Your data matches 31 different statistics following compositions of up to 3 maps.
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Matching statistic: St001344
(load all 6 compositions to match this statistic)
Mapping
Mp00255: Decorated permutations —lower permutation⟶ Permutations
St001344: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+,+] => [1,2] => 1
[-,+] => [2,1] => 1
[+,-] => [1,2] => 1
[-,-] => [1,2] => 1
[2,1] => [1,2] => 1
[+,+,+] => [1,2,3] => 1
[-,+,+] => [2,3,1] => 1
[+,-,+] => [1,3,2] => 1
[+,+,-] => [1,2,3] => 1
[-,-,+] => [3,1,2] => 1
[-,+,-] => [2,1,3] => 1
[+,-,-] => [1,2,3] => 1
[-,-,-] => [1,2,3] => 1
[+,3,2] => [1,2,3] => 1
[-,3,2] => [2,1,3] => 1
[2,1,+] => [1,3,2] => 1
[2,1,-] => [1,2,3] => 1
[2,3,1] => [1,2,3] => 1
[3,1,2] => [1,2,3] => 1
[3,+,1] => [2,1,3] => 1
[3,-,1] => [1,3,2] => 1
[+,+,+,+] => [1,2,3,4] => 1
[-,+,+,+] => [2,3,4,1] => 1
[+,-,+,+] => [1,3,4,2] => 1
[+,+,-,+] => [1,2,4,3] => 1
[+,+,+,-] => [1,2,3,4] => 1
[-,-,+,+] => [3,4,1,2] => 1
[-,+,-,+] => [2,4,1,3] => 1
[-,+,+,-] => [2,3,1,4] => 1
[+,-,-,+] => [1,4,2,3] => 1
[+,-,+,-] => [1,3,2,4] => 1
[+,+,-,-] => [1,2,3,4] => 1
[-,-,-,+] => [4,1,2,3] => 1
[-,-,+,-] => [3,1,2,4] => 1
[-,+,-,-] => [2,1,3,4] => 1
[+,-,-,-] => [1,2,3,4] => 1
[-,-,-,-] => [1,2,3,4] => 1
[+,+,4,3] => [1,2,3,4] => 1
[-,+,4,3] => [2,3,1,4] => 1
[+,-,4,3] => [1,3,2,4] => 1
[-,-,4,3] => [3,1,2,4] => 1
[+,3,2,+] => [1,2,4,3] => 1
[-,3,2,+] => [2,4,1,3] => 1
[+,3,2,-] => [1,2,3,4] => 1
[-,3,2,-] => [2,1,3,4] => 1
[+,3,4,2] => [1,2,3,4] => 1
[-,3,4,2] => [2,1,3,4] => 1
[+,4,2,3] => [1,2,3,4] => 1
[-,4,2,3] => [2,3,1,4] => 1
[+,4,+,2] => [1,3,2,4] => 1
Description
The neighbouring number of a permutation. For a permutation $\pi$, this is $$\min \big(\big\{|\pi(k)-\pi(k+1)|:k\in\{1,\ldots,n-1\}\big\}\cup \big\{|\pi(1) - \pi(n)|\big\}\big).$$
Matching statistic: St001771
(load all 12 compositions to match this statistic)
Mapping
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001771: Signed permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Values
[+,+] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[-,+] => [2,1] => [2,1] => [2,1] => 0 = 1 - 1
[+,-] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[-,-] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[+,+,+] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[-,+,+] => [2,3,1] => [2,1,3] => [2,1,3] => 0 = 1 - 1
[+,-,+] => [1,3,2] => [3,1,2] => [3,1,2] => 0 = 1 - 1
[+,+,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[-,-,+] => [3,1,2] => [1,3,2] => [1,3,2] => 0 = 1 - 1
[-,+,-] => [2,1,3] => [2,3,1] => [2,3,1] => 0 = 1 - 1
[+,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[-,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[+,3,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[-,3,2] => [2,1,3] => [2,3,1] => [2,3,1] => 0 = 1 - 1
[2,1,+] => [1,3,2] => [3,1,2] => [3,1,2] => 0 = 1 - 1
[2,1,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[3,+,1] => [2,1,3] => [2,3,1] => [2,3,1] => 0 = 1 - 1
[3,-,1] => [1,3,2] => [3,1,2] => [3,1,2] => 0 = 1 - 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,+,+,+] => [2,3,4,1] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[+,-,+,+] => [1,3,4,2] => [3,1,2,4] => [3,1,2,4] => 0 = 1 - 1
[+,+,-,+] => [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 0 = 1 - 1
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,-,+,+] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[-,+,-,+] => [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 0 = 1 - 1
[-,+,+,-] => [2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 0 = 1 - 1
[+,-,-,+] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 0 = 1 - 1
[+,-,+,-] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,-,-,+] => [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[-,-,+,-] => [3,1,2,4] => [1,3,4,2] => [1,3,4,2] => 0 = 1 - 1
[-,+,-,-] => [2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 0 = 1 - 1
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[+,+,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,+,4,3] => [2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 0 = 1 - 1
[+,-,4,3] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[-,-,4,3] => [3,1,2,4] => [1,3,4,2] => [1,3,4,2] => 0 = 1 - 1
[+,3,2,+] => [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 0 = 1 - 1
[-,3,2,+] => [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 0 = 1 - 1
[+,3,2,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,3,2,-] => [2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 0 = 1 - 1
[+,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,3,4,2] => [2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 0 = 1 - 1
[+,4,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,4,2,3] => [2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 0 = 1 - 1
[+,4,+,2] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[-,+,+,+,+] => [2,3,4,5,1] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[+,-,+,+,+] => [1,3,4,5,2] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 1 - 1
[+,+,-,+,+] => [1,2,4,5,3] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 - 1
[+,+,+,-,+] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[-,-,+,+,+] => [3,4,5,1,2] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 1 - 1
[-,+,-,+,+] => [2,4,5,1,3] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1 - 1
[-,+,+,-,+] => [2,3,5,1,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 1 - 1
[-,+,+,+,-] => [2,3,4,1,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 1 - 1
[+,-,-,+,+] => [1,4,5,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 1 - 1
[+,-,+,-,+] => [1,3,5,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 2 - 1
[-,-,+,+,-] => [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1 - 1
[-,+,-,-,+] => [2,5,1,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 1 - 1
[-,+,-,+,-] => [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 2 - 1
[-,+,+,-,-] => [2,3,1,4,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 1 - 1
[-,+,-,-,-] => [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1 - 1
[-,+,+,5,4] => [2,3,4,1,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 1 - 1
[-,-,+,5,4] => [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1 - 1
[-,+,-,5,4] => [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 2 - 1
[+,+,4,3,+] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[-,+,4,3,+] => [2,3,5,1,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 1 - 1
[+,-,4,3,+] => [1,3,5,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 2 - 1
[-,+,4,3,-] => [2,3,1,4,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 1 - 1
[-,+,4,5,3] => [2,3,1,4,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 1 - 1
[-,+,5,3,4] => [2,3,4,1,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 1 - 1
[-,-,5,3,4] => [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1 - 1
[-,+,5,+,3] => [2,4,3,1,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 1 - 1
[+,+,5,-,3] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[-,+,5,-,3] => [2,3,1,5,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 1 - 1
[+,-,5,-,3] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 1 - 1
[-,-,5,-,3] => [3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 1 - 1
[+,3,2,+,+] => [1,2,4,5,3] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 - 1
[-,3,2,+,+] => [2,4,5,1,3] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1 - 1
[-,3,2,-,+] => [2,5,1,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 1 - 1
[-,3,2,+,-] => [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 2 - 1
[-,3,2,-,-] => [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1 - 1
[-,3,2,5,4] => [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 2 - 1
[-,3,4,2,+] => [2,5,1,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 1 - 1
