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Your data matches 56 different statistics following compositions of up to 3 maps.
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Matching statistic: St001377
St001377: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => -1
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 2
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 0
[1,4,3,2] => 2
[2,1,3,4] => 0
[2,1,4,3] => 2
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => -1
[2,4,3,1] => 1
[3,1,2,4] => -1
[3,1,4,2] => 1
[3,2,1,4] => 0
[3,2,4,1] => 0
[3,4,1,2] => -2
[3,4,2,1] => 0
[4,1,2,3] => -2
[4,1,3,2] => 0
[4,2,1,3] => -1
[4,2,3,1] => -1
[4,3,1,2] => -2
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 3
[1,2,4,3,5] => 2
[1,2,4,5,3] => 2
[1,2,5,3,4] => 1
[1,2,5,4,3] => 4
[1,3,2,4,5] => 1
[1,3,2,5,4] => 4
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 0
[1,3,5,4,2] => 3
[1,4,2,3,5] => 0
[1,4,2,5,3] => 3
[1,4,3,2,5] => 2
[1,4,3,5,2] => 2
[1,4,5,2,3] => -1
Description
The major index minus the number of inversions of a permutation.
This is, the difference between [[St000004]] and [[St000018]].
Matching statistic: St000145
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000145: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 75%●distinct values known / distinct values provided: 60%
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000145: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 75%●distinct values known / distinct values provided: 60%
Values
[1] => [1,0]
=> []
=> ? = 0
[1,2] => [1,0,1,0]
=> [1]
=> 0
[2,1] => [1,1,0,0]
=> []
=> ? = 0
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> -1
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 0
[3,1,2] => [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0}
[3,2,1] => [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> -1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 0
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> -1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> -2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> -2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 0
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> -1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> -1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> 0
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> 0
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {-2,0,0,0,0,2}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {-2,0,0,0,0,2}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {-2,0,0,0,0,2}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {-2,0,0,0,0,2}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {-2,0,0,0,0,2}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {-2,0,0,0,0,2}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> -1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 0
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> -1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> -2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> -2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 0
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> -1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 0
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> -1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> -2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> -2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 0
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> -1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 0
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> -1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> -2
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> -2
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> -3
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> -3
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> -3
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> -3
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> -3
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> -3
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 0
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-2,-2,-2,-1,-1,0,0,0,0,1,1,2,2,2,3,3,4,4,4}
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-2,-2,-2,-1,-1,0,0,0,0,1,1,2,2,2,3,3,4,4,4}
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-2,-2,-2,-1,-1,0,0,0,0,1,1,2,2,2,3,3,4,4,4}
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-2,-2,-2,-1,-1,0,0,0,0,1,1,2,2,2,3,3,4,4,4}
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-2,-2,-2,-1,-1,0,0,0,0,1,1,2,2,2,3,3,4,4,4}
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-2,-2,-2,-1,-1,0,0,0,0,1,1,2,2,2,3,3,4,4,4}
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-2,-2,-2,-1,-1,0,0,0,0,1,1,2,2,2,3,3,4,4,4}
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-2,-2,-2,-1,-1,0,0,0,0,1,1,2,2,2,3,3,4,4,4}
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-2,-2,-2,-1,-1,0,0,0,0,1,1,2,2,2,3,3,4,4,4}
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-2,-2,-2,-1,-1,0,0,0,0,1,1,2,2,2,3,3,4,4,4}
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-2,-2,-2,-1,-1,0,0,0,0,1,1,2,2,2,3,3,4,4,4}
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-2,-2,-2,-1,-1,0,0,0,0,1,1,2,2,2,3,3,4,4,4}
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-2,-2,-2,-1,-1,0,0,0,0,1,1,2,2,2,3,3,4,4,4}
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-2,-2,-2,-1,-1,0,0,0,0,1,1,2,2,2,3,3,4,4,4}
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-2,-2,-2,-1,-1,0,0,0,0,1,1,2,2,2,3,3,4,4,4}
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-2,-2,-2,-1,-1,0,0,0,0,1,1,2,2,2,3,3,4,4,4}
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-2,-2,-2,-1,-1,0,0,0,0,1,1,2,2,2,3,3,4,4,4}
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-2,-2,-2,-1,-1,0,0,0,0,1,1,2,2,2,3,3,4,4,4}
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-2,-2,-2,-1,-1,0,0,0,0,1,1,2,2,2,3,3,4,4,4}
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-2,-2,-2,-1,-1,0,0,0,0,1,1,2,2,2,3,3,4,4,4}
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-2,-2,-2,-1,-1,0,0,0,0,1,1,2,2,2,3,3,4,4,4}
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-2,-2,-2,-1,-1,0,0,0,0,1,1,2,2,2,3,3,4,4,4}
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-2,-2,-2,-1,-1,0,0,0,0,1,1,2,2,2,3,3,4,4,4}
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-2,-2,-2,-1,-1,0,0,0,0,1,1,2,2,2,3,3,4,4,4}
