Your data matches 41 different statistics following compositions of up to 3 maps.
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Matching statistic: St000650
(load all 3 compositions to match this statistic)
Mapping
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000650: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => [2,1] => 0
[2,1] => [2,1] => [2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [3,1,2] => 0
[2,1,3] => [2,1,3] => [2,1,3] => [2,3,1] => 0
[2,3,1] => [3,1,2] => [3,2,1] => [1,2,3] => 0
[3,1,2] => [2,3,1] => [3,1,2] => [1,3,2] => 0
[3,2,1] => [3,2,1] => [2,3,1] => [2,1,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 0
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 0
[1,3,4,2] => [1,4,2,3] => [1,4,3,2] => [4,1,2,3] => 0
[1,4,2,3] => [1,3,4,2] => [1,4,2,3] => [4,1,3,2] => 0
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => [4,2,1,3] => 0
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 0
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 0
[2,3,1,4] => [3,1,2,4] => [3,2,1,4] => [2,3,4,1] => 0
[2,3,4,1] => [4,1,2,3] => [4,3,2,1] => [1,2,3,4] => 0
[2,4,1,3] => [3,1,4,2] => [4,2,1,3] => [1,3,4,2] => 0
[2,4,3,1] => [4,1,3,2] => [3,4,2,1] => [2,1,3,4] => 0
[3,1,2,4] => [2,3,1,4] => [3,1,2,4] => [2,4,3,1] => 0
[3,1,4,2] => [2,4,1,3] => [4,3,1,2] => [1,2,4,3] => 0
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => [3,2,4,1] => 0
[3,2,4,1] => [4,2,1,3] => [2,4,3,1] => [3,1,2,4] => 0
[3,4,1,2] => [3,4,1,2] => [3,1,4,2] => [2,4,1,3] => 0
[3,4,2,1] => [4,3,1,2] => [4,2,3,1] => [1,3,2,4] => 0
[4,1,2,3] => [2,3,4,1] => [4,1,2,3] => [1,4,3,2] => 1
[4,1,3,2] => [2,4,3,1] => [3,4,1,2] => [2,1,4,3] => 1
[4,2,1,3] => [3,2,4,1] => [2,4,1,3] => [3,1,4,2] => 1
[4,2,3,1] => [4,2,3,1] => [2,3,4,1] => [3,2,1,4] => 1
[4,3,1,2] => [3,4,2,1] => [4,1,3,2] => [1,4,2,3] => 1
[4,3,2,1] => [4,3,2,1] => [3,2,4,1] => [2,3,1,4] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 0
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => 0
[1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,3,4] => [5,4,1,3,2] => 0
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => [5,4,2,1,3] => 0
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 0
[1,3,4,2,5] => [1,4,2,3,5] => [1,4,3,2,5] => [5,2,3,4,1] => 0
[1,3,4,5,2] => [1,5,2,3,4] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,3,5,2,4] => [1,4,2,5,3] => [1,5,3,2,4] => [5,1,3,4,2] => 0
[1,3,5,4,2] => [1,5,2,4,3] => [1,4,5,3,2] => [5,2,1,3,4] => 0
[1,4,2,3,5] => [1,3,4,2,5] => [1,4,2,3,5] => [5,2,4,3,1] => 0
[1,4,2,5,3] => [1,3,5,2,4] => [1,5,4,2,3] => [5,1,2,4,3] => 0
[1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => 0
[1,4,3,5,2] => [1,5,3,2,4] => [1,3,5,4,2] => [5,3,1,2,4] => 0
[1,4,5,2,3] => [1,4,5,2,3] => [1,4,2,5,3] => [5,2,4,1,3] => 0
[1,4,5,3,2] => [1,5,4,2,3] => [1,5,3,4,2] => [5,1,3,2,4] => 0
Description
The number of 3-rises of a permutation. A 3-rise of a permutation $\pi$ is an index $i$ such that $\pi(i)+3 = \pi(i+1)$. For 1-rises, or successions, see [[St000441]], for 2-rises see [[St000534]].
