Your data matches 4 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000771
Mapping
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => [1] => ([],1)
 => 1
[-] => [1] => [1] => ([],1)
 => 1
[-,+] => [2,1] => [1,1] => ([(0,1)],2)
 => 1
[-,+,+] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[+,-,+] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[2,1,+] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[3,-,1] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[-,+,+,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[+,-,+,+] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[+,+,-,+] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[+,3,2,+] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[+,4,-,2] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[-,4,-,2] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[2,1,+,+] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[2,4,-,1] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[3,1,2,+] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[3,+,1,+] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[3,-,1,+] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,1,-,2] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[4,-,1,3] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[4,-,+,1] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,+,-,1] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,3,1,2] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[4,3,2,1] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[-,+,+,+,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[+,-,+,+,+] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[+,+,-,+,+] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[+,+,+,-,+] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[+,+,4,3,+] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[+,+,5,-,3] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[-,+,5,-,3] => [2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[+,-,5,-,3] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[-,-,5,-,3] => [3,1,2,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[+,3,2,+,+] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[+,3,5,-,2] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[-,3,5,-,2] => [2,1,3,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[+,4,2,3,+] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[+,4,+,2,+] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[+,4,-,2,+] => [1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[-,4,-,2,+] => [2,5,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[+,5,2,-,3] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[-,5,2,-,3] => [2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[+,5,-,2,4] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[-,5,-,2,4] => [2,4,1,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[+,5,-,+,2] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[+,5,+,-,2] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[-,5,-,+,2] => [4,2,1,5,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[-,5,+,-,2] => [3,2,1,5,4] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[+,5,4,2,3] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[-,5,4,2,3] => [2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian $$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$ Its eigenvalues are $0,4,4,6$, so the statistic is $2$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St000774
Mapping
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000774: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => [1] => ([],1)
 => 1
[-] => [1] => [1] => ([],1)
 => 1
[-,+] => [2,1] => [1,1] => ([(0,1)],2)
 => 1
[-,+,+] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[+,-,+] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[2,1,+] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[3,-,1] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[-,+,+,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[+,-,+,+] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[+,+,-,+] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[+,3,2,+] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[+,4,-,2] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[-,4,-,2] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[2,1,+,+] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[2,4,-,1] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[3,1,2,+] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[3,+,1,+] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[3,-,1,+] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,1,-,2] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[4,-,1,3] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[4,-,+,1] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,+,-,1] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,3,1,2] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[4,3,2,1] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[-,+,+,+,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[+,-,+,+,+] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[+,+,-,+,+] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[+,+,+,-,+] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[+,+,4,3,+] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[+,+,5,-,3] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[-,+,5,-,3] => [2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[+,-,5,-,3] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[-,-,5,-,3] => [3,1,2,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[+,3,2,+,+] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[+,3,5,-,2] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[-,3,5,-,2] => [2,1,3,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[+,4,2,3,+] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[+,4,+,2,+] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[+,4,-,2,+] => [1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[-,4,-,2,+] => [2,5,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[+,5,2,-,3] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[-,5,2,-,3] => [2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[+,5,-,2,4] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[-,5,-,2,4] => [2,4,1,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[+,5,-,+,2] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[+,5,+,-,2] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[-,5,-,+,2] => [4,2,1,5,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[-,5,+,-,2] => [3,2,1,5,4] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[+,5,4,2,3] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[-,5,4,2,3] => [2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
Description
The maximal multiplicity of a Laplacian eigenvalue in a graph.
