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Your data matches 34 different statistics following compositions of up to 3 maps.
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Matching statistic: St000362
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Values
([],1)
=> 0
([],2)
=> 0
([(0,1)],2)
=> 1
([],3)
=> 0
([(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> 1
([(0,1),(0,2),(1,2)],3)
=> 2
([],4)
=> 0
([(2,3)],4)
=> 1
([(1,3),(2,3)],4)
=> 1
([(0,3),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> 2
([(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([],5)
=> 0
([(3,4)],5)
=> 1
([(2,4),(3,4)],5)
=> 1
([(1,4),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(1,4),(2,3)],5)
=> 2
([(1,4),(2,3),(3,4)],5)
=> 2
([(0,1),(2,4),(3,4)],5)
=> 2
([(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3
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
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%
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
([],2)
=> [2] => [2]
=> 0
([(0,1)],2)
=> [1,1] => [1,1]
=> 1
([],3)
=> [3] => [3]
=> 0
([(1,2)],3)
=> [2,1] => [2,1]
=> 1
([(0,2),(1,2)],3)
=> [2,1] => [2,1]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [1,1,1]
=> 2
([],4)
=> [4] => [4]
=> 0
([(2,3)],4)
=> [3,1] => [3,1]
=> 1
([(1,3),(2,3)],4)
=> [3,1] => [3,1]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [3,1] => [3,1]
=> 1
([(0,3),(1,2)],4)
=> [2,2] => [2,2]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => [2,2]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [2,1,1]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [2,1,1]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => [2,2]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [2,1,1]
=> 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1]
=> 3
([],5)
=> [5] => [5]
=> 0
([(3,4)],5)
=> [4,1] => [4,1]
=> 1
([(2,4),(3,4)],5)
=> [4,1] => [4,1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [4,1] => [4,1]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => [4,1]
=> 1
([(1,4),(2,3)],5)
=> [3,2] => [3,2]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [3,2] => [3,2]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [3,2] => [3,2]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => [3,2]
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1]
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [3,2]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => [3,2]
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1]
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1]
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [3,2]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => [3,2]
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [2,2,1]
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [2,1,1,1]
=> 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
([(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
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [2,2,1]
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> 3
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
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%
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
([],2)
=> [2] => [1,1,0,0]
=> 0
([(0,1)],2)
=> [1,1] => [1,0,1,0]
=> 1
([],3)
=> [3] => [1,1,1,0,0,0]
=> 0
([(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> 1
([(0,2),(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [1,0,1,0,1,0]
=> 2
([],4)
=> [4] => [1,1,1,1,0,0,0,0]
=> 0
([(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
([(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
([(0,3),(1,2)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
([(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
([],5)
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
([(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
([(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
([(1,4),(2,3)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
([(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
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(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
([(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
([(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
([(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
([(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
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
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)
(load all 2 compositions to match this statistic)
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%
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
([],2)
=> [2] => [1,1,0,0]
=> 0
([(0,1)],2)
=> [1,1] => [1,0,1,0]
=> 1
([],3)
=> [3] => [1,1,1,0,0,0]
=> 0
([(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> 1
([(0,2),(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [1,0,1,0,1,0]
=> 2
([],4)
=> [4] => [1,1,1,1,0,0,0,0]
=> 0
([(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
([(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
([(0,3),(1,2)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
([(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
([],5)
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
([(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
([(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
([(1,4),(2,3)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
([(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
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(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
([(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
([(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
([(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
([(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
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
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: St000171
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
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%
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
([],2)
=> [2] => [2] => ([],2)
=> 0
([(0,1)],2)
=> [1,1] => [1,1] => ([(0,1)],2)
=> 1
([],3)
=> [3] => [3] => ([],3)
=> 0
([(1,2)],3)
=> [2,1] => [1,2] => ([(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> [2,1] => [1,2] => ([(1,2)],3)
=> 1
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
([],4)
=> [4] => [4] => ([],4)
=> 0
([(2,3)],4)
=> [3,1] => [1,3] => ([(2,3)],4)
=> 1
([(1,3),(2,3)],4)
=> [3,1] => [1,3] => ([(2,3)],4)
=> 1
([(0,3),(1,3),(2,3)],4)
=> [3,1] => [1,3] => ([(2,3)],4)
=> 1
([(0,3),(1,2)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
([(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
([(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
([],5)
=> [5] => [5] => ([],5)
=> 0
([(3,4)],5)
=> [4,1] => [1,4] => ([(3,4)],5)
=> 1
([(2,4),(3,4)],5)
=> [4,1] => [1,4] => ([(3,4)],5)
=> 1
([(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,4] => ([(3,4)],5)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,4] => ([(3,4)],5)
=> 1
([(1,4),(2,3)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(1,4),(2,3),(3,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(0,1),(2,4),(3,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
([(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
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(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
([(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
([(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
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(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
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
Description
The degree of the graph.
