Your data matches 90 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001323
(load all 10 compositions to match this statistic)
Mapping
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001323: 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)
 => 0
[2] => ([],2)
 => 0
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[1,2] => ([(1,2)],3)
 => 0
[2,1] => ([(0,2),(1,2)],3)
 => 1
[3] => ([],3)
 => 0
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 0
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,3] => ([(2,3)],4)
 => 0
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[2,2] => ([(1,3),(2,3)],4)
 => 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[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
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 0
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4] => ([(3,4)],5)
 => 0
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,3] => ([(2,4),(3,4)],5)
 => 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[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
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[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)
 => 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[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)
 => 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 0
[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)
 => 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 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)
 => 2
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,5] => ([(4,5)],6)
 => 0
[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)
 => 1
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
Description
The independence gap of a graph. This is the difference between the independence number [[St000093]] and the minimal size of a maximally independent set of a graph. In particular, this statistic is $0$ for well covered graphs
Matching statistic: St000052
(load all 7 compositions to match this statistic)
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1,0]
 => 0
[1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 0
[2] => [1,1,0,0]
 => [1,0,1,0]
 => 0
[1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[1,2] => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 0
[2,1] => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 0
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 0
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 3
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 2
Description
The number of valleys of a Dyck path not on the x-axis. That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St000147
(load all 3 compositions to match this statistic)
Mapping
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1],[]]
 => []
 => 0
[1,1] => [[1,1],[]]
 => []
 => 0
[2] => [[2],[]]
 => []
 => 0
[1,1,1] => [[1,1,1],[]]
 => []
 => 0
[1,2] => [[2,1],[]]
 => []
 => 0
[2,1] => [[2,2],[1]]
 => [1]
 => 1
[3] => [[3],[]]
 => []
 => 0
[1,1,1,1] => [[1,1,1,1],[]]
 => []
 => 0
[1,1,2] => [[2,1,1],[]]
 => []
 => 0
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => 1
[1,3] => [[3,1],[]]
 => []
 => 0
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 1
[2,2] => [[3,2],[1]]
 => [1]
 => 1
[3,1] => [[3,3],[2]]
 => [2]
 => 2
[4] => [[4],[]]
 => []
 => 0
[1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => 0
[1,1,1,2] => [[2,1,1,1],[]]
 => []
 => 0
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 1
[1,1,3] => [[3,1,1],[]]
 => []
 => 0
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 1
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => 1
[1,3,1] => [[3,3,1],[2]]
 => [2]
 => 2
[1,4] => [[4,1],[]]
 => []
 => 0
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 1
[2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 2
[2,3] => [[4,2],[1]]
 => [1]
 => 1
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 2
[3,2] => [[4,3],[2]]
 => [2]
 => 2
[4,1] => [[4,4],[3]]
 => [3]
 => 3
[5] => [[5],[]]
 => []
 => 0
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => []
 => 0
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => 0
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => 1
[1,1,1,3] => [[3,1,1,1],[]]
 => []
 => 0
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 1
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => 1
[1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => 2
[1,1,4] => [[4,1,1],[]]
 => []
 => 0
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => 2
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => 2
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => 2
[1,4,1] => [[4,4,1],[3]]
 => [3]
 => 3
[1,5] => [[5,1],[]]
 => []
 => 0
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => 2
Description
The largest part of an integer partition.
Matching statistic: St000171
Mapping
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00250: Graphs —clique 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] => ([(0,1)],2)
 => ([],1)
 => 0
[2] => ([],2)
 => ([],2)
 => 0
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 0
[1,2] => ([(1,2)],3)
 => ([],2)
 => 0
[2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
[3] => ([],3)
 => ([],3)
 => 0
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[1,3] => ([(2,3)],4)
 => ([],3)
 => 0
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[2,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[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)
 => ([],1)
 => 0
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,4] => ([(3,4)],5)
 => ([],4)
 => 0
[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)],2)
 => 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,3] => ([(2,4),(3,4)],5)
 => ([(2,3)],4)
 => 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[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)
 => ([],1)
 => 0
[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)
 => ([],2)
 => 0
[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)],2)
 => 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => 0
[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)],2)
 => 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 1
[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),(1,2)],3)
 => 2
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([],4)
 => 0
[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)],2)
 => 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 1
[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),(1,2)],3)
 => 2
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3)],4)
 => 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)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,5] => ([(4,5)],6)
 => ([],5)
 => 0
[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)],2)
 => 1
[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)],3)
 => 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)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
Description
The degree of the graph. This is the maximal vertex degree of a graph.
Matching statistic: St000204
(load all 6 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
St000204: 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]
 => [[.,.],.]
 => 0
[2] => [1,1,0,0]
 => [.,[.,.]]
 => 0
[1,1,1] => [1,0,1,0,1,0]
 => [[[.,.],.],.]
 => 0
[1,2] => [1,0,1,1,0,0]
 => [[.,.],[.,.]]
 => 0
[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]
 => [[[[.,.],.],.],.]
 => 0
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [[[.,.],.],[.,.]]
