Your data matches 28 different statistics following compositions of up to 3 maps.
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Matching statistic: St000362
(load all 7 compositions to match this statistic)
Mapping
St000362: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => 0 = 1 - 1
([],2)
 => 0 = 1 - 1
([(0,1)],2)
 => 1 = 2 - 1
([],3)
 => 0 = 1 - 1
([(1,2)],3)
 => 1 = 2 - 1
([(0,2),(1,2)],3)
 => 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([],4)
 => 0 = 1 - 1
([(2,3)],4)
 => 1 = 2 - 1
([(1,3),(2,3)],4)
 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
 => 1 = 2 - 1
([(0,3),(1,2)],4)
 => 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
 => 2 = 3 - 1
([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
([],5)
 => 0 = 1 - 1
([(3,4)],5)
 => 1 = 2 - 1
([(2,4),(3,4)],5)
 => 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
([(1,4),(2,3)],5)
 => 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
 => 2 = 3 - 1
([(0,1),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => 2 = 3 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 3 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 3 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 3 = 4 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 3 = 4 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
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: St001176
Mapping
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => [1]
 => 0 = 1 - 1
([],2)
 => [2] => [2]
 => 0 = 1 - 1
([(0,1)],2)
 => [1,1] => [1,1]
 => 1 = 2 - 1
([],3)
 => [3] => [3]
 => 0 = 1 - 1
([(1,2)],3)
 => [2,1] => [2,1]
 => 1 = 2 - 1
([(0,2),(1,2)],3)
 => [2,1] => [2,1]
 => 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [1,1,1]
 => 2 = 3 - 1
([],4)
 => [4] => [4]
 => 0 = 1 - 1
([(2,3)],4)
 => [3,1] => [3,1]
 => 1 = 2 - 1
([(1,3),(2,3)],4)
 => [3,1] => [3,1]
 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
 => [3,1] => [3,1]
 => 1 = 2 - 1
([(0,3),(1,2)],4)
 => [2,2] => [2,2]
 => 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [2,2]
 => 2 = 3 - 1
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [2,1,1]
 => 2 = 3 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [2,1,1]
 => 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [2,2]
 => 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [2,1,1]
 => 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,1,1,1]
 => 3 = 4 - 1
([],5)
 => [5] => [5]
 => 0 = 1 - 1
([(3,4)],5)
 => [4,1] => [4,1]
 => 1 = 2 - 1
([(2,4),(3,4)],5)
 => [4,1] => [4,1]
 => 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
 => [4,1] => [4,1]
 => 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => [4,1]
 => 1 = 2 - 1
([(1,4),(2,3)],5)
 => [3,2] => [3,2]
 => 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
 => [3,2] => [3,2]
 => 2 = 3 - 1
([(0,1),(2,4),(3,4)],5)
 => [3,2] => [3,2]
 => 2 = 3 - 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [3,1,1]
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,2] => [3,2]
 => 2 = 3 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [3,1,1]
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [3,1,1]
 => 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [3,2]
 => 2 = 3 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2] => [3,2]
 => 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [3,1,1]
 => 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [3,1,1]
 => 2 = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [3,1,1]
 => 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [3,2]
 => 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [3,1,1]
 => 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [3,2] => [3,2]
 => 2 = 3 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => 3 = 4 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [2,2,1]
 => 3 = 4 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => 3 = 4 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => 3 = 4 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [2,1,1,1]
 => 3 = 4 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [2,1,1,1]
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [2,1,1,1]
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [2,2,1]
 => 3 = 4 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => 3 = 4 - 1
Description
The size of a partition minus its first part. This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St001227
Mapping
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
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] => [1,0]
 => 0 = 1 - 1
([],2)
 => [2] => [1,1,0,0]
 => 0 = 1 - 1
([(0,1)],2)
 => [1,1] => [1,0,1,0]
 => 1 = 2 - 1
([],3)
 => [3] => [1,1,1,0,0,0]
 => 0 = 1 - 1
([(1,2)],3)
 => [2,1] => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,2),(1,2)],3)
 => [2,1] => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [1,0,1,0,1,0]
 => 2 = 3 - 1
([],4)
 => [4] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
([(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(1,3),(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(1,2)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
([],5)
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
([(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(2,4),(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(1,4),(2,3)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 3 - 1