This is the maximal vertex degree of a graph.
Matching statistic: St000228
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%
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
([],2)
=> [2] => [2]
=> []
=> 0
([(0,1)],2)
=> [1,1] => [1,1]
=> [1]
=> 1
([],3)
=> [3] => [3]
=> []
=> 0
([(1,2)],3)
=> [2,1] => [2,1]
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [2,1] => [2,1]
=> [1]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [1,1,1]
=> [1,1]
=> 2
([],4)
=> [4] => [4]
=> []
=> 0
([(2,3)],4)
=> [3,1] => [3,1]
=> [1]
=> 1
([(1,3),(2,3)],4)
=> [3,1] => [3,1]
=> [1]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [3,1] => [3,1]
=> [1]
=> 1
([(0,3),(1,2)],4)
=> [2,2] => [2,2]
=> [2]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => [2,2]
=> [2]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [2,1,1]
=> [1,1]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [2,1,1]
=> [1,1]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => [2,2]
=> [2]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [2,1,1]
=> [1,1]
=> 2
([(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
([],5)
=> [5] => [5]
=> []
=> 0
([(3,4)],5)
=> [4,1] => [4,1]
=> [1]
=> 1
([(2,4),(3,4)],5)
=> [4,1] => [4,1]
=> [1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [4,1] => [4,1]
=> [1]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => [4,1]
=> [1]
=> 1
([(1,4),(2,3)],5)
=> [3,2] => [3,2]
=> [2]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [3,2] => [3,2]
=> [2]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [3,2] => [3,2]
=> [2]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1]
=> [1,1]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => [3,2]
=> [2]
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1]
=> [1,1]
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1]
=> [1,1]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [3,2]
=> [2]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => [3,2]
=> [2]
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1]
=> [1,1]
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1]
=> [1,1]
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1]
=> [1,1]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [3,2]
=> [2]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1]
=> [1,1]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => [3,2]
=> [2]
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> [2,1]
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> [2,1]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> [2,1]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [2,2,1]
=> [2,1]
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> [2,1]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> [2,1]
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> [2,1]
=> 3
([(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
([(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
([(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
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [2,2,1]
=> [2,1]
=> 3
([(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
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)
(load all 2 compositions to match this statistic)
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%
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
([],2)
=> [2] => [2] => ([],2)
=> 0
([(0,1)],2)
=> [1,1] => [1,1] => ([(0,1)],2)
=> 1
([],3)
=> [3] => [3] => ([],3)
=> 0
([(1,2)],3)
=> [2,1] => [1,2] => ([(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> [2,1] => [1,2] => ([(1,2)],3)
=> 1
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
([],4)
=> [4] => [4] => ([],4)
=> 0
([(2,3)],4)
=> [3,1] => [1,3] => ([(2,3)],4)
=> 1
([(1,3),(2,3)],4)
=> [3,1] => [1,3] => ([(2,3)],4)
=> 1
([(0,3),(1,3),(2,3)],4)
=> [3,1] => [1,3] => ([(2,3)],4)
=> 1
([(0,3),(1,2)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
([(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
([(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
([],5)
=> [5] => [5] => ([],5)
=> 0
([(3,4)],5)
=> [4,1] => [1,4] => ([(3,4)],5)
=> 1
([(2,4),(3,4)],5)
=> [4,1] => [1,4] => ([(3,4)],5)
=> 1
([(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,4] => ([(3,4)],5)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,4] => ([(3,4)],5)
=> 1
([(1,4),(2,3)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(1,4),(2,3),(3,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(0,1),(2,4),(3,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
([(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
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(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
([(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
([(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
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(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
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
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: St001265
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001265: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001265: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1] => [1,0]
=> 0
([],2)
=> [2] => [1,1] => [1,0,1,0]
=> 0
([(0,1)],2)
=> [1,1] => [2] => [1,1,0,0]
=> 1
([],3)
=> [3] => [1,1,1] => [1,0,1,0,1,0]
=> 0
([(1,2)],3)
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
([(0,2),(1,2)],3)
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> 2
([],4)
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
([(2,3)],4)
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
([(1,3),(2,3)],4)
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
([(0,3),(1,2)],4)
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 3
([],5)
=> [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
([(3,4)],5)
=> [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
([(2,4),(3,4)],5)
=> [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
([(1,4),(2,3)],5)
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
([(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]
=> 2
([(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]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
([(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]
=> 2
([(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]
=> 2
([(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]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
([(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]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
([(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]
=> 3
([(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]
=> 3
([(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]
=> 3
([(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]
=> 3
([(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]
=> 3
([(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]
=> 3
([(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]
=> 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
([(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]
=> 3
([(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]
=> 3
Description
The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra.
