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Your data matches 6 different statistics following compositions of up to 3 maps.
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Matching statistic: St001870
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(load all 4 compositions to match this statistic)
St001870: 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] => 2
[-1,2] => 1
[-1,-2] => 1
[2,1] => 0
[2,-1] => 2
[-2,1] => 1
[-2,-1] => 1
[1,2,3] => 0
[1,2,-3] => 2
[1,-2,3] => 2
[1,-2,-3] => 2
[-1,2,3] => 1
[-1,2,-3] => 3
[-1,-2,3] => 1
[-1,-2,-3] => 1
[1,3,2] => 0
[1,3,-2] => 2
[1,-3,2] => 2
[1,-3,-2] => 2
[-1,3,2] => 1
[-1,3,-2] => 3
[-1,-3,2] => 1
[-1,-3,-2] => 1
[2,1,3] => 0
[2,1,-3] => 2
[2,-1,3] => 2
[2,-1,-3] => 2
[-2,1,3] => 1
[-2,1,-3] => 3
[-2,-1,3] => 1
[-2,-1,-3] => 1
[2,3,1] => 0
[2,3,-1] => 2
[2,-3,1] => 2
[2,-3,-1] => 2
[-2,3,1] => 1
[-2,3,-1] => 3
[-2,-3,1] => 1
[-2,-3,-1] => 1
[3,1,2] => 0
[3,1,-2] => 2
[3,-1,2] => 2
[3,-1,-2] => 2
[-3,1,2] => 1
[-3,1,-2] => 3
[-3,-1,2] => 1
[-3,-1,-2] => 1
Description
The number of positive entries followed by a negative entry in a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, this is the number of positive entries followed by a negative entry in $\pi(-n),\dots,\pi(-1),\pi(1),\dots,\pi(n)$.
Matching statistic: St000288
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00271: Signed permutations —flag negatives to negatives⟶ Signed permutations
Mp00267: Signed permutations —signs⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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] => [1] => 0 => 0
[-1] => [-1] => 1 => 1
[1,2] => [1,2] => 00 => 0
[1,-2] => [-1,-2] => 11 => 2
[-1,2] => [1,-2] => 01 => 1
[-1,-2] => [-1,2] => 10 => 1
[2,1] => [2,1] => 00 => 0
[2,-1] => [-2,-1] => 11 => 2
[-2,1] => [2,-1] => 01 => 1
[-2,-1] => [-2,1] => 10 => 1
[1,2,3] => [1,2,3] => 000 => 0
[1,2,-3] => [-1,-2,3] => 110 => 2
[1,-2,3] => [1,-2,-3] => 011 => 2
[1,-2,-3] => [-1,2,-3] => 101 => 2
[-1,2,3] => [1,2,-3] => 001 => 1
[-1,2,-3] => [-1,-2,-3] => 111 => 3
[-1,-2,3] => [1,-2,3] => 010 => 1
[-1,-2,-3] => [-1,2,3] => 100 => 1
[1,3,2] => [1,3,2] => 000 => 0
[1,3,-2] => [-1,-3,2] => 110 => 2
[1,-3,2] => [1,-3,-2] => 011 => 2
[1,-3,-2] => [-1,3,-2] => 101 => 2
[-1,3,2] => [1,3,-2] => 001 => 1
[-1,3,-2] => [-1,-3,-2] => 111 => 3
[-1,-3,2] => [1,-3,2] => 010 => 1
[-1,-3,-2] => [-1,3,2] => 100 => 1
[2,1,3] => [2,1,3] => 000 => 0
[2,1,-3] => [-2,-1,3] => 110 => 2
[2,-1,3] => [2,-1,-3] => 011 => 2
[2,-1,-3] => [-2,1,-3] => 101 => 2
[-2,1,3] => [2,1,-3] => 001 => 1
[-2,1,-3] => [-2,-1,-3] => 111 => 3
[-2,-1,3] => [2,-1,3] => 010 => 1
[-2,-1,-3] => [-2,1,3] => 100 => 1
[2,3,1] => [2,3,1] => 000 => 0
[2,3,-1] => [-2,-3,1] => 110 => 2
[2,-3,1] => [2,-3,-1] => 011 => 2
[2,-3,-1] => [-2,3,-1] => 101 => 2
[-2,3,1] => [2,3,-1] => 001 => 1
[-2,3,-1] => [-2,-3,-1] => 111 => 3
[-2,-3,1] => [2,-3,1] => 010 => 1
[-2,-3,-1] => [-2,3,1] => 100 => 1
[3,1,2] => [3,1,2] => 000 => 0
[3,1,-2] => [-3,-1,2] => 110 => 2
[3,-1,2] => [3,-1,-2] => 011 => 2
[3,-1,-2] => [-3,1,-2] => 101 => 2
[-3,1,2] => [3,1,-2] => 001 => 1
[-3,1,-2] => [-3,-1,-2] => 111 => 3
[-3,-1,2] => [3,-1,2] => 010 => 1
