Your data matches 9 different statistics following compositions of up to 3 maps.
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Matching statistic: St000333
(load all 2 compositions to match this statistic)
Mapping
Mp00256: Decorated permutations —upper permutation⟶ Permutations
St000333: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => 0
[-] => [1] => 0
[+,+] => [1,2] => 0
[-,+] => [2,1] => 1
[+,-] => [1,2] => 0
[-,-] => [1,2] => 0
[2,1] => [2,1] => 1
[+,+,+] => [1,2,3] => 0
[-,+,+] => [2,3,1] => 1
[+,-,+] => [1,3,2] => 1
[+,+,-] => [1,2,3] => 0
[-,-,+] => [3,1,2] => 1
[-,+,-] => [2,1,3] => 2
[+,-,-] => [1,2,3] => 0
[-,-,-] => [1,2,3] => 0
[+,3,2] => [1,3,2] => 1
[-,3,2] => [3,1,2] => 1
[2,1,+] => [2,3,1] => 1
[2,1,-] => [2,1,3] => 2
[2,3,1] => [3,1,2] => 1
[3,1,2] => [2,3,1] => 1
[3,+,1] => [2,3,1] => 1
[3,-,1] => [3,1,2] => 1
[+,+,+,+] => [1,2,3,4] => 0
[-,+,+,+] => [2,3,4,1] => 1
[+,-,+,+] => [1,3,4,2] => 1
[+,+,-,+] => [1,2,4,3] => 1
[+,+,+,-] => [1,2,3,4] => 0
[-,-,+,+] => [3,4,1,2] => 1
[-,+,-,+] => [2,4,1,3] => 1
[-,+,+,-] => [2,3,1,4] => 2
[+,-,-,+] => [1,4,2,3] => 1
[+,-,+,-] => [1,3,2,4] => 2
[+,+,-,-] => [1,2,3,4] => 0
[-,-,-,+] => [4,1,2,3] => 1
[-,-,+,-] => [3,1,2,4] => 2
[-,+,-,-] => [2,1,3,4] => 2
[+,-,-,-] => [1,2,3,4] => 0
[-,-,-,-] => [1,2,3,4] => 0
[+,+,4,3] => [1,2,4,3] => 1
[-,+,4,3] => [2,4,1,3] => 1
[+,-,4,3] => [1,4,2,3] => 1
[-,-,4,3] => [4,1,2,3] => 1
[+,3,2,+] => [1,3,4,2] => 1
[-,3,2,+] => [3,4,1,2] => 1
[+,3,2,-] => [1,3,2,4] => 2
[-,3,2,-] => [3,1,2,4] => 2
[+,3,4,2] => [1,4,2,3] => 1
[-,3,4,2] => [4,1,2,3] => 1
[+,4,2,3] => [1,3,4,2] => 1
Description
The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. This descent set is denoted by $\operatorname{ZDer}(\sigma)$ in [1].
Matching statistic: St000340
Mapping
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => [1] => [1,0]
 => 0
[-] => [1] => [1] => [1,0]
 => 0
[+,+] => [1,2] => [1,2] => [1,0,1,0]
 => 0
[-,+] => [2,1] => [2,1] => [1,1,0,0]
 => 1
[+,-] => [1,2] => [1,2] => [1,0,1,0]
 => 0
[-,-] => [1,2] => [1,2] => [1,0,1,0]
 => 0
[2,1] => [2,1] => [2,1] => [1,1,0,0]
 => 1
[+,+,+] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 0
[-,+,+] => [2,3,1] => [3,2,1] => [1,1,1,0,0,0]
 => 1
[+,-,+] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => 1
[+,+,-] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 0
[-,-,+] => [3,1,2] => [3,1,2] => [1,1,1,0,0,0]
 => 1
[-,+,-] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[+,-,-] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 0
[-,-,-] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 0
[+,3,2] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => 1
[-,3,2] => [3,1,2] => [3,1,2] => [1,1,1,0,0,0]
 => 1
[2,1,+] => [2,3,1] => [3,2,1] => [1,1,1,0,0,0]
 => 1
[2,1,-] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[2,3,1] => [3,1,2] => [3,1,2] => [1,1,1,0,0,0]
 => 1
[3,1,2] => [2,3,1] => [3,2,1] => [1,1,1,0,0,0]
 => 1
[3,+,1] => [2,3,1] => [3,2,1] => [1,1,1,0,0,0]
 => 1
[3,-,1] => [3,1,2] => [3,1,2] => [1,1,1,0,0,0]
 => 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 0
[-,+,+,+] => [2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 1
[+,-,+,+] => [1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1
[+,+,-,+] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 1
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 0
[-,-,+,+] => [3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 1
[-,+,-,+] => [2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 1
[-,+,+,-] => [2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 2
[+,-,-,+] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 1
[+,-,+,-] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 2
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 0
[-,-,-,+] => [4,1,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 1
[-,-,+,-] => [3,1,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 2
[-,+,-,-] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 2
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 0
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 0
[+,+,4,3] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 1
[-,+,4,3] => [2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 1
[+,-,4,3] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 1
[-,-,4,3] => [4,1,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 1
[+,3,2,+] => [1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1
[-,3,2,+] => [3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 1
[+,3,2,-] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 2
[-,3,2,-] => [3,1,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 2
[+,3,4,2] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 1
[-,3,4,2] => [4,1,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 1
[+,4,2,3] => [1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1
Description
The number of non-final maximal constant sub-paths of length greater than one. This is the total number of occurrences of the patterns $110$ and $001$.
