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Your data matches 114 different statistics following compositions of up to 3 maps.
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Matching statistic: St000053
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 0
[1,2] => [2] => [1,1,0,0]
=> 0
[2,1] => [2] => [1,1,0,0]
=> 0
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [3] => [1,1,1,0,0,0]
=> 0
[2,3,1] => [3] => [1,1,1,0,0,0]
=> 0
[3,1,2] => [3] => [1,1,1,0,0,0]
=> 0
[3,2,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,3,4,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[2,1,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,1,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,3,1,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,3,4,1] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,4,1,3] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,4,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[3,1,2,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[3,1,4,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,2,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,2,4,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,4,1,2] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[3,4,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,1,2,3] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[4,1,3,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,2,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[4,2,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,3,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,3,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,4,5,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,5,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,5,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,3,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,2,5,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,4,5,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,2,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,4,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,2,3,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,4,2,5,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,2,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,5,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,5,2,3] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
Description
The number of valleys of the Dyck path.
Matching statistic: St000272
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000272: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000272: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 0
[1,2] => [2] => ([],2)
=> 0
[2,1] => [2] => ([],2)
=> 0
[1,2,3] => [3] => ([],3)
=> 0
[1,3,2] => [1,2] => ([(1,2)],3)
=> 1
[2,1,3] => [3] => ([],3)
=> 0
[2,3,1] => [3] => ([],3)
=> 0
[3,1,2] => [3] => ([],3)
=> 0
[3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,3,4] => [4] => ([],4)
=> 0
[1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[1,3,2,4] => [1,3] => ([(2,3)],4)
=> 1
[1,3,4,2] => [1,3] => ([(2,3)],4)
=> 1
[1,4,2,3] => [1,3] => ([(2,3)],4)
=> 1
[1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [4] => ([],4)
=> 0
[2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[2,3,1,4] => [4] => ([],4)
=> 0
[2,3,4,1] => [4] => ([],4)
=> 0
[2,4,1,3] => [4] => ([],4)
=> 0
[2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[3,1,2,4] => [4] => ([],4)
=> 0
[3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,4,1,2] => [4] => ([],4)
=> 0
[3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,1,2,3] => [4] => ([],4)
=> 0
[4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => [1,3] => ([(2,3)],4)
=> 1
[4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,2,3,4,5] => [5] => ([],5)
=> 0
[1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> 1
[1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,4,2,5] => [1,4] => ([(3,4)],5)
=> 1
[1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> 1
[1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> 1
[1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => [1,4] => ([(3,4)],5)
=> 1
[1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => [1,4] => ([(3,4)],5)
=> 1
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: St000306
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 0
[1,2] => [2] => [1,1,0,0]
=> 0
[2,1] => [2] => [1,1,0,0]
=> 0
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [3] => [1,1,1,0,0,0]
=> 0
[2,3,1] => [3] => [1,1,1,0,0,0]
=> 0
[3,1,2] => [3] => [1,1,1,0,0,0]
=> 0
[3,2,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,3,4,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[2,1,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,1,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,3,1,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,3,4,1] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,4,1,3] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,4,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[3,1,2,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[3,1,4,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,2,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,2,4,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,4,1,2] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[3,4,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,1,2,3] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[4,1,3,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,2,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[4,2,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,3,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,3,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,4,5,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,5,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,5,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,3,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,2,5,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,4,5,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,2,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,4,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,2,3,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,4,2,5,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,2,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,5,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,5,2,3] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
Description
The bounce count of a Dyck path.
For a Dyck path $D$ of length $2n$, this is the number of points $(i,i)$ for $1 \leq i < n$ that are touching points of the [[Mp00099|bounce path]] of $D$.