Matching statistic: St000649
(load all 7 compositions to match this statistic)
Mapping
Mp00066: Permutations —inverse⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000649: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [3,1,2] => [2,3,1] => 0
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[2,3,1] => [3,1,2] => [1,3,2] => [1,3,2] => 0
[3,1,2] => [2,3,1] => [3,2,1] => [3,2,1] => 0
[3,2,1] => [3,2,1] => [2,3,1] => [3,1,2] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => [2,3,1,4] => 0
[1,3,4,2] => [1,4,2,3] => [4,1,2,3] => [2,3,4,1] => 0
[1,4,2,3] => [1,3,4,2] => [3,1,4,2] => [2,4,1,3] => 0
[1,4,3,2] => [1,4,3,2] => [4,1,3,2] => [2,4,3,1] => 0
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,3,1,4] => [3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[2,3,4,1] => [4,1,2,3] => [1,4,2,3] => [1,3,4,2] => 0
[2,4,1,3] => [3,1,4,2] => [1,3,4,2] => [1,4,2,3] => 0
[2,4,3,1] => [4,1,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[3,1,2,4] => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,1,4,2] => [2,4,1,3] => [4,2,1,3] => [3,2,4,1] => 0
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => [3,1,2,4] => 0
[3,2,4,1] => [4,2,1,3] => [2,4,1,3] => [3,1,4,2] => 0
[3,4,1,2] => [3,4,1,2] => [4,3,1,2] => [3,4,2,1] => 0
[3,4,2,1] => [4,3,1,2] => [3,4,1,2] => [3,4,1,2] => 0
[4,1,2,3] => [2,3,4,1] => [3,2,4,1] => [4,2,1,3] => 1
[4,1,3,2] => [2,4,3,1] => [4,2,3,1] => [4,2,3,1] => 1
[4,2,1,3] => [3,2,4,1] => [2,3,4,1] => [4,1,2,3] => 1
[4,2,3,1] => [4,2,3,1] => [2,4,3,1] => [4,1,3,2] => 1
[4,3,1,2] => [3,4,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[4,3,2,1] => [4,3,2,1] => [3,4,2,1] => [4,3,1,2] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,5,3,4] => [1,2,4,5,3] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,5,3,4] => 0
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => [2,3,1,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [3,1,2,5,4] => [2,3,1,5,4] => 0
[1,3,4,2,5] => [1,4,2,3,5] => [4,1,2,3,5] => [2,3,4,1,5] => 0
[1,3,4,5,2] => [1,5,2,3,4] => [5,1,2,3,4] => [2,3,4,5,1] => 0
[1,3,5,2,4] => [1,4,2,5,3] => [4,1,2,5,3] => [2,3,5,1,4] => 0
[1,3,5,4,2] => [1,5,2,4,3] => [5,1,2,4,3] => [2,3,5,4,1] => 0
[1,4,2,3,5] => [1,3,4,2,5] => [3,1,4,2,5] => [2,4,1,3,5] => 0