Matching statistic: St000738
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%
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 = 0 + 1
([],2)
=> [2] => [2]
=> [[1,2]]
=> 1 = 0 + 1
([(0,1)],2)
=> [1,1] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
([],3)
=> [3] => [3]
=> [[1,2,3]]
=> 1 = 0 + 1
([(1,2)],3)
=> [2,1] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
([(0,2),(1,2)],3)
=> [2,1] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
([],4)
=> [4] => [4]
=> [[1,2,3,4]]
=> 1 = 0 + 1
([(2,3)],4)
=> [3,1] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> [3,1] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
=> [3,1] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
([(0,3),(1,2)],4)
=> [2,2] => [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
([(0,3),(1,2),(2,3)],4)
=> [2,2] => [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
([(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 = 3 + 1
([],5)
=> [5] => [5]
=> [[1,2,3,4,5]]
=> 1 = 0 + 1
([(3,4)],5)
=> [4,1] => [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
([(2,4),(3,4)],5)
=> [4,1] => [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
([(1,4),(2,4),(3,4)],5)
=> [4,1] => [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
([(1,4),(2,3)],5)
=> [3,2] => [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
([(1,4),(2,3),(3,4)],5)
=> [3,2] => [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
([(0,1),(2,4),(3,4)],5)
=> [3,2] => [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
([(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 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
([(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 = 2 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4 = 3 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4 = 3 + 1
([(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 = 3 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4 = 3 + 1
([(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 = 3 + 1
([(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 = 3 + 1
([(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 = 3 + 1
([(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 = 3 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 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]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
([(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 = 3 + 1
([(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 = 3 + 1
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
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%
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 = 0 + 1
([],2)
=> [2] => [2] => [1,1,0,0]
=> 1 = 0 + 1
([(0,1)],2)
=> [1,1] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
([],3)
=> [3] => [3] => [1,1,1,0,0,0]
=> 1 = 0 + 1
([(1,2)],3)
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
([(0,2),(1,2)],3)
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 2 + 1
([],4)
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
([(2,3)],4)
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
([(0,3),(1,2)],4)
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
([(0,3),(1,2),(2,3)],4)
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
([(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 = 2 + 1
([(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 = 3 + 1
([],5)
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
([(3,4)],5)
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
([(2,4),(3,4)],5)
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
([(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
([(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 + 1
([(1,4),(2,3)],5)
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
([(1,4),(2,3),(3,4)],5)
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
([(0,1),(2,4),(3,4)],5)
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
([(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 = 2 + 1
([(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 = 2 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
([(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 = 2 + 1
([(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 = 2 + 1
([(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 = 2 + 1
([(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 = 2 + 1
([(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 = 2 + 1
([(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 = 2 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
([(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 = 3 + 1
([(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 = 3 + 1
([(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 = 3 + 1
([(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 = 3 + 1
([(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 = 3 + 1
([(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 = 3 + 1
([(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 = 3 + 1
([(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 = 3 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 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] => [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
([(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 = 3 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 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 $A$.
The following 24 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001725The harmonious chromatic number of a graph. 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. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000259The diameter of a connected graph. St001060The distinguishing index of a graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000778The metric dimension of a graph. St001875The number of simple modules with projective dimension at most 1. St000454The largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St000120The number of left tunnels of a Dyck path. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St000741The Colin de Verdière graph invariant. St000264The girth of a graph, which is not a tree. 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 $eA$. St001812The biclique partition number of a graph. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000422The energy of a graph, if it is integral. 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. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
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