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Matching statistic: St000691
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Mp00267: Signed permutations —signs⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 => 0 = 2 - 2
[-1] => 1 => 0 = 2 - 2
[1,2] => 00 => 0 = 2 - 2
[1,-2] => 01 => 1 = 3 - 2
[-1,2] => 10 => 1 = 3 - 2
[-1,-2] => 11 => 0 = 2 - 2
[2,1] => 00 => 0 = 2 - 2
[2,-1] => 01 => 1 = 3 - 2
[-2,1] => 10 => 1 = 3 - 2
[-2,-1] => 11 => 0 = 2 - 2
[1,2,3] => 000 => 0 = 2 - 2
[1,2,-3] => 001 => 1 = 3 - 2
[1,-2,3] => 010 => 2 = 4 - 2
[1,-2,-3] => 011 => 1 = 3 - 2
[-1,2,3] => 100 => 1 = 3 - 2
[-1,2,-3] => 101 => 2 = 4 - 2
[-1,-2,3] => 110 => 1 = 3 - 2
[-1,-2,-3] => 111 => 0 = 2 - 2
[1,3,2] => 000 => 0 = 2 - 2
[1,3,-2] => 001 => 1 = 3 - 2
[1,-3,2] => 010 => 2 = 4 - 2
[1,-3,-2] => 011 => 1 = 3 - 2
[-1,3,2] => 100 => 1 = 3 - 2
[-1,3,-2] => 101 => 2 = 4 - 2
[-1,-3,2] => 110 => 1 = 3 - 2
[-1,-3,-2] => 111 => 0 = 2 - 2
[2,1,3] => 000 => 0 = 2 - 2
[2,1,-3] => 001 => 1 = 3 - 2
[2,-1,3] => 010 => 2 = 4 - 2
[2,-1,-3] => 011 => 1 = 3 - 2
[-2,1,3] => 100 => 1 = 3 - 2
[-2,1,-3] => 101 => 2 = 4 - 2
[-2,-1,3] => 110 => 1 = 3 - 2
[-2,-1,-3] => 111 => 0 = 2 - 2
[2,3,1] => 000 => 0 = 2 - 2
[2,3,-1] => 001 => 1 = 3 - 2
[2,-3,1] => 010 => 2 = 4 - 2
[2,-3,-1] => 011 => 1 = 3 - 2
[-2,3,1] => 100 => 1 = 3 - 2
[-2,3,-1] => 101 => 2 = 4 - 2
[-2,-3,1] => 110 => 1 = 3 - 2
[-2,-3,-1] => 111 => 0 = 2 - 2
[3,1,2] => 000 => 0 = 2 - 2
[3,1,-2] => 001 => 1 = 3 - 2
[3,-1,2] => 010 => 2 = 4 - 2
[3,-1,-2] => 011 => 1 = 3 - 2
[-3,1,2] => 100 => 1 = 3 - 2
[-3,1,-2] => 101 => 2 = 4 - 2
[-3,-1,2] => 110 => 1 = 3 - 2
[-3,-1,-2] => 111 => 0 = 2 - 2
Description
The number of changes of a binary word.
This is the number of indices $i$ such that $w_i \neq w_{i+1}$.
Matching statistic: St001486
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(load all 4 compositions to match this statistic)
Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 => [2] => 2
[-1] => 1 => [1,1] => 2
[1,2] => 00 => [3] => 2
[1,-2] => 01 => [2,1] => 3
[-1,2] => 10 => [1,2] => 3
[-1,-2] => 11 => [1,1,1] => 2
[2,1] => 00 => [3] => 2
[2,-1] => 01 => [2,1] => 3
[-2,1] => 10 => [1,2] => 3
[-2,-1] => 11 => [1,1,1] => 2
[1,2,3] => 000 => [4] => 2
[1,2,-3] => 001 => [3,1] => 3
[1,-2,3] => 010 => [2,2] => 4
[1,-2,-3] => 011 => [2,1,1] => 3
[-1,2,3] => 100 => [1,3] => 3
[-1,2,-3] => 101 => [1,2,1] => 4
[-1,-2,3] => 110 => [1,1,2] => 3
[-1,-2,-3] => 111 => [1,1,1,1] => 2
[1,3,2] => 000 => [4] => 2
[1,3,-2] => 001 => [3,1] => 3
[1,-3,2] => 010 => [2,2] => 4
[1,-3,-2] => 011 => [2,1,1] => 3
[-1,3,2] => 100 => [1,3] => 3
[-1,3,-2] => 101 => [1,2,1] => 4
[-1,-3,2] => 110 => [1,1,2] => 3
[-1,-3,-2] => 111 => [1,1,1,1] => 2
[2,1,3] => 000 => [4] => 2
[2,1,-3] => 001 => [3,1] => 3
[2,-1,3] => 010 => [2,2] => 4
[2,-1,-3] => 011 => [2,1,1] => 3
[-2,1,3] => 100 => [1,3] => 3
[-2,1,-3] => 101 => [1,2,1] => 4
[-2,-1,3] => 110 => [1,1,2] => 3
[-2,-1,-3] => 111 => [1,1,1,1] => 2
[2,3,1] => 000 => [4] => 2
[2,3,-1] => 001 => [3,1] => 3
[2,-3,1] => 010 => [2,2] => 4
[2,-3,-1] => 011 => [2,1,1] => 3
[-2,3,1] => 100 => [1,3] => 3
[-2,3,-1] => 101 => [1,2,1] => 4
[-2,-3,1] => 110 => [1,1,2] => 3
[-2,-3,-1] => 111 => [1,1,1,1] => 2
[3,1,2] => 000 => [4] => 2
[3,1,-2] => 001 => [3,1] => 3
[3,-1,2] => 010 => [2,2] => 4
[3,-1,-2] => 011 => [2,1,1] => 3
[-3,1,2] => 100 => [1,3] => 3
[-3,1,-2] => 101 => [1,2,1] => 4
[-3,-1,2] => 110 => [1,1,2] => 3
[-3,-1,-2] => 111 => [1,1,1,1] => 2
Description
The number of corners of the ribbon associated with an integer composition.
We associate a ribbon shape to a composition $c=(c_1,\dots,c_n)$ with $c_i$ cells in the $i$-th row from bottom to top, such that the cells in two rows overlap in precisely one cell.
This statistic records the total number of corners of the ribbon shape.
