Your data matches 186 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000171
(load all 5 compositions to match this statistic)
Mapping
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)
 => 0
[1,1] => ([(0,1)],2)
 => 1
[2] => ([],2)
 => 0
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,2] => ([(1,2)],3)
 => 1
[2,1] => ([(0,2),(1,2)],3)
 => 2
[3] => ([],3)
 => 0
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,3] => ([(2,3)],4)
 => 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[2,2] => ([(1,3),(2,3)],4)
 => 2
[3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[4] => ([],4)
 => 0
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4] => ([(3,4)],5)
 => 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[2,3] => ([(2,4),(3,4)],5)
 => 2
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[5] => ([],5)
 => 0
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 2
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,5] => ([(4,5)],6)
 => 1
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
Description
The degree of the graph. This is the maximal vertex degree of a graph.
Matching statistic: St000987
(load all 5 compositions to match this statistic)
Mapping
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)
 => 0
[1,1] => ([(0,1)],2)
 => 1
[2] => ([],2)
 => 0
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,2] => ([(1,2)],3)
 => 1
[2,1] => ([(0,2),(1,2)],3)
 => 2
[3] => ([],3)
 => 0
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,3] => ([(2,3)],4)
 => 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[2,2] => ([(1,3),(2,3)],4)
 => 2
[3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[4] => ([],4)
 => 0
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4] => ([(3,4)],5)
 => 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[2,3] => ([(2,4),(3,4)],5)
 => 2
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[5] => ([],5)
 => 0
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 2
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,5] => ([(4,5)],6)
 => 1
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
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: St001227
(load all 17 compositions to match this statistic)
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => 0
[1,1] => [1,0,1,0]
 => 1
[2] => [1,1,0,0]
 => 0
[1,1,1] => [1,0,1,0,1,0]
 => 2
[1,2] => [1,0,1,1,0,0]
 => 2
[2,1] => [1,1,0,0,1,0]
 => 1
[3] => [1,1,1,0,0,0]
 => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
[1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
[1,3] => [1,0,1,1,1,0,0,0]
 => 3
[2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[2,2] => [1,1,0,0,1,1,0,0]
 => 2
[3,1] => [1,1,1,0,0,0,1,0]
 => 1
[4] => [1,1,1,1,0,0,0,0]
 => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 4
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 4
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 4
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 4
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 4
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 4
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 5
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 5
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 5
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 5
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 5
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 5
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 5
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 5
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 5
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 5
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 5
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 5
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 4
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 4
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 4
Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
Matching statistic: St001291
(load all 23 compositions to match this statistic)
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => 1 = 0 + 1
[1,1] => [1,0,1,0]
 => 2 = 1 + 1
[2] => [1,1,0,0]
 => 1 = 0 + 1
[1,1,1] => [1,0,1,0,1,0]
 => 3 = 2 + 1
[1,2] => [1,0,1,1,0,0]
 => 2 = 1 + 1
[2,1] => [1,1,0,0,1,0]
 => 3 = 2 + 1
[3] => [1,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,3] => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[3,1] => [1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[4] => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 5 = 4 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 5 = 4 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4 = 3 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 5 = 4 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 5 = 4 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 5 = 4 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 6 = 5 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 6 = 5 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 5 = 4 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 6 = 5 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 3 = 2 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 6 = 5 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 6 = 5 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 5 = 4 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 6 = 5 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 1 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 6 = 5 + 1
Description
The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. Let $A$ be the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]]. This statistics is the number of indecomposable summands of $D(A) \otimes D(A)$, where $D(A)$ is the natural dual of $A$.
Matching statistic: St001725
(load all 6 compositions to match this statistic)
Mapping
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 = 0 + 1
[1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[2] => ([],2)
 => 1 = 0 + 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[2,1] => ([(0,2),(1,2)],3)
 => 3 = 2 + 1
[3] => ([],3)
 => 1 = 0 + 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4] => ([],4)
 => 1 = 0 + 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[1,4] => ([(3,4)],5)
 => 2 = 1 + 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 4 + 1
[5] => ([],5)
 => 1 = 0 + 1
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 3 + 1
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 3 + 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[1,5] => ([(4,5)],6)
 => 2 = 1 + 1
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
Description
The harmonious chromatic number of a graph. A harmonious colouring is a proper vertex colouring such that any pair of colours appears at most once on adjacent vertices.