Matching statistic: St000362
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000362: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000362: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 0
[1,2] => [2] => ([],2)
=> 0
[2,1] => [2] => ([],2)
=> 0
[1,2,3] => [3] => ([],3)
=> 0
[1,3,2] => [1,2] => ([(1,2)],3)
=> 1
[2,1,3] => [3] => ([],3)
=> 0
[2,3,1] => [3] => ([],3)
=> 0
[3,1,2] => [3] => ([],3)
=> 0
[3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,3,4] => [4] => ([],4)
=> 0
[1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[1,3,2,4] => [1,3] => ([(2,3)],4)
=> 1
[1,3,4,2] => [1,3] => ([(2,3)],4)
=> 1
[1,4,2,3] => [1,3] => ([(2,3)],4)
=> 1
[1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [4] => ([],4)
=> 0
[2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[2,3,1,4] => [4] => ([],4)
=> 0
[2,3,4,1] => [4] => ([],4)
=> 0
[2,4,1,3] => [4] => ([],4)
=> 0
[2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[3,1,2,4] => [4] => ([],4)
=> 0
[3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,4,1,2] => [4] => ([],4)
=> 0
[3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,1,2,3] => [4] => ([],4)
=> 0
[4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => [1,3] => ([(2,3)],4)
=> 1
[4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,2,3,4,5] => [5] => ([],5)
=> 0
[1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> 1
[1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,4,2,5] => [1,4] => ([(3,4)],5)
=> 1
[1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> 1
[1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> 1
[1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => [1,4] => ([(3,4)],5)
=> 1
[1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => [1,4] => ([(3,4)],5)
=> 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: St000536
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000536: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000536: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 0
[1,2] => [2] => ([],2)
=> 0
[2,1] => [2] => ([],2)
=> 0
[1,2,3] => [3] => ([],3)
=> 0
[1,3,2] => [1,2] => ([(1,2)],3)
=> 1
[2,1,3] => [3] => ([],3)
=> 0
[2,3,1] => [3] => ([],3)
=> 0
[3,1,2] => [3] => ([],3)
=> 0
[3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,3,4] => [4] => ([],4)
=> 0
[1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[1,3,2,4] => [1,3] => ([(2,3)],4)
=> 1
[1,3,4,2] => [1,3] => ([(2,3)],4)
=> 1
[1,4,2,3] => [1,3] => ([(2,3)],4)
=> 1
[1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [4] => ([],4)
=> 0
[2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[2,3,1,4] => [4] => ([],4)
=> 0
[2,3,4,1] => [4] => ([],4)
=> 0
[2,4,1,3] => [4] => ([],4)
=> 0
[2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[3,1,2,4] => [4] => ([],4)
=> 0
[3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,4,1,2] => [4] => ([],4)
=> 0
[3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,1,2,3] => [4] => ([],4)
=> 0
[4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => [1,3] => ([(2,3)],4)
=> 1
[4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,2,3,4,5] => [5] => ([],5)
=> 0
[1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> 1
[1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,4,2,5] => [1,4] => ([(3,4)],5)
=> 1
[1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> 1
[1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> 1
[1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => [1,4] => ([(3,4)],5)
=> 1
[1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => [1,4] => ([(3,4)],5)
=> 1
Description
The pathwidth of a graph.
Matching statistic: St001169
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001169: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001169: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 0
[1,2] => [2] => [1,1,0,0]
=> 0
[2,1] => [2] => [1,1,0,0]
=> 0
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [3] => [1,1,1,0,0,0]
=> 0
[2,3,1] => [3] => [1,1,1,0,0,0]
=> 0
[3,1,2] => [3] => [1,1,1,0,0,0]
=> 0
[3,2,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,3,4,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[2,1,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,1,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,3,1,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,3,4,1] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,4,1,3] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,4,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[3,1,2,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[3,1,4,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,2,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,2,4,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,4,1,2] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[3,4,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,1,2,3] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[4,1,3,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,2,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[4,2,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,3,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,3,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,4,5,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,5,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,5,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,3,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,2,5,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,4,5,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,2,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,4,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,2,3,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,4,2,5,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,2,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,5,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,5,2,3] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
Description
Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St001197
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001197: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001197: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 0
[1,2] => [2] => [1,1,0,0]
=> 0
[2,1] => [2] => [1,1,0,0]
=> 0
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [3] => [1,1,1,0,0,0]
=> 0
[2,3,1] => [3] => [1,1,1,0,0,0]
=> 0
[3,1,2] => [3] => [1,1,1,0,0,0]
=> 0
[3,2,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,3,4,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[2,1,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,1,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,3,1,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,3,4,1] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,4,1,3] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,4,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[3,1,2,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[3,1,4,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,2,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,2,4,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,4,1,2] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[3,4,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,1,2,3] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[4,1,3,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,2,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[4,2,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,3,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,3,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,4,5,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,5,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,5,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,3,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,2,5,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,4,5,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,2,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,4,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,2,3,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,4,2,5,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,2,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,5,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,5,2,3] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
Description
The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001205
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001205: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001205: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 0
[1,2] => [2] => [1,1,0,0]
=> 0
[2,1] => [2] => [1,1,0,0]
=> 0
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [3] => [1,1,1,0,0,0]
=> 0
[2,3,1] => [3] => [1,1,1,0,0,0]
=> 0
[3,1,2] => [3] => [1,1,1,0,0,0]
=> 0
[3,2,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,3,4,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[2,1,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,1,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,3,1,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,3,4,1] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,4,1,3] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,4,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[3,1,2,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[3,1,4,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,2,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,2,4,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,4,1,2] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[3,4,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,1,2,3] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[4,1,3,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,2,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[4,2,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,3,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,3,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,4,5,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,5,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,5,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,3,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,2,5,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,4,5,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,2,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,4,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,2,3,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,4,2,5,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,2,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,5,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,5,2,3] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
Description
The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
See http://www.findstat.org/DyckPaths/NakayamaAlgebras for the definition of Nakayama algebra and the relation to Dyck paths.