Matching statistic: St000010
Mp00267: Signed permutations —signs⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 => [1] => [1]
=> 1 = 2 - 1
[-1] => 1 => [1] => [1]
=> 1 = 2 - 1
[1,2] => 00 => [2] => [2]
=> 1 = 2 - 1
[1,-2] => 01 => [1,1] => [1,1]
=> 2 = 3 - 1
[-1,2] => 10 => [1,1] => [1,1]
=> 2 = 3 - 1
[-1,-2] => 11 => [2] => [2]
=> 1 = 2 - 1
[2,1] => 00 => [2] => [2]
=> 1 = 2 - 1
[2,-1] => 01 => [1,1] => [1,1]
=> 2 = 3 - 1
[-2,1] => 10 => [1,1] => [1,1]
=> 2 = 3 - 1
[-2,-1] => 11 => [2] => [2]
=> 1 = 2 - 1
[1,2,3] => 000 => [3] => [3]
=> 1 = 2 - 1
[1,2,-3] => 001 => [2,1] => [2,1]
=> 2 = 3 - 1
[1,-2,3] => 010 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[1,-2,-3] => 011 => [1,2] => [2,1]
=> 2 = 3 - 1
[-1,2,3] => 100 => [1,2] => [2,1]
=> 2 = 3 - 1
[-1,2,-3] => 101 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[-1,-2,3] => 110 => [2,1] => [2,1]
=> 2 = 3 - 1
[-1,-2,-3] => 111 => [3] => [3]
=> 1 = 2 - 1
[1,3,2] => 000 => [3] => [3]
=> 1 = 2 - 1
[1,3,-2] => 001 => [2,1] => [2,1]
=> 2 = 3 - 1
[1,-3,2] => 010 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[1,-3,-2] => 011 => [1,2] => [2,1]
=> 2 = 3 - 1
[-1,3,2] => 100 => [1,2] => [2,1]
=> 2 = 3 - 1
[-1,3,-2] => 101 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[-1,-3,2] => 110 => [2,1] => [2,1]
=> 2 = 3 - 1
[-1,-3,-2] => 111 => [3] => [3]
=> 1 = 2 - 1
[2,1,3] => 000 => [3] => [3]
=> 1 = 2 - 1
[2,1,-3] => 001 => [2,1] => [2,1]
=> 2 = 3 - 1
[2,-1,3] => 010 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[2,-1,-3] => 011 => [1,2] => [2,1]
=> 2 = 3 - 1
[-2,1,3] => 100 => [1,2] => [2,1]
=> 2 = 3 - 1
[-2,1,-3] => 101 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[-2,-1,3] => 110 => [2,1] => [2,1]
=> 2 = 3 - 1
[-2,-1,-3] => 111 => [3] => [3]
=> 1 = 2 - 1
[2,3,1] => 000 => [3] => [3]
=> 1 = 2 - 1
[2,3,-1] => 001 => [2,1] => [2,1]
=> 2 = 3 - 1
[2,-3,1] => 010 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[2,-3,-1] => 011 => [1,2] => [2,1]
=> 2 = 3 - 1
[-2,3,1] => 100 => [1,2] => [2,1]
=> 2 = 3 - 1
[-2,3,-1] => 101 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[-2,-3,1] => 110 => [2,1] => [2,1]
=> 2 = 3 - 1
[-2,-3,-1] => 111 => [3] => [3]
=> 1 = 2 - 1
[3,1,2] => 000 => [3] => [3]
=> 1 = 2 - 1
[3,1,-2] => 001 => [2,1] => [2,1]
=> 2 = 3 - 1
[3,-1,2] => 010 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[3,-1,-2] => 011 => [1,2] => [2,1]
=> 2 = 3 - 1
[-3,1,2] => 100 => [1,2] => [2,1]
=> 2 = 3 - 1
[-3,1,-2] => 101 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[-3,-1,2] => 110 => [2,1] => [2,1]
=> 2 = 3 - 1
[-3,-1,-2] => 111 => [3] => [3]
=> 1 = 2 - 1
Description
The length of the partition.
Matching statistic: St000097
Mp00267: Signed permutations —signs⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 => [1] => ([],1)
=> 1 = 2 - 1
[-1] => 1 => [1] => ([],1)
=> 1 = 2 - 1
[1,2] => 00 => [2] => ([],2)
=> 1 = 2 - 1
[1,-2] => 01 => [1,1] => ([(0,1)],2)
=> 2 = 3 - 1
[-1,2] => 10 => [1,1] => ([(0,1)],2)
=> 2 = 3 - 1
[-1,-2] => 11 => [2] => ([],2)
=> 1 = 2 - 1
[2,1] => 00 => [2] => ([],2)
=> 1 = 2 - 1
[2,-1] => 01 => [1,1] => ([(0,1)],2)
=> 2 = 3 - 1
[-2,1] => 10 => [1,1] => ([(0,1)],2)
=> 2 = 3 - 1
[-2,-1] => 11 => [2] => ([],2)
=> 1 = 2 - 1
[1,2,3] => 000 => [3] => ([],3)
=> 1 = 2 - 1
[1,2,-3] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,-2,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[1,-2,-3] => 011 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[-1,2,3] => 100 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[-1,2,-3] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[-1,-2,3] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[-1,-2,-3] => 111 => [3] => ([],3)
=> 1 = 2 - 1
[1,3,2] => 000 => [3] => ([],3)
=> 1 = 2 - 1
[1,3,-2] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,-3,2] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[1,-3,-2] => 011 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[-1,3,2] => 100 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[-1,3,-2] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[-1,-3,2] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[-1,-3,-2] => 111 => [3] => ([],3)
=> 1 = 2 - 1
[2,1,3] => 000 => [3] => ([],3)
=> 1 = 2 - 1
[2,1,-3] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[2,-1,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[2,-1,-3] => 011 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[-2,1,3] => 100 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[-2,1,-3] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[-2,-1,3] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[-2,-1,-3] => 111 => [3] => ([],3)
=> 1 = 2 - 1
[2,3,1] => 000 => [3] => ([],3)
=> 1 = 2 - 1
[2,3,-1] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[2,-3,1] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[2,-3,-1] => 011 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[-2,3,1] => 100 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[-2,3,-1] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[-2,-3,1] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[-2,-3,-1] => 111 => [3] => ([],3)
=> 1 = 2 - 1
[3,1,2] => 000 => [3] => ([],3)
=> 1 = 2 - 1
[3,1,-2] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[3,-1,2] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[3,-1,-2] => 011 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[-3,1,2] => 100 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[-3,1,-2] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[-3,-1,2] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[-3,-1,-2] => 111 => [3] => ([],3)