Matching statistic: St000638
Mapping
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000638: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => [1] => [1] => 1 = 0 + 1
[-] => [1] => [1] => [1] => 1 = 0 + 1
[+,+] => [1,2] => [1,2] => [1,2] => 1 = 0 + 1
[-,+] => [2,1] => [2,1] => [2,1] => 2 = 1 + 1
[+,-] => [1,2] => [1,2] => [1,2] => 1 = 0 + 1
[-,-] => [1,2] => [1,2] => [1,2] => 1 = 0 + 1
[2,1] => [2,1] => [2,1] => [2,1] => 2 = 1 + 1
[+,+,+] => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[-,+,+] => [2,3,1] => [3,2,1] => [2,3,1] => 2 = 1 + 1
[+,-,+] => [1,3,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[+,+,-] => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[-,-,+] => [3,1,2] => [3,1,2] => [3,2,1] => 2 = 1 + 1
[-,+,-] => [2,1,3] => [2,1,3] => [2,1,3] => 3 = 2 + 1
[+,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[-,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[+,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[-,3,2] => [3,1,2] => [3,1,2] => [3,2,1] => 2 = 1 + 1
[2,1,+] => [2,3,1] => [3,2,1] => [2,3,1] => 2 = 1 + 1
[2,1,-] => [2,1,3] => [2,1,3] => [2,1,3] => 3 = 2 + 1
[2,3,1] => [3,1,2] => [3,1,2] => [3,2,1] => 2 = 1 + 1
[3,1,2] => [2,3,1] => [3,2,1] => [2,3,1] => 2 = 1 + 1
[3,+,1] => [2,3,1] => [3,2,1] => [2,3,1] => 2 = 1 + 1
[3,-,1] => [3,1,2] => [3,1,2] => [3,2,1] => 2 = 1 + 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[-,+,+,+] => [2,3,4,1] => [4,2,3,1] => [2,3,4,1] => 2 = 1 + 1
[+,-,+,+] => [1,3,4,2] => [1,4,3,2] => [1,3,4,2] => 2 = 1 + 1
[+,+,-,+] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[-,-,+,+] => [3,4,1,2] => [4,1,3,2] => [3,4,2,1] => 2 = 1 + 1
[-,+,-,+] => [2,4,1,3] => [4,2,1,3] => [2,4,3,1] => 2 = 1 + 1
[-,+,+,-] => [2,3,1,4] => [3,2,1,4] => [2,3,1,4] => 3 = 2 + 1
[+,-,-,+] => [1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 2 = 1 + 1
[+,-,+,-] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 3 = 2 + 1
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[-,-,-,+] => [4,1,2,3] => [4,1,2,3] => [4,3,2,1] => 2 = 1 + 1
[-,-,+,-] => [3,1,2,4] => [3,1,2,4] => [3,2,1,4] => 3 = 2 + 1
[-,+,-,-] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 3 = 2 + 1
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[+,+,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[-,+,4,3] => [2,4,1,3] => [4,2,1,3] => [2,4,3,1] => 2 = 1 + 1
[+,-,4,3] => [1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 2 = 1 + 1
[-,-,4,3] => [4,1,2,3] => [4,1,2,3] => [4,3,2,1] => 2 = 1 + 1
[+,3,2,+] => [1,3,4,2] => [1,4,3,2] => [1,3,4,2] => 2 = 1 + 1
[-,3,2,+] => [3,4,1,2] => [4,1,3,2] => [3,4,2,1] => 2 = 1 + 1
[+,3,2,-] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 3 = 2 + 1
[-,3,2,-] => [3,1,2,4] => [3,1,2,4] => [3,2,1,4] => 3 = 2 + 1
[+,3,4,2] => [1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 2 = 1 + 1
[-,3,4,2] => [4,1,2,3] => [4,1,2,3] => [4,3,2,1] => 2 = 1 + 1
[+,4,2,3] => [1,3,4,2] => [1,4,3,2] => [1,3,4,2] => 2 = 1 + 1
Description
The number of up-down runs of a permutation. An '''up-down run''' of a permutation $\pi=\pi_{1}\pi_{2}\cdots\pi_{n}$ is either a maximal monotone consecutive subsequence or $\pi_{1}$ if 1 is a descent of $\pi$. For example, the up-down runs of $\pi=85712643$ are $8$, $85$, $57$, $71$, $126$, and $643$.