[-3,-1,-2] => [-3,1,2] => 100 => 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000453
Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000453: 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
St000453: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 => [2] => ([],2)
=> 1 = 0 + 1
[-1] => 1 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,2] => 00 => [3] => ([],3)
=> 1 = 0 + 1
[1,-2] => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[-1,2] => 10 => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[-1,-2] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,1] => 00 => [3] => ([],3)
=> 1 = 0 + 1
[2,-1] => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[-2,1] => 10 => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[-2,-1] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2,3] => 000 => [4] => ([],4)
=> 1 = 0 + 1
[1,2,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,-2,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,-2,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[-1,2,3] => 100 => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[-1,2,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[-1,-2,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[-1,-2,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,3,2] => 000 => [4] => ([],4)
=> 1 = 0 + 1
[1,3,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,-3,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,-3,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[-1,3,2] => 100 => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[-1,3,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[-1,-3,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[-1,-3,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3] => 000 => [4] => ([],4)
=> 1 = 0 + 1
[2,1,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,-1,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,-1,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[-2,1,3] => 100 => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[-2,1,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[-2,-1,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[-2,-1,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,1] => 000 => [4] => ([],4)
=> 1 = 0 + 1
[2,3,-1] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,-3,1] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,-3,-1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[-2,3,1] => 100 => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[-2,3,-1] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[-2,-3,1] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[-2,-3,-1] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,2] => 000 => [4] => ([],4)
=> 1 = 0 + 1
[3,1,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,-1,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,-1,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[-3,1,2] => 100 => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[-3,1,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[-3,-1,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[-3,-1,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
Description
The number of distinct Laplacian eigenvalues of a graph.