[-,3,4,2,-] => [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1 - 1
[-,3,4,5,2] => [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1 - 1
[-,3,5,2,4] => [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 2 - 1
[-,3,5,+,2] => [4,2,1,3,5] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 1 - 1
[+,3,5,-,2] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[-,3,5,-,2] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 1 - 1
[+,4,2,3,+] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[-,4,2,3,+] => [2,3,5,1,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 1 - 1
[-,4,2,3,-] => [2,3,1,4,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 1 - 1
[-,4,2,5,3] => [2,3,1,4,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 1 - 1
[+,4,+,2,+] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 1 - 1
[-,4,+,2,+] => [3,2,5,1,4] => [3,5,2,4,1] => [3,5,2,4,1] => ? = 1 - 1
[+,4,-,2,+] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
Description
The number of occurrences of the signed pattern 1-2 in a signed permutation. This is the number of pairs $1\leq i < j\leq n$ such that $0 < \pi(i) < -\pi(j)$.
Matching statistic: St001870
(load all 10 compositions to match this statistic)
Mapping
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001870: Signed permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Values
[+,+] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[-,+] => [2,1] => [2,1] => [2,1] => 0 = 1 - 1
[+,-] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[-,-] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[+,+,+] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[-,+,+] => [2,3,1] => [2,1,3] => [2,1,3] => 0 = 1 - 1
[+,-,+] => [1,3,2] => [3,1,2] => [3,1,2] => 0 = 1 - 1
[+,+,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[-,-,+] => [3,1,2] => [1,3,2] => [1,3,2] => 0 = 1 - 1
[-,+,-] => [2,1,3] => [2,3,1] => [2,3,1] => 0 = 1 - 1
[+,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[-,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[+,3,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[-,3,2] => [2,1,3] => [2,3,1] => [2,3,1] => 0 = 1 - 1
[2,1,+] => [1,3,2] => [3,1,2] => [3,1,2] => 0 = 1 - 1
[2,1,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[3,+,1] => [2,1,3] => [2,3,1] => [2,3,1] => 0 = 1 - 1
[3,-,1] => [1,3,2] => [3,1,2] => [3,1,2] => 0 = 1 - 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,+,+,+] => [2,3,4,1] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[+,-,+,+] => [1,3,4,2] => [3,1,2,4] => [3,1,2,4] => 0 = 1 - 1
[+,+,-,+] => [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 0 = 1 - 1
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,-,+,+] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[-,+,-,+] => [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 0 = 1 - 1
[-,+,+,-] => [2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 0 = 1 - 1
[+,-,-,+] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 0 = 1 - 1
[+,-,+,-] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,-,-,+] => [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[-,-,+,-] => [3,1,2,4] => [1,3,4,2] => [1,3,4,2] => 0 = 1 - 1
[-,+,-,-] => [2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 0 = 1 - 1
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[+,+,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,+,4,3] => [2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 0 = 1 - 1
[+,-,4,3] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[-,-,4,3] => [3,1,2,4] => [1,3,4,2] => [1,3,4,2] => 0 = 1 - 1
[+,3,2,+] => [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 0 = 1 - 1
[-,3,2,+] => [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 0 = 1 - 1
[+,3,2,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,3,2,-] => [2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 0 = 1 - 1
[+,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,3,4,2] => [2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 0 = 1 - 1
[+,4,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,4,2,3] => [2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 0 = 1 - 1
[+,4,+,2] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[-,+,+,+,+] => [2,3,4,5,1] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[+,-,+,+,+] => [1,3,4,5,2] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 1 - 1
[+,+,-,+,+] => [1,2,4,5,3] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 - 1
[+,+,+,-,+] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[-,-,+,+,+] => [3,4,5,1,2] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 1 - 1
[-,+,-,+,+] => [2,4,5,1,3] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1 - 1
[-,+,+,-,+] => [2,3,5,1,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 1 - 1
[-,+,+,+,-] => [2,3,4,1,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 1 - 1
[+,-,-,+,+] => [1,4,5,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 1 - 1
[+,-,+,-,+] => [1,3,5,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 2 - 1
[-,-,+,+,-] => [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1 - 1
[-,+,-,-,+] => [2,5,1,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 1 - 1
[-,+,-,+,-] => [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 2 - 1
[-,+,+,-,-] => [2,3,1,4,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 1 - 1
[-,+,-,-,-] => [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1 - 1
[-,+,+,5,4] => [2,3,4,1,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 1 - 1
[-,-,+,5,4] => [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1 - 1
[-,+,-,5,4] => [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 2 - 1
[+,+,4,3,+] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[-,+,4,3,+] => [2,3,5,1,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 1 - 1
[+,-,4,3,+] => [1,3,5,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 2 - 1
[-,+,4,3,-] => [2,3,1,4,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 1 - 1
[-,+,4,5,3] => [2,3,1,4,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 1 - 1
[-,+,5,3,4] => [2,3,4,1,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 1 - 1
[-,-,5,3,4] => [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1 - 1
[-,+,5,+,3] => [2,4,3,1,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 1 - 1
[+,+,5,-,3] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[-,+,5,-,3] => [2,3,1,5,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 1 - 1
[+,-,5,-,3] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 1 - 1
[-,-,5,-,3] => [3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 1 - 1
[+,3,2,+,+] => [1,2,4,5,3] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 - 1
[-,3,2,+,+] => [2,4,5,1,3] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1 - 1
[-,3,2,-,+] => [2,5,1,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 1 - 1
[-,3,2,+,-] => [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 2 - 1
[-,3,2,-,-] => [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1 - 1
[-,3,2,5,4] => [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 2 - 1
[-,3,4,2,+] => [2,5,1,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 1 - 1
[-,3,4,2,-] => [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1 - 1
[-,3,4,5,2] => [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1 - 1
[-,3,5,2,4] => [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 2 - 1
[-,3,5,+,2] => [4,2,1,3,5] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 1 - 1
[+,3,5,-,2] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[-,3,5,-,2] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 1 - 1
[+,4,2,3,+] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[-,4,2,3,+] => [2,3,5,1,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 1 - 1
[-,4,2,3,-] => [2,3,1,4,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 1 - 1
[-,4,2,5,3] => [2,3,1,4,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 1 - 1