Matching statistic: St001199
(load all 2 compositions to match this statistic)
Mapping
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 82%●distinct values known / distinct values provided: 33%
Values
[+] => [1] => [1] => [1,0]
 => ? ∊ {0,0}
[-] => [1] => [1] => [1,0]
 => ? ∊ {0,0}
[+,+] => [1,2] => [2,1] => [1,1,0,0]
 => ? ∊ {0,0,0}
[-,+] => [2,1] => [1,2] => [1,0,1,0]
 => 1
[+,-] => [1,2] => [2,1] => [1,1,0,0]
 => ? ∊ {0,0,0}
[-,-] => [1,2] => [2,1] => [1,1,0,0]
 => ? ∊ {0,0,0}
[2,1] => [2,1] => [1,2] => [1,0,1,0]
 => 1
[+,+,+] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {0,0,0,0,2,2}
[-,+,+] => [2,3,1] => [1,3,2] => [1,0,1,1,0,0]
 => 1
[+,-,+] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
 => 1
[+,+,-] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {0,0,0,0,2,2}
[-,-,+] => [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
 => 1
[-,+,-] => [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
 => ? ∊ {0,0,0,0,2,2}
[+,-,-] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {0,0,0,0,2,2}
[-,-,-] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {0,0,0,0,2,2}
[+,3,2] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
 => 1
[-,3,2] => [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
 => 1
[2,1,+] => [2,3,1] => [1,3,2] => [1,0,1,1,0,0]
 => 1
[2,1,-] => [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
 => ? ∊ {0,0,0,0,2,2}
[2,3,1] => [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
 => 1
[3,1,2] => [2,3,1] => [1,3,2] => [1,0,1,1,0,0]
 => 1
[3,+,1] => [2,3,1] => [1,3,2] => [1,0,1,1,0,0]
 => 1
[3,-,1] => [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
 => 1
[+,+,+,+] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2}
[-,+,+,+] => [2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1
[+,-,+,+] => [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 1
[+,+,-,+] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 1
[+,+,+,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2}
[-,-,+,+] => [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 1
[-,+,-,+] => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 1
[-,+,+,-] => [2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2}
[+,-,-,+] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 1
[+,-,+,-] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2}
[+,+,-,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2}
[-,-,-,+] => [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 1
[-,-,+,-] => [3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2}
[-,+,-,-] => [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2}
[+,-,-,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2}
[-,-,-,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2}
[+,+,4,3] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 1
[-,+,4,3] => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 1
[+,-,4,3] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 1
[-,-,4,3] => [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 1
[+,3,2,+] => [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 1
[-,3,2,+] => [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 1
[+,3,2,-] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2}
[-,3,2,-] => [3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2}
[+,3,4,2] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 1
[-,3,4,2] => [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 1
[+,4,2,3] => [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 1
[-,4,2,3] => [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 1
[+,4,+,2] => [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 1
[-,4,+,2] => [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 1
[+,4,-,2] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 1
[-,4,-,2] => [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 1
[2,1,+,+] => [2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1
[2,1,-,+] => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 1
[2,1,+,-] => [2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,-,-] => [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,4,3] => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 1
[2,3,1,+] => [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 1
[2,3,1,-] => [3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,1] => [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 1
[2,4,1,3] => [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 1
[2,4,+,1] => [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 1
[2,4,-,1] => [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 1
[3,1,2,+] => [2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1
[3,1,2,-] => [2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,4,2] => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 1
[3,+,1,+] => [2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1
[3,-,1,+] => [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 1
[3,+,1,-] => [2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2}
[3,-,1,-] => [3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2}
[3,+,4,1] => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 1
[3,-,4,1] => [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 1
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 1
[3,4,2,1] => [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 1
[4,1,2,3] => [2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1
[+,+,+,+,+] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,+,+,+,-] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[-,+,+,+,-] => [2,3,4,1,5] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,-,+,+,-] => [1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,+,-,+,-] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,+,+,-,-] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[-,-,+,+,-] => [3,4,1,2,5] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[-,+,-,+,-] => [2,4,1,3,5] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[-,+,+,-,-] => [2,3,1,4,5] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,-,-,+,-] => [1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,-,+,-,-] => [1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,+,-,-,-] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[-,-,-,+,-] => [4,1,2,3,5] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[-,-,+,-,-] => [3,1,2,4,5] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[-,+,-,-,-] => [2,1,3,4,5] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,-,-,-,-] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[-,-,-,-,-] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,+,4,3,-] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[-,+,4,3,-] => [2,4,1,3,5] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,-,4,3,-] => [1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[-,-,4,3,-] => [4,1,2,3,5] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,3,2,+,-] => [1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St000260
(load all 10 compositions to match this statistic)
Mapping
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 69%●distinct values known / distinct values provided: 67%
Values
[+] => [1] => [1] => ([],1)
 => 0
[-] => [1] => [1] => ([],1)
 => 0
[+,+] => [1,2] => [1,2] => ([],2)
 => ? ∊ {0,0,0}
[-,+] => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[+,-] => [1,2] => [1,2] => ([],2)
 => ? ∊ {0,0,0}
[-,-] => [1,2] => [1,2] => ([],2)
 => ? ∊ {0,0,0}
[2,1] => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[+,+,+] => [1,2,3] => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,1,1,2,2}
[-,+,+] => [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[+,-,+] => [1,3,2] => [1,3,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,1,1,2,2}
[+,+,-] => [1,2,3] => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,1,1,2,2}
[-,-,+] => [3,1,2] => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[-,+,-] => [2,1,3] => [2,1,3] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,1,1,2,2}
[+,-,-] => [1,2,3] => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,1,1,2,2}
[-,-,-] => [1,2,3] => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,0,1,1,2,2}
[+,3,2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,1,1,2,2}
[-,3,2] => [3,1,2] => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[2,1,+] => [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[2,1,-] => [2,1,3] => [2,1,3] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,1,1,2,2}
[2,3,1] => [3,1,2] => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[3,1,2] => [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[3,+,1] => [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[3,-,1] => [3,1,2] => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,+,+,+] => [2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[+,-,+,+] => [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[+,+,-,+] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,-,+,+] => [3,4,1,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[-,+,-,+] => [2,4,1,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[-,+,+,-] => [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[+,-,-,+] => [1,4,2,3] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[+,-,+,-] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,-,-,+] => [4,1,2,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[-,-,+,-] => [3,1,2,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,+,-,-] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[+,+,4,3] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,+,4,3] => [2,4,1,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[+,-,4,3] => [1,4,2,3] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,-,4,3] => [4,1,2,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[+,3,2,+] => [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,3,2,+] => [3,4,1,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[+,3,2,-] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,3,2,-] => [3,1,2,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[+,3,4,2] => [1,4,2,3] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,3,4,2] => [4,1,2,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[+,4,2,3] => [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,4,2,3] => [3,4,1,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[+,4,+,2] => [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,4,+,2] => [3,4,1,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[+,4,-,2] => [1,4,2,3] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,4,-,2] => [4,1,2,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,1,+,+] => [2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[2,1,-,+] => [2,4,1,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[2,1,+,-] => [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,-,-] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,4,3] => [2,4,1,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[2,3,1,+] => [3,4,1,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[2,3,1,-] => [3,1,2,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,1] => [4,1,2,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,4,1,3] => [3,4,1,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[2,4,+,1] => [3,4,1,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[2,4,-,1] => [4,1,2,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[3,1,2,+] => [2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[3,1,2,-] => [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,4,2] => [2,4,1,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[3,+,1,+] => [2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[3,-,1,+] => [3,4,1,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[3,+,1,-] => [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[3,-,1,-] => [3,1,2,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[3,+,4,1] => [2,4,1,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[3,-,4,1] => [4,1,2,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[3,4,1,2] => [3,4,1,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[3,4,2,1] => [3,4,1,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,1,2,3] => [2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,1,+,2] => [2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,1,-,2] => [2,4,1,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,+,1,3] => [2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,-,1,3] => [3,4,1,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,+,+,1] => [2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,-,+,1] => [3,4,1,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,+,-,1] => [2,4,1,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,-,-,1] => [4,1,2,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,3,1,2] => [3,4,1,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,3,2,1] => [3,4,1,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[+,+,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,-,+,+,+] => [1,3,4,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,+,-,+,+] => [1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,+,+,-,+] => [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,+,+,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[-,+,+,+,-] => [2,3,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,-,-,+,+] => [1,4,5,2,3] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,-,+,-,+] => [1,3,5,2,4] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,-,+,+,-] => [1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,+,-,-,+] => [1,2,5,3,4] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,+,-,+,-] => [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,+,+,-,-] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
Description
The radius of a connected graph. This is the minimum eccentricity of any vertex.
Matching statistic: St001878
Mapping
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 67%
Values
[+] => [1] => ([],1)
 => ([],1)
 => ? ∊ {0,0}
[-] => [1] => ([],1)
 => ([],1)
 => ? ∊ {0,0}
[+,+] => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1}
[-,+] => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1}
[+,-] => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1}
[-,-] => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1}
[2,1] => [2,1] => ([],2)
 => ([],1)
 => ? ∊ {0,0,0,1,1}
[+,+,+] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[-,+,+] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[+,-,+] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[+,+,-] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[-,-,+] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[-,+,-] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[+,-,-] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[-,-,-] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[+,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,2,2}
[-,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,2,2}
[2,1,+] => [2,1,3] => ([(0,2),(1,2)],3)
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,2,2}
[2,1,-] => [2,1,3] => ([(0,2),(1,2)],3)
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,2,2}
[2,3,1] => [2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,1,2,2}
[3,1,2] => [3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,1,2,2}
[3,+,1] => [3,2,1] => ([],3)
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,2,2}
[3,-,1] => [3,2,1] => ([],3)
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,2,2}
[+,+,+,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[-,+,+,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[+,-,+,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[+,+,-,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[+,+,+,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[-,-,+,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[-,+,-,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[-,+,+,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[+,-,-,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[+,-,+,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[+,+,-,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[-,-,-,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[-,-,+,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[-,+,-,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[+,-,-,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[-,-,-,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[+,+,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[-,+,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[+,-,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[-,-,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[+,3,2,+] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[-,3,2,+] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[+,3,2,-] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[-,3,2,-] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[+,3,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[-,3,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[+,4,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[-,4,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[+,4,+,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[-,4,+,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[+,4,-,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[-,4,-,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,1,+,+] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,1,-,+] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,1,+,-] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,1,-,-] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,3,1,+] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,3,1,-] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[2,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[2,4,+,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,4,-,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,1,2,+] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,1,2,-] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[3,+,1,+] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,-,1,+] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,+,1,-] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,-,1,-] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,+,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,-,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[4,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[4,1,+,2] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[+,+,+,+,+] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[-,+,+,+,+] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[+,-,+,+,+] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[+,+,-,+,+] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[+,+,+,-,+] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[+,+,+,+,-] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[-,-,+,+,+] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[-,+,-,+,+] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[-,+,+,-,+] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[-,+,+,+,-] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[+,-,-,+,+] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[+,-,+,-,+] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[+,-,+,+,-] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[+,+,-,-,+] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[+,+,-,+,-] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[+,+,+,-,-] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[-,-,-,+,+] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[-,-,+,-,+] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[-,-,+,+,-] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[-,+,-,-,+] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[-,+,-,+,-] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St000454
Mapping
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => [1] => ([],1)
 => 0
[-] => [1] => [1] => ([],1)
 => 0
[+,+] => [1,2] => [1,2] => ([],2)
 => 0
[-,+] => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[+,-] => [1,2] => [1,2] => ([],2)
 => 0
[-,-] => [1,2] => [1,2] => ([],2)
 => 0
[2,1] => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[+,+,+] => [1,2,3] => [1,2,3] => ([],3)
 => 0
[-,+,+] => [2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
 => ? ∊ {1,1,1,1,2,2}
[+,-,+] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => ? ∊ {1,1,1,1,2,2}
[+,+,-] => [1,2,3] => [1,2,3] => ([],3)
 => 0
[-,-,+] => [3,1,2] => [1,3,2] => ([(1,2)],3)
 => 1
[-,+,-] => [2,1,3] => [2,1,3] => ([(1,2)],3)
 => 1
[+,-,-] => [1,2,3] => [1,2,3] => ([],3)
 => 0
[-,-,-] => [1,2,3] => [1,2,3] => ([],3)
 => 0
[+,3,2] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => ? ∊ {1,1,1,1,2,2}
[-,3,2] => [3,1,2] => [1,3,2] => ([(1,2)],3)
 => 1
[2,1,+] => [2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
 => ? ∊ {1,1,1,1,2,2}
[2,1,-] => [2,1,3] => [2,1,3] => ([(1,2)],3)
 => 1
[2,3,1] => [3,1,2] => [1,3,2] => ([(1,2)],3)
 => 1
[3,1,2] => [2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
 => ? ∊ {1,1,1,1,2,2}
[3,+,1] => [2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
 => ? ∊ {1,1,1,1,2,2}
[3,-,1] => [3,1,2] => [1,3,2] => ([(1,2)],3)
 => 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[-,+,+,+] => [2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {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}
[+,-,+,+] => [1,3,4,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {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}
[+,+,-,+] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {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}
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[-,-,+,+] => [3,4,1,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {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}
[-,+,-,+] => [2,4,1,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 1
[-,+,+,-] => [2,3,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {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}
[+,-,-,+] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ? ∊ {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}
[+,-,+,-] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ? ∊ {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}
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[-,-,-,+] => [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
 => 1
[-,-,+,-] => [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
 => 1
[-,+,-,-] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => 1
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[+,+,4,3] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {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}
[-,+,4,3] => [2,4,1,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 1
[+,-,4,3] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ? ∊ {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}
[-,-,4,3] => [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
 => 1
[+,3,2,+] => [1,3,4,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {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}
[-,3,2,+] => [3,4,1,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {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}
[+,3,2,-] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ? ∊ {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}
[-,3,2,-] => [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
 => 1
[+,3,4,2] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ? ∊ {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}
[-,3,4,2] => [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
 => 1
[+,4,2,3] => [1,3,4,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {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}
[-,4,2,3] => [3,4,1,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {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}
[+,4,+,2] => [1,3,4,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {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}
[-,4,+,2] => [3,4,1,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {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}
[+,4,-,2] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ? ∊ {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}
[-,4,-,2] => [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
 => 1
[2,1,+,+] => [2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {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}
[2,1,-,+] => [2,4,1,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 1
[2,1,+,-] => [2,3,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {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}
[2,1,-,-] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => 1
[2,1,4,3] => [2,4,1,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 1
[2,3,1,+] => [3,4,1,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {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}
[2,3,1,-] => [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
 => 1
[2,3,4,1] => [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
 => 1
[2,4,1,3] => [3,4,1,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {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}
[2,4,+,1] => [3,4,1,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {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}
[2,4,-,1] => [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
 => 1
[3,1,2,+] => [2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {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}
[3,1,2,-] => [2,3,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {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}
[3,1,4,2] => [2,4,1,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 1
[3,+,1,+] => [2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {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}
[3,-,1,+] => [3,4,1,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {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}
[3,+,1,-] => [2,3,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {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}
[3,-,1,-] => [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
 => 1
[3,+,4,1] => [2,4,1,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 1
[3,-,4,1] => [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
 => 1
[3,4,1,2] => [3,4,1,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {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}
[3,4,2,1] => [3,4,1,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {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}
[4,1,2,3] => [2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {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}
[4,1,+,2] => [2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {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}
[4,1,-,2] => [2,4,1,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 1
[4,+,1,3] => [2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {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}
[4,-,1,3] => [3,4,1,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {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}
[4,+,+,1] => [2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {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}
[4,-,+,1] => [3,4,1,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {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}
[4,+,-,1] => [2,4,1,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 1
[4,-,-,1] => [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
 => 1
[4,3,1,2] => [3,4,1,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {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}
[4,3,2,1] => [3,4,1,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {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}
[+,+,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[-,+,+,+,+] => [2,3,4,5,1] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[+,-,+,+,+] => [1,3,4,5,2] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,+,-,+,+] => [1,2,4,5,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,+,+,-,+] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[+,+,+,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[-,-,+,+,+] => [3,4,5,1,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[-,+,-,+,+] => [2,4,5,1,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[-,+,+,-,+] => [2,3,5,1,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[-,+,+,+,-] => [2,3,4,1,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[+,+,+,-,-] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[-,-,+,-,+] => [3,5,1,2,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 1
Description
The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001491
Mapping
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 33%
Values
[+] => [1] => [1] => => ? ∊ {0,0}
[-] => [1] => [1] => => ? ∊ {0,0}
[+,+] => [1,2] => [2,1] => 0 => ? ∊ {0,0,0}
[-,+] => [2,1] => [1,2] => 1 => 1
[+,-] => [1,2] => [2,1] => 0 => ? ∊ {0,0,0}
[-,-] => [1,2] => [2,1] => 0 => ? ∊ {0,0,0}
[2,1] => [2,1] => [1,2] => 1 => 1
[+,+,+] => [1,2,3] => [3,2,1] => 00 => ? ∊ {0,0,0,0,1,1,2,2}
[-,+,+] => [2,3,1] => [1,3,2] => 10 => 1
[+,-,+] => [1,3,2] => [2,3,1] => 00 => ? ∊ {0,0,0,0,1,1,2,2}
[+,+,-] => [1,2,3] => [3,2,1] => 00 => ? ∊ {0,0,0,0,1,1,2,2}
[-,-,+] => [3,1,2] => [2,1,3] => 01 => 1
[-,+,-] => [2,1,3] => [3,1,2] => 00 => ? ∊ {0,0,0,0,1,1,2,2}
[+,-,-] => [1,2,3] => [3,2,1] => 00 => ? ∊ {0,0,0,0,1,1,2,2}
[-,-,-] => [1,2,3] => [3,2,1] => 00 => ? ∊ {0,0,0,0,1,1,2,2}
[+,3,2] => [1,3,2] => [2,3,1] => 00 => ? ∊ {0,0,0,0,1,1,2,2}
[-,3,2] => [3,1,2] => [2,1,3] => 01 => 1
[2,1,+] => [2,3,1] => [1,3,2] => 10 => 1
[2,1,-] => [2,1,3] => [3,1,2] => 00 => ? ∊ {0,0,0,0,1,1,2,2}
[2,3,1] => [3,1,2] => [2,1,3] => 01 => 1
[3,1,2] => [2,3,1] => [1,3,2] => 10 => 1
[3,+,1] => [2,3,1] => [1,3,2] => 10 => 1
[3,-,1] => [3,1,2] => [2,1,3] => 01 => 1
[+,+,+,+] => [1,2,3,4] => [4,3,2,1] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,+,+,+] => [2,3,4,1] => [1,4,3,2] => 100 => 1
[+,-,+,+] => [1,3,4,2] => [2,4,3,1] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[+,+,-,+] => [1,2,4,3] => [3,4,2,1] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[+,+,+,-] => [1,2,3,4] => [4,3,2,1] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,-,+,+] => [3,4,1,2] => [2,1,4,3] => 010 => 1
[-,+,-,+] => [2,4,1,3] => [3,1,4,2] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,+,+,-] => [2,3,1,4] => [4,1,3,2] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[+,-,-,+] => [1,4,2,3] => [3,2,4,1] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[+,-,+,-] => [1,3,2,4] => [4,2,3,1] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[+,+,-,-] => [1,2,3,4] => [4,3,2,1] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,-,-,+] => [4,1,2,3] => [3,2,1,4] => 001 => 1
[-,-,+,-] => [3,1,2,4] => [4,2,1,3] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,+,-,-] => [2,1,3,4] => [4,3,1,2] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[+,-,-,-] => [1,2,3,4] => [4,3,2,1] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,-,-,-] => [1,2,3,4] => [4,3,2,1] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[+,+,4,3] => [1,2,4,3] => [3,4,2,1] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,+,4,3] => [2,4,1,3] => [3,1,4,2] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[+,-,4,3] => [1,4,2,3] => [3,2,4,1] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,-,4,3] => [4,1,2,3] => [3,2,1,4] => 001 => 1
[+,3,2,+] => [1,3,4,2] => [2,4,3,1] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,3,2,+] => [3,4,1,2] => [2,1,4,3] => 010 => 1
[+,3,2,-] => [1,3,2,4] => [4,2,3,1] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,3,2,-] => [3,1,2,4] => [4,2,1,3] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[+,3,4,2] => [1,4,2,3] => [3,2,4,1] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,3,4,2] => [4,1,2,3] => [3,2,1,4] => 001 => 1
[+,4,2,3] => [1,3,4,2] => [2,4,3,1] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,4,2,3] => [3,4,1,2] => [2,1,4,3] => 010 => 1
[+,4,+,2] => [1,3,4,2] => [2,4,3,1] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,4,+,2] => [3,4,1,2] => [2,1,4,3] => 010 => 1
[+,4,-,2] => [1,4,2,3] => [3,2,4,1] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,4,-,2] => [4,1,2,3] => [3,2,1,4] => 001 => 1
[2,1,+,+] => [2,3,4,1] => [1,4,3,2] => 100 => 1
[2,1,-,+] => [2,4,1,3] => [3,1,4,2] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,+,-] => [2,3,1,4] => [4,1,3,2] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,-,-] => [2,1,3,4] => [4,3,1,2] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,4,3] => [2,4,1,3] => [3,1,4,2] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,1,+] => [3,4,1,2] => [2,1,4,3] => 010 => 1
[2,3,1,-] => [3,1,2,4] => [4,2,1,3] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,1] => [4,1,2,3] => [3,2,1,4] => 001 => 1
[2,4,1,3] => [3,4,1,2] => [2,1,4,3] => 010 => 1
[2,4,+,1] => [3,4,1,2] => [2,1,4,3] => 010 => 1
[2,4,-,1] => [4,1,2,3] => [3,2,1,4] => 001 => 1
[3,1,2,+] => [2,3,4,1] => [1,4,3,2] => 100 => 1
[3,1,2,-] => [2,3,1,4] => [4,1,3,2] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,4,2] => [2,4,1,3] => [3,1,4,2] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[3,+,1,+] => [2,3,4,1] => [1,4,3,2] => 100 => 1
[3,-,1,+] => [3,4,1,2] => [2,1,4,3] => 010 => 1
[3,+,1,-] => [2,3,1,4] => [4,1,3,2] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[3,-,1,-] => [3,1,2,4] => [4,2,1,3] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[3,+,4,1] => [2,4,1,3] => [3,1,4,2] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[3,-,4,1] => [4,1,2,3] => [3,2,1,4] => 001 => 1
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 010 => 1
[3,4,2,1] => [3,4,1,2] => [2,1,4,3] => 010 => 1
[4,1,2,3] => [2,3,4,1] => [1,4,3,2] => 100 => 1
[4,1,+,2] => [2,3,4,1] => [1,4,3,2] => 100 => 1
[4,1,-,2] => [2,4,1,3] => [3,1,4,2] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[4,+,1,3] => [2,3,4,1] => [1,4,3,2] => 100 => 1
[4,-,1,3] => [3,4,1,2] => [2,1,4,3] => 010 => 1
[4,+,+,1] => [2,3,4,1] => [1,4,3,2] => 100 => 1
[4,-,+,1] => [3,4,1,2] => [2,1,4,3] => 010 => 1
[4,+,-,1] => [2,4,1,3] => [3,1,4,2] => 000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[4,-,-,1] => [4,1,2,3] => [3,2,1,4] => 001 => 1
[4,3,1,2] => [3,4,1,2] => [2,1,4,3] => 010 => 1
[4,3,2,1] => [3,4,1,2] => [2,1,4,3] => 010 => 1
[+,+,+,+,+] => [1,2,3,4,5] => [5,4,3,2,1] => 0000 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[-,+,+,+,+] => [2,3,4,5,1] => [1,5,4,3,2] => 1000 => 1
[+,-,+,+,+] => [1,3,4,5,2] => [2,5,4,3,1] => 0000 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[-,-,+,+,+] => [3,4,5,1,2] => [2,1,5,4,3] => 0100 => 1
[-,-,-,+,+] => [4,5,1,2,3] => [3,2,1,5,4] => 0010 => 1
[-,-,-,-,+] => [5,1,2,3,4] => [4,3,2,1,5] => 0001 => 1
[-,-,-,5,4] => [5,1,2,3,4] => [4,3,2,1,5] => 0001 => 1
[-,-,4,3,+] => [4,5,1,2,3] => [3,2,1,5,4] => 0010 => 1
[-,-,4,5,3] => [5,1,2,3,4] => [4,3,2,1,5] => 0001 => 1
[-,-,5,3,4] => [4,5,1,2,3] => [3,2,1,5,4] => 0010 => 1
[-,-,5,+,3] => [4,5,1,2,3] => [3,2,1,5,4] => 0010 => 1
[-,-,5,-,3] => [5,1,2,3,4] => [4,3,2,1,5] => 0001 => 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset. Let $A_n=K[x]/(x^n)$. We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
Matching statistic: St000456
(load all 2 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: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 33%
Values
[+] => [1] => [1] => ([],1)
 => ? ∊ {0,0}
[-] => [1] => [1] => ([],1)
 => ? ∊ {0,0}
[+,+] => [1,2] => [2] => ([],2)
 => ? ∊ {0,0,0}
[-,+] => [2,1] => [1,1] => ([(0,1)],2)
 => 1
[+,-] => [1,2] => [2] => ([],2)
 => ? ∊ {0,0,0}
[-,-] => [1,2] => [2] => ([],2)
 => ? ∊ {0,0,0}
[2,1] => [2,1] => [1,1] => ([(0,1)],2)
 => 1
[+,+,+] => [1,2,3] => [3] => ([],3)
 => ? ∊ {0,0,0,0,1,1,1,1,2,2}
[-,+,+] => [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)
 => ? ∊ {0,0,0,0,1,1,1,1,2,2}
[-,-,+] => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,1,1,1,1,2,2}
[-,+,-] => [2,1,3] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,1,1,1,1,2,2}
[+,-,-] => [1,2,3] => [3] => ([],3)
 => ? ∊ {0,0,0,0,1,1,1,1,2,2}
[-,-,-] => [1,2,3] => [3] => ([],3)
 => ? ∊ {0,0,0,0,1,1,1,1,2,2}
[+,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[-,3,2] => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,1,1,1,1,2,2}
[2,1,+] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[2,1,-] => [2,1,3] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,1,1,1,1,2,2}
[2,3,1] => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,1,1,1,1,2,2}
[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)
 => ? ∊ {0,0,0,0,1,1,1,1,2,2}
[+,+,+,+] => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,+,+,+] => [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)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,-,+,+] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,+,-,+] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,+,+,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[+,-,-,+] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[+,-,+,-] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[+,+,-,-] => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,-,-,+] => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,-,+,-] => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,+,-,-] => [2,1,3,4] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[+,-,-,-] => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,-,-,-] => [1,2,3,4] => [4] => ([],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[+,+,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)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[+,-,4,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,-,4,3] => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[+,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)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[+,3,2,-] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,3,2,-] => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[+,3,4,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,3,4,2] => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[+,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)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[+,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)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[+,4,-,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[-,4,-,2] => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[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)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,+,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,-,-] => [2,1,3,4] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,4,3] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,1,+] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,1,-] => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,1] => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,3] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,+,1] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,-,1] => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[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.