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Your data matches 50 different statistics following compositions of up to 3 maps.
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Matching statistic: St000288
(load all 57 compositions to match this statistic)

Mapping

Mp00267: Signed permutations —signs⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => 0 => 0
[-1] => 1 => 1
[1,2] => 00 => 0
[1,-2] => 01 => 1
[-1,2] => 10 => 1
[-1,-2] => 11 => 2
[2,1] => 00 => 0
[2,-1] => 01 => 1
[-2,1] => 10 => 1
[-2,-1] => 11 => 2
[1,2,3] => 000 => 0
[1,2,-3] => 001 => 1
[1,-2,3] => 010 => 1
[1,-2,-3] => 011 => 2
[-1,2,3] => 100 => 1
[-1,2,-3] => 101 => 2
[-1,-2,3] => 110 => 2
[-1,-2,-3] => 111 => 3
[1,3,2] => 000 => 0
[1,3,-2] => 001 => 1
[1,-3,2] => 010 => 1
[1,-3,-2] => 011 => 2
[-1,3,2] => 100 => 1
[-1,3,-2] => 101 => 2
[-1,-3,2] => 110 => 2
[-1,-3,-2] => 111 => 3
[2,1,3] => 000 => 0
[2,1,-3] => 001 => 1
[2,-1,3] => 010 => 1
[2,-1,-3] => 011 => 2
[-2,1,3] => 100 => 1
[-2,1,-3] => 101 => 2
[-2,-1,3] => 110 => 2
[-2,-1,-3] => 111 => 3
[2,3,1] => 000 => 0
[2,3,-1] => 001 => 1
[2,-3,1] => 010 => 1
[2,-3,-1] => 011 => 2
[-2,3,1] => 100 => 1
[-2,3,-1] => 101 => 2
[-2,-3,1] => 110 => 2
[-2,-3,-1] => 111 => 3
[3,1,2] => 000 => 0
[3,1,-2] => 001 => 1
[3,-1,2] => 010 => 1
[3,-1,-2] => 011 => 2
[-3,1,2] => 100 => 1
[-3,1,-2] => 101 => 2
[-3,-1,2] => 110 => 2
[-3,-1,-2] => 111 => 3

Description

The number of ones in a binary word. This is also known as the Hamming weight of the word.

Matching statistic: St000053

Mapping

Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to 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%

Values

[1] => 0 => [2] => [1,1,0,0]
 => 0
[-1] => 1 => [1,1] => [1,0,1,0]
 => 1
[1,2] => 00 => [3] => [1,1,1,0,0,0]
 => 0
[1,-2] => 01 => [2,1] => [1,1,0,0,1,0]
 => 1
[-1,2] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[-1,-2] => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[2,1] => 00 => [3] => [1,1,1,0,0,0]
 => 0
[2,-1] => 01 => [2,1] => [1,1,0,0,1,0]
 => 1
[-2,1] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[-2,-1] => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,-2,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,-2,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-1,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-1,2,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-1,-2,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-1,-2,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,3,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,-3,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,-3,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-1,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-1,3,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-1,-3,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-1,-3,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,1,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[2,-1,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[2,-1,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-2,1,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-2,1,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-2,-1,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-2,-1,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,3,-1] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[2,-3,1] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[2,-3,-1] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-2,3,1] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-2,3,-1] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-2,-3,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-2,-3,-1] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[3,1,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[3,-1,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[3,-1,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-3,1,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-3,1,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-3,-1,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-3,-1,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3

Description

The number of valleys of the Dyck path.

Matching statistic: St000272

Mapping

Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to 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%

Values

[1] => 0 => [2] => ([],2)
 => 0
[-1] => 1 => [1,1] => ([(0,1)],2)
 => 1
[1,2] => 00 => [3] => ([],3)
 => 0
[1,-2] => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[-1,2] => 10 => [1,2] => ([(1,2)],3)
 => 1
[-1,-2] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,1] => 00 => [3] => ([],3)
 => 0
[2,-1] => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[-2,1] => 10 => [1,2] => ([(1,2)],3)
 => 1
[-2,-1] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,2,3] => 000 => [4] => ([],4)
 => 0
[1,2,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,-2,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[1,-2,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-1,2,3] => 100 => [1,3] => ([(2,3)],4)
 => 1
[-1,2,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-1,-2,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[-1,-2,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,3,2] => 000 => [4] => ([],4)
 => 0
[1,3,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,-3,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[1,-3,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-1,3,2] => 100 => [1,3] => ([(2,3)],4)
 => 1
[-1,3,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-1,-3,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[-1,-3,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[2,1,3] => 000 => [4] => ([],4)
 => 0
[2,1,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,-1,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[2,-1,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-2,1,3] => 100 => [1,3] => ([(2,3)],4)
 => 1
[-2,1,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-2,-1,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[-2,-1,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[2,3,1] => 000 => [4] => ([],4)
 => 0
[2,3,-1] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,-3,1] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[2,-3,-1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-2,3,1] => 100 => [1,3] => ([(2,3)],4)
 => 1
[-2,3,-1] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-2,-3,1] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[-2,-3,-1] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,1,2] => 000 => [4] => ([],4)
 => 0
[3,1,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[3,-1,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[3,-1,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-3,1,2] => 100 => [1,3] => ([(2,3)],4)
 => 1
[-3,1,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-3,-1,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[-3,-1,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3

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

Mapping

Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to 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%

Values

[1] => 0 => [2] => [1,1,0,0]
 => 0
[-1] => 1 => [1,1] => [1,0,1,0]
 => 1
[1,2] => 00 => [3] => [1,1,1,0,0,0]
 => 0
[1,-2] => 01 => [2,1] => [1,1,0,0,1,0]
 => 1
[-1,2] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[-1,-2] => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[2,1] => 00 => [3] => [1,1,1,0,0,0]
 => 0
[2,-1] => 01 => [2,1] => [1,1,0,0,1,0]
 => 1
[-2,1] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[-2,-1] => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,-2,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,-2,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-1,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-1,2,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-1,-2,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-1,-2,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,3,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,-3,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,-3,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-1,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-1,3,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-1,-3,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-1,-3,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,1,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[2,-1,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[2,-1,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-2,1,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-2,1,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-2,-1,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-2,-1,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,3,-1] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[2,-3,1] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[2,-3,-1] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-2,3,1] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-2,3,-1] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-2,-3,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-2,-3,-1] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[3,1,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[3,-1,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[3,-1,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-3,1,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-3,1,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-3,-1,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-3,-1,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3

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

Mapping

Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to 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%

Values

[1] => 0 => [2] => ([],2)
 => 0
[-1] => 1 => [1,1] => ([(0,1)],2)
 => 1
[1,2] => 00 => [3] => ([],3)
 => 0
[1,-2] => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[-1,2] => 10 => [1,2] => ([(1,2)],3)
 => 1
[-1,-2] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,1] => 00 => [3] => ([],3)
 => 0
[2,-1] => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[-2,1] => 10 => [1,2] => ([(1,2)],3)
 => 1
[-2,-1] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,2,3] => 000 => [4] => ([],4)
 => 0
[1,2,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,-2,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[1,-2,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-1,2,3] => 100 => [1,3] => ([(2,3)],4)
 => 1
[-1,2,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-1,-2,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[-1,-2,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,3,2] => 000 => [4] => ([],4)
 => 0
[1,3,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,-3,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[1,-3,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-1,3,2] => 100 => [1,3] => ([(2,3)],4)
 => 1
[-1,3,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-1,-3,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[-1,-3,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[2,1,3] => 000 => [4] => ([],4)
 => 0
[2,1,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,-1,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[2,-1,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-2,1,3] => 100 => [1,3] => ([(2,3)],4)
 => 1
[-2,1,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-2,-1,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[-2,-1,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[2,3,1] => 000 => [4] => ([],4)
 => 0
[2,3,-1] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,-3,1] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[2,-3,-1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-2,3,1] => 100 => [1,3] => ([(2,3)],4)
 => 1
[-2,3,-1] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-2,-3,1] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[-2,-3,-1] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,1,2] => 000 => [4] => ([],4)
 => 0
[3,1,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[3,-1,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[3,-1,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-3,1,2] => 100 => [1,3] => ([(2,3)],4)
 => 1
[-3,1,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-3,-1,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[-3,-1,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 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: St000536

Mapping

Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to 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%

Values

[1] => 0 => [2] => ([],2)
 => 0
[-1] => 1 => [1,1] => ([(0,1)],2)
 => 1
[1,2] => 00 => [3] => ([],3)
 => 0
[1,-2] => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[-1,2] => 10 => [1,2] => ([(1,2)],3)
 => 1
[-1,-2] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,1] => 00 => [3] => ([],3)
 => 0
[2,-1] => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[-2,1] => 10 => [1,2] => ([(1,2)],3)
 => 1
[-2,-1] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,2,3] => 000 => [4] => ([],4)
 => 0
[1,2,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,-2,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[1,-2,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-1,2,3] => 100 => [1,3] => ([(2,3)],4)
 => 1
[-1,2,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-1,-2,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[-1,-2,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,3,2] => 000 => [4] => ([],4)
 => 0
[1,3,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,-3,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[1,-3,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-1,3,2] => 100 => [1,3] => ([(2,3)],4)
 => 1
[-1,3,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-1,-3,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[-1,-3,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[2,1,3] => 000 => [4] => ([],4)
 => 0
[2,1,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,-1,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[2,-1,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-2,1,3] => 100 => [1,3] => ([(2,3)],4)
 => 1
[-2,1,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-2,-1,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[-2,-1,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[2,3,1] => 000 => [4] => ([],4)
 => 0
[2,3,-1] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,-3,1] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[2,-3,-1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-2,3,1] => 100 => [1,3] => ([(2,3)],4)
 => 1
[-2,3,-1] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-2,-3,1] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[-2,-3,-1] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,1,2] => 000 => [4] => ([],4)
 => 0
[3,1,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[3,-1,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[3,-1,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-3,1,2] => 100 => [1,3] => ([(2,3)],4)
 => 1
[-3,1,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-3,-1,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[-3,-1,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3

Description

The pathwidth of a graph.

Matching statistic: St001169

Mapping

Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to 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%

Values

[1] => 0 => [2] => [1,1,0,0]
 => 0
[-1] => 1 => [1,1] => [1,0,1,0]
 => 1
[1,2] => 00 => [3] => [1,1,1,0,0,0]
 => 0
[1,-2] => 01 => [2,1] => [1,1,0,0,1,0]
 => 1
[-1,2] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[-1,-2] => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[2,1] => 00 => [3] => [1,1,1,0,0,0]
 => 0
[2,-1] => 01 => [2,1] => [1,1,0,0,1,0]
 => 1
[-2,1] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[-2,-1] => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,-2,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,-2,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-1,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-1,2,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-1,-2,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-1,-2,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,3,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,-3,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,-3,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-1,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-1,3,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-1,-3,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-1,-3,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,1,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[2,-1,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[2,-1,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-2,1,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-2,1,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-2,-1,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-2,-1,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,3,-1] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[2,-3,1] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[2,-3,-1] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-2,3,1] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-2,3,-1] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-2,-3,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-2,-3,-1] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[3,1,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[3,-1,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[3,-1,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-3,1,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-3,1,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-3,-1,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-3,-1,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3

Description

Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.

Matching statistic: St001197

Mapping

Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to 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%

Values

[1] => 0 => [2] => [1,1,0,0]
 => 0
[-1] => 1 => [1,1] => [1,0,1,0]
 => 1
[1,2] => 00 => [3] => [1,1,1,0,0,0]
 => 0
[1,-2] => 01 => [2,1] => [1,1,0,0,1,0]
 => 1
[-1,2] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[-1,-2] => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[2,1] => 00 => [3] => [1,1,1,0,0,0]
 => 0
[2,-1] => 01 => [2,1] => [1,1,0,0,1,0]
 => 1
[-2,1] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[-2,-1] => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,-2,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,-2,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-1,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-1,2,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-1,-2,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-1,-2,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,3,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,-3,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,-3,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-1,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-1,3,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-1,-3,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-1,-3,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,1,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[2,-1,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[2,-1,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-2,1,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-2,1,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-2,-1,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-2,-1,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,3,-1] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[2,-3,1] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[2,-3,-1] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-2,3,1] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-2,3,-1] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-2,-3,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-2,-3,-1] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[3,1,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[3,-1,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[3,-1,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-3,1,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-3,1,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-3,-1,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-3,-1,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3

Description

The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ .

Matching statistic: St001205

Mapping

Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to 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%

Values

[1] => 0 => [2] => [1,1,0,0]
 => 0
[-1] => 1 => [1,1] => [1,0,1,0]
 => 1
[1,2] => 00 => [3] => [1,1,1,0,0,0]
 => 0
[1,-2] => 01 => [2,1] => [1,1,0,0,1,0]
 => 1
[-1,2] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[-1,-2] => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[2,1] => 00 => [3] => [1,1,1,0,0,0]
 => 0
[2,-1] => 01 => [2,1] => [1,1,0,0,1,0]
 => 1
[-2,1] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[-2,-1] => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,-2,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,-2,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-1,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-1,2,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-1,-2,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-1,-2,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,3,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,-3,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,-3,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-1,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-1,3,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-1,-3,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-1,-3,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,1,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[2,-1,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[2,-1,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-2,1,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-2,1,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-2,-1,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-2,-1,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,3,-1] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[2,-3,1] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[2,-3,-1] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-2,3,1] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-2,3,-1] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-2,-3,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-2,-3,-1] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[3,1,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[3,-1,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[3,-1,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-3,1,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-3,1,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-3,-1,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-3,-1,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3

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

Mapping

Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to 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%

Values

[1] => 0 => [2] => [1,1,0,0]
 => 0
[-1] => 1 => [1,1] => [1,0,1,0]
 => 1
[1,2] => 00 => [3] => [1,1,1,0,0,0]
 => 0
[1,-2] => 01 => [2,1] => [1,1,0,0,1,0]
 => 1
[-1,2] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[-1,-2] => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[2,1] => 00 => [3] => [1,1,1,0,0,0]
 => 0
[2,-1] => 01 => [2,1] => [1,1,0,0,1,0]
 => 1
[-2,1] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[-2,-1] => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,-2,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,-2,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-1,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-1,2,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-1,-2,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-1,-2,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,3,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,-3,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,-3,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-1,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-1,3,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-1,-3,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-1,-3,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,1,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[2,-1,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[2,-1,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-2,1,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-2,1,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-2,-1,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-2,-1,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,3,-1] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[2,-3,1] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[2,-3,-1] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-2,3,1] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-2,3,-1] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-2,-3,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-2,-3,-1] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[3,1,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[3,-1,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[3,-1,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-3,1,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-3,1,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-3,-1,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-3,-1,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3

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.

The following 40 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001277The degeneracy of a graph. 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. St001372The length of a longest cyclic run of ones of a binary word. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001777The number of weak descents in an integer composition. 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. St000822The Hadwiger number of the graph. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. 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. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001812The biclique partition number of a graph. St001330The hat guessing number of a graph. St001870The number of positive entries followed by a negative entry in a signed permutation. St001430The number of positive entries in a signed permutation. St001429The number of negative entries in a signed permutation. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000937The number of positive values of the symmetric group character corresponding to the partition. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000941The number of characters of the symmetric group whose value on the partition is even.