[1,4,2,5,3] => [1,3,5,2,4] => [3,1,5,2,4] => [2,4,1,5,3] => 0
[1,4,3,2,5] => [1,4,3,2,5] => [4,1,3,2,5] => [2,4,3,1,5] => 0
[1,4,3,5,2] => [1,5,3,2,4] => [5,1,3,2,4] => [2,4,3,5,1] => 0
[1,4,5,2,3] => [1,4,5,2,3] => [4,1,5,2,3] => [2,4,5,1,3] => 0
[1,4,5,3,2] => [1,5,4,2,3] => [5,1,4,2,3] => [2,4,5,3,1] => 0
[2,4,6,5,7,1,3] => [6,1,7,2,4,3,5] => [1,6,7,2,4,3,5] => [1,4,6,5,7,2,3] => ? = 1
[2,5,1,3,6,7,4] => [3,1,4,7,2,5,6] => [1,3,4,7,2,5,6] => [1,5,2,3,6,7,4] => ? = 1
[2,5,1,3,7,4,6] => [3,1,4,6,2,7,5] => [1,3,4,6,2,7,5] => [1,5,2,3,7,4,6] => ? = 1
[2,5,1,6,4,7,3] => [3,1,7,5,2,4,6] => [1,3,7,5,2,4,6] => [1,5,2,6,4,7,3] => ? = 1
[2,5,1,6,7,3,4] => [3,1,6,7,2,4,5] => [1,3,6,7,2,4,5] => [1,5,2,6,7,3,4] => ? = 1
[2,5,1,7,4,3,6] => [3,1,6,5,2,7,4] => [1,3,6,5,2,7,4] => [1,5,2,7,4,3,6] => ? = 2
[2,5,1,7,6,4,3] => [3,1,7,6,2,5,4] => [1,3,7,6,2,5,4] => [1,5,2,7,6,4,3] => ? = 2
[2,5,4,1,6,7,3] => [4,1,7,3,2,5,6] => [1,4,7,3,2,5,6] => [1,5,4,2,6,7,3] => ? = 1
[2,5,4,1,7,3,6] => [4,1,6,3,2,7,5] => [1,4,6,3,2,7,5] => [1,5,4,2,7,3,6] => ? = 1
[2,5,4,6,3,7,1] => [7,1,5,3,2,4,6] => [1,7,5,3,2,4,6] => [1,5,4,6,3,7,2] => ? = 1
[2,5,4,6,7,1,3] => [6,1,7,3,2,4,5] => [1,6,7,3,2,4,5] => [1,5,4,6,7,2,3] => ? = 1
[2,5,4,7,3,1,6] => [6,1,5,3,2,7,4] => [1,6,5,3,2,7,4] => [1,5,4,7,3,2,6] => ? = 2
[2,5,4,7,6,3,1] => [7,1,6,3,2,5,4] => [1,7,6,3,2,5,4] => [1,5,4,7,6,3,2] => ? = 2
[2,5,6,1,3,7,4] => [4,1,5,7,2,3,6] => [1,4,5,7,2,3,6] => [1,5,6,2,3,7,4] => ? = 2
[2,5,6,1,7,4,3] => [4,1,7,6,2,3,5] => [1,4,7,6,2,3,5] => [1,5,6,2,7,4,3] => ? = 2
[2,5,6,3,4,7,1] => [7,1,4,5,2,3,6] => [1,7,4,5,2,3,6] => [1,5,6,3,4,7,2] => ? = 2
[2,5,6,3,7,1,4] => [6,1,4,7,2,3,5] => [1,6,4,7,2,3,5] => [1,5,6,3,7,2,4] => ? = 2
[2,5,6,7,3,4,1] => [7,1,5,6,2,3,4] => [1,7,5,6,2,3,4] => [1,5,6,7,3,4,2] => ? = 3
[2,5,6,7,4,1,3] => [6,1,7,5,2,3,4] => [1,6,7,5,2,3,4] => [1,5,6,7,4,2,3] => ? = 3
[2,6,1,3,4,7,5] => [3,1,4,5,7,2,6] => [1,3,4,5,7,2,6] => [1,6,2,3,4,7,5] => ? = 0
[2,6,1,3,7,5,4] => [3,1,4,7,6,2,5] => [1,3,4,7,6,2,5] => [1,6,2,3,7,5,4] => ? = 0
[2,6,1,5,3,7,4] => [3,1,5,7,4,2,6] => [1,3,5,7,4,2,6] => [1,6,2,5,3,7,4] => ? = 0
[2,6,1,5,7,4,3] => [3,1,7,6,4,2,5] => [1,3,7,6,4,2,5] => [1,6,2,5,7,4,3] => ? = 0
[2,6,1,7,3,4,5] => [3,1,5,6,7,2,4] => [1,3,5,6,7,2,4] => [1,6,2,7,3,4,5] => ? = 1