[1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> ? ∊ {-7,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> ? ∊ {-7,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> ? ∊ {-7,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> ? ∊ {-7,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> ? ∊ {-7,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> ? ∊ {-7,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> ? ∊ {-7,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> ? ∊ {-7,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,2,4,5,3,6] => [1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> ? ∊ {-7,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,2,4,5,6,3] => [1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> ? ∊ {-7,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,2,4,6,3,5] => [1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> ? ∊ {-7,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,2,4,6,5,3] => [1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> ? ∊ {-7,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> ? ∊ {-7,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,2,5,3,6,4] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> ? ∊ {-7,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> ? ∊ {-7,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,2,5,4,6,3] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> ? ∊ {-7,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-5,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
Description
The Dyson rank of a partition.
This rank is defined as the largest part minus the number of parts. It was introduced by Dyson [1] in connection to Ramanujan's partition congruences $$p(5n+4) \equiv 0 \pmod 5$$ and $$p(7n+6) \equiv 0 \pmod 7.$$
Matching statistic: St000772
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 27% ●values known / values provided: 33%●distinct values known / distinct values provided: 27%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 27% ●values known / values provided: 33%●distinct values known / distinct values provided: 27%
Values
[1] => [1] => ([],1)
=> 1 = 0 + 1
[1,2] => [2] => ([],2)
=> ? ∊ {0,0} + 1
[2,1] => [2] => ([],2)
=> ? ∊ {0,0} + 1
[1,2,3] => [3] => ([],3)
=> ? ∊ {-1,0,0,0,1} + 1
[1,3,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {-1,0,0,0,1} + 1
[2,1,3] => [3] => ([],3)
=> ? ∊ {-1,0,0,0,1} + 1
[2,3,1] => [3] => ([],3)
=> ? ∊ {-1,0,0,0,1} + 1
[3,1,2] => [3] => ([],3)
=> ? ∊ {-1,0,0,0,1} + 1
[3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,2,3,4] => [4] => ([],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,2,2,2} + 1
[1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,2,2,2} + 1
[1,3,2,4] => [1,3] => ([(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,2,2,2} + 1
[1,3,4,2] => [1,3] => ([(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,2,2,2} + 1
[1,4,2,3] => [1,3] => ([(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,2,2,2} + 1
[1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,1,3,4] => [4] => ([],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,2,2,2} + 1
[2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,2,2,2} + 1
[2,3,1,4] => [4] => ([],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,2,2,2} + 1
[2,3,4,1] => [4] => ([],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,2,2,2} + 1
[2,4,1,3] => [4] => ([],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,2,2,2} + 1
[2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,2,4] => [4] => ([],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,2,2,2} + 1
[3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,2,2,2} + 1
[3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,2,2,2} + 1
[3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,2,2,2} + 1
[3,4,1,2] => [4] => ([],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,2,2,2} + 1
[3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,1,2,3] => [4] => ([],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,2,2,2} + 1
[4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,2,2,2} + 1
[4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,3,1,2] => [1,3] => ([(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,2,2,2} + 1
[4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,3,4,2,5] => [1,4] => ([(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,2,3,5] => [1,4] => ([(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,4,5,2,3] => [1,4] => ([(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,5,2,3,4] => [1,4] => ([(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,5,3,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,5,4,2,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,1,3,4,5] => [5] => ([],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4,4,4} + 1
[2,1,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4,4,4} + 1
[2,1,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4,4,4} + 1
[2,1,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4,4,4} + 1
[2,1,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4,4,4} + 1
[2,1,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,3,1,4,5] => [5] => ([],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4,4,4} + 1
[2,3,1,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4,4,4} + 1
[2,3,5,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,4,5,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,5,1,4,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,5,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,5,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[3,1,5,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[3,2,5,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[3,4,5,2,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,5,1,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,5,2,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,5,4,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[4,1,5,3,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[4,2,5,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[4,3,5,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[4,5,1,3,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,5,2,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,5,3,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[5,1,2,4,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[5,1,3,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[5,1,4,3,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[5,2,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[5,2,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[5,3,1,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[5,3,2,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[5,3,4,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[5,4,1,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[5,4,2,3,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[5,4,3,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,2,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,2,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,2,6,5,4,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $1$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$.
