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Your data matches 239 different statistics following compositions of up to 3 maps.
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Matching statistic: St000288
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(load all 10 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: St000024
Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000024: 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
St000024: 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]
=> 1
[-1] => 1 => [1,1] => [1,0,1,0]
=> 0
[1,2] => 00 => [3] => [1,1,1,0,0,0]
=> 2
[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]
=> 0
[2,1] => 00 => [3] => [1,1,1,0,0,0]
=> 2
[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]
=> 0
[1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[1,2,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[1,-2,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,-2,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[-1,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[-1,2,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[-1,-2,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[-1,-2,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[1,3,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[1,-3,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,-3,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[-1,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[-1,3,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[-1,-3,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[-1,-3,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[2,1,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[2,-1,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,-1,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[-2,1,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[-2,1,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[-2,-1,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[-2,-1,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[2,3,-1] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[2,-3,1] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,-3,-1] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[-2,3,1] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[-2,3,-1] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[-2,-3,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[-2,-3,-1] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[3,1,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[3,-1,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,-1,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[-3,1,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[-3,1,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[-3,-1,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[-3,-1,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
Description
The number of double up and double down steps of a Dyck path.
In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
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: St000340
Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000340: 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
St000340: 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]
=> 1
[-1] => 1 => [1,1] => [1,0,1,0]
=> 0
[1,2] => 00 => [3] => [1,1,1,0,0,0]
=> 1
[1,-2] => 01 => [2,1] => [1,1,0,0,1,0]
=> 2
[-1,2] => 10 => [1,2] => [1,0,1,1,0,0]
=> 1
[-1,-2] => 11 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[2,1] => 00 => [3] => [1,1,1,0,0,0]
=> 1
[2,-1] => 01 => [2,1] => [1,1,0,0,1,0]
=> 2
[-2,1] => 10 => [1,2] => [1,0,1,1,0,0]
=> 1
[-2,-1] => 11 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[1,-2,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[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]
=> 1
[-1,-2,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,3,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[1,-3,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[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]
=> 1
[-1,-3,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,1,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[2,-1,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[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]
=> 1
[-2,-1,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,3,-1] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[2,-3,1] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[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]
=> 1
[-2,-3,-1] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 1
[3,1,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[3,-1,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[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]
=> 1
[-3,-1,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
Description
The number of non-final maximal constant sub-paths of length greater than one.
This is the total number of occurrences of the patterns $110$ and $001$.
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: St000394
Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000394: 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
St000394: 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]
=> 1
[-1] => 1 => [1,1] => [1,0,1,0]
=> 0
[1,2] => 00 => [3] => [1,1,1,0,0,0]
=> 2
[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]
=> 0
[2,1] => 00 => [3] => [1,1,1,0,0,0]
=> 2
[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]
=> 0
[1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[1,2,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[1,-2,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,-2,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[-1,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[-1,2,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[-1,-2,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[-1,-2,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[1,3,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[1,-3,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,-3,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[-1,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[-1,3,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[-1,-3,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[-1,-3,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[2,1,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[2,-1,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,-1,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[-2,1,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[-2,1,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[-2,-1,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[-2,-1,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[2,3,-1] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[2,-3,1] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,-3,-1] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[-2,3,1] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[-2,3,-1] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[-2,-3,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[-2,-3,-1] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3