Matching statistic: St000777
Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 83%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 83%
Values
[1] => 0 => [2] => ([],2)
=> ? = 0 + 1
[-1] => 1 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,2] => 00 => [3] => ([],3)
=> ? = 0 + 1
[1,-2] => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[-1,2] => 10 => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
[-1,-2] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,1] => 00 => [3] => ([],3)
=> ? = 0 + 1
[2,-1] => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[-2,1] => 10 => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
[-2,-1] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2,3] => 000 => [4] => ([],4)
=> ? = 0 + 1
[1,2,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,-2,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 + 1
[1,-2,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[-1,2,3] => 100 => [1,3] => ([(2,3)],4)
=> ? = 1 + 1
[-1,2,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[-1,-2,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[-1,-2,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,3,2] => 000 => [4] => ([],4)
=> ? = 0 + 1
[1,3,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,-3,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 + 1
[1,-3,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[-1,3,2] => 100 => [1,3] => ([(2,3)],4)
=> ? = 1 + 1
[-1,3,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[-1,-3,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[-1,-3,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3] => 000 => [4] => ([],4)
=> ? = 0 + 1
[2,1,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,-1,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 + 1
[2,-1,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[-2,1,3] => 100 => [1,3] => ([(2,3)],4)
=> ? = 1 + 1
[-2,1,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[-2,-1,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[-2,-1,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,1] => 000 => [4] => ([],4)
=> ? = 0 + 1
[2,3,-1] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,-3,1] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 + 1
[2,-3,-1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[-2,3,1] => 100 => [1,3] => ([(2,3)],4)
=> ? = 1 + 1
[-2,3,-1] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[-2,-3,1] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[-2,-3,-1] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,2] => 000 => [4] => ([],4)
=> ? = 0 + 1
[3,1,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,-1,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 + 1
[3,-1,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[-3,1,2] => 100 => [1,3] => ([(2,3)],4)