 => 0
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [[[.,.],[.,.]],.]
 => 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [[.,.],[.,[.,.]]]
 => 0
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [[.,[.,.]],[.,.]]
 => 1
[3,1] => [1,1,1,0,0,0,1,0]
 => [[.,[.,[.,.]]],.]
 => 2
[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]
 => [[[[[.,.],.],.],.],.]
 => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [[[[.,.],.],.],[.,.]]
 => 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [[[[.,.],.],[.,.]],.]
 => 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [[[.,.],.],[.,[.,.]]]
 => 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [[[[.,.],[.,.]],.],.]
 => 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],.]
 => 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [[.,.],[.,[.,[.,.]]]]
 => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [[[.,[.,.]],.],[.,.]]
 => 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [[[.,[.,.]],[.,.]],.]
 => 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[.,[.,.]],[.,[.,.]]]
 => 1
[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]
 => [[.,[.,[.,[.,.]]]],.]
 => 3
[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]
 => [[[[[[.,.],.],.],.],.],.]
 => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [[[[[.,.],.],.],.],[.,.]]
 => 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [[[[[.,.],.],.],[.,.]],.]
 => 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [[[[.,.],.],.],[.,[.,.]]]
 => 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [[[[[.,.],.],[.,.]],.],.]
 => 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [[[[.,.],.],[.,.]],[.,.]]
 => 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [[[[.,.],.],[.,[.,.]]],.]
 => 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,[.,[.,.]]]]
 => 0
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [[[[[.,.],[.,.]],.],.],.]
 => 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [[[[.,.],[.,.]],.],[.,.]]
 => 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [[[[.,.],[.,.]],[.,.]],.]
 => 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [[[.,.],[.,.]],[.,[.,.]]]
 => 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [[[[.,.],[.,[.,.]]],.],.]
 => 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [[[.,.],[.,[.,.]]],[.,.]]
 => 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [[[.,.],[.,[.,[.,.]]]],.]
 => 3
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[.,.],[.,[.,[.,[.,.]]]]]
 => 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [[[[[.,[.,.]],.],.],.],.]
 => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [[[[.,[.,.]],.],.],[.,.]]
 => 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [[[[.,[.,.]],.],[.,.]],.]
 => 2
Description
The number of internal nodes of a binary tree. That is, the total number of nodes of the tree minus [[St000203]]. A counting formula for the total number of internal nodes across all binary trees of size $n$ is given in [1]. This is equivalent to the number of internal triangles in all triangulations of an $(n+1)$-gon.
Matching statistic: St000272
Mapping
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00250: Graphs —clique graph⟶ 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)
 => ([],1)
 => 0
[2] => ([],2)
 => ([],2)
 => 0
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 0
[1,2] => ([(1,2)],3)
 => ([],2)
 => 0
[2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
[3] => ([],3)
 => ([],3)
 => 0
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[1,3] => ([(2,3)],4)
 => ([],3)
 => 0
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[2,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[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)
 => ([],1)
 => 0
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,4] => ([(3,4)],5)
 => ([],4)
 => 0
[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)],2)
 => 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,3] => ([(2,4),(3,4)],5)
 => ([(2,3)],4)
 => 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[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)
 => ([],1)
 => 0
[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)
 => ([],2)
 => 0
[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)],2)
 => 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => 0
[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)],2)
 => 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 1
[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),(1,2)],3)
 => 2
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([],4)
 => 0
[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)],2)
 => 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 1
[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),(1,2)],3)
 => 2
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3)],4)
 => 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)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,5] => ([(4,5)],6)
 => ([],5)
 => 0
[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)],2)
 => 1
[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)],3)
 => 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)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
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.
Matching statistic: St000362
Mapping
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00250: Graphs —clique graph⟶ Graphs
St000362: 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)
 => ([],1)
 => 0
[2] => ([],2)
 => ([],2)
 => 0
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 0
[1,2] => ([(1,2)],3)
 => ([],2)
 => 0
[2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
[3] => ([],3)
 => ([],3)
 => 0
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[1,3] => ([(2,3)],4)
 => ([],3)
 => 0
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[2,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[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)
 => ([],1)
 => 0
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,4] => ([(3,4)],5)
 => ([],4)
 => 0
[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)],2)
 => 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,3] => ([(2,4),(3,4)],5)
 => ([(2,3)],4)
 => 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[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)
 => ([],1)
 => 0
[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)
 => ([],2)
 => 0
[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)],2)
 => 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => 0
[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)],2)
 => 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 1
[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),(1,2)],3)
 => 2
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([],4)
 => 0
[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)],2)
 => 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 1
[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),(1,2)],3)
 => 2
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3)],4)
 => 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)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,5] => ([(4,5)],6)
 => ([],5)
 => 0
[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)],2)
 => 1
[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)],3)
 => 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)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
Description
The size of a minimal vertex cover 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 $S$. Finding a minimal vertex cover is an NP-hard optimization problem.