The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Matching statistic: St000771
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 33%
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 33%
Values
[1] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,2] => [1,2] => [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [2,1] => [1,2] => ([],2)
=> ? = 0 + 1
[1,2,3] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,3,2] => [1,3,2] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {-1,0,0,0} + 1
[2,1,3] => [2,1,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,3,1] => [3,2,1] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {-1,0,0,0} + 1
[3,1,2] => [3,2,1] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {-1,0,0,0} + 1
[3,2,1] => [3,2,1] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {-1,0,0,0} + 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,4,3] => [1,2,4,3] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,1,1} + 1
[1,3,2,4] => [1,3,2,4] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,3,4,2] => [1,4,3,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,1,1} + 1
[1,4,2,3] => [1,4,3,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,1,1} + 1
[1,4,3,2] => [1,4,3,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,1,1} + 1
[2,1,3,4] => [2,1,3,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,1,4,3] => [2,1,4,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,1,1} + 1
[2,3,1,4] => [3,2,1,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,3,4,1] => [4,2,3,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,1,1} + 1
[2,4,1,3] => [3,4,1,2] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,4,3,1] => [4,3,2,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,1,1} + 1
[3,1,2,4] => [3,2,1,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,1,4,2] => [4,2,3,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,1,1} + 1
[3,2,1,4] => [3,2,1,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,2,4,1] => [4,2,3,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,1,1} + 1
[3,4,1,2] => [4,3,2,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,1,1} + 1
[3,4,2,1] => [4,3,2,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,1,1} + 1
[4,1,2,3] => [4,2,3,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,1,1} + 1
[4,1,3,2] => [4,2,3,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,1,1} + 1
[4,2,1,3] => [4,3,2,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,1,1} + 1
[4,2,3,1] => [4,3,2,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,1,1} + 1
[4,3,1,2] => [4,3,2,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,1,1} + 1
[4,3,2,1] => [4,3,2,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,1,1} + 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4} + 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,4,5,3] => [1,2,5,4,3] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4} + 1
[1,2,5,3,4] => [1,2,5,4,3] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4} + 1
[1,2,5,4,3] => [1,2,5,4,3] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4} + 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4} + 1
[1,3,4,2,5] => [1,4,3,2,5] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,4,5,2] => [1,5,3,4,2] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4} + 1
[1,3,5,2,4] => [1,4,5,2,3] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[1,3,5,4,2] => [1,5,4,3,2] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4} + 1
[1,4,2,3,5] => [1,4,3,2,5] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,2,5,3] => [1,5,3,4,2] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4} + 1
[1,4,3,2,5] => [1,4,3,2,5] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,3,5,2] => [1,5,3,4,2] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4} + 1
[1,4,5,2,3] => [1,5,4,3,2] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4} + 1
[1,4,5,3,2] => [1,5,4,3,2] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4} + 1
[1,5,2,3,4] => [1,5,3,4,2] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4} + 1
[1,5,2,4,3] => [1,5,3,4,2] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4} + 1
[1,5,3,2,4] => [1,5,4,3,2] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4} + 1
[1,5,3,4,2] => [1,5,4,3,2] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4} + 1
[1,5,4,2,3] => [1,5,4,3,2] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4} + 1
[1,5,4,3,2] => [1,5,4,3,2] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4} + 1
[2,1,3,4,5] => [2,1,3,4,5] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4} + 1
[2,1,4,3,5] => [2,1,4,3,5] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,1,4,5,3] => [2,1,5,4,3] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4} + 1
[2,1,5,3,4] => [2,1,5,4,3] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4} + 1
[2,1,5,4,3] => [2,1,5,4,3] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4} + 1
[2,3,1,4,5] => [3,2,1,4,5] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,3,1,5,4] => [3,2,1,5,4] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4} + 1
[2,3,4,1,5] => [4,2,3,1,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,3,4,5,1] => [5,2,3,4,1] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4} + 1
[2,3,5,1,4] => [4,2,5,1,3] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,3,5,4,1] => [5,2,4,3,1] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4} + 1
[2,4,1,3,5] => [3,4,1,2,5] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,4,1,5,3] => [3,5,1,4,2] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[2,4,3,1,5] => [4,3,2,1,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[2,4,3,5,1] => [5,3,2,4,1] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4} + 1
[2,4,5,1,3] => [4,5,3,1,2] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,4,5,3,1] => [5,4,3,2,1] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4} + 1