([(0,1),(2,4),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 3 - 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 3 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 3 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 3 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 4 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 4 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 4 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 4 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 4 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 4 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 4 - 1
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: St001509
(load all 2 compositions to match this statistic)
Mapping
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001509: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => [1,0]
 => 0 = 1 - 1
([],2)
 => [2] => [1,1,0,0]
 => 0 = 1 - 1
([(0,1)],2)
 => [1,1] => [1,0,1,0]
 => 1 = 2 - 1
([],3)
 => [3] => [1,1,1,0,0,0]
 => 0 = 1 - 1
([(1,2)],3)
 => [2,1] => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,2),(1,2)],3)
 => [2,1] => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [1,0,1,0,1,0]
 => 2 = 3 - 1
([],4)
 => [4] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
([(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(1,3),(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(1,2)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
([],5)
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
([(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(2,4),(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(1,4),(2,3)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 3 - 1
([(0,1),(2,4),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 3 - 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 3 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 3 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 3 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 4 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 4 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 4 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 4 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 4 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 4 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 4 - 1
Description
The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. Given two lattice paths $U,L$ from $(0,0)$ to $(d,n-d)$, [1] describes a bijection between lattice paths weakly between $U$ and $L$ and subsets of $\{1,\dots,n\}$ such that the set of all such subsets gives the standard complex of the lattice path matroid $M[U,L]$. This statistic gives the cardinality of the image of this bijection when a Dyck path is considered as a path weakly below the diagonal and relative to the trivial lower boundary.
Matching statistic: St000738
Mapping
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: 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] => [2]
 => [[1,2]]
 => 1
([(0,1)],2)
 => [1,1] => [1,1]
 => [[1],[2]]
 => 2
([],3)
 => [3] => [3]
 => [[1,2,3]]
 => 1
([(1,2)],3)
 => [2,1] => [2,1]
 => [[1,3],[2]]
 => 2
([(0,2),(1,2)],3)
 => [2,1] => [2,1]
 => [[1,3],[2]]
 => 2
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [1,1,1]
 => [[1],[2],[3]]
 => 3
([],4)
 => [4] => [4]
 => [[1,2,3,4]]
 => 1
([(2,3)],4)
 => [3,1] => [3,1]
 => [[1,3,4],[2]]
 => 2
([(1,3),(2,3)],4)
 => [3,1] => [3,1]
 => [[1,3,4],[2]]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,1] => [3,1]
 => [[1,3,4],[2]]
 => 2
([(0,3),(1,2)],4)
 => [2,2] => [2,2]
 => [[1,2],[3,4]]
 => 3
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [2,2]
 => [[1,2],[3,4]]
 => 3
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [2,1,1]
 => [[1,4],[2],[3]]
 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [2,1,1]
 => [[1,4],[2],[3]]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [2,2]
 => [[1,2],[3,4]]
 => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [2,1,1]
 => [[1,4],[2],[3]]
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 4
([],5)
 => [5] => [5]
 => [[1,2,3,4,5]]
 => 1
([(3,4)],5)
 => [4,1] => [4,1]
 => [[1,3,4,5],[2]]
 => 2
([(2,4),(3,4)],5)
 => [4,1] => [4,1]
 => [[1,3,4,5],[2]]
 => 2
([(1,4),(2,4),(3,4)],5)
 => [4,1] => [4,1]
 => [[1,3,4,5],[2]]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => [4,1]
 => [[1,3,4,5],[2]]
 => 2
([(1,4),(2,3)],5)
 => [3,2] => [3,2]
 => [[1,2,5],[3,4]]
 => 3
([(1,4),(2,3),(3,4)],5)
 => [3,2] => [3,2]
 => [[1,2,5],[3,4]]
 => 3
([(0,1),(2,4),(3,4)],5)
 => [3,2] => [3,2]
 => [[1,2,5],[3,4]]
 => 3
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,2] => [3,2]
 => [[1,2,5],[3,4]]
 => 3
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 3
([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [3,2]
 => [[1,2,5],[3,4]]
 => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2] => [3,2]
 => [[1,2,5],[3,4]]
 => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [3,2]
 => [[1,2,5],[3,4]]
 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => [3,2] => [3,2]
 => [[1,2,5],[3,4]]
 => 3
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 4
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 4
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 4
Description
The first entry in the last row of a standard tableau. For the last entry in the first row, see [[St000734]].