[+,4,+,2,+] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 1 - 1
[-,4,+,2,+] => [3,2,5,1,4] => [3,5,2,4,1] => [3,5,2,4,1] => ? = 1 - 1
[+,4,-,2,+] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
Description
The number of positive entries followed by a negative entry in a signed permutation. For a signed permutation $\pi\in\mathfrak H_n$, this is the number of positive entries followed by a negative entry in $\pi(-n),\dots,\pi(-1),\pi(1),\dots,\pi(n)$.
Matching statistic: St001895
(load all 12 compositions to match this statistic)
Mapping
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001895: Signed permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Values
[+,+] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[-,+] => [2,1] => [2,1] => [2,1] => 0 = 1 - 1
[+,-] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[-,-] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[+,+,+] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[-,+,+] => [2,3,1] => [2,1,3] => [2,1,3] => 0 = 1 - 1
[+,-,+] => [1,3,2] => [3,1,2] => [3,1,2] => 0 = 1 - 1
[+,+,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[-,-,+] => [3,1,2] => [1,3,2] => [1,3,2] => 0 = 1 - 1
[-,+,-] => [2,1,3] => [2,3,1] => [2,3,1] => 0 = 1 - 1
[+,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[-,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[+,3,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[-,3,2] => [2,1,3] => [2,3,1] => [2,3,1] => 0 = 1 - 1
[2,1,+] => [1,3,2] => [3,1,2] => [3,1,2] => 0 = 1 - 1
[2,1,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[3,+,1] => [2,1,3] => [2,3,1] => [2,3,1] => 0 = 1 - 1
[3,-,1] => [1,3,2] => [3,1,2] => [3,1,2] => 0 = 1 - 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,+,+,+] => [2,3,4,1] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[+,-,+,+] => [1,3,4,2] => [3,1,2,4] => [3,1,2,4] => 0 = 1 - 1
[+,+,-,+] => [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 0 = 1 - 1
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,-,+,+] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[-,+,-,+] => [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 0 = 1 - 1
[-,+,+,-] => [2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 0 = 1 - 1
[+,-,-,+] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 0 = 1 - 1
[+,-,+,-] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,-,-,+] => [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[-,-,+,-] => [3,1,2,4] => [1,3,4,2] => [1,3,4,2] => 0 = 1 - 1
[-,+,-,-] => [2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 0 = 1 - 1
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[+,+,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,+,4,3] => [2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 0 = 1 - 1
[+,-,4,3] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[-,-,4,3] => [3,1,2,4] => [1,3,4,2] => [1,3,4,2] => 0 = 1 - 1
[+,3,2,+] => [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 0 = 1 - 1
[-,3,2,+] => [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 0 = 1 - 1
[+,3,2,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,3,2,-] => [2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 0 = 1 - 1
[+,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,3,4,2] => [2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 0 = 1 - 1
[+,4,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[-,4,2,3] => [2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 0 = 1 - 1
[+,4,+,2] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[-,+,+,+,+] => [2,3,4,5,1] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[+,-,+,+,+] => [1,3,4,5,2] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 1 - 1
[+,+,-,+,+] => [1,2,4,5,3] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 - 1
[+,+,+,-,+] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[-,-,+,+,+] => [3,4,5,1,2] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 1 - 1
[-,+,-,+,+] => [2,4,5,1,3] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1 - 1
[-,+,+,-,+] => [2,3,5,1,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 1 - 1
[-,+,+,+,-] => [2,3,4,1,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 1 - 1
[+,-,-,+,+] => [1,4,5,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 1 - 1
[+,-,+,-,+] => [1,3,5,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 2 - 1
[-,-,+,+,-] => [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1 - 1
[-,+,-,-,+] => [2,5,1,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 1 - 1
[-,+,-,+,-] => [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 2 - 1
[-,+,+,-,-] => [2,3,1,4,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 1 - 1
[-,+,-,-,-] => [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1 - 1
[-,+,+,5,4] => [2,3,4,1,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 1 - 1
[-,-,+,5,4] => [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1 - 1
[-,+,-,5,4] => [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 2 - 1
[+,+,4,3,+] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[-,+,4,3,+] => [2,3,5,1,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 1 - 1
[+,-,4,3,+] => [1,3,5,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 2 - 1
[-,+,4,3,-] => [2,3,1,4,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 1 - 1
[-,+,4,5,3] => [2,3,1,4,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 1 - 1
[-,+,5,3,4] => [2,3,4,1,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 1 - 1
[-,-,5,3,4] => [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1 - 1
[-,+,5,+,3] => [2,4,3,1,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 1 - 1
[+,+,5,-,3] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[-,+,5,-,3] => [2,3,1,5,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 1 - 1
[+,-,5,-,3] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 1 - 1
[-,-,5,-,3] => [3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 1 - 1
[+,3,2,+,+] => [1,2,4,5,3] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 - 1
[-,3,2,+,+] => [2,4,5,1,3] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1 - 1
[-,3,2,-,+] => [2,5,1,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 1 - 1
[-,3,2,+,-] => [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 2 - 1
[-,3,2,-,-] => [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1 - 1
[-,3,2,5,4] => [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 2 - 1
[-,3,4,2,+] => [2,5,1,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 1 - 1
[-,3,4,2,-] => [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1 - 1
[-,3,4,5,2] => [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1 - 1
[-,3,5,2,4] => [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 2 - 1
[-,3,5,+,2] => [4,2,1,3,5] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 1 - 1
[+,3,5,-,2] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[-,3,5,-,2] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 1 - 1
[+,4,2,3,+] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[-,4,2,3,+] => [2,3,5,1,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 1 - 1
[-,4,2,3,-] => [2,3,1,4,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 1 - 1
[-,4,2,5,3] => [2,3,1,4,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 1 - 1
[+,4,+,2,+] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 1 - 1
[-,4,+,2,+] => [3,2,5,1,4] => [3,5,2,4,1] => [3,5,2,4,1] => ? = 1 - 1
[+,4,-,2,+] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
Description
The oddness of a signed permutation. The direct sum of two signed permutations $\sigma\in\mathfrak H_k$ and $\tau\in\mathfrak H_m$ is the signed permutation in $\mathfrak H_{k+m}$ obtained by concatenating $\sigma$ with the result of increasing the absolute value of every entry in $\tau$ by $k$. This statistic records the number of blocks with an odd number of signs in the direct sum decomposition of a signed permutation.