[2,5,1,3,4] => [3,5,1,4,2] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[2,5,1,4,3] => [3,5,1,4,2] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[2,5,3,1,4] => [4,5,3,1,2] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,5,3,4,1] => [5,4,3,2,1] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4} + 1
[2,5,4,1,3] => [4,5,3,1,2] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,5,4,3,1] => [5,4,3,2,1] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4} + 1
[3,1,2,4,5] => [3,2,1,4,5] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,1,4,2,5] => [4,2,3,1,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,1,5,2,4] => [4,2,5,1,3] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[3,2,1,4,5] => [3,2,1,4,5] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,2,4,1,5] => [4,2,3,1,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,2,5,1,4] => [4,2,5,1,3] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[3,4,1,2,5] => [4,3,2,1,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[3,4,2,1,5] => [4,3,2,1,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[3,5,1,2,4] => [4,5,3,1,2] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[3,5,2,1,4] => [4,5,3,1,2] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[4,1,2,3,5] => [4,2,3,1,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,1,3,2,5] => [4,2,3,1,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,2,1,3,5] => [4,3,2,1,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[4,2,3,1,5] => [4,3,2,1,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[4,3,1,2,5] => [4,3,2,1,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[4,3,2,1,5] => [4,3,2,1,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [2,3,4,6,5,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [2,3,5,4,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $2$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St000477
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000477: Integer partitions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 53%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000477: Integer partitions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 53%
Values
[1] => [1,0]
=> [[1],[]]
=> []
=> ? = 0
[1,2] => [1,0,1,0]
=> [[1,1],[]]
=> []
=> ? ∊ {0,0}
[2,1] => [1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {-1,0,0,0,0,1}
[1,3,2] => [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {-1,0,0,0,0,1}
[2,1,3] => [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> ? ∊ {-1,0,0,0,0,1}
[2,3,1] => [1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {-1,0,0,0,0,1}
[3,1,2] => [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {-1,0,0,0,0,1}
[3,2,1] => [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {-1,0,0,0,0,1}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> -1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> -1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,4,4,4}
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> -1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 0
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> -1
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> -1
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> -2
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 2
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 3
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 2
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 2
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 0
[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 0
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> -1
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> -1
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> -1
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 2
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> -1
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 2
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 2
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 2
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 2
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 2
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 2
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 2
[1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> -1
[1,2,4,5,3,6] => [1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> 2
[1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1
[1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> -1
[1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 0
[1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> -1
[1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> -1
[1,3,4,2,5,6] => [1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> -2
[1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> 2
[1,3,4,5,2,6] => [1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> 3
[1,3,4,6,2,5] => [1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> 2
[1,3,4,6,5,2] => [1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> 2
[1,3,5,2,4,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> 0
[1,3,5,4,2,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> 0
[1,3,5,6,2,4] => [1,0,1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3,1],[1,1]]
=> [1,1]
=> -1
[1,3,5,6,4,2] => [1,0,1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3,1],[1,1]]
=> [1,1]
=> -1
[1,4,2,3,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1]]
=> [1,1]
=> -1
[1,4,2,5,3,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> [2]
=> 2
[1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1]]
=> [1,1]
=> -1
[1,4,3,5,2,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> [2]
=> 2
[1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> [2]
=> 2
[1,5,2,4,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> [2]
=> 2
Description
The weight of a partition according to Alladi.