Matching statistic: St000010
(load all 3 compositions to match this statistic)
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => []
 => 0
[1,1] => [1,0,1,0]
 => [1]
 => 1
[2] => [1,1,0,0]
 => []
 => 0
[1,1,1] => [1,0,1,0,1,0]
 => [2,1]
 => 2
[1,2] => [1,0,1,1,0,0]
 => [1,1]
 => 2
[2,1] => [1,1,0,0,1,0]
 => [2]
 => 1
[3] => [1,1,1,0,0,0]
 => []
 => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 3
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 3
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 3
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 3
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 2
[2,2] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 2
[3,1] => [1,1,1,0,0,0,1,0]
 => [3]
 => 1
[4] => [1,1,1,1,0,0,0,0]
 => []
 => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => 4
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => 4
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => 4
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => 4
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => 4
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => 4
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 4
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => 3
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => 3
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 3
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 3
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => 5
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => 5
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => 5
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => 5
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => 5
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => 5
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2,1]
 => 5
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => 5
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => 5
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => 5
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => 5
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1,1]
 => 5
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [4,4,1,1,1]
 => 5
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [5,1,1,1,1]
 => 5
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => 5
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2]
 => 4
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2]
 => 4
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2]
 => 4
Description
The length of the partition.
Matching statistic: St000051
(load all 3 compositions to match this statistic)
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000051: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [.,.]
 => 0
[1,1] => [1,0,1,0]
 => [[.,.],.]
 => 1
[2] => [1,1,0,0]
 => [.,[.,.]]
 => 0
[1,1,1] => [1,0,1,0,1,0]
 => [[[.,.],.],.]
 => 2
[1,2] => [1,0,1,1,0,0]
 => [[.,.],[.,.]]
 => 1
[2,1] => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => 2
[3] => [1,1,1,0,0,0]
 => [.,[.,[.,.]]]
 => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [[[[.,.],.],.],.]
 => 3
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [[[.,.],.],[.,.]]
 => 2
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [[[.,.],[.,.]],.]
 => 3
[1,3] => [1,0,1,1,1,0,0,0]
 => [[.,.],[.,[.,.]]]
 => 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => 3
[2,2] => [1,1,0,0,1,1,0,0]
 => [[.,[.,.]],[.,.]]
 => 2
[3,1] => [1,1,1,0,0,0,1,0]
 => [[.,[.,[.,.]]],.]
 => 3
[4] => [1,1,1,1,0,0,0,0]
 => [.,[.,[.,[.,.]]]]
 => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [[[[[.,.],.],.],.],.]
 => 4
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [[[[.,.],.],.],[.,.]]
 => 3
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [[[[.,.],.],[.,.]],.]
 => 4
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [[[.,.],.],[.,[.,.]]]
 => 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [[[[.,.],[.,.]],.],.]
 => 4
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => 3
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],.]
 => 4
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [[.,.],[.,[.,[.,.]]]]
 => 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => 4
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [[[.,[.,.]],.],[.,.]]
 => 3
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [[[.,[.,.]],[.,.]],.]
 => 4
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[.,[.,.]],[.,[.,.]]]
 => 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => 4
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [[.,[.,[.,.]]],[.,.]]
 => 3
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [[.,[.,[.,[.,.]]]],.]
 => 4
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [[[[[[.,.],.],.],.],.],.]
 => 5
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [[[[[.,.],.],.],.],[.,.]]
 => 4
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [[[[[.,.],.],.],[.,.]],.]
 => 5
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [[[[.,.],.],.],[.,[.,.]]]
 => 3
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [[[[[.,.],.],[.,.]],.],.]
 => 5
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [[[[.,.],.],[.,.]],[.,.]]
 => 4
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [[[[.,.],.],[.,[.,.]]],.]
 => 5
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,[.,[.,.]]]]
 => 2
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [[[[[.,.],[.,.]],.],.],.]
 => 5
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [[[[.,.],[.,.]],.],[.,.]]
 => 4
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [[[[.,.],[.,.]],[.,.]],.]
 => 5
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [[[.,.],[.,.]],[.,[.,.]]]