=> 1 = 2 - 1
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000288
Mp00267: Signed permutations —signs⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 => [1] => 1 => 1 = 2 - 1
[-1] => 1 => [1] => 1 => 1 = 2 - 1
[1,2] => 00 => [2] => 10 => 1 = 2 - 1
[1,-2] => 01 => [1,1] => 11 => 2 = 3 - 1
[-1,2] => 10 => [1,1] => 11 => 2 = 3 - 1
[-1,-2] => 11 => [2] => 10 => 1 = 2 - 1
[2,1] => 00 => [2] => 10 => 1 = 2 - 1
[2,-1] => 01 => [1,1] => 11 => 2 = 3 - 1
[-2,1] => 10 => [1,1] => 11 => 2 = 3 - 1
[-2,-1] => 11 => [2] => 10 => 1 = 2 - 1
[1,2,3] => 000 => [3] => 100 => 1 = 2 - 1
[1,2,-3] => 001 => [2,1] => 101 => 2 = 3 - 1
[1,-2,3] => 010 => [1,1,1] => 111 => 3 = 4 - 1
[1,-2,-3] => 011 => [1,2] => 110 => 2 = 3 - 1
[-1,2,3] => 100 => [1,2] => 110 => 2 = 3 - 1
[-1,2,-3] => 101 => [1,1,1] => 111 => 3 = 4 - 1
[-1,-2,3] => 110 => [2,1] => 101 => 2 = 3 - 1
[-1,-2,-3] => 111 => [3] => 100 => 1 = 2 - 1
[1,3,2] => 000 => [3] => 100 => 1 = 2 - 1
[1,3,-2] => 001 => [2,1] => 101 => 2 = 3 - 1
[1,-3,2] => 010 => [1,1,1] => 111 => 3 = 4 - 1
[1,-3,-2] => 011 => [1,2] => 110 => 2 = 3 - 1
[-1,3,2] => 100 => [1,2] => 110 => 2 = 3 - 1
[-1,3,-2] => 101 => [1,1,1] => 111 => 3 = 4 - 1
[-1,-3,2] => 110 => [2,1] => 101 => 2 = 3 - 1
[-1,-3,-2] => 111 => [3] => 100 => 1 = 2 - 1
[2,1,3] => 000 => [3] => 100 => 1 = 2 - 1
[2,1,-3] => 001 => [2,1] => 101 => 2 = 3 - 1
[2,-1,3] => 010 => [1,1,1] => 111 => 3 = 4 - 1
[2,-1,-3] => 011 => [1,2] => 110 => 2 = 3 - 1
[-2,1,3] => 100 => [1,2] => 110 => 2 = 3 - 1
[-2,1,-3] => 101 => [1,1,1] => 111 => 3 = 4 - 1
[-2,-1,3] => 110 => [2,1] => 101 => 2 = 3 - 1
[-2,-1,-3] => 111 => [3] => 100 => 1 = 2 - 1
[2,3,1] => 000 => [3] => 100 => 1 = 2 - 1
[2,3,-1] => 001 => [2,1] => 101 => 2 = 3 - 1
[2,-3,1] => 010 => [1,1,1] => 111 => 3 = 4 - 1
[2,-3,-1] => 011 => [1,2] => 110 => 2 = 3 - 1
[-2,3,1] => 100 => [1,2] => 110 => 2 = 3 - 1
[-2,3,-1] => 101 => [1,1,1] => 111 => 3 = 4 - 1
[-2,-3,1] => 110 => [2,1] => 101 => 2 = 3 - 1
[-2,-3,-1] => 111 => [3] => 100 => 1 = 2 - 1
[3,1,2] => 000 => [3] => 100 => 1 = 2 - 1
[3,1,-2] => 001 => [2,1] => 101 => 2 = 3 - 1
[3,-1,2] => 010 => [1,1,1] => 111 => 3 = 4 - 1
[3,-1,-2] => 011 => [1,2] => 110 => 2 = 3 - 1
[-3,1,2] => 100 => [1,2] => 110 => 2 = 3 - 1
[-3,1,-2] => 101 => [1,1,1] => 111 => 3 = 4 - 1
[-3,-1,2] => 110 => [2,1] => 101 => 2 = 3 - 1
[-3,-1,-2] => 111 => [3] => 100 => 1 = 2 - 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St001581
Mp00267: Signed permutations —signs⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001581: Graphs ⟶ ℤResult quality: 88% ●values known / values provided: 98%●distinct values known / distinct values provided: 88%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001581: Graphs ⟶ ℤResult quality: 88% ●values known / values provided: 98%●distinct values known / distinct values provided: 88%
Values
[1] => 0 => [1] => ([],1)
=> 1 = 2 - 1
[-1] => 1 => [1] => ([],1)
=> 1 = 2 - 1
[1,2] => 00 => [2] => ([],2)
=> 1 = 2 - 1
[1,-2] => 01 => [1,1] => ([(0,1)],2)
=> 2 = 3 - 1
[-1,2] => 10 => [1,1] => ([(0,1)],2)
=> 2 = 3 - 1
[-1,-2] => 11 => [2] => ([],2)
=> 1 = 2 - 1
[2,1] => 00 => [2] => ([],2)
=> 1 = 2 - 1
[2,-1] => 01 => [1,1] => ([(0,1)],2)
=> 2 = 3 - 1
[-2,1] => 10 => [1,1] => ([(0,1)],2)
=> 2 = 3 - 1
[-2,-1] => 11 => [2] => ([],2)
=> 1 = 2 - 1
[1,2,3] => 000 => [3] => ([],3)
=> 1 = 2 - 1
[1,2,-3] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,-2,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[1,-2,-3] => 011 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[-1,2,3] => 100 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[-1,2,-3] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[-1,-2,3] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[-1,-2,-3] => 111 => [3] => ([],3)
=> 1 = 2 - 1
[1,3,2] => 000 => [3] => ([],3)
=> 1 = 2 - 1
[1,3,-2] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,-3,2] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[1,-3,-2] => 011 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[-1,3,2] => 100 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[-1,3,-2] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[-1,-3,2] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[-1,-3,-2] => 111 => [3] => ([],3)
=> 1 = 2 - 1
[2,1,3] => 000 => [3] => ([],3)
=> 1 = 2 - 1
[2,1,-3] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[2,-1,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[2,-1,-3] => 011 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[-2,1,3] => 100 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[-2,1,-3] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[-2,-1,3] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[-2,-1,-3] => 111 => [3] => ([],3)
=> 1 = 2 - 1
[2,3,1] => 000 => [3] => ([],3)
=> 1 = 2 - 1
[2,3,-1] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[2,-3,1] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[2,-3,-1] => 011 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[-2,3,1] => 100 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[-2,3,-1] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[-2,-3,1] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[-2,-3,-1] => 111 => [3] => ([],3)
=> 1 = 2 - 1
[3,1,2] => 000 => [3] => ([],3)
=> 1 = 2 - 1
[3,1,-2] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[3,-1,2] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[3,-1,-2] => 011 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[-3,1,2] => 100 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[-3,1,-2] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[-3,-1,2] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[-3,-1,-2] => 111 => [3] => ([],3)