Matching statistic: St001291
Mapping
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
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] => [1] => [1,0]
 => 1
([],2)
 => [2] => [2] => [1,1,0,0]
 => 1
([(0,1)],2)
 => [1,1] => [1,1] => [1,0,1,0]
 => 2
([],3)
 => [3] => [3] => [1,1,1,0,0,0]
 => 1
([(1,2)],3)
 => [2,1] => [1,2] => [1,0,1,1,0,0]
 => 2
([(0,2),(1,2)],3)
 => [2,1] => [1,2] => [1,0,1,1,0,0]
 => 2
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
 => 3
([],4)
 => [4] => [4] => [1,1,1,1,0,0,0,0]
 => 1
([(2,3)],4)
 => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
([(1,3),(2,3)],4)
 => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
([(0,3),(1,2)],4)
 => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 3
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 3
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
([],5)
 => [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1
([(3,4)],5)
 => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2
([(2,4),(3,4)],5)
 => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2
([(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2
([(1,4),(2,3)],5)
 => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
([(1,4),(2,3),(3,4)],5)
 => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
([(0,1),(2,4),(3,4)],5)
 => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3
([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4
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 2 compositions to match this statistic)
Mapping
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ 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] => ([],2)
 => 1
([(0,1)],2)
 => [1,1] => [1,1] => ([(0,1)],2)
 => 2
([],3)
 => [3] => [3] => ([],3)
 => 1
([(1,2)],3)
 => [2,1] => [1,2] => ([(1,2)],3)
 => 2
([(0,2),(1,2)],3)
 => [2,1] => [1,2] => ([(1,2)],3)
 => 2
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
([],4)
 => [4] => [4] => ([],4)
 => 1
([(2,3)],4)
 => [3,1] => [1,3] => ([(2,3)],4)
 => 2
([(1,3),(2,3)],4)
 => [3,1] => [1,3] => ([(2,3)],4)
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,1] => [1,3] => ([(2,3)],4)
 => 2
([(0,3),(1,2)],4)
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 3
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 3
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([],5)
 => [5] => [5] => ([],5)
 => 1
([(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => 2
([(2,4),(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => 2
([(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => 2
([(1,4),(2,3)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 3
([(1,4),(2,3),(3,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 3
([(0,1),(2,4),(3,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 3
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 3
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3
([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 3
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(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 2 compositions to match this statistic)
Mapping
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ 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] => ([],1)
 => 0 = 1 - 1
([],2)
 => [2] => [2] => ([],2)
 => 0 = 1 - 1
([(0,1)],2)
 => [1,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
([],3)
 => [3] => [3] => ([],3)
 => 0 = 1 - 1
([(1,2)],3)
 => [2,1] => [1,2] => ([(1,2)],3)
 => 1 = 2 - 1
([(0,2),(1,2)],3)
 => [2,1] => [1,2] => ([(1,2)],3)
 => 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([],4)
 => [4] => [4] => ([],4)
 => 0 = 1 - 1
([(2,3)],4)
 => [3,1] => [1,3] => ([(2,3)],4)
 => 1 = 2 - 1
([(1,3),(2,3)],4)
 => [3,1] => [1,3] => ([(2,3)],4)
 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
 => [3,1] => [1,3] => ([(2,3)],4)
 => 1 = 2 - 1
([(0,3),(1,2)],4)
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 3 - 1
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
([],5)
 => [5] => [5] => ([],5)
 => 0 = 1 - 1
([(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => 1 = 2 - 1
([(2,4),(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => 1 = 2 - 1
([(1,4),(2,3)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,1),(2,4),(3,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 3 - 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 3 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(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: St000228
Mapping
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => [1]
 => []
 => 0 = 1 - 1
([],2)
 => [2] => [2]
 => []
 => 0 = 1 - 1
([(0,1)],2)
 => [1,1] => [1,1]
 => [1]
 => 1 = 2 - 1
([],3)
 => [3] => [3]
 => []
 => 0 = 1 - 1
([(1,2)],3)
 => [2,1] => [2,1]
 => [1]
 => 1 = 2 - 1
([(0,2),(1,2)],3)
 => [2,1] => [2,1]
 => [1]
 => 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [1,1,1]
 => [1,1]
 => 2 = 3 - 1
([],4)
 => [4] => [4]
 => []
 => 0 = 1 - 1
([(2,3)],4)
 => [3,1] => [3,1]
 => [1]
 => 1 = 2 - 1