[2,6,1,7,4,5,3] => [3,1,7,5,6,2,4] => [1,3,7,5,6,2,4] => [1,6,2,7,4,5,3] => ? = 1
[2,6,4,1,3,7,5] => [4,1,5,3,7,2,6] => [1,4,5,3,7,2,6] => [1,6,4,2,3,7,5] => ? = 0
[2,6,4,1,7,5,3] => [4,1,7,3,6,2,5] => [1,4,7,3,6,2,5] => [1,6,4,2,7,5,3] => ? = 0
[2,6,4,5,1,7,3] => [5,1,7,3,4,2,6] => [1,5,7,3,4,2,6] => [1,6,4,5,2,7,3] => ? = 0
[2,6,4,5,7,3,1] => [7,1,6,3,4,2,5] => [1,7,6,3,4,2,5] => [1,6,4,5,7,3,2] => ? = 0
[2,6,4,7,1,3,5] => [5,1,6,3,7,2,4] => [1,5,6,3,7,2,4] => [1,6,4,7,2,3,5] => ? = 1
[2,6,4,7,3,5,1] => [7,1,5,3,6,2,4] => [1,7,5,3,6,2,4] => [1,6,4,7,3,5,2] => ? = 1
[2,6,5,1,4,7,3] => [4,1,7,5,3,2,6] => [1,4,7,5,3,2,6] => [1,6,5,2,4,7,3] => ? = 0
[2,6,5,1,7,3,4] => [4,1,6,7,3,2,5] => [1,4,6,7,3,2,5] => [1,6,5,2,7,3,4] => ? = 0
[2,6,5,3,1,7,4] => [5,1,4,7,3,2,6] => [1,5,4,7,3,2,6] => [1,6,5,3,2,7,4] => ? = 0
[2,6,5,3,7,4,1] => [7,1,4,6,3,2,5] => [1,7,4,6,3,2,5] => [1,6,5,3,7,4,2] => ? = 0
[2,6,5,7,1,4,3] => [5,1,7,6,3,2,4] => [1,5,7,6,3,2,4] => [1,6,5,7,2,4,3] => ? = 1
[2,6,5,7,4,3,1] => [7,1,6,5,3,2,4] => [1,7,6,5,3,2,4] => [1,6,5,7,4,3,2] => ? = 1
[3,1,4,5,6,7,2] => [2,7,1,3,4,5,6] => [7,2,1,3,4,5,6] => [3,2,4,5,6,7,1] => ? = 0
[3,1,4,5,7,2,6] => [2,6,1,3,4,7,5] => [6,2,1,3,4,7,5] => [3,2,4,5,7,1,6] => ? = 0
[3,1,4,6,2,7,5] => [2,5,1,3,7,4,6] => [5,2,1,3,7,4,6] => [3,2,4,6,1,7,5] => ? = 0
[3,1,4,6,7,5,2] => [2,7,1,3,6,4,5] => [7,2,1,3,6,4,5] => [3,2,4,6,7,5,1] => ? = 0
[3,1,4,7,2,5,6] => [2,5,1,3,6,7,4] => [5,2,1,3,6,7,4] => [3,2,4,7,1,5,6] => ? = 1
[3,1,4,7,6,2,5] => [2,6,1,3,7,5,4] => [6,2,1,3,7,5,4] => [3,2,4,7,6,1,5] => ? = 1
[3,1,5,2,6,7,4] => [2,4,1,7,3,5,6] => [4,2,1,7,3,5,6] => [3,2,5,1,6,7,4] => ? = 0
[3,1,5,2,7,4,6] => [2,4,1,6,3,7,5] => [4,2,1,6,3,7,5] => [3,2,5,1,7,4,6] => ? = 0
[3,1,5,6,4,7,2] => [2,7,1,5,3,4,6] => [7,2,1,5,3,4,6] => [3,2,5,6,4,7,1] => ? = 0
[3,1,5,6,7,2,4] => [2,6,1,7,3,4,5] => [6,2,1,7,3,4,5] => [3,2,5,6,7,1,4] => ? = 0
[3,1,5,7,4,2,6] => [2,6,1,5,3,7,4] => [6,2,1,5,3,7,4] => [3,2,5,7,4,1,6] => ? = 1
[3,1,5,7,6,4,2] => [2,7,1,6,3,5,4] => [7,2,1,6,3,5,4] => [3,2,5,7,6,4,1] => ? = 1
[3,1,6,2,4,7,5] => [2,4,1,5,7,3,6] => [4,2,1,5,7,3,6] => [3,2,6,1,4,7,5] => ? = 1
Description
The number of 3-excedences of a permutation. This is the number of positions $1\leq i\leq n$ such that $\sigma(i)=i+3$.