Matching statistic: St000260
Mapping
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 50%
Values
[+,+] => [1,2] => [2] => ([],2)
 => ? = 1
[-,+] => [2,1] => [1,1] => ([(0,1)],2)
 => 1
[+,-] => [1,2] => [2] => ([],2)
 => ? = 1
[-,-] => [1,2] => [2] => ([],2)
 => ? = 1
[2,1] => [2,1] => [1,1] => ([(0,1)],2)
 => 1
[+,+,+] => [1,2,3] => [3] => ([],3)
 => ? = 1
[-,+,+] => [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)
 => ? = 1
[-,-,+] => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? = 1
[-,+,-] => [2,1,3] => [1,2] => ([(1,2)],3)
 => ? = 1
[+,-,-] => [1,2,3] => [3] => ([],3)
 => ? = 1
[-,-,-] => [1,2,3] => [3] => ([],3)
 => ? = 1
[+,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[-,3,2] => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? = 1
[2,1,+] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[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)
 => 1
[3,+,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[3,-,1] => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? = 1
[+,+,+,+] => [1,2,3,4] => [4] => ([],4)
 => ? = 1
[-,+,+,+] => [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)
 => ? = 1
[-,-,+,+] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[-,+,-,+] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[-,+,+,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[+,-,-,+] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[+,-,+,-] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[+,+,-,-] => [1,2,3,4] => [4] => ([],4)
 => ? = 1
[-,-,-,+] => [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)
 => ? = 1
[-,-,-,-] => [1,2,3,4] => [4] => ([],4)
 => ? = 1
[+,+,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)
 => ? = 1
[+,-,4,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[-,-,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)
 => 1
[-,3,2,+] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[+,3,2,-] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[-,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)
 => ? = 1
[-,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)
 => 1
[-,4,2,3] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[+,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)
 => ? = 1
[+,4,-,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[-,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)
 => 1
[2,1,-,+] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[2,1,+,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[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)
 => ? = 1
[2,3,1,+] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[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)
 => ? = 1
[2,4,+,1] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[2,4,-,1] => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => ? = 1
[3,1,2,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[3,1,2,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[3,1,4,2] => [2,4,1,3] => [2,2] => ([(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 radius of a connected graph. This is the minimum eccentricity of any vertex.
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: 50%
Values
[+,+] => [1,2] => [2] => ([],2)
 => ? = 1
[-,+] => [2,1] => [1,1] => ([(0,1)],2)
 => 1
[+,-] => [1,2] => [2] => ([],2)
 => ? = 1
[-,-] => [1,2] => [2] => ([],2)
 => ? = 1
[2,1] => [2,1] => [1,1] => ([(0,1)],2)
 => 1
[+,+,+] => [1,2,3] => [3] => ([],3)
 => ? = 1
[-,+,+] => [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)
 => ? = 1
[-,-,+] => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? = 1
[-,+,-] => [2,1,3] => [1,2] => ([(1,2)],3)
 => ? = 1
[+,-,-] => [1,2,3] => [3] => ([],3)
 => ? = 1
[-,-,-] => [1,2,3] => [3] => ([],3)
 => ? = 1
[+,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[-,3,2] => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? = 1
[2,1,+] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[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)
 => 1
[3,+,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[3,-,1] => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? = 1
[+,+,+,+] => [1,2,3,4] => [4] => ([],4)
 => ? = 1
[-,+,+,+] => [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)
 => ? = 1
[-,-,+,+] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[-,+,-,+] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[-,+,+,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[+,-,-,+] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[+,-,+,-] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[+,+,-,-] => [1,2,3,4] => [4] => ([],4)
 => ? = 1
[-,-,-,+] => [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)
 => ? = 1
[-,-,-,-] => [1,2,3,4] => [4] => ([],4)
 => ? = 1
[+,+,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)
 => ? = 1
[+,-,4,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[-,-,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)
 => 1
[-,3,2,+] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[+,3,2,-] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[-,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)
 => ? = 1
[-,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)
 => 1
[-,4,2,3] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[+,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)
 => ? = 1
[+,4,-,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[-,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)
 => 1
[2,1,-,+] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[2,1,+,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[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)
 => ? = 1
[2,3,1,+] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[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)
 => ? = 1
[2,4,+,1] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[2,4,-,1] => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => ? = 1
[3,1,2,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[3,1,2,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[3,1,4,2] => [2,4,1,3] => [2,2] => ([(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.