Matching statistic: St000509
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000509: Integer partitions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 60%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000509: Integer partitions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 60%
Values
[1] => [1,0]
=> [[1],[]]
=> []
=> ? = 0
[1,2] => [1,0,1,0]
=> [[1,1],[]]
=> []
=> ? ∊ {0,0}
[2,1] => [1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {-1,0,0,0,0,1}
[1,3,2] => [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {-1,0,0,0,0,1}
[2,1,3] => [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> ? ∊ {-1,0,0,0,0,1}
[2,3,1] => [1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {-1,0,0,0,0,1}
[3,1,2] => [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {-1,0,0,0,0,1}
[3,2,1] => [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {-1,0,0,0,0,1}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> -1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> -1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> -3
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> -1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 0
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> -1
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> -1
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 0
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 1
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 3
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 1
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 1
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 0
[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 0
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> -1
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> -1
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> -1
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 1
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> -1
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 1
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 1
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 1
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 1
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 1
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 1
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 1
[1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> -1
[1,2,4,5,3,6] => [1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> 1
[1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> -3
[1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> -1
[1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 0
[1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> -1
[1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> -1
[1,3,4,2,5,6] => [1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> 0
[1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> 1
[1,3,4,5,2,6] => [1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> 3
[1,3,4,6,2,5] => [1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> 1
[1,3,4,6,5,2] => [1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> 1
[1,3,5,2,4,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> 0
[1,3,5,4,2,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> 0
[1,3,5,6,2,4] => [1,0,1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3,1],[1,1]]
=> [1,1]
=> -1
[1,3,5,6,4,2] => [1,0,1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3,1],[1,1]]
=> [1,1]
=> -1
[1,4,2,3,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1]]
=> [1,1]
=> -1
[1,4,2,5,3,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> [2]
=> 1
[1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1]]
=> [1,1]
=> -1
[1,4,3,5,2,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> [2]
=> 1
[1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> [2]
=> 1
[1,5,2,4,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> [2]
=> 1
Description
The diagonal index (content) of a partition.
The '''diagonal index''' of the cell at row $r$ and column $c$ of a partition is $c - r$; this is sometimes called the '''content''' of the cell. The '''diagonal index of a partition''' is the sum of the diagonal index of each cell of the partition.
Matching statistic: St000454
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 40%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 40%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 0
[1,2] => [2] => ([],2)
=> ([],2)
=> 0
[2,1] => [2] => ([],2)
=> ([],2)
=> 0
[1,2,3] => [3] => ([],3)
=> ([],3)
=> 0
[1,3,2] => [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[2,1,3] => [3] => ([],3)
=> ([],3)
=> 0
[2,3,1] => [3] => ([],3)
=> ([],3)
=> 0
[3,1,2] => [3] => ([],3)
=> ([],3)
=> 0
[3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = -1
[1,2,3,4] => [4] => ([],4)
=> ([],4)
=> 0
[1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,2,2,2}
[1,3,2,4] => [1,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,3,4,2] => [1,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,4,2,3] => [1,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,2,2,2}
[2,1,3,4] => [4] => ([],4)
=> ([],4)
=> 0
[2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,2,2,2}
[2,3,1,4] => [4] => ([],4)
=> ([],4)
=> 0
[2,3,4,1] => [4] => ([],4)
=> ([],4)
=> 0
[2,4,1,3] => [4] => ([],4)
=> ([],4)
=> 0
[2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,2,2,2}
[3,1,2,4] => [4] => ([],4)
=> ([],4)
=> 0
[3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,2,2,2}
[3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,2,2,2}
[3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,2,2,2}
[3,4,1,2] => [4] => ([],4)
=> ([],4)
=> 0
[3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,2,2,2}
[4,1,2,3] => [4] => ([],4)
=> ([],4)
=> 0
[4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,2,2,2}
[4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,2,2,2}
[4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,2,2,2}
[4,3,1,2] => [1,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,2,2,2}
[1,2,3,4,5] => [5] => ([],5)
=> ([],5)
=> 0
[1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1
[1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[1,3,4,2,5] => [1,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1
[1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1
[1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1
[1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[1,4,2,3,5] => [1,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1
[1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[1,4,5,2,3] => [1,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1
[1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[1,5,2,3,4] => [1,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1
[1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[1,5,3,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[1,5,4,2,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2