=> 1 = 2 - 1
[5,7,-4,2,3,6,-8,1] => 00100010 => [2,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[-4,3,-5,2,-8,-6,1,7] => 10101100 => [1,1,1,1,2,2] => ([(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 - 1
[4,3,-5,2,8,-7,1,6] => 00100100 => [2,1,2,1,2] => ([(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[-4,3,-7,-5,2,6,-8,1] => 10110010 => [1,1,2,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 - 1
[4,3,7,-6,2,5,-8,1] => 00010010 => [3,1,2,1,1] => ([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[3,-4,-8,-6,2,-7,1,5] => 01110100 => [1,3,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[5,8,-4,3,-7,1,2,6] => 00101000 => [2,1,1,1,3] => ([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[4,-7,-5,6,-8,-3,1,2] => 01101100 => [1,2,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[-7,3,-5,6,-8,-4,1,2] => 10101100 => [1,1,1,1,2,2] => ([(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 - 1
[-6,-4,3,5,-7,2,-8,1] => 11001010 => [2,2,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 - 1
[5,6,3,4,-7,2,-8,1] => 00001010 => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[-7,-6,3,-8,4,-5,1,2] => 11010100 => [2,1,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 - 1
[-7,4,-6,5,-8,-3,1,2] => 10101100 => [1,1,1,1,2,2] => ([(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 - 1
[3,-7,-5,-6,2,4,-8,1] => 01110010 => [1,3,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[-7,-6,2,-5,3,4,-8,1] => 11010010 => [2,1,1,2,1,1] => ([(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 - 1
[4,2,-7,8,-6,1,3,5] => 00101000 => [2,1,1,1,3] => ([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[3,7,-6,2,4,5,-8,1] => 00100010 => [2,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[6,8,-3,2,4,-7,1,5] => 00100100 => [2,1,2,1,2] => ([(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[-8,5,-6,-3,2,-7,1,4] => 10110100 => [1,1,2,1,1,2] => ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 - 1
[-6,5,-7,-3,2,4,-8,1] => 10110010 => [1,1,2,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 - 1
[5,-6,-7,-4,2,3,-8,1] => 01110010 => [1,3,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[-8,-6,5,-4,2,-7,1,3] => 11010100 => [2,1,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 - 1
[-7,2,-4,6,-8,-5,1,3] => 10101100 => [1,1,1,1,2,2] => ([(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 - 1
[4,7,6,-5,2,3,-8,1] => 00010010 => [3,1,2,1,1] => ([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[5,7,-6,4,2,3,-8,1] => 00100010 => [2,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[4,-8,-5,2,-6,-7,1,3] => 01101100 => [1,2,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[4,-8,-5,-7,1,-6,2,3] => 01110100 => [1,3,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[5,8,-6,2,-7,1,3,4] => 00101000 => [2,1,1,1,3] => ([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[-6,5,-7,4,-8,-2,1,3] => 10101100 => [1,1,1,1,2,2] => ([(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 - 1
[5,-6,-7,4,-8,-3,1,2] => 01101100 => [1,2,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[6,-7,-4,-8,3,-5,1,2] => 01110100 => [1,3,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[5,7,-8,4,3,-6,1,2] => 00100100 => [2,1,2,1,2] => ([(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[5,8,-6,4,-7,3,1,2] => 00101000 => [2,1,1,1,3] => ([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[-5,4,-6,3,-7,2,-8,1] => 10101010 => [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 9 - 1
[5,-7,-3,2,8,-4,1,6] => 01100100 => [1,2,2,1,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[3,8,-6,2,-7,-5,-4,1] => 00101110 => [2,1,1,3,1] => ([(0,7),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[5,8,-7,-3,2,-6,-4,1] => 00110110 => [2,2,1,2,1] => ([(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[3,8,-7,-4,2,-6,1,5] => 00110100 => [2,2,1,1,2] => ([(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[5,-8,-7,3,-4,1,2,6] => 01101000 => [1,2,1,1,3] => ([(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[3,8,-7,6,-5,-4,1,2] => 00101100 => [2,1,1,2,2] => ([(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[3,8,-7,-6,5,-4,1,2] => 00110100 => [2,2,1,1,2] => ([(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[3,8,-7,-6,2,5,-4,1] => 00110010 => [2,2,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[4,7,8,-6,-3,2,-5,1] => 00011010 => [3,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[3,7,8,-6,4,-5,1,2] => 00010100 => [3,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[-5,8,-7,3,-6,-4,1,2] => 10101100 => [1,1,1,1,2,2] => ([(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 - 1
[-4,3,-7,8,-6,1,2,5] => 10101000 => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 - 1
[-3,7,8,-6,2,-5,1,4] => 10010100 => [1,2,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 - 1
[3,7,8,-6,2,4,-5,1] => 00010010 => [3,1,2,1,1] => ([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[2,7,8,-6,3,-5,1,4] => 00010100 => [3,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[5,-8,-7,2,-4,1,3,6] => 01101000 => [1,2,1,1,3] => ([(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
Description
The achromatic number of a graph.
This is the maximal number of colours of a proper colouring, such that for any pair of colours there are two adjacent vertices with these colours.