 => 3
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [[[[.,.],[.,[.,.]]],.],.]
 => 5
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [[[.,.],[.,[.,.]]],[.,.]]
 => 4
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [[[.,.],[.,[.,[.,.]]]],.]
 => 5
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[.,.],[.,[.,[.,[.,.]]]]]
 => 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [[[[[.,[.,.]],.],.],.],.]
 => 5
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [[[[.,[.,.]],.],.],[.,.]]
 => 4
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [[[[.,[.,.]],.],[.,.]],.]
 => 5
Description
The size of the left subtree of a binary tree.
Matching statistic: St000141
(load all 5 compositions to match this statistic)
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1] => 0
[1,1] => [1,0,1,0]
 => [2,1] => 1
[2] => [1,1,0,0]
 => [1,2] => 0
[1,1,1] => [1,0,1,0,1,0]
 => [3,2,1] => 2
[1,2] => [1,0,1,1,0,0]
 => [2,3,1] => 1
[2,1] => [1,1,0,0,1,0]
 => [3,1,2] => 2
[3] => [1,1,1,0,0,0]
 => [1,2,3] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 3
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 2
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 3
[1,3] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 3
[2,2] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 2
[3,1] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3
[4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => 4
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => 3
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => 4
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => 4
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => 3
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => 4
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => 4
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => 3
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => 4
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => 4
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => 3
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 4
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => 5
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,2,1] => 4
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,2,1] => 5
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,2,1] => 3
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [6,5,3,4,2,1] => 5
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,6,3,4,2,1] => 4
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [6,3,4,5,2,1] => 5
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,2,1] => 2
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,3,1] => 5
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [5,6,4,2,3,1] => 4
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,5,2,3,1] => 5
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [4,5,6,2,3,1] => 3
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [6,5,2,3,4,1] => 5
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [5,6,2,3,4,1] => 4
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,5,1] => 5
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,1,2] => 5
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,1,2] => 4
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,1,2] => 5
Description
The maximum drop size of a permutation. The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St000147
(load all 3 compositions to match this statistic)
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => []
 => 0
[1,1] => [1,0,1,0]
 => [1]
 => 1
[2] => [1,1,0,0]
 => []
 => 0
[1,1,1] => [1,0,1,0,1,0]
 => [2,1]
 => 2
[1,2] => [1,0,1,1,0,0]
 => [1,1]
 => 1
[2,1] => [1,1,0,0,1,0]
 => [2]
 => 2
[3] => [1,1,1,0,0,0]
 => []
 => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 3
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 2
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 3
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 3
[2,2] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 2
[3,1] => [1,1,1,0,0,0,1,0]
 => [3]
 => 3
[4] => [1,1,1,1,0,0,0,0]
 => []
 => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => 4
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => 3
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => 4
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => 4
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => 3
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => 4
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => 4
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => 3
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 4
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => 4
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 3
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 4
[5] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => 5
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => 4
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => 5
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => 3
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => 5
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => 4
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => 5
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2,1]
 => 2
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => 5
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => 4
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => 5
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => 3
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1,1]
 => 5
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [4,4,1,1,1]
 => 4
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [5,1,1,1,1]
 => 5
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2]
 => 5
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2]
 => 4
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2]
 => 5
Description
The largest part of an integer partition.
Matching statistic: St000272
Mapping
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00147: Graphs —square⟶ Graphs
St000272: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 0
[1,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[2] => ([],2)
 => ([],2)
 => 0
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 1
[2,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[3] => ([],3)
 => ([],3)
 => 0
[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)
 => 3
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],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)
 => 3
[1,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 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)
 => 3
[2,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4] => ([],4)
 => ([],4)
 => 0
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,1,2] => ([(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)
 => 3
[1,1,2,1] => ([(0,4),(1,2),(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)
 => 4
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,2,1,1] => ([(0,3),(0,4),(1,2),(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)
 => 4
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,3,1] => ([(0,4),(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)
 => 4
[1,4] => ([(3,4)],5)
 => ([(3,4)],5)
 => 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(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)
 => 4
[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)
 => 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)
 => 4
[2,3] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 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)
 => 4
[3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],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)
 => 4
[5] => ([],5)
 => ([],5)
 => 0
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => 2
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,5] => ([(4,5)],6)
 => ([(4,5)],6)
 => 1
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
Description
The treewidth of a graph. A graph has treewidth zero if and only if it has no edges. A connected graph has treewidth at most one if and only if it is a tree. A connected graph has treewidth at most two if and only if it is a series-parallel graph.