Matching statistic: St001719
(load all 5 compositions to match this statistic)
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 25%
Values
[1,2] => [1,0,1,0]
 => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,1] => [1,1,0,0]
 => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,2,3] => [1,0,1,0,1,0]
 => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 1 = 0 + 1
[1,3,2] => [1,0,1,1,0,0]
 => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 1 = 0 + 1
[2,1,3] => [1,1,0,0,1,0]
 => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 1 = 0 + 1
[2,3,1] => [1,1,0,1,0,0]
 => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 1 = 0 + 1
[3,1,2] => [1,1,1,0,0,0]
 => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => 1 = 0 + 1
[3,2,1] => [1,1,1,0,0,0]
 => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => 1 = 0 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => 1 = 0 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => 1 = 0 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => 1 = 0 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => 1 = 0 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => 1 = 0 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => 1 = 0 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => 1 = 0 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
 => 1 = 0 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => 1 = 0 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => 1 = 0 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => 1 = 0 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => 1 = 0 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => 1 = 0 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => 1 = 0 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => 1 = 0 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => 1 = 0 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 1 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 1 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 1 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 1 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 1 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 1 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
 => ? = 0 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => 1 = 0 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ? = 0 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1 = 0 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1 = 0 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1 = 0 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ? = 0 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1 = 0 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
 => ? = 0 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ? = 0 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1 = 0 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1 = 0 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
 => ? = 0 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ? = 0 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
 => ? = 0 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ? = 0 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 0 + 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 0 + 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ? = 1 + 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ? = 1 + 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ? = 1 + 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ? = 1 + 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ? = 1 + 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ? = 1 + 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => 1 = 0 + 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
 => ? = 0 + 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1 = 0 + 1
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1 = 0 + 1
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
 => ? = 0 + 1
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
 => ? = 0 + 1
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1 = 0 + 1
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1 = 0 + 1
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ? = 0 + 1
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
 => ? = 0 + 1
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => 1 = 0 + 1
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => 1 = 0 + 1
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ? = 0 + 1
[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
 => [1,3,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 0 + 1
[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ? = 0 + 1
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
 => [1,3,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 0 + 1
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
 => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
 => ? = 0 + 1
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
 => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
 => ? = 0 + 1
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 1
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 1
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 1
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 1
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 1
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 1
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1 = 0 + 1
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
 => ? = 0 + 1
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1 = 0 + 1
[3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
 => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => 1 = 0 + 1
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
 => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
 => ? = 0 + 1
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
 => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
 => ? = 0 + 1
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1 = 0 + 1
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
 => ? = 0 + 1
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1 = 0 + 1
[3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => 1 = 0 + 1
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
 => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
 => ? = 0 + 1
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
 => ? = 0 + 1
[3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 0 + 1
[3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
 => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
 => ? = 0 + 1
[3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 0 + 1
[3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
 => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
 => ? = 0 + 1
[1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 1 = 0 + 1
[1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [2,4,6,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 1 = 0 + 1
[1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [2,4,6,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 1 = 0 + 1
[1,2,4,5,6,3] => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [2,4,1,5,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => 1 = 0 + 1
Description
The number of shortest chains of small intervals from the bottom to the top in a lattice. An interval $[a, b]$ in a lattice is small if $b$ is a join of elements covering $a$.
Matching statistic: St001720
(load all 4 compositions to match this statistic)
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001720: Lattices ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 25%
Values
[1,2] => [1,0,1,0]
 => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 0 + 2
[2,1] => [1,1,0,0]
 => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 0 + 2
[1,2,3] => [1,0,1,0,1,0]
 => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,3,2] => [1,0,1,1,0,0]
 => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[2,1,3] => [1,1,0,0,1,0]
 => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[2,3,1] => [1,1,0,1,0,0]
 => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[3,1,2] => [1,1,1,0,0,0]
 => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => 2 = 0 + 2
[3,2,1] => [1,1,1,0,0,0]
 => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => 2 = 0 + 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => 2 = 0 + 2
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => 2 = 0 + 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => 2 = 0 + 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => 2 = 0 + 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => 2 = 0 + 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => 2 = 0 + 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => 2 = 0 + 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
 => 2 = 0 + 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => 2 = 0 + 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => 2 = 0 + 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => 2 = 0 + 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => 2 = 0 + 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => 2 = 0 + 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => 2 = 0 + 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => 2 = 0 + 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => 2 = 0 + 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 1 + 2
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 1 + 2
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 1 + 2
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 1 + 2
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 1 + 2
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 1 + 2
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
 => ? = 0 + 2
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => 2 = 0 + 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ? = 0 + 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 2 = 0 + 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 2 = 0 + 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ? = 0 + 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 2 = 0 + 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
 => ? = 0 + 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ? = 0 + 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 2 = 0 + 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 2 = 0 + 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
 => ? = 0 + 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ? = 0 + 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
 => ? = 0 + 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ? = 0 + 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 0 + 2
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 0 + 2
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ? = 1 + 2
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ? = 1 + 2
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ? = 1 + 2
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ? = 1 + 2
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ? = 1 + 2
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ? = 1 + 2
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => 2 = 0 + 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
 => ? = 0 + 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 2 = 0 + 2
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 2 = 0 + 2
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
 => ? = 0 + 2
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
 => ? = 0 + 2
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 2 = 0 + 2
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ? = 0 + 2
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
 => ? = 0 + 2
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => 2 = 0 + 2
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => 2 = 0 + 2
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ? = 0 + 2
[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
 => [1,3,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 0 + 2
[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ? = 0 + 2
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
 => [1,3,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 0 + 2
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
 => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
 => ? = 0 + 2
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
 => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
 => ? = 0 + 2
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 2
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 2
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 2
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 2
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 2
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 2
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 2 = 0 + 2
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
 => ? = 0 + 2
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 2 = 0 + 2
[3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
 => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => 2 = 0 + 2
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
 => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
 => ? = 0 + 2
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
 => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
 => ? = 0 + 2
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 2 = 0 + 2
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
 => ? = 0 + 2
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 2 = 0 + 2
[3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => 2 = 0 + 2
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
 => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
 => ? = 0 + 2
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
 => ? = 0 + 2
[3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 0 + 2
[3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
 => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
 => ? = 0 + 2
[3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 0 + 2
[3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
 => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
 => ? = 0 + 2
[1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 2 = 0 + 2
[1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [2,4,6,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 2 = 0 + 2
[1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [2,4,6,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 2 = 0 + 2
[1,2,4,5,6,3] => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [2,4,1,5,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => 2 = 0 + 2
Description
The minimal length of a chain of small intervals in a lattice. An interval $[a, b]$ is small if $b$ is a join of elements covering $a$.
Matching statistic: St001964
(load all 8 compositions to match this statistic)
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 25%
Values
[1,2] => [1,2] => ([(0,1)],2)
 => 0
[2,1] => [1,2] => ([(0,1)],2)
 => 0
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 0
[1,3,2] => [1,2,3] => ([(0,2),(2,1)],3)
 => 0
[2,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 0
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 0
[3,1,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[3,2,1] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,2,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,3,2,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,4,2,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0
[1,4,3,2] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0
[2,1,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 0
[2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 0
[2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 0
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 0
[2,4,1,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0
[2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0
[3,1,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 0
[3,1,4,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 0
[3,2,1,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 0
[3,2,4,1] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 0
[3,4,1,2] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 0
[3,4,2,1] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 0
[4,1,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1
[4,1,3,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 1
[4,2,1,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1
[4,2,3,1] => [1,4,2,3] => ([(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] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 1
[4,3,2,1] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 1
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,2,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,2,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,2,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[1,2,5,4,3] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[1,3,2,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,3,4,5,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[1,3,5,4,2] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[1,4,2,3,5] => [1,2,4,3,5] => ([(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)
 => ? = 0
[1,4,2,5,3] => [1,2,4,5,3] => ([(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)
 => ? = 0
[1,4,3,2,5] => [1,2,4,3,5] => ([(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)
 => ? = 0
[1,4,3,5,2] => [1,2,4,5,3] => ([(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)
 => ? = 0
[1,4,5,2,3] => [1,2,4,3,5] => ([(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)
 => ? = 0
[1,4,5,3,2] => [1,2,4,3,5] => ([(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)
 => ? = 0
[1,5,2,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 1
[1,5,2,4,3] => [1,2,5,3,4] => ([(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
[1,5,3,2,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 1
[1,5,3,4,2] => [1,2,5,3,4] => ([(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
[1,5,4,2,3] => [1,2,5,3,4] => ([(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
[1,5,4,3,2] => [1,2,5,3,4] => ([(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
[2,1,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[2,1,5,3,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[2,1,5,4,3] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[2,3,5,1,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[2,3,5,4,1] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[2,4,1,3,5] => [1,2,4,3,5] => ([(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)
 => ? = 0
[2,4,1,5,3] => [1,2,4,5,3] => ([(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)
 => ? = 0
[2,4,3,1,5] => [1,2,4,3,5] => ([(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)
 => ? = 0
[2,4,3,5,1] => [1,2,4,5,3] => ([(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)
 => ? = 0
[2,4,5,1,3] => [1,2,4,3,5] => ([(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)
 => ? = 0
[2,4,5,3,1] => [1,2,4,3,5] => ([(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)
 => ? = 0
[2,5,1,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 1
[2,5,1,4,3] => [1,2,5,3,4] => ([(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
[2,5,3,1,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 1
[2,5,3,4,1] => [1,2,5,3,4] => ([(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
[2,5,4,1,3] => [1,2,5,3,4] => ([(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
[2,5,4,3,1] => [1,2,5,3,4] => ([(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
[3,1,2,4,5] => [1,3,2,4,5] => ([(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)
 => ? = 0
[3,1,2,5,4] => [1,3,2,4,5] => ([(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)
 => ? = 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
Description
The interval resolution global dimension of a poset. This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
Matching statistic: St000259
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 25%
Values
[1,2] => [1,2] => ([],2)
 => ([],1)
 => 0
[2,1] => [1,2] => ([],2)
 => ([],1)
 => 0
[1,2,3] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[1,3,2] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[2,1,3] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[2,3,1] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[3,1,2] => [1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 0
[3,2,1] => [1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 0
[1,2,3,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,2,4,3] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,3,2,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,3,4,2] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[2,1,3,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,1,4,3] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,3,1,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,3,4,1] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[1,5,4,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,1,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,5,3,1] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[2,5,1,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
Description
The diameter of a connected graph. This is the greatest distance between any pair of vertices.