=> ? = 1 + 1
[-3,1,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[-3,-1,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[-3,-1,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,1] => 000 => [4] => ([],4)
=> ? = 0 + 1
[3,2,-1] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,-2,1] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 + 1
[3,-2,-1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[-3,2,1] => 100 => [1,3] => ([(2,3)],4)
=> ? = 1 + 1
[-3,2,-1] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[-3,-2,1] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[-3,-2,-1] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,3,4] => 0000 => [5] => ([],5)
=> ? = 0 + 1
[1,2,3,-4] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,-3,4] => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,2,-3,-4] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,-2,3,4] => 0100 => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[1,-2,3,-4] => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,-2,-3,4] => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,-2,-3,-4] => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[-1,2,3,4] => 1000 => [1,4] => ([(3,4)],5)
=> ? = 1 + 1
[-1,2,3,-4] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[-1,2,-3,4] => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[-1,2,-3,-4] => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[-1,-2,3,4] => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[-1,-2,3,-4] => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[-1,-2,-3,4] => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[-1,-2,-3,-4] => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,4,3] => 0000 => [5] => ([],5)
=> ? = 0 + 1
[1,2,4,-3] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,-4,3] => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,2,-4,-3] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,-2,4,3] => 0100 => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[1,-2,4,-3] => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,-2,-4,3] => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,-2,-4,-3] => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[-1,2,4,3] => 1000 => [1,4] => ([(3,4)],5)
=> ? = 1 + 1
[-1,2,4,-3] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[-1,2,-4,3] => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[-1,2,-4,-3] => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[-1,-2,4,3] => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[-1,-2,4,-3] => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[-1,-2,-4,3] => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[-1,-2,-4,-3] => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,2,4] => 0000 => [5] => ([],5)
=> ? = 0 + 1