Matching statistic: St001889
(load all 2 compositions to match this statistic)
Mapping
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00168: Signed permutations —Kreweras complement⟶ Signed permutations
St001889: Signed permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 50%
Values
[+,+] => [1,2] => [1,2] => [2,-1] => 0 = 1 - 1
[-,+] => [2,1] => [2,1] => [-1,2] => 0 = 1 - 1
[+,-] => [1,2] => [1,2] => [2,-1] => 0 = 1 - 1
[-,-] => [1,2] => [1,2] => [2,-1] => 0 = 1 - 1
[2,1] => [2,1] => [2,1] => [-1,2] => 0 = 1 - 1
[+,+,+] => [1,2,3] => [1,2,3] => [2,3,-1] => 0 = 1 - 1
[-,+,+] => [2,3,1] => [2,3,1] => [-1,2,3] => 0 = 1 - 1
[+,-,+] => [1,3,2] => [1,3,2] => [2,-1,3] => 0 = 1 - 1
[+,+,-] => [1,2,3] => [1,2,3] => [2,3,-1] => 0 = 1 - 1
[-,-,+] => [3,1,2] => [3,1,2] => [3,-1,2] => 0 = 1 - 1
[-,+,-] => [2,1,3] => [2,1,3] => [3,2,-1] => 0 = 1 - 1
[+,-,-] => [1,2,3] => [1,2,3] => [2,3,-1] => 0 = 1 - 1
[-,-,-] => [1,2,3] => [1,2,3] => [2,3,-1] => 0 = 1 - 1
[+,3,2] => [1,3,2] => [1,3,2] => [2,-1,3] => 0 = 1 - 1
[-,3,2] => [3,1,2] => [3,1,2] => [3,-1,2] => 0 = 1 - 1
[2,1,+] => [2,3,1] => [2,3,1] => [-1,2,3] => 0 = 1 - 1
[2,1,-] => [2,1,3] => [2,1,3] => [3,2,-1] => 0 = 1 - 1
[2,3,1] => [3,1,2] => [3,1,2] => [3,-1,2] => 0 = 1 - 1
[3,1,2] => [2,3,1] => [2,3,1] => [-1,2,3] => 0 = 1 - 1
[3,+,1] => [2,3,1] => [2,3,1] => [-1,2,3] => 0 = 1 - 1
[3,-,1] => [3,1,2] => [3,1,2] => [3,-1,2] => 0 = 1 - 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 0 = 1 - 1
[-,+,+,+] => [2,3,4,1] => [2,3,4,1] => [-1,2,3,4] => 0 = 1 - 1
[+,-,+,+] => [1,3,4,2] => [1,3,4,2] => [2,-1,3,4] => 0 = 1 - 1
[+,+,-,+] => [1,2,4,3] => [1,2,4,3] => [2,3,-1,4] => 0 = 1 - 1
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 0 = 1 - 1
[-,-,+,+] => [3,4,1,2] => [3,4,1,2] => [4,-1,2,3] => 0 = 1 - 1
[-,+,-,+] => [2,4,1,3] => [2,4,1,3] => [4,2,-1,3] => 0 = 1 - 1
[-,+,+,-] => [2,3,1,4] => [2,3,1,4] => [4,2,3,-1] => 0 = 1 - 1
[+,-,-,+] => [1,4,2,3] => [1,4,2,3] => [2,4,-1,3] => 0 = 1 - 1
[+,-,+,-] => [1,3,2,4] => [1,3,2,4] => [2,4,3,-1] => 0 = 1 - 1
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 0 = 1 - 1
[-,-,-,+] => [4,1,2,3] => [4,1,2,3] => [3,4,-1,2] => 0 = 1 - 1
[-,-,+,-] => [3,1,2,4] => [3,1,2,4] => [3,4,2,-1] => 0 = 1 - 1
[-,+,-,-] => [2,1,3,4] => [2,1,3,4] => [3,2,4,-1] => 0 = 1 - 1
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 0 = 1 - 1
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 0 = 1 - 1
[+,+,4,3] => [1,2,4,3] => [1,2,4,3] => [2,3,-1,4] => 0 = 1 - 1
[-,+,4,3] => [2,4,1,3] => [2,4,1,3] => [4,2,-1,3] => 0 = 1 - 1
[+,-,4,3] => [1,4,2,3] => [1,4,2,3] => [2,4,-1,3] => 0 = 1 - 1
[-,-,4,3] => [4,1,2,3] => [4,1,2,3] => [3,4,-1,2] => 0 = 1 - 1
[+,3,2,+] => [1,3,4,2] => [1,3,4,2] => [2,-1,3,4] => 0 = 1 - 1
[-,3,2,+] => [3,4,1,2] => [3,4,1,2] => [4,-1,2,3] => 0 = 1 - 1
[+,3,2,-] => [1,3,2,4] => [1,3,2,4] => [2,4,3,-1] => 0 = 1 - 1
[-,3,2,-] => [3,1,2,4] => [3,1,2,4] => [3,4,2,-1] => 0 = 1 - 1
[+,3,4,2] => [1,4,2,3] => [1,4,2,3] => [2,4,-1,3] => 0 = 1 - 1
[-,3,4,2] => [4,1,2,3] => [4,1,2,3] => [3,4,-1,2] => 0 = 1 - 1
[+,4,2,3] => [1,3,4,2] => [1,3,4,2] => [2,-1,3,4] => 0 = 1 - 1
[-,4,2,3] => [3,4,1,2] => [3,4,1,2] => [4,-1,2,3] => 0 = 1 - 1
[+,4,+,2] => [1,3,4,2] => [1,3,4,2] => [2,-1,3,4] => 0 = 1 - 1
[+,+,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 1 - 1
[+,-,+,+,+] => [1,3,4,5,2] => [1,3,4,5,2] => [2,-1,3,4,5] => ? = 1 - 1
[+,+,-,+,+] => [1,2,4,5,3] => [1,2,4,5,3] => [2,3,-1,4,5] => ? = 1 - 1
[+,+,+,-,+] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,-1,5] => ? = 1 - 1
[+,+,+,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 1 - 1
[-,-,+,+,+] => [3,4,5,1,2] => [3,4,5,1,2] => [5,-1,2,3,4] => ? = 1 - 1