Matching statistic: St000260
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 25%
Values
[1,2] => [1,2] => ([],2)
 => ([],1)
 => 0
[2,1] => [1,2] => ([],2)
 => ([],1)
 => 0
[1,2,3] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[1,3,2] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[2,1,3] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[2,3,1] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[3,1,2] => [1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 0
[3,2,1] => [1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 0
[1,2,3,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,2,4,3] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,3,2,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,3,4,2] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[2,1,3,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,1,4,3] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,3,1,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,3,4,1] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[1,5,4,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,1,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,5,3,1] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[2,5,1,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
Description
The radius of a connected graph. This is the minimum eccentricity of any vertex.
Matching statistic: St000302
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000302: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 25%
Values
[1,2] => [1,2] => ([],2)
 => ([],1)
 => 0
[2,1] => [1,2] => ([],2)
 => ([],1)
 => 0
[1,2,3] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[1,3,2] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[2,1,3] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[2,3,1] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[3,1,2] => [1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 0
[3,2,1] => [1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 0
[1,2,3,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,2,4,3] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,3,2,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,3,4,2] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[2,1,3,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,1,4,3] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,3,1,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,3,4,1] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[1,5,4,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,1,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,5,3,1] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[2,5,1,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
Description
The determinant of the distance matrix of a connected graph.
Matching statistic: St000466
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000466: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 25%
Values
[1,2] => [1,2] => ([],2)
 => ([],1)
 => 0
[2,1] => [1,2] => ([],2)
 => ([],1)
 => 0
[1,2,3] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[1,3,2] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[2,1,3] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[2,3,1] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[3,1,2] => [1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 0
[3,2,1] => [1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 0
[1,2,3,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,2,4,3] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,3,2,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,3,4,2] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[2,1,3,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,1,4,3] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,3,1,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,3,4,1] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[1,5,4,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,1,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,5,3,1] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[2,5,1,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
Description
The Gutman (or modified Schultz) index of a connected graph. This is $$\sum_{\{u,v\}\subseteq V} d(u)d(v)d(u,v)$$ where $d(u)$ is the degree of vertex $u$ and $d(u,v)$ is the distance between vertices $u$ and $v$. For trees on $n$ vertices, the modified Schultz index is related to the Wiener index via $S^\ast(T)=4W(T)-(n-1)(2n-1)$ [1].
Matching statistic: St000467
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000467: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 25%
Values
[1,2] => [1,2] => ([],2)
 => ([],1)
 => 0
[2,1] => [1,2] => ([],2)
 => ([],1)
 => 0
[1,2,3] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[1,3,2] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[2,1,3] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[2,3,1] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[3,1,2] => [1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 0
[3,2,1] => [1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 0
[1,2,3,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,2,4,3] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,3,2,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,3,4,2] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[2,1,3,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,1,4,3] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,3,1,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,3,4,1] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[1,5,4,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,1,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,5,3,1] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[2,5,1,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
Description
The hyper-Wiener index of a connected graph. This is $$ \sum_{\{u,v\}\subseteq V} d(u,v)+d(u,v)^2. $$
The following 31 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St001864The number of excedances of a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000914The sum of the values of the Möbius function of a poset. 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. St001845The number of join irreducibles minus the rank of a lattice. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001821The sorting index of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.