[1,3,2,-4] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,-2,4] => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,3,-2,-4] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,-3,2,4] => 0100 => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[1,-3,2,-4] => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,-3,-2,4] => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,-3,-2,-4] => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[-1,3,2,4] => 1000 => [1,4] => ([(3,4)],5)
=> ? = 1 + 1
[-1,3,2,-4] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St001430
Mp00271: Signed permutations —flag negatives to negatives⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001430: Signed permutations ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 100%
Mp00244: Signed permutations —bar⟶ Signed permutations
St001430: Signed permutations ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [-1] => 0
[-1] => [-1] => [1] => 1
[1,2] => [1,2] => [-1,-2] => 0
[1,-2] => [-1,-2] => [1,2] => 2
[-1,2] => [1,-2] => [-1,2] => 1
[-1,-2] => [-1,2] => [1,-2] => 1
[2,1] => [2,1] => [-2,-1] => 0
[2,-1] => [-2,-1] => [2,1] => 2
[-2,1] => [2,-1] => [-2,1] => 1
[-2,-1] => [-2,1] => [2,-1] => 1
[1,2,3] => [1,2,3] => [-1,-2,-3] => 0
[1,2,-3] => [-1,-2,3] => [1,2,-3] => 2
[1,-2,3] => [1,-2,-3] => [-1,2,3] => 2
[1,-2,-3] => [-1,2,-3] => [1,-2,3] => 2
[-1,2,3] => [1,2,-3] => [-1,-2,3] => 1
[-1,2,-3] => [-1,-2,-3] => [1,2,3] => 3
[-1,-2,3] => [1,-2,3] => [-1,2,-3] => 1
[-1,-2,-3] => [-1,2,3] => [1,-2,-3] => 1
[1,3,2] => [1,3,2] => [-1,-3,-2] => 0
[1,3,-2] => [-1,-3,2] => [1,3,-2] => 2
[1,-3,2] => [1,-3,-2] => [-1,3,2] => 2
[1,-3,-2] => [-1,3,-2] => [1,-3,2] => 2
[-1,3,2] => [1,3,-2] => [-1,-3,2] => 1
[-1,3,-2] => [-1,-3,-2] => [1,3,2] => 3
[-1,-3,2] => [1,-3,2] => [-1,3,-2] => 1
[-1,-3,-2] => [-1,3,2] => [1,-3,-2] => 1
[2,1,3] => [2,1,3] => [-2,-1,-3] => 0
[2,1,-3] => [-2,-1,3] => [2,1,-3] => 2
[2,-1,3] => [2,-1,-3] => [-2,1,3] => 2
[2,-1,-3] => [-2,1,-3] => [2,-1,3] => 2
[-2,1,3] => [2,1,-3] => [-2,-1,3] => 1
[-2,1,-3] => [-2,-1,-3] => [2,1,3] => 3
[-2,-1,3] => [2,-1,3] => [-2,1,-3] => 1
[-2,-1,-3] => [-2,1,3] => [2,-1,-3] => 1
[2,3,1] => [2,3,1] => [-2,-3,-1] => 0
[2,3,-1] => [-2,-3,1] => [2,3,-1] => 2
[2,-3,1] => [2,-3,-1] => [-2,3,1] => 2
[2,-3,-1] => [-2,3,-1] => [2,-3,1] => 2
[-2,3,1] => [2,3,-1] => [-2,-3,1] => 1
[-2,3,-1] => [-2,-3,-1] => [2,3,1] => 3
[-2,-3,1] => [2,-3,1] => [-2,3,-1] => 1
[-2,-3,-1] => [-2,3,1] => [2,-3,-1] => 1
[3,1,2] => [3,1,2] => [-3,-1,-2] => 0
[3,1,-2] => [-3,-1,2] => [3,1,-2] => 2
[3,-1,2] => [3,-1,-2] => [-3,1,2] => 2
[3,-1,-2] => [-3,1,-2] => [3,-1,2] => 2
[-3,1,2] => [3,1,-2] => [-3,-1,2] => 1
[-3,1,-2] => [-3,-1,-2] => [3,1,2] => 3
[-3,-1,2] => [3,-1,2] => [-3,1,-2] => 1
[-3,-1,-2] => [-3,1,2] => [3,-1,-2] => 1