The following 176 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000316The number of non-left-to-right-maxima of a permutation. 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. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. 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. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001644The dimension of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001962The proper pathwidth of a graph. St001971The number of negative eigenvalues of the adjacency matrix of the graph. St000026The position of the first return of a Dyck path. St000054The first entry of the permutation. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000363The number of minimal vertex covers of a graph. St000740The last entry of a permutation. St000822The Hadwiger number of the graph. 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. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001330The hat guessing number of a graph. St001389The number of partitions of the same length below the given integer partition. St001494The Alon-Tarsi number of a graph. St001497The position of the largest weak excedence of a permutation. 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. St001963The tree-depth of a graph. St000013The height of a Dyck path. St000019The cardinality of the support of a permutation. St000024The number of double up and double down steps of a Dyck path. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000067The inversion number of the alternating sign matrix. St000133The "bounce" of a permutation. St000204The number of internal nodes of a binary tree. St000209Maximum difference of elements in cycles. St000224The sorting index of a permutation. St000304The load of a permutation. St000356The number of occurrences of the pattern 13-2. St000377The dinv defect of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000459The hook length of the base cell of a partition. St000497The lcb statistic of a set partition. St000572The dimension exponent of a set partition. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001176The size of a partition minus its first part. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001726The number of visible inversions of a permutation. St001841The number of inversions of a set partition. St000011The number of touch points (or returns) of a Dyck path. St000025The number of initial rises of a Dyck path. St000058The order of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000093The cardinality of a maximal independent set of vertices of a graph. St000240The number of indices that are not small excedances. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000443The number of long tunnels of a Dyck path. St000501The size of the first part in the decomposition of a permutation. St000505The biggest entry in the block containing the 1. St000507The number of ascents of a standard tableau. St000676The number of odd rises of a Dyck path. St000692Babson and Steingrímsson's statistic of a permutation. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000730The maximal arc length of a set partition. St000734The last entry in the first row of a standard tableau. St000738The first entry in the last row of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000808The number of up steps of the associated bargraph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001462The number of factors of a standard tableaux under concatenation. St001481The minimal height of a peak of a Dyck path. St001778The largest greatest common divisor of an element and its image in a permutation. St001806The upper middle entry of a permutation. St001809The index of the step at the first peak of maximal height in a Dyck path. St000439The position of the first down step of a Dyck path. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000653The last descent of a permutation. St001480The number of simple summands of the module J^2/J^3. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000216The absolute length of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000503The maximal difference between two elements in a common block. St000728The dimension of a set partition. St000795The mad of a permutation. St000809The reduced reflection length of the permutation. St000874The position of the last double rise in a Dyck path. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001077The prefix exchange distance of a permutation. St001723The differential of a graph. St001724The 2-packing differential of a graph. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000444The length of the maximal rise of a Dyck path. St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000784The maximum of the length and the largest part of the integer partition. St000288The number of ones in a binary word. St000733The row containing the largest entry of a standard tableau. St001082The number of boxed occurrences of 123 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001812The biclique partition number of a graph. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000840The number of closers smaller than the largest opener in a perfect matching. St000651The maximal size of a rise in a permutation. St000028The number of stack-sorts needed to sort a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000799The number of occurrences of the vincular pattern |213 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000652The maximal difference between successive positions of a permutation. St001742The difference of the maximal and the minimal degree in a graph. 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. St000800The number of occurrences of the vincular pattern |231 in a permutation. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000060The greater neighbor of the maximum. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000235The number of indices that are not cyclical small weak excedances. St001769The reflection length of a signed permutation. St000005The bounce statistic of a Dyck path. St001557The number of inversions of the second entry of a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001060The distinguishing index of a graph. St000632The jump number of the poset. St000307The number of rowmotion orbits of a poset. 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. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000317The cycle descent number of a permutation. St000624The normalized sum of the minimal distances to a greater element. St000682The Grundy value of Welter's game on a binary word. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000091The descent variation of a composition. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph.