Your data matches 11 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001567
Mapping
St001567: Decorated permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => 1
[-] => 0
[+,+] => 2
[-,+] => 1
[+,-] => 1
[-,-] => 0
[2,1] => 1
[+,+,+] => 3
[-,+,+] => 2
[+,-,+] => 2
[+,+,-] => 2
[-,-,+] => 1
[-,+,-] => 1
[+,-,-] => 1
[-,-,-] => 0
[+,3,2] => 2
[-,3,2] => 1
[2,1,+] => 2
[2,1,-] => 1
[2,3,1] => 2
[3,1,2] => 1
[3,+,1] => 2
[3,-,1] => 1
[+,+,+,+] => 4
[-,+,+,+] => 3
[+,-,+,+] => 3
[+,+,-,+] => 3
[+,+,+,-] => 3
[-,-,+,+] => 2
[-,+,-,+] => 2
[-,+,+,-] => 2
[+,-,-,+] => 2
[+,-,+,-] => 2
[+,+,-,-] => 2
[-,-,-,+] => 1
[-,-,+,-] => 1
[-,+,-,-] => 1
[+,-,-,-] => 1
[-,-,-,-] => 0
[+,+,4,3] => 3
[-,+,4,3] => 2
[+,-,4,3] => 2
[-,-,4,3] => 1
[+,3,2,+] => 3
[-,3,2,+] => 2
[+,3,2,-] => 2
[-,3,2,-] => 1
[+,3,4,2] => 3
[-,3,4,2] => 2
[+,4,2,3] => 2
Description
The number of weak exceedances of a decorated permutation. A '''weak exceedance''' of a decorated permutation $\tau$ is a position $i$ such that $i > \tau(i)$ or $i = \tau(i)$ and $\tau(i)$ has positive decoration.
Matching statistic: St001542
Mapping
St001542: Decorated permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => ? ∊ {0,1}
[-] => ? ∊ {0,1}
[+,+] => 0
[-,+] => 1
[+,-] => 1
[-,-] => 2
[2,1] => 1
[+,+,+] => 0
[-,+,+] => 1
[+,-,+] => 1
[+,+,-] => 1
[-,-,+] => 2
[-,+,-] => 2
[+,-,-] => 2
[-,-,-] => 3
[+,3,2] => 1
[-,3,2] => 2
[2,1,+] => 1
[2,1,-] => 2
[2,3,1] => 2
[3,1,2] => 1
[3,+,1] => 1
[3,-,1] => 2
[+,+,+,+] => 0
[-,+,+,+] => 1
[+,-,+,+] => 1
[+,+,-,+] => 1
[+,+,+,-] => 1
[-,-,+,+] => 2
[-,+,-,+] => 2
[-,+,+,-] => 2
[+,-,-,+] => 2
[+,-,+,-] => 2
[+,+,-,-] => 2
[-,-,-,+] => 3
[-,-,+,-] => 3
[-,+,-,-] => 3
[+,-,-,-] => 3
[-,-,-,-] => 4
[+,+,4,3] => 1
[-,+,4,3] => 2
[+,-,4,3] => 2
[-,-,4,3] => 3
[+,3,2,+] => 1
[-,3,2,+] => 2
[+,3,2,-] => 2
[-,3,2,-] => 3
[+,3,4,2] => 2
[-,3,4,2] => 3
[+,4,2,3] => 1
[-,4,2,3] => 2
[+,4,+,2] => 1
Description
The dimension of the subspace of the complex vector space for the associated Grassmannian. Given an affine permutation, this is $$\frac{1}{n} \sum^n_{i=1} (f(i)-i)$$ This value is seen as $k$ in the notation Gr($k,n$), ($k,n$)-bounded affine permutations, ($k,n$)-Grassmann necklaces, and ($k,n$)-Le diagrams.