[-,+,-,+,+] => [2,4,5,1,3] => [2,4,5,1,3] => [5,2,-1,3,4] => ? = 1 - 1
[-,+,+,-,+] => [2,3,5,1,4] => [2,3,5,1,4] => [5,2,3,-1,4] => ? = 1 - 1
[-,+,+,+,-] => [2,3,4,1,5] => [2,3,4,1,5] => [5,2,3,4,-1] => ? = 1 - 1
[+,-,-,+,+] => [1,4,5,2,3] => [1,4,5,2,3] => [2,5,-1,3,4] => ? = 1 - 1
[+,-,+,-,+] => [1,3,5,2,4] => [1,3,5,2,4] => [2,5,3,-1,4] => ? = 2 - 1
[+,-,+,+,-] => [1,3,4,2,5] => [1,3,4,2,5] => [2,5,3,4,-1] => ? = 1 - 1
[+,+,-,-,+] => [1,2,5,3,4] => [1,2,5,3,4] => [2,3,5,-1,4] => ? = 1 - 1
[+,+,-,+,-] => [1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,-1] => ? = 1 - 1
[+,+,+,-,-] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 1 - 1
[-,-,-,+,+] => [4,5,1,2,3] => [4,5,1,2,3] => [4,5,-1,2,3] => ? = 1 - 1
[-,-,+,-,+] => [3,5,1,2,4] => [3,5,1,2,4] => [4,5,2,-1,3] => ? = 1 - 1
[-,-,+,+,-] => [3,4,1,2,5] => [3,4,1,2,5] => [4,5,2,3,-1] => ? = 1 - 1
[-,+,-,-,+] => [2,5,1,3,4] => [2,5,1,3,4] => [4,2,5,-1,3] => ? = 1 - 1
[-,+,-,+,-] => [2,4,1,3,5] => [2,4,1,3,5] => [4,2,5,3,-1] => ? = 2 - 1
[-,+,+,-,-] => [2,3,1,4,5] => [2,3,1,4,5] => [4,2,3,5,-1] => ? = 1 - 1
[+,-,-,-,+] => [1,5,2,3,4] => [1,5,2,3,4] => [2,4,5,-1,3] => ? = 1 - 1
[+,-,-,+,-] => [1,4,2,3,5] => [1,4,2,3,5] => [2,4,5,3,-1] => ? = 1 - 1
[+,-,+,-,-] => [1,3,2,4,5] => [1,3,2,4,5] => [2,4,3,5,-1] => ? = 1 - 1
[+,+,-,-,-] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 1 - 1
[-,-,-,-,+] => [5,1,2,3,4] => [5,1,2,3,4] => [3,4,5,-1,2] => ? = 1 - 1
[-,-,-,+,-] => [4,1,2,3,5] => [4,1,2,3,5] => [3,4,5,2,-1] => ? = 1 - 1
[-,-,+,-,-] => [3,1,2,4,5] => [3,1,2,4,5] => [3,4,2,5,-1] => ? = 1 - 1
[-,+,-,-,-] => [2,1,3,4,5] => [2,1,3,4,5] => [3,2,4,5,-1] => ? = 1 - 1
[+,-,-,-,-] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 1 - 1
[-,-,-,-,-] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 1 - 1
[+,+,+,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,-1,5] => ? = 1 - 1
[-,+,+,5,4] => [2,3,5,1,4] => [2,3,5,1,4] => [5,2,3,-1,4] => ? = 1 - 1
[+,-,+,5,4] => [1,3,5,2,4] => [1,3,5,2,4] => [2,5,3,-1,4] => ? = 1 - 1
[+,+,-,5,4] => [1,2,5,3,4] => [1,2,5,3,4] => [2,3,5,-1,4] => ? = 1 - 1
[-,-,+,5,4] => [3,5,1,2,4] => [3,5,1,2,4] => [4,5,2,-1,3] => ? = 1 - 1
[-,+,-,5,4] => [2,5,1,3,4] => [2,5,1,3,4] => [4,2,5,-1,3] => ? = 2 - 1
[+,-,-,5,4] => [1,5,2,3,4] => [1,5,2,3,4] => [2,4,5,-1,3] => ? = 1 - 1
[-,-,-,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => [3,4,5,-1,2] => ? = 1 - 1
[+,+,4,3,+] => [1,2,4,5,3] => [1,2,4,5,3] => [2,3,-1,4,5] => ? = 1 - 1
[-,+,4,3,+] => [2,4,5,1,3] => [2,4,5,1,3] => [5,2,-1,3,4] => ? = 1 - 1
[+,-,4,3,+] => [1,4,5,2,3] => [1,4,5,2,3] => [2,5,-1,3,4] => ? = 2 - 1
[+,+,4,3,-] => [1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,-1] => ? = 1 - 1
[-,-,4,3,+] => [4,5,1,2,3] => [4,5,1,2,3] => [4,5,-1,2,3] => ? = 1 - 1
[-,+,4,3,-] => [2,4,1,3,5] => [2,4,1,3,5] => [4,2,5,3,-1] => ? = 1 - 1
[+,-,4,3,-] => [1,4,2,3,5] => [1,4,2,3,5] => [2,4,5,3,-1] => ? = 1 - 1
[-,-,4,3,-] => [4,1,2,3,5] => [4,1,2,3,5] => [3,4,5,2,-1] => ? = 1 - 1
[+,+,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => [2,3,5,-1,4] => ? = 1 - 1
[-,+,4,5,3] => [2,5,1,3,4] => [2,5,1,3,4] => [4,2,5,-1,3] => ? = 1 - 1
[+,-,4,5,3] => [1,5,2,3,4] => [1,5,2,3,4] => [2,4,5,-1,3] => ? = 1 - 1
Description
The size of the connectivity set of a signed permutation. According to [1], the connectivity set of a signed permutation $w\in\mathfrak H_n$ is $n$ minus the number of generators appearing in any reduced word for $w$. The connectivity set can be defined for arbitrary Coxeter systems. For permutations, see [[St000234]]. For the number of connected elements in a Coxeter system see [[St001888]].