Matching statistic: St000454
Mapping
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00250: Graphs —clique graph⟶ Graphs
St000454: 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)
 => ([],1)
 => 0
[2] => ([],2)
 => ([],2)
 => 0
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 0
[1,2] => ([(1,2)],3)
 => ([],2)
 => 0
[2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
[3] => ([],3)
 => ([],3)
 => 0
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[1,3] => ([(2,3)],4)
 => ([],3)
 => 0
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[2,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[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)
 => ([],1)
 => 0
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,4] => ([(3,4)],5)
 => ([],4)
 => 0
[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)],2)
 => 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,3] => ([(2,4),(3,4)],5)
 => ([(2,3)],4)
 => 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[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)
 => ([],1)
 => 0
[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)
 => ([],2)
 => 0
[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)],2)
 => 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => 0
[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)],2)
 => 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 1
[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),(1,2)],3)
 => 2
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([],4)
 => 0
[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)],2)
 => 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 1
[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),(1,2)],3)
 => 2
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3)],4)
 => 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)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,5] => ([(4,5)],6)
 => ([],5)
 => 0
[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)],2)
 => 1
[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)],3)
 => 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)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
Description
The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St000536
Mapping
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00250: Graphs —clique graph⟶ Graphs
St000536: 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)
 => ([],1)
 => 0
[2] => ([],2)
 => ([],2)
 => 0
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 0
[1,2] => ([(1,2)],3)
 => ([],2)
 => 0
[2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
[3] => ([],3)
 => ([],3)
 => 0
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[1,3] => ([(2,3)],4)
 => ([],3)
 => 0
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[2,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[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)
 => ([],1)
 => 0
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,4] => ([(3,4)],5)
 => ([],4)
 => 0
[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)],2)
 => 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,3] => ([(2,4),(3,4)],5)
 => ([(2,3)],4)
 => 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[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)
 => ([],1)
 => 0
[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)
 => ([],2)
 => 0
[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)],2)
 => 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => 0
[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)],2)
 => 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 1
[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),(1,2)],3)
 => 2
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([],4)
 => 0
[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)],2)
 => 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 1
[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),(1,2)],3)
 => 2
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3)],4)
 => 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)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,5] => ([(4,5)],6)
 => ([],5)
 => 0
[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)],2)
 => 1
[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)],3)
 => 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)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
Description
The pathwidth of a graph.
Matching statistic: St000987
Mapping
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00250: Graphs —clique 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] => ([(0,1)],2)
 => ([],1)
 => 0
[2] => ([],2)
 => ([],2)
 => 0
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 0
[1,2] => ([(1,2)],3)
 => ([],2)
 => 0
[2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
[3] => ([],3)
 => ([],3)
 => 0
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[1,3] => ([(2,3)],4)
 => ([],3)
 => 0
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[2,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[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)
 => ([],1)
 => 0
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,4] => ([(3,4)],5)
 => ([],4)
 => 0
[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)],2)
 => 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,3] => ([(2,4),(3,4)],5)
 => ([(2,3)],4)
 => 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[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)
 => ([],1)
 => 0
[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)
 => ([],2)
 => 0
[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)],2)
 => 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => 0
[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)],2)
 => 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 1
[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),(1,2)],3)
 => 2
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([],4)
 => 0
[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)],2)
 => 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 1
[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),(1,2)],3)
 => 2
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3)],4)
 => 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)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,5] => ([(4,5)],6)
 => ([],5)
 => 0
[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)],2)
 => 1
[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)],3)
 => 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)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
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.
The following 80 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001120The length of a longest path in a graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001644The dimension of a graph. St001962The proper pathwidth of a graph. St001971The number of negative eigenvalues of the adjacency matrix of the graph. 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. 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. 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. St001725The harmonious chromatic number of a graph. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St000010The length of the partition. St000676The number of odd rises of a Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St000093The cardinality of a maximal independent set of vertices of a graph. St000288The number of ones in a binary word. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St000822The Hadwiger number of the graph. St001330The hat guessing number of a graph. St000497The lcb statistic of a set partition. St000572The dimension exponent of a set partition. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001812The biclique partition number of a graph. St000065The number of entries equal to -1 in an alternating sign matrix. St001642The Prague dimension of a graph. St000356The number of occurrences of the pattern 13-2. St000463The number of admissible inversions of a permutation. St000225Difference between largest and smallest parts in a partition. St000734The last entry in the first row of a standard tableau. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000358The number of occurrences of the pattern 31-2. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000145The Dyson rank of a partition. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. St000119The number of occurrences of the pattern 321 in a permutation. St000233The number of nestings of a set partition. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St001411The number of patterns 321 or 3412 in a permutation. St001727The number of invisible inversions of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000317The cycle descent number of a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001552The number of inversions between excedances and fixed points of a permutation. St000711The number of big exceedences of a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St001674The number of vertices of the largest induced star graph in the graph. St001645The pebbling number of a connected graph. St001435The number of missing boxes in the first row. St000299The number of nonisomorphic vertex-induced subtrees. St001960The number of descents of a permutation minus one if its first entry is not one. St001881The number of factors of a lattice as a Cartesian product of lattices.