Matching statistic: St001118
Mapping
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001118: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 95%●distinct values known / distinct values provided: 67%
Values
[+] => [1] => [1] => ([],1)
 => ? ∊ {0,1}
[-] => [1] => [1] => ([],1)
 => ? ∊ {0,1}
[+,+] => [1,2] => [2] => ([],2)
 => ? ∊ {0,1,2}
[-,+] => [2,1] => [1,1] => ([(0,1)],2)
 => 1
[+,-] => [1,2] => [2] => ([],2)
 => ? ∊ {0,1,2}
[-,-] => [1,2] => [2] => ([],2)
 => ? ∊ {0,1,2}
[2,1] => [2,1] => [1,1] => ([(0,1)],2)
 => 1
[+,+,+] => [1,2,3] => [3] => ([],3)
 => ? ∊ {0,1,2,3}
[-,+,+] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2
[+,-,+] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => 2
[+,+,-] => [1,2,3] => [3] => ([],3)
 => ? ∊ {0,1,2,3}
[-,-,+] => [3,1,2] => [1,2] => ([(1,2)],3)
 => 1
[-,+,-] => [2,1,3] => [1,2] => ([(1,2)],3)
 => 1
[+,-,-] => [1,2,3] => [3] => ([],3)
 => ? ∊ {0,1,2,3}
[-,-,-] => [1,2,3] => [3] => ([],3)
 => ? ∊ {0,1,2,3}
[+,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => 2
[-,3,2] => [3,1,2] => [1,2] => ([(1,2)],3)
 => 1
[2,1,+] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2
[2,1,-] => [2,1,3] => [1,2] => ([(1,2)],3)
 => 1
[2,3,1] => [3,1,2] => [1,2] => ([(1,2)],3)
 => 1
[3,1,2] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2
[3,+,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2
[3,-,1] => [3,1,2] => [1,2] => ([(1,2)],3)
 => 1
[+,+,+,+] => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {0,1,2,3,4}
[-,+,+,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[+,-,+,+] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[+,+,-,+] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[+,+,+,-] => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {0,1,2,3,4}
[-,-,+,+] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[-,+,-,+] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[-,+,+,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[+,-,-,+] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[+,-,+,-] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[+,+,-,-] => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {0,1,2,3,4}
[-,-,-,+] => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => 1
[-,-,+,-] => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => 1
[-,+,-,-] => [2,1,3,4] => [1,3] => ([(2,3)],4)
 => 1
[+,-,-,-] => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {0,1,2,3,4}
[-,-,-,-] => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {0,1,2,3,4}
[+,+,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[-,+,4,3] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[+,-,4,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[-,-,4,3] => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => 1
[+,3,2,+] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[-,3,2,+] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[+,3,2,-] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[-,3,2,-] => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => 1
[+,3,4,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[-,3,4,2] => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => 1
[+,4,2,3] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[-,4,2,3] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[+,4,+,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[-,4,+,2] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[+,4,-,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[-,4,-,2] => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => 1
[2,1,+,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[2,1,-,+] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[2,1,+,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[2,1,-,-] => [2,1,3,4] => [1,3] => ([(2,3)],4)
 => 1
[2,1,4,3] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[2,3,1,+] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[2,3,1,-] => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => 1
[2,3,4,1] => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => 1
[2,4,1,3] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[+,+,+,+,+] => [1,2,3,4,5] => [5] => ([],5)
 => ? ∊ {0,1,2,3,4,5}
[+,+,+,+,-] => [1,2,3,4,5] => [5] => ([],5)
 => ? ∊ {0,1,2,3,4,5}
[+,+,+,-,-] => [1,2,3,4,5] => [5] => ([],5)
 => ? ∊ {0,1,2,3,4,5}
[+,+,-,-,-] => [1,2,3,4,5] => [5] => ([],5)
 => ? ∊ {0,1,2,3,4,5}
[+,-,-,-,-] => [1,2,3,4,5] => [5] => ([],5)
 => ? ∊ {0,1,2,3,4,5}
[-,-,-,-,-] => [1,2,3,4,5] => [5] => ([],5)
 => ? ∊ {0,1,2,3,4,5}
Description
The acyclic chromatic index of a graph. An acyclic edge coloring of a graph is a proper colouring of the edges of a graph such that the union of the edges colored with any two given colours is a forest. The smallest number of colours such that such a colouring exists is the acyclic chromatic index.
Matching statistic: St001060
(load all 11 compositions to match this statistic)
Mapping
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 57%●distinct values known / distinct values provided: 50%
Values
[+] => [1] => [1] => ([],1)
 => ? ∊ {0,1}
[-] => [1] => [1] => ([],1)
 => ? ∊ {0,1}
[+,+] => [1,2] => [2] => ([],2)
 => ? ∊ {0,1,1,1,2}
[-,+] => [2,1] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,2}
[+,-] => [1,2] => [2] => ([],2)
 => ? ∊ {0,1,1,1,2}
[-,-] => [1,2] => [2] => ([],2)
 => ? ∊ {0,1,1,1,2}
[2,1] => [1,2] => [2] => ([],2)
 => ? ∊ {0,1,1,1,2}
[+,+,+] => [1,2,3] => [3] => ([],3)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3}
[-,+,+] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2
[+,-,+] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => 2
[+,+,-] => [1,2,3] => [3] => ([],3)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3}
[-,-,+] => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3}
[-,+,-] => [2,1,3] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3}
[+,-,-] => [1,2,3] => [3] => ([],3)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3}
[-,-,-] => [1,2,3] => [3] => ([],3)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3}
[+,3,2] => [1,2,3] => [3] => ([],3)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3}
[-,3,2] => [2,1,3] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3}
[2,1,+] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => 2
[2,1,-] => [1,2,3] => [3] => ([],3)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3}
[2,3,1] => [1,2,3] => [3] => ([],3)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3}
[3,1,2] => [1,2,3] => [3] => ([],3)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3}
[3,+,1] => [2,1,3] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3}
[3,-,1] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => 2
[+,+,+,+] => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4}
[-,+,+,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[+,-,+,+] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[+,+,-,+] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[+,+,+,-] => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4}
[-,-,+,+] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[-,+,-,+] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[-,+,+,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[+,-,-,+] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[+,-,+,-] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[+,+,-,-] => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4}
[-,-,-,+] => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4}
[-,-,+,-] => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4}
[-,+,-,-] => [2,1,3,4] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4}
[+,-,-,-] => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4}
[-,-,-,-] => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4}
[+,+,4,3] => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4}
[-,+,4,3] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[+,-,4,3] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[-,-,4,3] => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4}
[+,3,2,+] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[-,3,2,+] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[+,3,2,-] => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4}
[-,3,2,-] => [2,1,3,4] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4}
[+,3,4,2] => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4}
[-,3,4,2] => [2,1,3,4] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4}
[+,4,2,3] => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4}
[-,4,2,3] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[+,4,+,2] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[-,4,+,2] => [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
[+,4,-,2] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[-,4,-,2] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,1,+,+] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[2,1,-,+] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[2,1,+,-] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[2,1,-,-] => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4}
[2,1,4,3] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[2,3,1,+] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[2,3,1,-] => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4}
[2,3,4,1] => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4}
[2,4,1,3] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[2,4,+,1] => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4}
[2,4,-,1] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[3,1,2,+] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[3,1,2,-] => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4}
[3,1,4,2] => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4}
[3,+,1,+] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,-,1,+] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,+,1,-] => [2,1,3,4] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4}
[3,-,1,-] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[3,+,4,1] => [2,1,3,4] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4}
[3,-,4,1] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[3,4,1,2] => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4}
[3,4,2,1] => [2,1,3,4] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4}
[4,1,2,3] => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4}
[4,1,+,2] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[4,1,-,2] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[4,+,1,3] => [2,1,3,4] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4}
[4,-,1,3] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[4,+,+,1] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[4,-,+,1] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,+,-,1] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,-,-,1] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[4,3,1,2] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[4,3,2,1] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[+,+,+,+,+] => [1,2,3,4,5] => [5] => ([],5)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[-,+,+,+,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[+,-,+,+,+] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[+,+,-,+,+] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[+,+,+,-,+] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[+,+,+,+,-] => [1,2,3,4,5] => [5] => ([],5)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[-,-,+,+,+] => [3,4,5,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[-,+,-,+,+] => [2,4,5,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[-,+,+,-,+] => [2,3,5,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[-,+,+,+,-] => [2,3,4,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[+,+,+,-,-] => [1,2,3,4,5] => [5] => ([],5)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[-,-,-,+,+] => [4,5,1,2,3] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
Description
The distinguishing index of a graph. This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism. If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St001198
Mapping
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 42%●distinct values known / distinct values provided: 33%
Values
[+] => [1] => [1] => [1,0]
 => ? ∊ {0,1}
[-] => [1] => [1] => [1,0]
 => ? ∊ {0,1}
[+,+] => [1,2] => [2,1] => [1,1,0,0]
 => ? ∊ {0,1,1,1}
[-,+] => [2,1] => [1,2] => [1,0,1,0]
 => 2
[+,-] => [1,2] => [2,1] => [1,1,0,0]
 => ? ∊ {0,1,1,1}
[-,-] => [1,2] => [2,1] => [1,1,0,0]
 => ? ∊ {0,1,1,1}
[2,1] => [1,2] => [2,1] => [1,1,0,0]
 => ? ∊ {0,1,1,1}
[+,+,+] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,3}
[-,+,+] => [2,3,1] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[+,-,+] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,3}
[+,+,-] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,3}
[-,-,+] => [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
 => 2
[-,+,-] => [2,1,3] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[+,-,-] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,3}
[-,-,-] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,3}
[+,3,2] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,3}
[-,3,2] => [2,1,3] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[2,1,+] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,3}
[2,1,-] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,3}
[2,3,1] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,3}
[3,1,2] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,3}
[3,+,1] => [2,1,3] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[3,-,1] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,3}
[+,+,+,+] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,+,+,+] => [2,3,4,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 2
[+,-,+,+] => [1,3,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[+,+,-,+] => [1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[+,+,+,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,-,+,+] => [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2
[-,+,-,+] => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 2
[-,+,+,-] => [2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 2
[+,-,-,+] => [1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[+,-,+,-] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[+,+,-,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,-,-,+] => [4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2
[-,-,+,-] => [3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 2
[-,+,-,-] => [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2
[+,-,-,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,-,-,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[+,+,4,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,+,4,3] => [2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 2
[+,-,4,3] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,-,4,3] => [3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 2
[+,3,2,+] => [1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,3,2,+] => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 2
[+,3,2,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,3,2,-] => [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2
[+,3,4,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,3,4,2] => [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2
[+,4,2,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,4,2,3] => [2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 2
[+,4,+,2] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,4,+,2] => [3,2,1,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 3
[+,4,-,2] => [1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,4,-,2] => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 2
[2,1,+,+] => [1,3,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,1,-,+] => [1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,1,+,-] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,1,-,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,1,4,3] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,3,1,+] => [1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,3,1,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,4,+,1] => [3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 2
[2,4,-,1] => [1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,1,2,+] => [1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,1,2,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,1,4,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,+,1,+] => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 2
[3,-,1,+] => [1,4,3,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,+,1,-] => [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2
[3,-,1,-] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,+,4,1] => [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2
[3,-,4,1] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,4,2,1] => [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2
[4,+,1,3] => [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2
[4,+,+,1] => [2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 2
[4,-,+,1] => [3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2
[4,+,-,1] => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 2
[4,3,2,1] => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 2
[-,+,+,+,+] => [2,3,4,5,1] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 2
[-,-,+,+,+] => [3,4,5,1,2] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[-,+,-,+,+] => [2,4,5,1,3] => [4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => 2
[-,+,+,-,+] => [2,3,5,1,4] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
 => 2
[-,+,+,+,-] => [2,3,4,1,5] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => 2
[-,-,-,+,+] => [4,5,1,2,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[-,-,+,-,+] => [3,5,1,2,4] => [3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
 => 2
[-,-,+,+,-] => [3,4,1,2,5] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => 2
[-,+,-,-,+] => [2,5,1,3,4] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
 => 2
[-,+,-,+,-] => [2,4,1,3,5] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => 2
[-,+,+,-,-] => [2,3,1,4,5] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => 2
[-,-,-,-,+] => [5,1,2,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 2
[-,-,-,+,-] => [4,1,2,3,5] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 2
[-,-,+,-,-] => [3,1,2,4,5] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => 2
[-,+,-,-,-] => [2,1,3,4,5] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 2
[-,+,+,5,4] => [2,3,4,1,5] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => 2
[-,-,+,5,4] => [3,4,1,2,5] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => 2
[-,+,-,5,4] => [2,4,1,3,5] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => 2
[-,-,-,5,4] => [4,1,2,3,5] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 2
Description
The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001200
Mapping
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 42%●distinct values known / distinct values provided: 33%
Values
[+] => [1] => [1] => [1,0]
 => ? ∊ {0,1}
[-] => [1] => [1] => [1,0]
 => ? ∊ {0,1}
[+,+] => [1,2] => [2,1] => [1,1,0,0]
 => ? ∊ {0,1,1,1}
[-,+] => [2,1] => [1,2] => [1,0,1,0]
 => 2
[+,-] => [1,2] => [2,1] => [1,1,0,0]
 => ? ∊ {0,1,1,1}
[-,-] => [1,2] => [2,1] => [1,1,0,0]
 => ? ∊ {0,1,1,1}
[2,1] => [1,2] => [2,1] => [1,1,0,0]
 => ? ∊ {0,1,1,1}
[+,+,+] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,3}
[-,+,+] => [2,3,1] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[+,-,+] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,3}
[+,+,-] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,3}
[-,-,+] => [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
 => 2
[-,+,-] => [2,1,3] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[+,-,-] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,3}
[-,-,-] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,3}
[+,3,2] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,3}
[-,3,2] => [2,1,3] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[2,1,+] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,3}
[2,1,-] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,3}
[2,3,1] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,3}
[3,1,2] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,3}
[3,+,1] => [2,1,3] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[3,-,1] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,3}
[+,+,+,+] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,+,+,+] => [2,3,4,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 2
[+,-,+,+] => [1,3,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[+,+,-,+] => [1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[+,+,+,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,-,+,+] => [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2
[-,+,-,+] => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 2
[-,+,+,-] => [2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 2
[+,-,-,+] => [1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[+,-,+,-] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[+,+,-,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,-,-,+] => [4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2
[-,-,+,-] => [3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 2
[-,+,-,-] => [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2
[+,-,-,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,-,-,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[+,+,4,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,+,4,3] => [2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 2
[+,-,4,3] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,-,4,3] => [3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 2
[+,3,2,+] => [1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,3,2,+] => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 2
[+,3,2,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,3,2,-] => [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2
[+,3,4,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,3,4,2] => [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2
[+,4,2,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,4,2,3] => [2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 2
[+,4,+,2] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,4,+,2] => [3,2,1,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 3
[+,4,-,2] => [1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,4,-,2] => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 2
[2,1,+,+] => [1,3,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,1,-,+] => [1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,1,+,-] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,1,-,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,1,4,3] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,3,1,+] => [1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,3,1,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,4,+,1] => [3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 2
[2,4,-,1] => [1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,1,2,+] => [1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,1,2,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,1,4,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,+,1,+] => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 2
[3,-,1,+] => [1,4,3,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,+,1,-] => [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2
[3,-,1,-] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,+,4,1] => [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2
[3,-,4,1] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,4,2,1] => [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2
[4,+,1,3] => [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2
[4,+,+,1] => [2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 2
[4,-,+,1] => [3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2
[4,+,-,1] => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 2
[4,3,2,1] => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 2
[-,+,+,+,+] => [2,3,4,5,1] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 2
[-,-,+,+,+] => [3,4,5,1,2] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[-,+,-,+,+] => [2,4,5,1,3] => [4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => 2
[-,+,+,-,+] => [2,3,5,1,4] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
 => 2
[-,+,+,+,-] => [2,3,4,1,5] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => 2
[-,-,-,+,+] => [4,5,1,2,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[-,-,+,-,+] => [3,5,1,2,4] => [3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
 => 2
[-,-,+,+,-] => [3,4,1,2,5] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => 2
[-,+,-,-,+] => [2,5,1,3,4] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
 => 2
[-,+,-,+,-] => [2,4,1,3,5] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => 2
[-,+,+,-,-] => [2,3,1,4,5] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => 2
[-,-,-,-,+] => [5,1,2,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 2
[-,-,-,+,-] => [4,1,2,3,5] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 2
[-,-,+,-,-] => [3,1,2,4,5] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => 2
[-,+,-,-,-] => [2,1,3,4,5] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 2
[-,+,+,5,4] => [2,3,4,1,5] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => 2
[-,-,+,5,4] => [3,4,1,2,5] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => 2
[-,+,-,5,4] => [2,4,1,3,5] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => 2
[-,-,-,5,4] => [4,1,2,3,5] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 2
Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001206
Mapping
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 42%●distinct values known / distinct values provided: 33%
Values
[+] => [1] => [1] => [1,0]
 => ? ∊ {0,1}
[-] => [1] => [1] => [1,0]
 => ? ∊ {0,1}
[+,+] => [1,2] => [2,1] => [1,1,0,0]
 => ? ∊ {0,1,1,1}
[-,+] => [2,1] => [1,2] => [1,0,1,0]
 => 2
[+,-] => [1,2] => [2,1] => [1,1,0,0]
 => ? ∊ {0,1,1,1}
[-,-] => [1,2] => [2,1] => [1,1,0,0]
 => ? ∊ {0,1,1,1}
[2,1] => [1,2] => [2,1] => [1,1,0,0]
 => ? ∊ {0,1,1,1}
[+,+,+] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,3}
[-,+,+] => [2,3,1] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[+,-,+] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,3}
[+,+,-] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,3}
[-,-,+] => [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
 => 2
[-,+,-] => [2,1,3] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[+,-,-] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,3}
[-,-,-] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,3}
[+,3,2] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,3}
[-,3,2] => [2,1,3] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[2,1,+] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,3}
[2,1,-] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,3}
[2,3,1] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,3}
[3,1,2] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,3}
[3,+,1] => [2,1,3] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[3,-,1] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,3}
[+,+,+,+] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,+,+,+] => [2,3,4,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 2
[+,-,+,+] => [1,3,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[+,+,-,+] => [1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[+,+,+,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,-,+,+] => [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2
[-,+,-,+] => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 2
[-,+,+,-] => [2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 2
[+,-,-,+] => [1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[+,-,+,-] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[+,+,-,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,-,-,+] => [4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2
[-,-,+,-] => [3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 2
[-,+,-,-] => [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2
[+,-,-,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,-,-,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[+,+,4,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,+,4,3] => [2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 2
[+,-,4,3] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,-,4,3] => [3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 2
[+,3,2,+] => [1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,3,2,+] => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 2
[+,3,2,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,3,2,-] => [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2
[+,3,4,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,3,4,2] => [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2
[+,4,2,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,4,2,3] => [2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 2
[+,4,+,2] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,4,+,2] => [3,2,1,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 3
[+,4,-,2] => [1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,4,-,2] => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 2
[2,1,+,+] => [1,3,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,1,-,+] => [1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,1,+,-] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,1,-,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,1,4,3] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,3,1,+] => [1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,3,1,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,4,+,1] => [3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 2
[2,4,-,1] => [1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,1,2,+] => [1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,1,2,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,1,4,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,+,1,+] => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 2
[3,-,1,+] => [1,4,3,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,+,1,-] => [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2
[3,-,1,-] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,+,4,1] => [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2
[3,-,4,1] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,4,2,1] => [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2
[4,+,1,3] => [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2
[4,+,+,1] => [2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 2
[4,-,+,1] => [3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2
[4,+,-,1] => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 2
[4,3,2,1] => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 2
[-,+,+,+,+] => [2,3,4,5,1] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 2
[-,-,+,+,+] => [3,4,5,1,2] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[-,+,-,+,+] => [2,4,5,1,3] => [4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => 2
[-,+,+,-,+] => [2,3,5,1,4] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
 => 2
[-,+,+,+,-] => [2,3,4,1,5] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => 2
[-,-,-,+,+] => [4,5,1,2,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[-,-,+,-,+] => [3,5,1,2,4] => [3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
 => 2
[-,-,+,+,-] => [3,4,1,2,5] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => 2
[-,+,-,-,+] => [2,5,1,3,4] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
 => 2
[-,+,-,+,-] => [2,4,1,3,5] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => 2
[-,+,+,-,-] => [2,3,1,4,5] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => 2
[-,-,-,-,+] => [5,1,2,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 2
[-,-,-,+,-] => [4,1,2,3,5] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 2
[-,-,+,-,-] => [3,1,2,4,5] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => 2
[-,+,-,-,-] => [2,1,3,4,5] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 2
[-,+,+,5,4] => [2,3,4,1,5] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => 2
[-,-,+,5,4] => [3,4,1,2,5] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => 2
[-,+,-,5,4] => [2,4,1,3,5] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => 2
[-,-,-,5,4] => [4,1,2,3,5] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 2
Description
The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.
Matching statistic: St000264
(load all 10 compositions to match this statistic)
Mapping
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 33%
Values
[+] => [1] => [1] => ([],1)
 => ? ∊ {0,1}
[-] => [1] => [1] => ([],1)
 => ? ∊ {0,1}
[+,+] => [1,2] => [1,2] => ([],2)
 => ? ∊ {0,1,1,1,2}
[-,+] => [2,1] => [2,1] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,2}
[+,-] => [1,2] => [1,2] => ([],2)
 => ? ∊ {0,1,1,1,2}
[-,-] => [1,2] => [1,2] => ([],2)
 => ? ∊ {0,1,1,1,2}
[2,1] => [1,2] => [1,2] => ([],2)
 => ? ∊ {0,1,1,1,2}
[+,+,+] => [1,2,3] => [1,2,3] => ([],3)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[-,+,+] => [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[+,-,+] => [1,3,2] => [1,3,2] => ([(1,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[+,+,-] => [1,2,3] => [1,2,3] => ([],3)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[-,-,+] => [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[-,+,-] => [2,1,3] => [2,1,3] => ([(1,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[+,-,-] => [1,2,3] => [1,2,3] => ([],3)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[-,-,-] => [1,2,3] => [1,2,3] => ([],3)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[+,3,2] => [1,2,3] => [1,2,3] => ([],3)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[-,3,2] => [2,1,3] => [2,1,3] => ([(1,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[2,1,+] => [1,3,2] => [1,3,2] => ([(1,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[2,1,-] => [1,2,3] => [1,2,3] => ([],3)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[2,3,1] => [1,2,3] => [1,2,3] => ([],3)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[3,1,2] => [1,2,3] => [1,2,3] => ([],3)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[3,+,1] => [2,1,3] => [2,1,3] => ([(1,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[3,-,1] => [1,3,2] => [1,3,2] => ([(1,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[-,+,+,+] => [2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[+,-,+,+] => [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
[+,+,-,+] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[-,-,+,+] => [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[-,+,-,+] => [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[-,+,+,-] => [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 3
[+,-,-,+] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[+,-,+,-] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[-,-,-,+] => [4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[-,-,+,-] => [3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[-,+,-,-] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[+,+,4,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[-,+,4,3] => [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 3
[+,-,4,3] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[-,-,4,3] => [3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[+,3,2,+] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[-,3,2,+] => [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[+,3,2,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[-,3,2,-] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[+,3,4,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[-,3,4,2] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[+,4,2,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[-,4,2,3] => [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 3
[+,4,+,2] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[-,4,+,2] => [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[+,4,-,2] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[-,4,-,2] => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,1,+,+] => [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
[2,1,-,+] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,1,+,-] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,1,-,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,1,4,3] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[4,-,1,3] => [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
[4,+,+,1] => [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 3
[4,-,+,1] => [3,1,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 4
[-,+,+,+,+] => [2,3,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[+,-,+,+,+] => [1,3,4,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[+,+,-,+,+] => [1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3
[-,-,+,+,+] => [3,4,5,1,2] => [5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[-,+,-,+,+] => [2,4,5,1,3] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 3
[-,+,+,-,+] => [2,3,5,1,4] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[-,+,+,+,-] => [2,3,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[+,-,-,+,+] => [1,4,5,2,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[+,-,+,-,+] => [1,3,5,2,4] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[+,-,+,+,-] => [1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 3
[-,-,-,+,+] => [4,5,1,2,3] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[-,-,+,-,+] => [3,5,1,2,4] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[-,-,+,+,-] => [3,4,1,2,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[-,+,-,-,+] => [2,5,1,3,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[-,+,-,+,-] => [2,4,1,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[-,+,+,-,-] => [2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => 3
[-,+,+,5,4] => [2,3,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[+,-,+,5,4] => [1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 3
[-,-,+,5,4] => [3,4,1,2,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[-,+,-,5,4] => [2,4,1,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[-,+,4,3,+] => [2,3,5,1,4] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[+,-,4,3,+] => [1,3,5,2,4] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[-,-,4,3,+] => [3,5,1,2,4] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[-,+,4,3,-] => [2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => 3
[-,+,4,5,3] => [2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => 3
[-,+,5,3,4] => [2,3,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[+,-,5,3,4] => [1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 3
[-,-,5,3,4] => [3,4,1,2,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[-,+,5,+,3] => [2,4,3,1,5] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[-,+,5,-,3] => [2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 3
[+,3,2,+,+] => [1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3
[-,3,2,+,+] => [2,4,5,1,3] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 3
[-,3,2,-,+] => [2,5,1,3,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[-,3,2,+,-] => [2,4,1,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[-,3,2,5,4] => [2,4,1,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[-,3,4,2,+] => [2,5,1,3,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[-,3,5,2,4] => [2,4,1,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
Description
The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.
Matching statistic: St001879
(load all 3 compositions to match this statistic)
Mapping
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%
Values
[+] => [1] => [] => ([],0)
 => ? ∊ {0,1}
[-] => [1] => [] => ([],0)
 => ? ∊ {0,1}
[+,+] => [1,2] => [1] => ([],1)
 => ? ∊ {0,1,1,1,2}
[-,+] => [1,2] => [1] => ([],1)
 => ? ∊ {0,1,1,1,2}
[+,-] => [1,2] => [1] => ([],1)
 => ? ∊ {0,1,1,1,2}
[-,-] => [1,2] => [1] => ([],1)
 => ? ∊ {0,1,1,1,2}
[2,1] => [2,1] => [1] => ([],1)
 => ? ∊ {0,1,1,1,2}
[+,+,+] => [1,2,3] => [1,2] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[-,+,+] => [1,2,3] => [1,2] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[+,-,+] => [1,2,3] => [1,2] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[+,+,-] => [1,2,3] => [1,2] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[-,-,+] => [1,2,3] => [1,2] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[-,+,-] => [1,2,3] => [1,2] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[+,-,-] => [1,2,3] => [1,2] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[-,-,-] => [1,2,3] => [1,2] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[+,3,2] => [1,3,2] => [1,2] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[-,3,2] => [1,3,2] => [1,2] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[2,1,+] => [2,1,3] => [2,1] => ([],2)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[2,1,-] => [2,1,3] => [2,1] => ([],2)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[2,3,1] => [2,3,1] => [2,1] => ([],2)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[3,1,2] => [3,1,2] => [1,2] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[3,+,1] => [3,2,1] => [2,1] => ([],2)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[3,-,1] => [3,2,1] => [2,1] => ([],2)
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[+,+,+,+] => [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2
[-,+,+,+] => [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2
[+,-,+,+] => [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2
[+,+,-,+] => [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2
[+,+,+,-] => [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2
[-,-,+,+] => [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2
[-,+,-,+] => [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2
[-,+,+,-] => [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2
[+,-,-,+] => [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2
[+,-,+,-] => [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2
[+,+,-,-] => [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2
[-,-,-,+] => [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2
[-,-,+,-] => [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2
[-,+,-,-] => [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2
[+,-,-,-] => [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2
[-,-,-,-] => [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2
[+,+,4,3] => [1,2,4,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2
[-,+,4,3] => [1,2,4,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2
[+,-,4,3] => [1,2,4,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2
[-,-,4,3] => [1,2,4,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2
[+,3,2,+] => [1,3,2,4] => [1,3,2] => ([(0,1),(0,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,3,2,+] => [1,3,2,4] => [1,3,2] => ([(0,1),(0,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[+,3,2,-] => [1,3,2,4] => [1,3,2] => ([(0,1),(0,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,3,2,-] => [1,3,2,4] => [1,3,2] => ([(0,1),(0,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[+,3,4,2] => [1,3,4,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,3,4,2] => [1,3,4,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[+,4,2,3] => [1,4,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2
[-,4,2,3] => [1,4,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2
[+,4,+,2] => [1,4,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,4,+,2] => [1,4,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[+,4,-,2] => [1,4,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,4,-,2] => [1,4,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,1,+,+] => [2,1,3,4] => [2,1,3] => ([(0,2),(1,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,1,-,+] => [2,1,3,4] => [2,1,3] => ([(0,2),(1,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,1,+,-] => [2,1,3,4] => [2,1,3] => ([(0,2),(1,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,1,-,-] => [2,1,3,4] => [2,1,3] => ([(0,2),(1,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,1,4,3] => [2,1,4,3] => [2,1,3] => ([(0,2),(1,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,3,1,+] => [2,3,1,4] => [2,3,1] => ([(1,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,3,1,-] => [2,3,1,4] => [2,3,1] => ([(1,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,3,4,1] => [2,3,4,1] => [2,3,1] => ([(1,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,4,1,3] => [2,4,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,4,+,1] => [2,4,3,1] => [2,3,1] => ([(1,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,4,-,1] => [2,4,3,1] => [2,3,1] => ([(1,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,1,2,+] => [3,1,2,4] => [3,1,2] => ([(1,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,1,2,-] => [3,1,2,4] => [3,1,2] => ([(1,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,1,4,2] => [3,1,4,2] => [3,1,2] => ([(1,2)],3)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,+,1,+] => [3,2,1,4] => [3,2,1] => ([],3)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,-,1,+] => [3,2,1,4] => [3,2,1] => ([],3)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,+,1,-] => [3,2,1,4] => [3,2,1] => ([],3)
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[4,1,2,3] => [4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2
[+,+,+,+,+] => [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3
[-,+,+,+,+] => [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3
[+,-,+,+,+] => [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3
[+,+,-,+,+] => [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3
[+,+,+,-,+] => [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3
[+,+,+,+,-] => [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3
[-,-,+,+,+] => [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3
[-,+,-,+,+] => [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3
[-,+,+,-,+] => [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3
[-,+,+,+,-] => [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3
[+,-,-,+,+] => [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3
[+,-,+,-,+] => [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3
[+,-,+,+,-] => [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3
[+,+,-,-,+] => [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3
[+,+,-,+,-] => [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3
[+,+,+,-,-] => [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3
[-,-,-,+,+] => [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3
[-,-,+,-,+] => [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3
[-,-,+,+,-] => [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3
[-,+,-,-,+] => [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3
[-,+,-,+,-] => [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3
[-,+,+,-,-] => [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3
[+,-,-,-,+] => [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3
[+,-,-,+,-] => [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3
[+,-,+,-,-] => [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3
[+,+,-,-,-] => [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3
[-,-,-,-,+] => [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St000456
(load all 4 compositions to match this statistic)
Mapping
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 17%
Values
[+] => [1] => [1] => ([],1)
 => ? ∊ {0,1}
[-] => [1] => [1] => ([],1)
 => ? ∊ {0,1}
[+,+] => [1,2] => [2] => ([],2)
 => ? ∊ {0,1,2}
[-,+] => [2,1] => [1,1] => ([(0,1)],2)
 => 1
[+,-] => [1,2] => [2] => ([],2)
 => ? ∊ {0,1,2}
[-,-] => [1,2] => [2] => ([],2)
 => ? ∊ {0,1,2}
[2,1] => [2,1] => [1,1] => ([(0,1)],2)
 => 1
[+,+,+] => [1,2,3] => [3] => ([],3)
 => ? ∊ {0,1,2,2,2,2,2,2,2,3}
[-,+,+] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[+,-,+] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[+,+,-] => [1,2,3] => [3] => ([],3)
 => ? ∊ {0,1,2,2,2,2,2,2,2,3}
[-,-,+] => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,1,2,2,2,2,2,2,2,3}
[-,+,-] => [2,1,3] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,1,2,2,2,2,2,2,2,3}
[+,-,-] => [1,2,3] => [3] => ([],3)
 => ? ∊ {0,1,2,2,2,2,2,2,2,3}
[-,-,-] => [1,2,3] => [3] => ([],3)
 => ? ∊ {0,1,2,2,2,2,2,2,2,3}
[+,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[-,3,2] => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,1,2,2,2,2,2,2,2,3}
[2,1,+] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[2,1,-] => [2,1,3] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,1,2,2,2,2,2,2,2,3}
[2,3,1] => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,1,2,2,2,2,2,2,2,3}
[3,1,2] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[3,+,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[3,-,1] => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,1,2,2,2,2,2,2,2,3}
[+,+,+,+] => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,+,+,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[+,-,+,+] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[+,+,-,+] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[+,+,+,-] => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,-,+,+] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,+,-,+] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,+,+,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[+,-,-,+] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[+,-,+,-] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[+,+,-,-] => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,-,-,+] => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,-,+,-] => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,+,-,-] => [2,1,3,4] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[+,-,-,-] => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,-,-,-] => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[+,+,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[-,+,4,3] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[+,-,4,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,-,4,3] => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[+,3,2,+] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[-,3,2,+] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[+,3,2,-] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,3,2,-] => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[+,3,4,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,3,4,2] => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[+,4,2,3] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[-,4,2,3] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[+,4,+,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[-,4,+,2] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[+,4,-,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[-,4,-,2] => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,1,+,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,1,-,+] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,1,+,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,1,-,-] => [2,1,3,4] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,1,4,3] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,3,1,+] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,3,1,-] => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,3,4,1] => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,4,1,3] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,4,+,1] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,4,-,1] => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,1,2,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[3,+,1,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,1,2,3] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,1,+,2] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,+,1,3] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,+,+,1] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[-,+,+,+,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[+,-,+,+,+] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[+,+,-,+,+] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[+,+,+,-,+] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[+,+,+,5,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[+,+,4,3,+] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[+,+,5,3,4] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[+,+,5,+,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[+,3,2,+,+] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[+,4,2,3,+] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[+,4,+,2,+] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[+,5,2,3,4] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[+,5,2,+,3] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[+,5,+,2,4] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[+,5,+,+,2] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[2,1,+,+,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[3,1,2,+,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[3,+,1,+,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[4,1,2,3,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[4,1,+,2,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[4,+,1,3,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[4,+,+,1,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[5,1,2,3,4] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[5,1,2,+,3] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[5,1,+,2,4] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[5,1,+,+,2] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[5,+,1,3,4] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[5,+,1,+,3] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
Description
The monochromatic index of a connected graph. This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path. For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
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St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.