Matching statistic: St000982
(load all 4 compositions to match this statistic)
Mapping
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => => ? = 1
[-] => [1] => => ? = 1
[-,+] => [2,1] => 1 => 1
[-,+,+] => [2,3,1] => 01 => 1
[+,-,+] => [1,3,2] => 01 => 1
[2,1,+] => [1,3,2] => 01 => 1
[3,-,1] => [1,3,2] => 01 => 1
[-,+,+,+] => [2,3,4,1] => 001 => 2
[+,-,+,+] => [1,3,4,2] => 001 => 2
[+,+,-,+] => [1,2,4,3] => 001 => 2
[+,3,2,+] => [1,2,4,3] => 001 => 2
[+,4,-,2] => [1,2,4,3] => 001 => 2
[-,4,-,2] => [2,1,4,3] => 101 => 1
[2,1,+,+] => [1,3,4,2] => 001 => 2
[2,4,-,1] => [1,2,4,3] => 001 => 2
[3,1,2,+] => [1,2,4,3] => 001 => 2
[3,+,1,+] => [2,1,4,3] => 101 => 1
[3,-,1,+] => [1,4,3,2] => 011 => 2
[4,1,-,2] => [1,2,4,3] => 001 => 2
[4,-,1,3] => [1,3,4,2] => 001 => 2
[4,-,+,1] => [3,1,4,2] => 101 => 1
[4,+,-,1] => [2,1,4,3] => 101 => 1
[4,3,1,2] => [1,2,4,3] => 001 => 2
[4,3,2,1] => [2,1,4,3] => 101 => 1
[-,+,+,+,+] => [2,3,4,5,1] => 0001 => 3
[+,-,+,+,+] => [1,3,4,5,2] => 0001 => 3
[+,+,-,+,+] => [1,2,4,5,3] => 0001 => 3
[+,+,+,-,+] => [1,2,3,5,4] => 0001 => 3
[+,+,4,3,+] => [1,2,3,5,4] => 0001 => 3
[+,+,5,-,3] => [1,2,3,5,4] => 0001 => 3
[-,+,5,-,3] => [2,3,1,5,4] => 0101 => 1
[+,-,5,-,3] => [1,3,2,5,4] => 0101 => 1
[-,-,5,-,3] => [3,1,2,5,4] => 1001 => 2
[+,3,2,+,+] => [1,2,4,5,3] => 0001 => 3
[+,3,5,-,2] => [1,2,3,5,4] => 0001 => 3
[-,3,5,-,2] => [2,1,3,5,4] => 1001 => 2
[+,4,2,3,+] => [1,2,3,5,4] => 0001 => 3
[+,4,+,2,+] => [1,3,2,5,4] => 0101 => 1
[+,4,-,2,+] => [1,2,5,4,3] => 0011 => 2
[-,4,-,2,+] => [2,5,1,4,3] => 0101 => 1
[+,5,2,-,3] => [1,2,3,5,4] => 0001 => 3
[-,5,2,-,3] => [2,3,1,5,4] => 0101 => 1
[+,5,-,2,4] => [1,2,4,5,3] => 0001 => 3
[-,5,-,2,4] => [2,4,1,5,3] => 0101 => 1
[+,5,-,+,2] => [1,4,2,5,3] => 0101 => 1
[+,5,+,-,2] => [1,3,2,5,4] => 0101 => 1
[-,5,-,+,2] => [4,2,1,5,3] => 1101 => 2
[-,5,+,-,2] => [3,2,1,5,4] => 1101 => 2
[+,5,4,2,3] => [1,2,3,5,4] => 0001 => 3
[-,5,4,2,3] => [2,3,1,5,4] => 0101 => 1
[+,5,4,3,2] => [1,3,2,5,4] => 0101 => 1
[-,5,4,3,2] => [3,2,1,5,4] => 1101 => 2
Description
The length of the longest constant subword.