Matching statistic: St000908
(load all 2 compositions to match this statistic)
Mapping
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000908: Posets ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 50%
Values
[+,+] => [1,2] => [2,1] => ([(0,1)],2)
 => 1
[-,+] => [1,2] => [2,1] => ([(0,1)],2)
 => 1
[+,-] => [1,2] => [2,1] => ([(0,1)],2)
 => 1
[-,-] => [1,2] => [2,1] => ([(0,1)],2)
 => 1
[2,1] => [2,1] => [1,2] => ([(0,1)],2)
 => 1
[+,+,+] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[-,+,+] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[+,-,+] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[+,+,-] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[-,-,+] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[-,+,-] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[+,-,-] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[-,-,-] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[+,3,2] => [1,3,2] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[-,3,2] => [1,3,2] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,1,+] => [2,1,3] => [3,2,1] => ([(0,2),(2,1)],3)
 => 1
[2,1,-] => [2,1,3] => [3,2,1] => ([(0,2),(2,1)],3)
 => 1
[2,3,1] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1
[3,1,2] => [3,1,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[3,+,1] => [3,2,1] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[3,-,1] => [3,2,1] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[+,+,+,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[-,+,+,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[+,-,+,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[+,+,-,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[+,+,+,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[-,-,+,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[-,+,-,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[-,+,+,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[+,-,-,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[+,-,+,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[+,+,-,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[-,-,-,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[-,-,+,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[-,+,-,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[+,-,-,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[-,-,-,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[+,+,4,3] => [1,2,4,3] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1
[-,+,4,3] => [1,2,4,3] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1
[+,-,4,3] => [1,2,4,3] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1
[-,-,4,3] => [1,2,4,3] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1
[+,3,2,+] => [1,3,2,4] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1
[-,3,2,+] => [1,3,2,4] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1
[+,3,2,-] => [1,3,2,4] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1
[-,3,2,-] => [1,3,2,4] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1
[+,3,4,2] => [1,3,4,2] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[-,3,4,2] => [1,3,4,2] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[+,4,2,3] => [1,4,2,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1
[-,4,2,3] => [1,4,2,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1
[+,4,+,2] => [1,4,3,2] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 1
[-,4,+,2] => [1,4,3,2] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 1
[+,4,-,2] => [1,4,3,2] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 1
[4,1,+,2] => [4,1,3,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1
[4,1,-,2] => [4,1,3,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1
[+,+,+,+,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[-,+,+,+,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[+,-,+,+,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[+,+,-,+,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[+,+,+,-,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[+,+,+,+,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[-,-,+,+,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[-,+,-,+,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[-,+,+,-,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[-,+,+,+,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[+,-,-,+,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[+,-,+,-,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[+,-,+,+,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[+,+,-,-,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[+,+,-,+,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[+,+,+,-,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[-,-,-,+,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[-,-,+,-,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[-,-,+,+,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[-,+,-,-,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[-,+,-,+,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[-,+,+,-,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[+,-,-,-,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[+,-,-,+,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[+,-,+,-,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[+,+,-,-,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[-,-,-,-,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[-,-,-,+,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[-,-,+,-,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[-,+,-,-,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[+,-,-,-,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[-,-,-,-,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[+,+,+,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1
[-,+,+,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1
[+,-,+,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1
[+,+,-,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1
[-,-,+,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1
[-,+,-,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[+,-,-,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1
[-,-,-,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1
[+,+,4,3,+] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 1
[-,+,4,3,+] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 1
[+,-,4,3,+] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 2
[+,+,4,3,-] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 1
[-,-,4,3,+] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 1
[-,+,4,3,-] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 1
Description
The length of the shortest maximal antichain in a poset.
Matching statistic: St000914
(load all 2 compositions to match this statistic)
Mapping
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000914: Posets ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 50%
Values
[+,+] => [1,2] => [2,1] => ([(0,1)],2)
 => 1
[-,+] => [1,2] => [2,1] => ([(0,1)],2)
 => 1
[+,-] => [1,2] => [2,1] => ([(0,1)],2)
 => 1
[-,-] => [1,2] => [2,1] => ([(0,1)],2)
 => 1
[2,1] => [2,1] => [1,2] => ([(0,1)],2)
 => 1
[+,+,+] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[-,+,+] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[+,-,+] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[+,+,-] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[-,-,+] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[-,+,-] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[+,-,-] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[-,-,-] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[+,3,2] => [1,3,2] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[-,3,2] => [1,3,2] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,1,+] => [2,1,3] => [3,2,1] => ([(0,2),(2,1)],3)
 => 1
[2,1,-] => [2,1,3] => [3,2,1] => ([(0,2),(2,1)],3)
 => 1
[2,3,1] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1
[3,1,2] => [3,1,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[3,+,1] => [3,2,1] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[3,-,1] => [3,2,1] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[+,+,+,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[-,+,+,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[+,-,+,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[+,+,-,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[+,+,+,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[-,-,+,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[-,+,-,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[-,+,+,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[+,-,-,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[+,-,+,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[+,+,-,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[-,-,-,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[-,-,+,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[-,+,-,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[+,-,-,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[-,-,-,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[+,+,4,3] => [1,2,4,3] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1
[-,+,4,3] => [1,2,4,3] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1
[+,-,4,3] => [1,2,4,3] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1
[-,-,4,3] => [1,2,4,3] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1
[+,3,2,+] => [1,3,2,4] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1
[-,3,2,+] => [1,3,2,4] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1
[+,3,2,-] => [1,3,2,4] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1
[-,3,2,-] => [1,3,2,4] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1
[+,3,4,2] => [1,3,4,2] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[-,3,4,2] => [1,3,4,2] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[+,4,2,3] => [1,4,2,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1
[-,4,2,3] => [1,4,2,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1
[+,4,+,2] => [1,4,3,2] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 1
[-,4,+,2] => [1,4,3,2] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 1
[+,4,-,2] => [1,4,3,2] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 1
[4,1,+,2] => [4,1,3,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1
[4,1,-,2] => [4,1,3,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1
[+,+,+,+,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[-,+,+,+,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[+,-,+,+,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[+,+,-,+,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[+,+,+,-,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[+,+,+,+,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[-,-,+,+,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[-,+,-,+,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[-,+,+,-,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[-,+,+,+,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[+,-,-,+,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[+,-,+,-,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[+,-,+,+,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[+,+,-,-,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[+,+,-,+,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[+,+,+,-,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[-,-,-,+,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[-,-,+,-,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[-,-,+,+,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[-,+,-,-,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[-,+,-,+,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[-,+,+,-,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[+,-,-,-,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[+,-,-,+,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[+,-,+,-,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[+,+,-,-,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[-,-,-,-,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[-,-,-,+,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[-,-,+,-,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[-,+,-,-,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[+,-,-,-,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[-,-,-,-,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[+,+,+,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1
[-,+,+,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1
[+,-,+,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1
[+,+,-,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1
[-,-,+,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1
[-,+,-,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[+,-,-,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1
[-,-,-,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1
[+,+,4,3,+] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 1
[-,+,4,3,+] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 1
[+,-,4,3,+] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 2
[+,+,4,3,-] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 1
[-,-,4,3,+] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 1
[-,+,4,3,-] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 1
Description
The sum of the values of the Möbius function of a poset. The Möbius function $\mu$ of a finite poset is defined as $$\mu (x,y)=\begin{cases} 1& \text{if }x = y\\ -\sum _{z: x\leq z < y}\mu (x,z)& \text{for }x < y\\ 0&\text{otherwise}. \end{cases} $$ Since $\mu(x,y)=0$ whenever $x\not\leq y$, this statistic is $$ \sum_{x\leq y} \mu(x,y). $$ If the poset has a minimal or a maximal element, then the definition implies immediately that the statistic equals $1$. Moreover, the statistic equals the sum of the statistics of the connected components. This statistic is also called the magnitude of a poset.
Matching statistic: St001301
(load all 2 compositions to match this statistic)
Mapping
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001301: Posets ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 50%
Values
[+,+] => [1,2] => [2,1] => ([(0,1)],2)
 => 0 = 1 - 1
[-,+] => [1,2] => [2,1] => ([(0,1)],2)
 => 0 = 1 - 1
[+,-] => [1,2] => [2,1] => ([(0,1)],2)
 => 0 = 1 - 1
[-,-] => [1,2] => [2,1] => ([(0,1)],2)
 => 0 = 1 - 1
[2,1] => [2,1] => [1,2] => ([(0,1)],2)
 => 0 = 1 - 1
[+,+,+] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[-,+,+] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[+,-,+] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[+,+,-] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[-,-,+] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[-,+,-] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[+,-,-] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[-,-,-] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[+,3,2] => [1,3,2] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[-,3,2] => [1,3,2] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[2,1,+] => [2,1,3] => [3,2,1] => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[2,1,-] => [2,1,3] => [3,2,1] => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[2,3,1] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[3,1,2] => [3,1,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[3,+,1] => [3,2,1] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[3,-,1] => [3,2,1] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[+,+,+,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 0 = 1 - 1
[-,+,+,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 0 = 1 - 1
[+,-,+,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 0 = 1 - 1
[+,+,-,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 0 = 1 - 1
[+,+,+,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 0 = 1 - 1
[-,-,+,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 0 = 1 - 1
[-,+,-,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 0 = 1 - 1
[-,+,+,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 0 = 1 - 1
[+,-,-,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 0 = 1 - 1
[+,-,+,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 0 = 1 - 1
[+,+,-,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 0 = 1 - 1
[-,-,-,+] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 0 = 1 - 1
[-,-,+,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 0 = 1 - 1
[-,+,-,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 0 = 1 - 1
[+,-,-,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 0 = 1 - 1
[-,-,-,-] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 0 = 1 - 1
[+,+,4,3] => [1,2,4,3] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 0 = 1 - 1
[-,+,4,3] => [1,2,4,3] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 0 = 1 - 1
[+,-,4,3] => [1,2,4,3] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 0 = 1 - 1
[-,-,4,3] => [1,2,4,3] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 0 = 1 - 1
[+,3,2,+] => [1,3,2,4] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 0 = 1 - 1
[-,3,2,+] => [1,3,2,4] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 0 = 1 - 1
[+,3,2,-] => [1,3,2,4] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 0 = 1 - 1
[-,3,2,-] => [1,3,2,4] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 0 = 1 - 1
[+,3,4,2] => [1,3,4,2] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 0 = 1 - 1
[-,3,4,2] => [1,3,4,2] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 0 = 1 - 1
[+,4,2,3] => [1,4,2,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1 - 1
[-,4,2,3] => [1,4,2,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1 - 1
[+,4,+,2] => [1,4,3,2] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 0 = 1 - 1
[-,4,+,2] => [1,4,3,2] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 0 = 1 - 1
[+,4,-,2] => [1,4,3,2] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 0 = 1 - 1
[4,1,+,2] => [4,1,3,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1 - 1
[4,1,-,2] => [4,1,3,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1 - 1
[+,+,+,+,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[-,+,+,+,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[+,-,+,+,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[+,+,-,+,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[+,+,+,-,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[+,+,+,+,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[-,-,+,+,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[-,+,-,+,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[-,+,+,-,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[-,+,+,+,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[+,-,-,+,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[+,-,+,-,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2 - 1
[+,-,+,+,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[+,+,-,-,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[+,+,-,+,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[+,+,+,-,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[-,-,-,+,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[-,-,+,-,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[-,-,+,+,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[-,+,-,-,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[-,+,-,+,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2 - 1
[-,+,+,-,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[+,-,-,-,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[+,-,-,+,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[+,-,+,-,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[+,+,-,-,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[-,-,-,-,+] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[-,-,-,+,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[-,-,+,-,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[-,+,-,-,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[+,-,-,-,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[-,-,-,-,-] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[+,+,+,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1 - 1
[-,+,+,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1 - 1
[+,-,+,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1 - 1
[+,+,-,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1 - 1
[-,-,+,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1 - 1
[-,+,-,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2 - 1
[+,-,-,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1 - 1
[-,-,-,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1 - 1
[+,+,4,3,+] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 1 - 1
[-,+,4,3,+] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 1 - 1
[+,-,4,3,+] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 2 - 1
[+,+,4,3,-] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 1 - 1
[-,-,4,3,+] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 1 - 1
[-,+,4,3,-] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 1 - 1
Description
The first Betti number of the order complex associated with the poset. The order complex of a poset is the simplicial complex whose faces are the chains of the poset. This statistic is the rank of the first homology group of the order complex.
The following 21 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001768The number of reduced words of a signed permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001927Sparre Andersen's number of positives of a signed permutation. St001964The interval resolution global dimension of a poset. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001857The number of edges in the reduced word graph of a signed permutation. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000084The number of subtrees. St000328The maximum number of child nodes in a tree. St001926Sparre Andersen's position of the maximum of a signed permutation.