Matching statistic: St001225
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001225: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001225: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 0
[1,2] => [2] => [1,1,0,0]
=> 0
[2,1] => [2] => [1,1,0,0]
=> 0
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [3] => [1,1,1,0,0,0]
=> 0
[2,3,1] => [3] => [1,1,1,0,0,0]
=> 0
[3,1,2] => [3] => [1,1,1,0,0,0]
=> 0
[3,2,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,3,4,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[2,1,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,1,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,3,1,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,3,4,1] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,4,1,3] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,4,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[3,1,2,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[3,1,4,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,2,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,2,4,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,4,1,2] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[3,4,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,1,2,3] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[4,1,3,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,2,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[4,2,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,3,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,3,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,4,5,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,5,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,5,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,3,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,2,5,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,4,5,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,2,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,4,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,2,3,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,4,2,5,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,2,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,5,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,5,2,3] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
Description
The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra.
Matching statistic: St001277
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001277: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001277: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 0
[1,2] => [2] => ([],2)
=> 0
[2,1] => [2] => ([],2)
=> 0
[1,2,3] => [3] => ([],3)
=> 0
[1,3,2] => [1,2] => ([(1,2)],3)
=> 1
[2,1,3] => [3] => ([],3)
=> 0
[2,3,1] => [3] => ([],3)
=> 0
[3,1,2] => [3] => ([],3)
=> 0
[3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,3,4] => [4] => ([],4)
=> 0
[1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[1,3,2,4] => [1,3] => ([(2,3)],4)
=> 1
[1,3,4,2] => [1,3] => ([(2,3)],4)
=> 1
[1,4,2,3] => [1,3] => ([(2,3)],4)
=> 1
[1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [4] => ([],4)
=> 0
[2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[2,3,1,4] => [4] => ([],4)
=> 0
[2,3,4,1] => [4] => ([],4)
=> 0
[2,4,1,3] => [4] => ([],4)
=> 0
[2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[3,1,2,4] => [4] => ([],4)
=> 0
[3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,4,1,2] => [4] => ([],4)
=> 0
[3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,1,2,3] => [4] => ([],4)
=> 0
[4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => [1,3] => ([(2,3)],4)
=> 1
[4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,2,3,4,5] => [5] => ([],5)
=> 0
[1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> 1
[1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,4,2,5] => [1,4] => ([(3,4)],5)
=> 1
[1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> 1
[1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> 1
[1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => [1,4] => ([(3,4)],5)
=> 1
[1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => [1,4] => ([(3,4)],5)
=> 1
Description
The degeneracy of a graph.
The degeneracy of a graph $G$ is the maximum of the minimum degrees of the (vertex induced) subgraphs of $G$.
The following 104 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001358The largest degree of a regular subgraph of a graph. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000288The number of ones in a binary word. St000822The Hadwiger number of the graph. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001116The game chromatic number of a graph. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
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. St001963The tree-depth of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000024The number of double up and double down steps of a Dyck path. St000157The number of descents of a standard tableau. St000171The degree of the graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000454The largest eigenvalue of a graph if it is integral. St000741The Colin de Verdière graph invariant. St000778The metric dimension of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001270The bandwidth of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001391The disjunction number of a graph. St001644The dimension of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001812The biclique partition number of a graph. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St000087The number of induced subgraphs. St000093The cardinality of a maximal independent set of vertices of a graph. St000147The largest part of an integer partition. St000228The size of a partition. St000286The number of connected components of the complement of a graph. St000363The number of minimal vertex covers of a graph. St000443The number of long tunnels of a Dyck path. St000469The distinguishing number of a graph. St000636The hull number of a graph. St000722The number of different neighbourhoods in a graph. St000733The row containing the largest entry of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000926The clique-coclique number of a graph. St000935The number of ordered refinements of an integer partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000312The number of leaves in a graph. St000445The number of rises of length 1 of a Dyck path. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000711The number of big exceedences of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000691The number of changes of a binary word. St000710The number of big deficiencies of a permutation. St001480The number of simple summands of the module J^2/J^3. St000619The number of cyclic descents of a permutation. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001427The number of descents of a signed permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001964The interval resolution global dimension of a poset. St000307The number of rowmotion orbits of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001896The number of right descents of a signed permutations.
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