Matching statistic: St000983
Mapping
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => => => ? = 1
[-] => [1] => => => ? = 1
[-,+] => [2,1] => 1 => 1 => 1
[-,+,+] => [2,3,1] => 01 => 00 => 1
[+,-,+] => [1,3,2] => 01 => 00 => 1
[2,1,+] => [1,3,2] => 01 => 00 => 1
[3,-,1] => [1,3,2] => 01 => 00 => 1
[-,+,+,+] => [2,3,4,1] => 001 => 011 => 2
[+,-,+,+] => [1,3,4,2] => 001 => 011 => 2
[+,+,-,+] => [1,2,4,3] => 001 => 011 => 2
[+,3,2,+] => [1,2,4,3] => 001 => 011 => 2
[+,4,-,2] => [1,2,4,3] => 001 => 011 => 2
[-,4,-,2] => [2,1,4,3] => 101 => 111 => 1
[2,1,+,+] => [1,3,4,2] => 001 => 011 => 2
[2,4,-,1] => [1,2,4,3] => 001 => 011 => 2
[3,1,2,+] => [1,2,4,3] => 001 => 011 => 2
[3,+,1,+] => [2,1,4,3] => 101 => 111 => 1
[3,-,1,+] => [1,4,3,2] => 011 => 001 => 2
[4,1,-,2] => [1,2,4,3] => 001 => 011 => 2
[4,-,1,3] => [1,3,4,2] => 001 => 011 => 2
[4,-,+,1] => [3,1,4,2] => 101 => 111 => 1
[4,+,-,1] => [2,1,4,3] => 101 => 111 => 1
[4,3,1,2] => [1,2,4,3] => 001 => 011 => 2
[4,3,2,1] => [2,1,4,3] => 101 => 111 => 1
[-,+,+,+,+] => [2,3,4,5,1] => 0001 => 0100 => 3
[+,-,+,+,+] => [1,3,4,5,2] => 0001 => 0100 => 3
[+,+,-,+,+] => [1,2,4,5,3] => 0001 => 0100 => 3
[+,+,+,-,+] => [1,2,3,5,4] => 0001 => 0100 => 3
[+,+,4,3,+] => [1,2,3,5,4] => 0001 => 0100 => 3
[+,+,5,-,3] => [1,2,3,5,4] => 0001 => 0100 => 3
[-,+,5,-,3] => [2,3,1,5,4] => 0101 => 0000 => 1
[+,-,5,-,3] => [1,3,2,5,4] => 0101 => 0000 => 1
[-,-,5,-,3] => [3,1,2,5,4] => 1001 => 1100 => 2
[+,3,2,+,+] => [1,2,4,5,3] => 0001 => 0100 => 3
[+,3,5,-,2] => [1,2,3,5,4] => 0001 => 0100 => 3
[-,3,5,-,2] => [2,1,3,5,4] => 1001 => 1100 => 2
[+,4,2,3,+] => [1,2,3,5,4] => 0001 => 0100 => 3
[+,4,+,2,+] => [1,3,2,5,4] => 0101 => 0000 => 1
[+,4,-,2,+] => [1,2,5,4,3] => 0011 => 0110 => 2
[-,4,-,2,+] => [2,5,1,4,3] => 0101 => 0000 => 1
[+,5,2,-,3] => [1,2,3,5,4] => 0001 => 0100 => 3
[-,5,2,-,3] => [2,3,1,5,4] => 0101 => 0000 => 1
[+,5,-,2,4] => [1,2,4,5,3] => 0001 => 0100 => 3
[-,5,-,2,4] => [2,4,1,5,3] => 0101 => 0000 => 1
[+,5,-,+,2] => [1,4,2,5,3] => 0101 => 0000 => 1
[+,5,+,-,2] => [1,3,2,5,4] => 0101 => 0000 => 1
[-,5,-,+,2] => [4,2,1,5,3] => 1101 => 1000 => 2
[-,5,+,-,2] => [3,2,1,5,4] => 1101 => 1000 => 2
[+,5,4,2,3] => [1,2,3,5,4] => 0001 => 0100 => 3
[-,5,4,2,3] => [2,3,1,5,4] => 0101 => 0000 => 1
[+,5,4,3,2] => [1,3,2,5,4] => 0101 => 0000 => 1
[-,5,4,3,2] => [3,2,1,5,4] => 1101 => 1000 => 2
Description
The length of the longest alternating subword. This is the length of the longest consecutive subword of the form $010...$ or of the form $101...$.