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Your data matches 45 different statistics following compositions of up to 3 maps.
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Matching statistic: St001429
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
St001429: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[-1] => 1
[1,2] => 0
[1,-2] => 1
[-1,2] => 1
[-1,-2] => 2
[2,1] => 0
[2,-1] => 1
[-2,1] => 1
[-2,-1] => 2
[1,2,3] => 0
[1,2,-3] => 1
[1,-2,3] => 1
[1,-2,-3] => 2
[-1,2,3] => 1
[-1,2,-3] => 2
[-1,-2,3] => 2
[-1,-2,-3] => 3
[1,3,2] => 0
[1,3,-2] => 1
[1,-3,2] => 1
[1,-3,-2] => 2
[-1,3,2] => 1
[-1,3,-2] => 2
[-1,-3,2] => 2
[-1,-3,-2] => 3
[2,1,3] => 0
[2,1,-3] => 1
[2,-1,3] => 1
[2,-1,-3] => 2
[-2,1,3] => 1
[-2,1,-3] => 2
[-2,-1,3] => 2
[-2,-1,-3] => 3
[2,3,1] => 0
[2,3,-1] => 1
[2,-3,1] => 1
[2,-3,-1] => 2
[-2,3,1] => 1
[-2,3,-1] => 2
[-2,-3,1] => 2
[-2,-3,-1] => 3
[3,1,2] => 0
[3,1,-2] => 1
[3,-1,2] => 1
[3,-1,-2] => 2
[-3,1,2] => 1
[-3,1,-2] => 2
[-3,-1,2] => 2
[-3,-1,-2] => 3
Description
The number of negative entries in a signed permutation.
Matching statistic: St000288
(load all 56 compositions to match this statistic)
(load all 56 compositions to match this statistic)
Mp00267: Signed permutations —signs⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
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%
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
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%
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
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%
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
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%
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
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%
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: St001142
Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001142: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001142: 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 projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001169
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%
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
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%
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 $eA$.
The following 35 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. 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. 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. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001812The biclique partition number of a graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001330The hat guessing number of a graph. St001430The number of positive entries in a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation.
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