[1,2,3,4,-5] => [-1,-2,3,4,5] => [1,2,-3,-4,-5] => ? = 2
[1,2,3,-4,5] => [1,-2,-3,4,5] => [-1,2,3,-4,-5] => ? = 2
[1,2,3,-4,-5] => [-1,2,-3,4,5] => [1,-2,3,-4,-5] => ? = 2
[1,2,-3,4,5] => [1,2,-3,-4,5] => [-1,-2,3,4,-5] => ? = 2
[1,2,-3,-4,5] => [1,-2,3,-4,5] => [-1,2,-3,4,-5] => ? = 2
[1,2,-3,-4,-5] => [-1,2,3,-4,5] => [1,-2,-3,4,-5] => ? = 2
[1,-2,3,4,5] => [1,2,3,-4,-5] => [-1,-2,-3,4,5] => ? = 2
[1,-2,3,4,-5] => [-1,-2,3,-4,-5] => [1,2,-3,4,5] => ? = 4
[1,-2,3,-4,-5] => [-1,2,-3,-4,-5] => [1,-2,3,4,5] => ? = 4
[1,-2,-3,4,5] => [1,2,-3,4,-5] => [-1,-2,3,-4,5] => ? = 2
[1,-2,-3,4,-5] => [-1,-2,-3,4,-5] => [1,2,3,-4,5] => ? = 4
[1,-2,-3,-4,5] => [1,-2,3,4,-5] => [-1,2,-3,-4,5] => ? = 2
[1,-2,-3,-4,-5] => [-1,2,3,4,-5] => [1,-2,-3,-4,5] => ? = 2
[-1,2,3,4,5] => [1,2,3,4,-5] => [-1,-2,-3,-4,5] => ? = 1
[-1,2,3,4,-5] => [-1,-2,3,4,-5] => [1,2,-3,-4,5] => ? = 3
[-1,2,3,-4,5] => [1,-2,-3,4,-5] => [-1,2,3,-4,5] => ? = 3
[-1,2,3,-4,-5] => [-1,2,-3,4,-5] => [1,-2,3,-4,5] => ? = 3
[-1,2,-3,4,5] => [1,2,-3,-4,-5] => [-1,-2,3,4,5] => ? = 3
[-1,2,-3,-4,5] => [1,-2,3,-4,-5] => [-1,2,-3,4,5] => ? = 3
[-1,2,-3,-4,-5] => [-1,2,3,-4,-5] => [1,-2,-3,4,5] => ? = 3
[-1,-2,3,4,5] => [1,2,3,-4,5] => [-1,-2,-3,4,-5] => ? = 1
[-1,-2,3,4,-5] => [-1,-2,3,-4,5] => [1,2,-3,4,-5] => ? = 3
[-1,-2,3,-4,5] => [1,-2,-3,-4,5] => [-1,2,3,4,-5] => ? = 3
[-1,-2,3,-4,-5] => [-1,2,-3,-4,5] => [1,-2,3,4,-5] => ? = 3
[-1,-2,-3,4,5] => [1,2,-3,4,5] => [-1,-2,3,-4,-5] => ? = 1
[-1,-2,-3,4,-5] => [-1,-2,-3,4,5] => [1,2,3,-4,-5] => ? = 3
[-1,-2,-3,-4,5] => [1,-2,3,4,5] => [-1,2,-3,-4,-5] => ? = 1
[-1,-2,-3,-4,-5] => [-1,2,3,4,5] => [1,-2,-3,-4,-5] => ? = 1
[1,2,3,5,-4] => [-1,-2,3,5,4] => [1,2,-3,-5,-4] => ? = 2
[1,2,3,-5,4] => [1,-2,-3,5,4] => [-1,2,3,-5,-4] => ? = 2
[1,2,3,-5,-4] => [-1,2,-3,5,4] => [1,-2,3,-5,-4] => ? = 2
[1,2,-3,5,4] => [1,2,-3,-5,4] => [-1,-2,3,5,-4] => ? = 2
[1,2,-3,-5,4] => [1,-2,3,-5,4] => [-1,2,-3,5,-4] => ? = 2
[1,2,-3,-5,-4] => [-1,2,3,-5,4] => [1,-2,-3,5,-4] => ? = 2
[1,-2,3,5,4] => [1,2,3,-5,-4] => [-1,-2,-3,5,4] => ? = 2
[1,-2,3,5,-4] => [-1,-2,3,-5,-4] => [1,2,-3,5,4] => ? = 4
[1,-2,3,-5,-4] => [-1,2,-3,-5,-4] => [1,-2,3,5,4] => ? = 4
[1,-2,-3,5,4] => [1,2,-3,5,-4] => [-1,-2,3,-5,4] => ? = 2
[1,-2,-3,-5,4] => [1,-2,3,5,-4] => [-1,2,-3,-5,4] => ? = 2
[1,-2,-3,-5,-4] => [-1,2,3,5,-4] => [1,-2,-3,-5,4] => ? = 2
[-1,2,3,5,4] => [1,2,3,5,-4] => [-1,-2,-3,-5,4] => ? = 1
[-1,2,3,5,-4] => [-1,-2,3,5,-4] => [1,2,-3,-5,4] => ? = 3
[-1,2,3,-5,4] => [1,-2,-3,5,-4] => [-1,2,3,-5,4] => ? = 3
[-1,2,3,-5,-4] => [-1,2,-3,5,-4] => [1,-2,3,-5,4] => ? = 3
[-1,2,-3,5,4] => [1,2,-3,-5,-4] => [-1,-2,3,5,4] => ? = 3
[-1,2,-3,-5,4] => [1,-2,3,-5,-4] => [-1,2,-3,5,4] => ? = 3
[-1,2,-3,-5,-4] => [-1,2,3,-5,-4] => [1,-2,-3,5,4] => ? = 3
[-1,-2,3,5,4] => [1,2,3,-5,4] => [-1,-2,-3,5,-4] => ? = 1