[1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[2,1,3,4,5] => [5] => ([],5)
=> ([],5)
=> 0
[2,1,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[2,1,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[2,1,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[2,1,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[2,1,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[2,3,1,4,5] => [5] => ([],5)
=> ([],5)
=> 0
[2,3,1,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[2,3,4,1,5] => [5] => ([],5)
=> ([],5)
=> 0
[2,3,4,5,1] => [5] => ([],5)
=> ([],5)
=> 0
[2,3,5,1,4] => [5] => ([],5)
=> ([],5)
=> 0
[2,3,5,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[2,4,1,3,5] => [5] => ([],5)
=> ([],5)
=> 0
[2,4,1,5,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[2,4,3,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[2,4,3,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[2,4,5,1,3] => [5] => ([],5)
=> ([],5)
=> 0
[2,4,5,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[2,5,1,3,4] => [5] => ([],5)
=> ([],5)
=> 0
[2,5,1,4,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[2,5,3,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[2,5,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[2,5,4,1,3] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[2,5,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[3,1,2,4,5] => [5] => ([],5)
=> ([],5)
=> 0
[3,1,2,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[3,1,4,2,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[3,1,4,5,2] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[3,1,5,2,4] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[3,1,5,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[3,2,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[3,2,1,5,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,2,4,1,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[3,2,4,5,1] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[3,2,5,1,4] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[3,2,5,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4}
[3,4,1,2,5] => [5] => ([],5)
=> ([],5)
=> 0
[3,4,5,1,2] => [5] => ([],5)
=> ([],5)
=> 0
[3,4,5,2,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[3,5,1,2,4] => [5] => ([],5)
=> ([],5)
=> 0
[3,5,1,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[3,5,2,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[4,1,2,3,5] => [5] => ([],5)
=> ([],5)
=> 0
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001605
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 20%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 20%
Values
[1] => [] => []
=> ?
=> ? = 0
[1,2] => [1] => [1]
=> []
=> ? ∊ {0,0}
[2,1] => [1] => [1]
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,2] => [1,1]
=> [1]
=> ? ∊ {-1,0,0,0,0,1}
[1,3,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {-1,0,0,0,0,1}
[2,1,3] => [2,1] => [2]
=> []
=> ? ∊ {-1,0,0,0,0,1}
[2,3,1] => [2,1] => [2]
=> []
=> ? ∊ {-1,0,0,0,0,1}
[3,1,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {-1,0,0,0,0,1}
[3,2,1] => [2,1] => [2]
=> []
=> ? ∊ {-1,0,0,0,0,1}
[1,2,3,4] => [1,2,3] => [1,1,1]
=> [1,1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
[1,2,4,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
[1,3,2,4] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
[1,3,4,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
[1,4,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
[1,4,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
[2,1,3,4] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
[2,1,4,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
[2,3,1,4] => [2,3,1] => [3]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
[2,3,4,1] => [2,3,1] => [3]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
[2,4,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
[2,4,3,1] => [2,3,1] => [3]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
[3,1,2,4] => [3,1,2] => [3]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
[3,1,4,2] => [3,1,2] => [3]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
[3,2,1,4] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
[3,2,4,1] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
[3,4,1,2] => [3,1,2] => [3]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
[3,4,2,1] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
[4,1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
[4,1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
[4,2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
[4,2,3,1] => [2,3,1] => [3]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
[4,3,1,2] => [3,1,2] => [3]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
[4,3,2,1] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
[1,2,3,4,5] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 2
[1,2,3,5,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 2
[1,2,4,3,5] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,2,4,5,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,2,5,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 2
[1,2,5,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,3,2,4,5] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,3,2,5,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,3,4,2,5] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,3,4,5,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,3,5,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,3,5,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,4,2,3,5] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,4,2,5,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,4,3,2,5] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,4,3,5,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,4,5,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,4,5,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,5,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 2
[1,5,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,5,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[5,1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 2
[1,2,3,4,5,6] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 6
[1,2,3,4,6,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 6
[1,2,3,5,4,6] => [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,2,3,5,6,4] => [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,2,3,6,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 6
[1,2,3,6,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,2,4,3,5,6] => [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,2,4,3,6,5] => [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,2,4,6,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,2,5,4,3,6] => [1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,2,5,4,6,3] => [1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,2,5,6,4,3] => [1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,2,6,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 6