([(1,3),(2,3)],4)
 => [3,1] => [3,1]
 => [1]
 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
 => [3,1] => [3,1]
 => [1]
 => 1 = 2 - 1
([(0,3),(1,2)],4)
 => [2,2] => [2,2]
 => [2]
 => 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [2,2]
 => [2]
 => 2 = 3 - 1
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [2,1,1]
 => [1,1]
 => 2 = 3 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [2,1,1]
 => [1,1]
 => 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [2,2]
 => [2]
 => 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [2,1,1]
 => [1,1]
 => 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,1,1,1]
 => [1,1,1]
 => 3 = 4 - 1
([],5)
 => [5] => [5]
 => []
 => 0 = 1 - 1
([(3,4)],5)
 => [4,1] => [4,1]
 => [1]
 => 1 = 2 - 1
([(2,4),(3,4)],5)
 => [4,1] => [4,1]
 => [1]
 => 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
 => [4,1] => [4,1]
 => [1]
 => 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => [4,1]
 => [1]
 => 1 = 2 - 1
([(1,4),(2,3)],5)
 => [3,2] => [3,2]
 => [2]
 => 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
 => [3,2] => [3,2]
 => [2]
 => 2 = 3 - 1
([(0,1),(2,4),(3,4)],5)
 => [3,2] => [3,2]
 => [2]
 => 2 = 3 - 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [3,1,1]
 => [1,1]
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,2] => [3,2]
 => [2]
 => 2 = 3 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [3,1,1]
 => [1,1]
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [3,1,1]
 => [1,1]
 => 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [3,2]
 => [2]
 => 2 = 3 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2] => [3,2]
 => [2]
 => 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [3,1,1]
 => [1,1]
 => 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [3,1,1]
 => [1,1]
 => 2 = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [3,1,1]
 => [1,1]
 => 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [3,2]
 => [2]
 => 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [3,1,1]
 => [1,1]
 => 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [3,2] => [3,2]
 => [2]
 => 2 = 3 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => [2,1]
 => 3 = 4 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => [2,1]
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => [2,1]
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [2,2,1]
 => [2,1]
 => 3 = 4 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => [2,1]
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => [2,1]
 => 3 = 4 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => [2,1]
 => 3 = 4 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [2,1,1,1]
 => [1,1,1]
 => 3 = 4 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [2,1,1,1]
 => [1,1,1]
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [2,1,1,1]
 => [1,1,1]
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [2,2,1]
 => [2,1]
 => 3 = 4 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => [2,1]
 => 3 = 4 - 1
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St000987
(load all 2 compositions to match this statistic)
Mapping
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ 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] => ([],1)
 => 0 = 1 - 1
([],2)
 => [2] => [2] => ([],2)
 => 0 = 1 - 1
([(0,1)],2)
 => [1,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
([],3)
 => [3] => [3] => ([],3)
 => 0 = 1 - 1
([(1,2)],3)
 => [2,1] => [1,2] => ([(1,2)],3)
 => 1 = 2 - 1
([(0,2),(1,2)],3)
 => [2,1] => [1,2] => ([(1,2)],3)
 => 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([],4)
 => [4] => [4] => ([],4)
 => 0 = 1 - 1
([(2,3)],4)
 => [3,1] => [1,3] => ([(2,3)],4)
 => 1 = 2 - 1
([(1,3),(2,3)],4)
 => [3,1] => [1,3] => ([(2,3)],4)
 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
 => [3,1] => [1,3] => ([(2,3)],4)
 => 1 = 2 - 1
([(0,3),(1,2)],4)
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 3 - 1
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
([],5)
 => [5] => [5] => ([],5)
 => 0 = 1 - 1
([(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => 1 = 2 - 1
([(2,4),(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => 1 = 2 - 1
([(1,4),(2,3)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,1),(2,4),(3,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 3 - 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 3 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(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.
The following 18 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001108The 2-dynamic chromatic number of a graph. St001494The Alon-Tarsi number of a graph. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001316The domatic number of a graph. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000264The girth of a graph, which is not a tree. St000422The energy of a graph, if it is integral. St001875The number of simple modules with projective dimension at most 1. St001060The distinguishing index of a graph.