Matching statistic: St000011
Mp00267: Signed permutations —signs⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1] => 0 => [1] => [1,0]
=> 1 = 2 - 1
[-1] => 1 => [1] => [1,0]
=> 1 = 2 - 1
[1,2] => 00 => [2] => [1,1,0,0]
=> 1 = 2 - 1
[1,-2] => 01 => [1,1] => [1,0,1,0]
=> 2 = 3 - 1
[-1,2] => 10 => [1,1] => [1,0,1,0]
=> 2 = 3 - 1
[-1,-2] => 11 => [2] => [1,1,0,0]
=> 1 = 2 - 1
[2,1] => 00 => [2] => [1,1,0,0]
=> 1 = 2 - 1
[2,-1] => 01 => [1,1] => [1,0,1,0]
=> 2 = 3 - 1
[-2,1] => 10 => [1,1] => [1,0,1,0]
=> 2 = 3 - 1
[-2,-1] => 11 => [2] => [1,1,0,0]
=> 1 = 2 - 1
[1,2,3] => 000 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,2,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[1,-2,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[1,-2,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-1,2,3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-1,2,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[-1,-2,3] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[-1,-2,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,3,2] => 000 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,3,-2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[1,-3,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[1,-3,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-1,3,2] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-1,3,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[-1,-3,2] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[-1,-3,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[2,1,3] => 000 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[2,1,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,-1,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[2,-1,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-2,1,3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-2,1,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[-2,-1,3] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[-2,-1,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[2,3,1] => 000 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[2,3,-1] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,-3,1] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[2,-3,-1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-2,3,1] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-2,3,-1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[-2,-3,1] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[-2,-3,-1] => 111 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[3,1,2] => 000 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[3,1,-2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[3,-1,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[3,-1,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-3,1,2] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-3,1,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[-3,-1,2] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[-3,-1,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[-3,2,-8,-6,-4,1,5,7] => 10111000 => [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 5 - 1
[3,2,8,-6,-5,1,4,7] => 00011000 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[5,-3,2,4,8,-7,1,6] => 01000100 => [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 6 - 1
[-7,-5,-3,2,4,6,-8,1] => 11100010 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 5 - 1
[5,7,-4,2,3,6,-8,1] => 00100010 => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 6 - 1
[-4,5,2,3,8,-7,1,6] => 10000100 => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 5 - 1
[2,-8,-4,-6,-5,1,3,7] => 01111000 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[-8,-6,1,-5,-4,2,3,7] => 11011000 => [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5 - 1
[-3,6,8,-5,1,2,4,7] => 10010000 => [1,2,1,4] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[-5,4,-8,-6,-2,1,3,7] => 10111000 => [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 5 - 1
[4,-5,-8,-6,-3,1,2,7] => 01111000 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[3,-8,-5,-6,-4,1,2,7] => 01111000 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[-6,4,3,8,-5,1,2,7] => 10001000 => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 5 - 1
[4,3,-5,2,8,-7,1,6] => 00100100 => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 6 - 1
[-4,3,-7,-5,2,6,-8,1] => 10110010 => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 7 - 1
[4,3,7,-6,2,5,-8,1] => 00010010 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 6 - 1
[3,-4,-8,-6,2,-7,1,5] => 01110100 => [1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 6 - 1
[5,-3,4,8,-7,1,2,6] => 01001000 => [1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 6 - 1
[-5,2,4,8,-7,1,3,6] => 10001000 => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 5 - 1
[4,-7,-5,6,-8,-3,1,2] => 01101100 => [1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 6 - 1
[-4,3,6,-7,-8,-5,1,2] => 10011100 => [1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 5 - 1
[-4,3,5,8,7,-6,1,2] => 10000100 => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 5 - 1
[3,-8,5,7,4,-6,1,2] => 01000100 => [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 6 - 1
[8,-6,2,4,-7,1,3,5] => 01001000 => [1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 6 - 1
[4,-8,-6,-3,-7,1,2,5] => 01111000 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[-4,3,-8,-6,-7,1,2,5] => 10111000 => [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 5 - 1
[-7,6,8,1,3,-5,2,4] => 10000100 => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 5 - 1
[5,-6,3,8,2,-7,1,4] => 01000100 => [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 6 - 1
[3,-7,-5,-6,2,4,-8,1] => 01110010 => [1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 6 - 1
[-5,2,6,8,3,-7,1,4] => 10000100 => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 5 - 1
[-5,7,8,-6,2,1,3,4] => 10010000 => [1,2,1,4] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[-4,-8,-7,1,-6,2,3,5] => 11101000 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 5 - 1
[-5,2,8,-7,1,3,4,6] => 10010000 => [1,2,1,4] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[-4,-8,-7,2,-5,1,3,6] => 11101000 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 5 - 1
[-8,-4,-6,2,3,-7,1,5] => 11100100 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 5 - 1
[3,7,-6,2,4,5,-8,1] => 00100010 => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 6 - 1
[6,8,-3,2,4,-7,1,5] => 00100100 => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 6 - 1
[2,-8,-7,-3,-6,1,4,5] => 01111000 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[3,-7,2,8,-5,1,4,6] => 01001000 => [1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 6 - 1
[-3,2,7,8,-4,1,5,6] => 10001000 => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 5 - 1
[-7,6,-8,-4,-2,1,3,5] => 10111000 => [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 5 - 1
[6,-7,-8,-4,-3,1,2,5] => 01111000 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[-7,-8,-3,2,5,-6,1,4] => 11100100 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 5 - 1
[-6,5,-7,-3,2,4,-8,1] => 10110010 => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 7 - 1
[5,-6,-7,-4,2,3,-8,1] => 01110010 => [1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 6 - 1
[-8,4,7,2,5,-6,1,3] => 10000100 => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 5 - 1
[-7,6,1,-8,-5,-4,2,3] => 10011100 => [1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 5 - 1
[6,-7,-3,-8,-5,1,2,4] => 01111000 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[7,2,6,-8,-5,1,3,4] => 00011000 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[7,-8,2,5,-6,1,3,4] => 01001000 => [1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 6 - 1
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000098
Mp00267: Signed permutations —signs⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[1] => 0 => [1] => ([],1)
=> 1 = 2 - 1
[-1] => 1 => [1] => ([],1)
=> 1 = 2 - 1
[1,2] => 00 => [2] => ([],2)
=> 1 = 2 - 1
[1,-2] => 01 => [1,1] => ([(0,1)],2)
=> 2 = 3 - 1
[-1,2] => 10 => [1,1] => ([(0,1)],2)
=> 2 = 3 - 1
[-1,-2] => 11 => [2] => ([],2)
=> 1 = 2 - 1
[2,1] => 00 => [2] => ([],2)
=> 1 = 2 - 1
[2,-1] => 01 => [1,1] => ([(0,1)],2)
=> 2 = 3 - 1
[-2,1] => 10 => [1,1] => ([(0,1)],2)
=> 2 = 3 - 1
[-2,-1] => 11 => [2] => ([],2)
=> 1 = 2 - 1
[1,2,3] => 000 => [3] => ([],3)
=> 1 = 2 - 1
[1,2,-3] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,-2,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[1,-2,-3] => 011 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[-1,2,3] => 100 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[-1,2,-3] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[-1,-2,3] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[-1,-2,-3] => 111 => [3] => ([],3)
=> 1 = 2 - 1
[1,3,2] => 000 => [3] => ([],3)
=> 1 = 2 - 1
[1,3,-2] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,-3,2] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[1,-3,-2] => 011 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[-1,3,2] => 100 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[-1,3,-2] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[-1,-3,2] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[-1,-3,-2] => 111 => [3] => ([],3)
=> 1 = 2 - 1
[2,1,3] => 000 => [3] => ([],3)
=> 1 = 2 - 1
[2,1,-3] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[2,-1,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[2,-1,-3] => 011 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[-2,1,3] => 100 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[-2,1,-3] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[-2,-1,3] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[-2,-1,-3] => 111 => [3] => ([],3)
=> 1 = 2 - 1
[2,3,1] => 000 => [3] => ([],3)
=> 1 = 2 - 1
[2,3,-1] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[2,-3,1] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[2,-3,-1] => 011 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[-2,3,1] => 100 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[-2,3,-1] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[-2,-3,1] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[-2,-3,-1] => 111 => [3] => ([],3)
=> 1 = 2 - 1
[3,1,2] => 000 => [3] => ([],3)
=> 1 = 2 - 1
[3,1,-2] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[3,-1,2] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[3,-1,-2] => 011 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[-3,1,2] => 100 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[-3,1,-2] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[-3,-1,2] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[-3,-1,-2] => 111 => [3] => ([],3)
=> 1 = 2 - 1
[5,6,7,8,3,4,1,2] => 00000000 => [8] => ([],8)
=> ? = 2 - 1
[5,6,7,8,3,1,2,4] => 00000000 => [8] => ([],8)
=> ? = 2 - 1
[5,6,7,8,4,1,2,3] => 00000000 => [8] => ([],8)
=> ? = 2 - 1
[5,6,7,8,1,4,2,3] => 00000000 => [8] => ([],8)
=> ? = 2 - 1
[7,8,5,6,3,4,1,2] => 00000000 => [8] => ([],8)
=> ? = 2 - 1
[-8,-6,-4,-2,1,3,5,7] => 11110000 => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 3 - 1
[6,8,-4,-3,1,2,5,7] => 00110000 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[-3,2,-8,-6,-4,1,5,7] => 10111000 => [1,1,3,3] => ([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[3,2,8,-6,-5,1,4,7] => 00011000 => [3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[-5,-3,2,4,-8,-6,1,7] => 11001100 => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[5,-3,2,4,8,-7,1,6] => 01000100 => [1,1,3,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[-4,5,2,3,8,-7,1,6] => 10000100 => [1,4,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[4,5,2,3,-8,-6,1,7] => 00001100 => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[2,-8,-4,-6,-5,1,3,7] => 01111000 => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[-8,-6,1,-5,-4,2,3,7] => 11011000 => [2,1,2,3] => ([(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[-3,6,8,-5,1,2,4,7] => 10010000 => [1,2,1,4] => ([(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[2,8,-6,-5,1,3,4,7] => 00110000 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[-5,4,-8,-6,-2,1,3,7] => 10111000 => [1,1,3,3] => ([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[4,-5,-8,-6,-3,1,2,7] => 01111000 => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[3,-8,-5,-6,-4,1,2,7] => 01111000 => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[-6,4,3,8,-5,1,2,7] => 10001000 => [1,3,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[-4,3,-5,2,-8,-6,1,7] => 10101100 => [1,1,1,1,2,2] => ([(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 - 1
[4,3,-5,2,8,-7,1,6] => 00100100 => [2,1,2,1,2] => ([(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[3,-4,-8,-6,2,-7,1,5] => 01110100 => [1,3,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[8,4,-7,-6,3,1,2,5] => 00110000 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[-5,4,-7,3,8,1,2,6] => 10100000 => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[5,8,-4,3,-7,1,2,6] => 00101000 => [2,1,1,1,3] => ([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[5,-3,4,8,-7,1,2,6] => 01001000 => [1,1,2,1,3] => ([(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[-5,2,4,8,-7,1,3,6] => 10001000 => [1,3,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[-7,-5,4,6,-8,-2,1,3] => 11001100 => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[4,-7,-5,6,-8,-3,1,2] => 01101100 => [1,2,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[-7,3,-5,6,-8,-4,1,2] => 10101100 => [1,1,1,1,2,2] => ([(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 - 1
[-4,3,6,-7,-8,-5,1,2] => 10011100 => [1,2,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[-4,3,5,8,7,-6,1,2] => 10000100 => [1,4,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[4,-8,6,-7,3,5,1,2] => 01010000 => [1,1,1,1,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[3,-8,5,7,4,-6,1,2] => 01000100 => [1,1,3,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[-7,-6,3,-8,4,-5,1,2] => 11010100 => [2,1,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 - 1
[-7,-6,5,3,-8,-4,1,2] => 11001100 => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[-7,4,-6,5,-8,-3,1,2] => 10101100 => [1,1,1,1,2,2] => ([(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 - 1
[6,7,4,5,-8,-2,1,3] => 00001100 => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[8,-6,2,4,-7,1,3,5] => 01001000 => [1,1,2,1,3] => ([(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[4,-8,-6,-3,-7,1,2,5] => 01111000 => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[-4,3,-8,-6,-7,1,2,5] => 10111000 => [1,1,3,3] => ([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[-7,6,8,1,3,-5,2,4] => 10000100 => [1,4,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[-7,-6,5,8,3,1,2,4] => 11000000 => [2,6] => ([(5,7),(6,7)],8)
=> ? = 3 - 1
[5,-6,3,8,2,-7,1,4] => 01000100 => [1,1,3,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[-5,2,6,8,3,-7,1,4] => 10000100 => [1,4,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[-5,7,8,-6,2,1,3,4] => 10010000 => [1,2,1,4] => ([(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[5,6,7,1,8,2,3,4] => 00000000 => [8] => ([],8)
=> ? = 2 - 1
[-4,-8,-7,1,-6,2,3,5] => 11101000 => [3,1,1,3] => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
Description
The chromatic number of a graph.
The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.
Matching statistic: St001028
Mp00267: Signed permutations —signs⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001028: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 91%●distinct values known / distinct values provided: 88%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001028: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 91%●distinct values known / distinct values provided: 88%
Values
[1] => 0 => [1] => [1,0]
=> 2
[-1] => 1 => [1] => [1,0]
=> 2
[1,2] => 00 => [2] => [1,1,0,0]
=> 2
[1,-2] => 01 => [1,1] => [1,0,1,0]
=> 3
[-1,2] => 10 => [1,1] => [1,0,1,0]
=> 3
[-1,-2] => 11 => [2] => [1,1,0,0]
=> 2
[2,1] => 00 => [2] => [1,1,0,0]
=> 2
[2,-1] => 01 => [1,1] => [1,0,1,0]
=> 3
[-2,1] => 10 => [1,1] => [1,0,1,0]
=> 3
[-2,-1] => 11 => [2] => [1,1,0,0]
=> 2
[1,2,3] => 000 => [3] => [1,1,1,0,0,0]
=> 2
[1,2,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[1,-2,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[1,-2,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-1,2,3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 3
[-1,2,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[-1,-2,3] => 110 => [2,1] => [1,1,0,0,1,0]
=> 3
[-1,-2,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[1,3,2] => 000 => [3] => [1,1,1,0,0,0]
=> 2
[1,3,-2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[1,-3,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[1,-3,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-1,3,2] => 100 => [1,2] => [1,0,1,1,0,0]
=> 3
[-1,3,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[-1,-3,2] => 110 => [2,1] => [1,1,0,0,1,0]
=> 3
[-1,-3,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[2,1,3] => 000 => [3] => [1,1,1,0,0,0]
=> 2
[2,1,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[2,-1,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[2,-1,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-2,1,3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 3
[-2,1,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[-2,-1,3] => 110 => [2,1] => [1,1,0,0,1,0]
=> 3
[-2,-1,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[2,3,1] => 000 => [3] => [1,1,1,0,0,0]
=> 2
[2,3,-1] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[2,-3,1] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[2,-3,-1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-2,3,1] => 100 => [1,2] => [1,0,1,1,0,0]
=> 3
[-2,3,-1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[-2,-3,1] => 110 => [2,1] => [1,1,0,0,1,0]
=> 3
[-2,-3,-1] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[3,1,2] => 000 => [3] => [1,1,1,0,0,0]
=> 2
[3,1,-2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[3,-1,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[3,-1,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-3,1,2] => 100 => [1,2] => [1,0,1,1,0,0]
=> 3
[-3,1,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[-3,-1,2] => 110 => [2,1] => [1,1,0,0,1,0]
=> 3
[-3,-1,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[5,6,7,8,3,4,1,2] => 00000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2
[5,6,7,8,3,1,2,4] => 00000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2
[5,6,7,8,4,1,2,3] => 00000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2
[5,6,7,8,1,4,2,3] => 00000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2
[7,8,5,6,3,4,1,2] => 00000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2
[-8,-6,-4,-2,1,3,5,7] => 11110000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 3
[6,8,-4,-3,1,2,5,7] => 00110000 => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[-3,2,-8,-6,-4,1,5,7] => 10111000 => [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 5
[3,2,8,-6,-5,1,4,7] => 00011000 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 4
[-5,-3,2,4,-8,-6,1,7] => 11001100 => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 5
[5,-3,2,4,8,-7,1,6] => 01000100 => [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 6
[-7,-5,-3,2,4,6,-8,1] => 11100010 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 5
[5,7,-4,2,3,6,-8,1] => 00100010 => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 6
[-4,5,2,3,8,-7,1,6] => 10000100 => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 5
[4,5,2,3,-8,-6,1,7] => 00001100 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[2,-8,-4,-6,-5,1,3,7] => 01111000 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 4
[-8,-6,1,-5,-4,2,3,7] => 11011000 => [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
[-3,6,8,-5,1,2,4,7] => 10010000 => [1,2,1,4] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 5
[2,8,-6,-5,1,3,4,7] => 00110000 => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[-5,4,-8,-6,-2,1,3,7] => 10111000 => [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 5
[4,-5,-8,-6,-3,1,2,7] => 01111000 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 4
[3,-8,-5,-6,-4,1,2,7] => 01111000 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 4
[-6,4,3,8,-5,1,2,7] => 10001000 => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 5
[-4,3,-5,2,-8,-6,1,7] => 10101100 => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 7
[4,3,-5,2,8,-7,1,6] => 00100100 => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 6
[-4,3,-7,-5,2,6,-8,1] => 10110010 => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 7
[4,3,7,-6,2,5,-8,1] => 00010010 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 6
[3,-4,-8,-6,2,-7,1,5] => 01110100 => [1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 6
[8,4,-7,-6,3,1,2,5] => 00110000 => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[-5,4,-7,3,8,1,2,6] => 10100000 => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[5,8,-4,3,-7,1,2,6] => 00101000 => [2,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 6
[5,-3,4,8,-7,1,2,6] => 01001000 => [1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 6
[-5,2,4,8,-7,1,3,6] => 10001000 => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 5
[-7,-5,4,6,-8,-2,1,3] => 11001100 => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 5
[4,-7,-5,6,-8,-3,1,2] => 01101100 => [1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 6
[-7,3,-5,6,-8,-4,1,2] => 10101100 => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 7
[-4,3,6,-7,-8,-5,1,2] => 10011100 => [1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 5
[-4,3,5,8,7,-6,1,2] => 10000100 => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 5
[4,-8,6,-7,3,5,1,2] => 01010000 => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 6
[-6,-4,3,5,-7,2,-8,1] => 11001010 => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 7
[5,6,3,4,-7,2,-8,1] => 00001010 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 6
[3,-8,5,7,4,-6,1,2] => 01000100 => [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 6
[-7,-6,3,-8,4,-5,1,2] => 11010100 => [2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 7
[-7,-6,5,3,-8,-4,1,2] => 11001100 => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 5
[-7,4,-6,5,-8,-3,1,2] => 10101100 => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 7
[6,7,4,5,-8,-2,1,3] => 00001100 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[8,-6,2,4,-7,1,3,5] => 01001000 => [1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 6
[4,-8,-6,-3,-7,1,2,5] => 01111000 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 4
[-4,3,-8,-6,-7,1,2,5] => 10111000 => [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 5
[-7,6,8,1,3,-5,2,4] => 10000100 => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 5
Description
Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001068
Mp00267: Signed permutations —signs⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 91%●distinct values known / distinct values provided: 88%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 91%●distinct values known / distinct values provided: 88%
Values
[1] => 0 => [1] => [1,0]
=> 1 = 2 - 1
[-1] => 1 => [1] => [1,0]
=> 1 = 2 - 1
[1,2] => 00 => [2] => [1,1,0,0]
=> 1 = 2 - 1
[1,-2] => 01 => [1,1] => [1,0,1,0]
=> 2 = 3 - 1
[-1,2] => 10 => [1,1] => [1,0,1,0]
=> 2 = 3 - 1
[-1,-2] => 11 => [2] => [1,1,0,0]
=> 1 = 2 - 1
[2,1] => 00 => [2] => [1,1,0,0]
=> 1 = 2 - 1
[2,-1] => 01 => [1,1] => [1,0,1,0]
=> 2 = 3 - 1
[-2,1] => 10 => [1,1] => [1,0,1,0]
=> 2 = 3 - 1
[-2,-1] => 11 => [2] => [1,1,0,0]
=> 1 = 2 - 1
[1,2,3] => 000 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,2,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[1,-2,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[1,-2,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-1,2,3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-1,2,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[-1,-2,3] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[-1,-2,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,3,2] => 000 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,3,-2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[1,-3,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[1,-3,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-1,3,2] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-1,3,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[-1,-3,2] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[-1,-3,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[2,1,3] => 000 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[2,1,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,-1,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[2,-1,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-2,1,3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-2,1,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[-2,-1,3] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[-2,-1,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[2,3,1] => 000 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[2,3,-1] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,-3,1] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[2,-3,-1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-2,3,1] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-2,3,-1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[-2,-3,1] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[-2,-3,-1] => 111 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[3,1,2] => 000 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[3,1,-2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[3,-1,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[3,-1,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-3,1,2] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-3,1,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[-3,-1,2] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[-3,-1,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[5,6,7,8,3,4,1,2] => 00000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[5,6,7,8,3,1,2,4] => 00000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[5,6,7,8,4,1,2,3] => 00000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[5,6,7,8,1,4,2,3] => 00000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[7,8,5,6,3,4,1,2] => 00000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[-8,-6,-4,-2,1,3,5,7] => 11110000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 3 - 1
[6,8,-4,-3,1,2,5,7] => 00110000 => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 1
[-3,2,-8,-6,-4,1,5,7] => 10111000 => [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 5 - 1
[3,2,8,-6,-5,1,4,7] => 00011000 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[-5,-3,2,4,-8,-6,1,7] => 11001100 => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 5 - 1
[5,-3,2,4,8,-7,1,6] => 01000100 => [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 6 - 1
[-7,-5,-3,2,4,6,-8,1] => 11100010 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 5 - 1
[5,7,-4,2,3,6,-8,1] => 00100010 => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 6 - 1
[-4,5,2,3,8,-7,1,6] => 10000100 => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 5 - 1
[4,5,2,3,-8,-6,1,7] => 00001100 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4 - 1
[2,-8,-4,-6,-5,1,3,7] => 01111000 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[-8,-6,1,-5,-4,2,3,7] => 11011000 => [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5 - 1
[-3,6,8,-5,1,2,4,7] => 10010000 => [1,2,1,4] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[2,8,-6,-5,1,3,4,7] => 00110000 => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 1
[-5,4,-8,-6,-2,1,3,7] => 10111000 => [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 5 - 1
[4,-5,-8,-6,-3,1,2,7] => 01111000 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[3,-8,-5,-6,-4,1,2,7] => 01111000 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[-6,4,3,8,-5,1,2,7] => 10001000 => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 5 - 1
[-4,3,-5,2,-8,-6,1,7] => 10101100 => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 7 - 1
[4,3,-5,2,8,-7,1,6] => 00100100 => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 6 - 1
[-4,3,-7,-5,2,6,-8,1] => 10110010 => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 7 - 1
[4,3,7,-6,2,5,-8,1] => 00010010 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 6 - 1
[3,-4,-8,-6,2,-7,1,5] => 01110100 => [1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 6 - 1
[8,4,-7,-6,3,1,2,5] => 00110000 => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 1
[-5,4,-7,3,8,1,2,6] => 10100000 => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5 - 1
[5,8,-4,3,-7,1,2,6] => 00101000 => [2,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 6 - 1
[5,-3,4,8,-7,1,2,6] => 01001000 => [1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 6 - 1
[-5,2,4,8,-7,1,3,6] => 10001000 => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 5 - 1
[-7,-5,4,6,-8,-2,1,3] => 11001100 => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 5 - 1
[4,-7,-5,6,-8,-3,1,2] => 01101100 => [1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 6 - 1
[-7,3,-5,6,-8,-4,1,2] => 10101100 => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 7 - 1
[-4,3,6,-7,-8,-5,1,2] => 10011100 => [1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 5 - 1
[-4,3,5,8,7,-6,1,2] => 10000100 => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 5 - 1
[4,-8,6,-7,3,5,1,2] => 01010000 => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 6 - 1
[-6,-4,3,5,-7,2,-8,1] => 11001010 => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 7 - 1
[5,6,3,4,-7,2,-8,1] => 00001010 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 6 - 1
[3,-8,5,7,4,-6,1,2] => 01000100 => [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 6 - 1
[-7,-6,3,-8,4,-5,1,2] => 11010100 => [2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 7 - 1
[-7,-6,5,3,-8,-4,1,2] => 11001100 => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 5 - 1
[-7,4,-6,5,-8,-3,1,2] => 10101100 => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 7 - 1
[6,7,4,5,-8,-2,1,3] => 00001100 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4 - 1
[8,-6,2,4,-7,1,3,5] => 01001000 => [1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 6 - 1
[4,-8,-6,-3,-7,1,2,5] => 01111000 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[-4,3,-8,-6,-7,1,2,5] => 10111000 => [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 5 - 1
[-7,6,8,1,3,-5,2,4] => 10000100 => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 5 - 1
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
The following 38 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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:
St001494The Alon-Tarsi number of a graph. St000053The number of valleys of the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001971The number of negative eigenvalues of the adjacency matrix of the graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St000306The bounce count of a Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000015The number of peaks of a Dyck path. St000388The number of orbits of vertices of a graph under automorphisms. St000822The Hadwiger number of the graph. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. 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. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001352The number of internal nodes in the modular decomposition of a graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001812The biclique partition number of a graph. St001330The hat guessing number of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001488The number of corners of a skew partition. St000307The number of rowmotion orbits of a poset. St000632The jump number of the poset.
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