[-1,-2,3,5,-4] => [-1,-2,3,-5,4] => [1,2,-3,5,-4] => ? = 3
[-1,-2,3,-5,4] => [1,-2,-3,-5,4] => [-1,2,3,5,-4] => ? = 3
Description
The number of positive entries in a signed permutation.
Matching statistic: St001429
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00188: Signed permutations —complement⟶ Signed permutations
Mp00271: Signed permutations —flag negatives to negatives⟶ Signed permutations
St001429: Signed permutations ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 100%
Mp00271: Signed permutations —flag negatives to negatives⟶ Signed permutations
St001429: Signed permutations ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[-1] => [-1] => [-1] => 1
[1,2] => [2,1] => [2,1] => 0
[1,-2] => [2,-1] => [-2,-1] => 2
[-1,2] => [-2,1] => [2,-1] => 1
[-1,-2] => [-2,-1] => [-2,1] => 1
[2,1] => [1,2] => [1,2] => 0
[2,-1] => [1,-2] => [-1,-2] => 2
[-2,1] => [-1,2] => [1,-2] => 1
[-2,-1] => [-1,-2] => [-1,2] => 1
[1,2,3] => [3,2,1] => [3,2,1] => 0
[1,2,-3] => [3,2,-1] => [-3,-2,1] => 2
[1,-2,3] => [3,-2,1] => [3,-2,-1] => 2
[1,-2,-3] => [3,-2,-1] => [-3,2,-1] => 2
[-1,2,3] => [-3,2,1] => [3,2,-1] => 1
[-1,2,-3] => [-3,2,-1] => [-3,-2,-1] => 3
[-1,-2,3] => [-3,-2,1] => [3,-2,1] => 1
[-1,-2,-3] => [-3,-2,-1] => [-3,2,1] => 1
[1,3,2] => [3,1,2] => [3,1,2] => 0
[1,3,-2] => [3,1,-2] => [-3,-1,2] => 2
[1,-3,2] => [3,-1,2] => [3,-1,-2] => 2
[1,-3,-2] => [3,-1,-2] => [-3,1,-2] => 2
[-1,3,2] => [-3,1,2] => [3,1,-2] => 1
[-1,3,-2] => [-3,1,-2] => [-3,-1,-2] => 3
[-1,-3,2] => [-3,-1,2] => [3,-1,2] => 1
[-1,-3,-2] => [-3,-1,-2] => [-3,1,2] => 1
[2,1,3] => [2,3,1] => [2,3,1] => 0
[2,1,-3] => [2,3,-1] => [-2,-3,1] => 2
[2,-1,3] => [2,-3,1] => [2,-3,-1] => 2
[2,-1,-3] => [2,-3,-1] => [-2,3,-1] => 2
[-2,1,3] => [-2,3,1] => [2,3,-1] => 1
[-2,1,-3] => [-2,3,-1] => [-2,-3,-1] => 3
[-2,-1,3] => [-2,-3,1] => [2,-3,1] => 1
[-2,-1,-3] => [-2,-3,-1] => [-2,3,1] => 1
[2,3,1] => [2,1,3] => [2,1,3] => 0
[2,3,-1] => [2,1,-3] => [-2,-1,3] => 2
[2,-3,1] => [2,-1,3] => [2,-1,-3] => 2
[2,-3,-1] => [2,-1,-3] => [-2,1,-3] => 2
[-2,3,1] => [-2,1,3] => [2,1,-3] => 1
[-2,3,-1] => [-2,1,-3] => [-2,-1,-3] => 3
[-2,-3,1] => [-2,-1,3] => [2,-1,3] => 1
[-2,-3,-1] => [-2,-1,-3] => [-2,1,3] => 1
[3,1,2] => [1,3,2] => [1,3,2] => 0
[3,1,-2] => [1,3,-2] => [-1,-3,2] => 2
[3,-1,2] => [1,-3,2] => [1,-3,-2] => 2
[3,-1,-2] => [1,-3,-2] => [-1,3,-2] => 2
[-3,1,2] => [-1,3,2] => [1,3,-2] => 1
[-3,1,-2] => [-1,3,-2] => [-1,-3,-2] => 3
[-3,-1,2] => [-1,-3,2] => [1,-3,2] => 1
[-3,-1,-2] => [-1,-3,-2] => [-1,3,2] => 1
[1,2,3,4,-5] => [5,4,3,2,-1] => [-5,-4,3,2,1] => ? = 2
[1,2,3,-4,5] => [5,4,3,-2,1] => [5,-4,-3,2,1] => ? = 2
[1,2,3,-4,-5] => [5,4,3,-2,-1] => [-5,4,-3,2,1] => ? = 2
[1,2,-3,4,5] => [5,4,-3,2,1] => [5,4,-3,-2,1] => ? = 2
[1,2,-3,-4,5] => [5,4,-3,-2,1] => [5,-4,3,-2,1] => ? = 2
