4] => [[1,3,4],[2]]
 => 4
[3,1,2,4] => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => [[1,2],[3],[4]]
 => 2
[3,1,4,2] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => [[1,2,4],[3]]
 => 4
[3,2,1,4] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => [[1,2,3],[4]]
 => 3
[3,2,4,1] => [[[.,.],[.,.]],.]
 => [1,3,2,4] => [[1,2,4],[3]]
 => 4
[3,4,1,2] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => [[1,3],[2,4]]
 => 3
[3,4,2,1] => [[[.,[.,.]],.],.]
 => [2,1,3,4] => [[1,3,4],[2]]
 => 4
[4,1,2,3] => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => [[1,2],[3],[4]]
 => 2
[4,1,3,2] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => [[1,2,4],[3]]
 => 4
[4,2,1,3] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => [[1,2,3],[4]]
 => 3
[4,2,3,1] => [[[.,.],[.,.]],.]
 => [1,3,2,4] => [[1,2,4],[3]]
 => 4
[4,3,1,2] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => [[1,2,3],[4]]
 => 3
[4,3,2,1] => [[[[.,.],.],.],.]
 => [1,2,3,4] => [[1,2,3,4]]
 => 4
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
 => 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [[1,5],[2],[3],[4]]
 => 5
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [[1,4],[2],[3],[5]]
 => 4
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [[1,5],[2],[3],[4]]
 => 5
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [[1,4],[2],[3],[5]]
 => 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [[1,4,5],[2],[3]]
 => 5
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [[1,3],[2],[4],[5]]
 => 3
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [[1,3,5],[2],[4]]
 => 5
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [[1,4],[2,5],[3]]
 => 4
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [[1,5],[2],[3],[4]]
 => 5
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [[1,4],[2,5],[3]]
 => 4
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [[1,4,5],[2],[3]]
 => 5
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [[1,3],[2],[4],[5]]
 => 3
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [[1,3,5],[2],[4]]
 => 5
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [[1,3,4],[2],[5]]
 => 4
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [[1,3,5],[2],[4]]
 => 5
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [[1,4],[2,5],[3]]
 => 4
Description
The last entry in the first row of a standard tableau.
The following 60 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000839The largest opener of a set partition. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001389The number of partitions of the same length below the given integer partition. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001883The mutual visibility number of a graph. St000272The treewidth of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000362The size of a minimal vertex cover of a graph. St000454The largest eigenvalue of a graph if it is integral. St000536The pathwidth of a graph. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St001120The length of a longest path in a graph. St001644The dimension of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001971The number of negative eigenvalues of the adjacency matrix of the graph. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St001721The degree of a binary word. St000653The last descent of a permutation. St001330The hat guessing number of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000740The last entry of a permutation. St001497The position of the largest weak excedence of a permutation. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001962The proper pathwidth of a graph. St000844The size of the largest block in the direct sum decomposition of a permutation. St000054The first entry of the permutation. St000141The maximum drop size of a permutation. St000019The cardinality of the support of a permutation. St001645The pebbling number of a connected graph. St000501The size of the first part in the decomposition of a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000822The Hadwiger number of the graph. St000051The size of the left subtree of a binary tree. St000316The number of non-left-to-right-maxima of a permutation. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St001812The biclique partition number of a graph. St000840The number of closers smaller than the largest opener in a perfect matching. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001742The difference of the maximal and the minimal degree in a graph. 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. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001877Number of indecomposable injective modules with projective dimension 2.