[3,1,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[3,-1,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,-1,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[-3,1,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[-3,1,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[-3,-1,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[-3,-1,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
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: St000691
Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 => [2] => 10 => 1
[-1] => 1 => [1,1] => 11 => 0
[1,2] => 00 => [3] => 100 => 1
[1,-2] => 01 => [2,1] => 101 => 2
[-1,2] => 10 => [1,2] => 110 => 1
[-1,-2] => 11 => [1,1,1] => 111 => 0
[2,1] => 00 => [3] => 100 => 1
[2,-1] => 01 => [2,1] => 101 => 2
[-2,1] => 10 => [1,2] => 110 => 1
[-2,-1] => 11 => [1,1,1] => 111 => 0
[1,2,3] => 000 => [4] => 1000 => 1
[1,2,-3] => 001 => [3,1] => 1001 => 2
[1,-2,3] => 010 => [2,2] => 1010 => 3
[1,-2,-3] => 011 => [2,1,1] => 1011 => 2
[-1,2,3] => 100 => [1,3] => 1100 => 1
[-1,2,-3] => 101 => [1,2,1] => 1101 => 2
[-1,-2,3] => 110 => [1,1,2] => 1110 => 1
[-1,-2,-3] => 111 => [1,1,1,1] => 1111 => 0
[1,3,2] => 000 => [4] => 1000 => 1
[1,3,-2] => 001 => [3,1] => 1001 => 2
[1,-3,2] => 010 => [2,2] => 1010 => 3
[1,-3,-2] => 011 => [2,1,1] => 1011 => 2
[-1,3,2] => 100 => [1,3] => 1100 => 1
[-1,3,-2] => 101 => [1,2,1] => 1101 => 2
[-1,-3,2] => 110 => [1,1,2] => 1110 => 1
[-1,-3,-2] => 111 => [1,1,1,1] => 1111 => 0
[2,1,3] => 000 => [4] => 1000 => 1
[2,1,-3] => 001 => [3,1] => 1001 => 2
[2,-1,3] => 010 => [2,2] => 1010 => 3
[2,-1,-3] => 011 => [2,1,1] => 1011 => 2
[-2,1,3] => 100 => [1,3] => 1100 => 1
[-2,1,-3] => 101 => [1,2,1] => 1101 => 2
[-2,-1,3] => 110 => [1,1,2] => 1110 => 1
[-2,-1,-3] => 111 => [1,1,1,1] => 1111 => 0
[2,3,1] => 000 => [4] => 1000 => 1
[2,3,-1] => 001 => [3,1] => 1001 => 2
[2,-3,1] => 010 => [2,2] => 1010 => 3
[2,-3,-1] => 011 => [2,1,1] => 1011 => 2
[-2,3,1] => 100 => [1,3] => 1100 => 1
[-2,3,-1] => 101 => [1,2,1] => 1101 => 2
[-2,-3,1] => 110 => [1,1,2] => 1110 => 1
[-2,-3,-1] => 111 => [1,1,1,1] => 1111 => 0
[3,1,2] => 000 => [4] => 1000 => 1
[3,1,-2] => 001 => [3,1] => 1001 => 2
[3,-1,2] => 010 => [2,2] => 1010 => 3
[3,-1,-2] => 011 => [2,1,1] => 1011 => 2
[-3,1,2] => 100 => [1,3] => 1100 => 1
[-3,1,-2] => 101 => [1,2,1] => 1101 => 2
[-3,-1,2] => 110 => [1,1,2] => 1110 => 1
[-3,-1,-2] => 111 => [1,1,1,1] => 1111 => 0
Description
The number of changes of a binary word.
This is the number of indices $i$ such that $w_i \neq w_{i+1}$.
The following 229 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. St001480The number of simple summands of the module J^2/J^3. 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. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000443The number of long tunnels of a Dyck path. St000453The number of distinct Laplacian eigenvalues of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000822The Hadwiger number of the graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one 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:
St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number 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. St001674The number of vertices of the largest induced star graph in the 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. 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. 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. St000444The length of the maximal rise of a Dyck path. St000681The Grundy value of Chomp on Ferrers diagrams. St000939The number of characters of the symmetric group whose value on the partition is positive. St000675The number of centered multitunnels of a Dyck path. St000744The length of the path to the largest entry in a standard Young tableau. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000937The number of positive values of the symmetric group character corresponding to the partition. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001637The number of (upper) dissectors of a poset. St001060The distinguishing index of a graph. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001360The number of covering relations in Young's lattice below a partition. St001933The largest multiplicity of a part in an integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St000993The multiplicity of the largest part of an integer partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001330The hat guessing number of a graph. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000144The pyramid weight of the Dyck path. St000294The number of distinct factors of a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000393The number of strictly increasing runs in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000420The number of Dyck paths that are weakly above a Dyck path. St000439The position of the first down step of a Dyck path. St000518The number of distinct subsequences in a binary word. St000532The total number of rook placements on a Ferrers board. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000674The number of hills of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000759The smallest missing part in an integer partition. St000762The sum of the positions of the weak records of an integer composition. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000983The length of the longest alternating subword. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001400The total number of Littlewood-Richardson tableaux of given shape. St001437The flex of a binary word. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001523The degree of symmetry of a Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001530The depth of a Dyck path. St001658The total number of rook placements on a Ferrers board. St001733The number of weak left to right maxima of a Dyck path. St001814The number of partitions interlacing the given partition. St001885The number of binary words with the same proper border set. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001568The smallest positive integer that does not appear twice in the partition. St001870The number of positive entries followed by a negative entry in a signed permutation. St001430The number of positive entries in a signed permutation. St000454The largest eigenvalue of a graph if it is integral. St000264The girth of a graph, which is not a tree. St001429The number of negative entries in a signed permutation. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000782The indicator function of whether a given perfect matching is an L & P matching. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St000735The last entry on the main diagonal of a standard tableau. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001498The normalised height of a Nakayama algebra with magnitude 1. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001383The BG-rank of an integer partition. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000478Another weight of a partition according to Alladi. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000929The constant term of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001099The 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 leaf labelled binary trees. St001100The 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 leaf labelled trees. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition.
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