[1,2,-3,-4,-5] => [5,4,-3,-2,-1] => [-5,4,3,-2,1] => ? = 2
[1,-2,3,4,5] => [5,-4,3,2,1] => [5,4,3,-2,-1] => ? = 2
[1,-2,3,4,-5] => [5,-4,3,2,-1] => [-5,-4,3,-2,-1] => ? = 4
[1,-2,3,-4,5] => [5,-4,3,-2,1] => [5,-4,-3,-2,-1] => ? = 4
[1,-2,3,-4,-5] => [5,-4,3,-2,-1] => [-5,4,-3,-2,-1] => ? = 4
[1,-2,-3,4,5] => [5,-4,-3,2,1] => [5,4,-3,2,-1] => ? = 2
[1,-2,-3,4,-5] => [5,-4,-3,2,-1] => [-5,-4,-3,2,-1] => ? = 4
[1,-2,-3,-4,5] => [5,-4,-3,-2,1] => [5,-4,3,2,-1] => ? = 2
[1,-2,-3,-4,-5] => [5,-4,-3,-2,-1] => [-5,4,3,2,-1] => ? = 2
[-1,2,3,4,-5] => [-5,4,3,2,-1] => [-5,-4,3,2,-1] => ? = 3
[-1,2,3,-4,5] => [-5,4,3,-2,1] => [5,-4,-3,2,-1] => ? = 3
[-1,2,3,-4,-5] => [-5,4,3,-2,-1] => [-5,4,-3,2,-1] => ? = 3
[-1,2,-3,4,5] => [-5,4,-3,2,1] => [5,4,-3,-2,-1] => ? = 3
[-1,2,-3,-4,5] => [-5,4,-3,-2,1] => [5,-4,3,-2,-1] => ? = 3
[-1,2,-3,-4,-5] => [-5,4,-3,-2,-1] => [-5,4,3,-2,-1] => ? = 3
[-1,-2,3,4,5] => [-5,-4,3,2,1] => [5,4,3,-2,1] => ? = 1
[-1,-2,3,4,-5] => [-5,-4,3,2,-1] => [-5,-4,3,-2,1] => ? = 3
[-1,-2,3,-4,5] => [-5,-4,3,-2,1] => [5,-4,-3,-2,1] => ? = 3
[-1,-2,3,-4,-5] => [-5,-4,3,-2,-1] => [-5,4,-3,-2,1] => ? = 3
[-1,-2,-3,4,5] => [-5,-4,-3,2,1] => [5,4,-3,2,1] => ? = 1
[-1,-2,-3,4,-5] => [-5,-4,-3,2,-1] => [-5,-4,-3,2,1] => ? = 3
[-1,-2,-3,-4,5] => [-5,-4,-3,-2,1] => [5,-4,3,2,1] => ? = 1
[-1,-2,-3,-4,-5] => [-5,-4,-3,-2,-1] => [-5,4,3,2,1] => ? = 1
[1,2,-3,5,4] => [5,4,-3,1,2] => [5,4,-3,-1,2] => ? = 2
[1,2,-3,5,-4] => [5,4,-3,1,-2] => [-5,-4,-3,-1,2] => ? = 4
[1,2,-3,-5,4] => [5,4,-3,-1,2] => [5,-4,3,-1,2] => ? = 2
[1,2,-3,-5,-4] => [5,4,-3,-1,-2] => [-5,4,3,-1,2] => ? = 2
[1,-2,3,5,4] => [5,-4,3,1,2] => [5,4,3,-1,-2] => ? = 2
[1,-2,3,5,-4] => [5,-4,3,1,-2] => [-5,-4,3,-1,-2] => ? = 4
[1,-2,3,-5,4] => [5,-4,3,-1,2] => [5,-4,-3,-1,-2] => ? = 4
[1,-2,3,-5,-4] => [5,-4,3,-1,-2] => [-5,4,-3,-1,-2] => ? = 4
[1,-2,-3,5,4] => [5,-4,-3,1,2] => [5,4,-3,1,-2] => ? = 2
[1,-2,-3,5,-4] => [5,-4,-3,1,-2] => [-5,-4,-3,1,-2] => ? = 4
[1,-2,-3,-5,4] => [5,-4,-3,-1,2] => [5,-4,3,1,-2] => ? = 2
[1,-2,-3,-5,-4] => [5,-4,-3,-1,-2] => [-5,4,3,1,-2] => ? = 2
[-1,2,3,5,-4] => [-5,4,3,1,-2] => [-5,-4,3,1,-2] => ? = 3
[-1,2,3,-5,4] => [-5,4,3,-1,2] => [5,-4,-3,1,-2] => ? = 3
[-1,2,3,-5,-4] => [-5,4,3,-1,-2] => [-5,4,-3,1,-2] => ? = 3
[-1,2,-3,5,4] => [-5,4,-3,1,2] => [5,4,-3,-1,-2] => ? = 3
[-1,2,-3,-5,4] => [-5,4,-3,-1,2] => [5,-4,3,-1,-2] => ? = 3
[-1,2,-3,-5,-4] => [-5,4,-3,-1,-2] => [-5,4,3,-1,-2] => ? = 3
[-1,-2,3,5,-4] => [-5,-4,3,1,-2] => [-5,-4,3,-1,2] => ? = 3
[-1,-2,3,-5,4] => [-5,-4,3,-1,2] => [5,-4,-3,-1,2] => ? = 3
[-1,-2,3,-5,-4] => [-5,-4,3,-1,-2] => [-5,4,-3,-1,2] => ? = 3
[-1,-2,-3,5,4] => [-5,-4,-3,1,2] => [5,4,-3,1,2] => ? = 1
Description
The number of negative entries in a signed permutation.
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