[1,2,6,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,2,6,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,2,6,5,4,3] => [1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,3,2,4,5,6] => [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,3,2,4,6,5] => [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,3,2,5,4,6] => [1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 1
[1,3,2,5,6,4] => [1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 1
[1,3,2,6,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,3,2,6,5,4] => [1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 1
[1,3,6,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,3,6,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 1
[1,4,3,2,5,6] => [1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,4,3,2,6,5] => [1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,4,3,6,2,5] => [1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,4,5,2,3,6] => [1,4,5,2,3] => [2,2,1]
=> [2,1]
=> 1
[1,4,5,2,6,3] => [1,4,5,2,3] => [2,2,1]
=> [2,1]
=> 1
[1,4,5,6,2,3] => [1,4,5,2,3] => [2,2,1]
=> [2,1]
=> 1
[1,4,6,3,2,5] => [1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,4,6,5,2,3] => [1,4,5,2,3] => [2,2,1]
=> [2,1]
=> 1
[1,5,3,4,2,6] => [1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,5,3,4,6,2] => [1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,5,3,6,4,2] => [1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,5,4,3,2,6] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,5,4,3,6,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,5,4,6,3,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,5,6,3,4,2] => [1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,5,6,4,3,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,6,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 6
[1,6,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,6,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,6,2,5,4,3] => [1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,6,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 2
Description
The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the cyclic group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St000478
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000478: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 27%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000478: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 27%
Values
[1] => ([],1)
=> [1]
=> []
=> ? = 0
[1,2] => ([],2)
=> [1,1]
=> [1]
=> ? ∊ {0,0}
[2,1] => ([(0,1)],2)
=> [2]
=> []
=> ? ∊ {0,0}
[1,2,3] => ([],3)
=> [1,1,1]
=> [1,1]
=> 0
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> [1]
=> ? ∊ {-1,0,0,0,1}
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> [1]
=> ? ∊ {-1,0,0,0,1}
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {-1,0,0,0,1}
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {-1,0,0,0,1}
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {-1,0,0,0,1}
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 0
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 0
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 0
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 0
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[2,1,3,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 0
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 0
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1
[1,2,3,4,5,6] => ([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[1,2,3,4,6,5] => ([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,2,3,5,4,6] => ([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3,5,6] => ([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 0
[1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
[1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
[1,2,4,6,5,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
[1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
[1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
[1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
[1,2,5,4,6,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
[1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
[1,2,5,6,4,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
[1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
[1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
[1,2,6,4,3,5] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
Description
Another weight of a partition according to Alladi.
According to Theorem 3.4 (Alladi 2012) in [1]
$$
\sum_{\pi\in GG_1(r)} w_1(\pi)
$$
equals the number of partitions of $r$ whose odd parts are all distinct. $GG_1(r)$ is the set of partitions of $r$ where consecutive entries differ by at least $2$, and consecutive even entries differ by at least $4$.
Matching statistic: St000566
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 27%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 27%
Values
[1] => ([],1)
=> [1]
=> []
=> ? = 0
[1,2] => ([],2)
=> [1,1]
=> [1]
=> ? ∊ {0,0}
[2,1] => ([(0,1)],2)
=> [2]
=> []
=> ? ∊ {0,0}
[1,2,3] => ([],3)
=> [1,1,1]
=> [1,1]
=> 0
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> [1]
=> ? ∊ {-1,0,0,0,1}
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> [1]
=> ? ∊ {-1,0,0,0,1}
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {-1,0,0,0,1}
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {-1,0,0,0,1}
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {-1,0,0,0,1}
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 0
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 0
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 0
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {-2,-2,-2,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,2,2,2}
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[2,1,3,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 1
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4}
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1
[1,2,3,4,5,6] => ([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[1,2,3,4,6,5] => ([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,2,3,5,4,6] => ([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3,5,6] => ([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 1
[1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
[1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
[1,2,4,6,5,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
[1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
[1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
[1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
[1,2,5,4,6,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
[1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
[1,2,5,6,4,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
[1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
[1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
[1,2,6,4,3,5] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
Description
The number of ways to select a row of a Ferrers shape and two cells in this row. Equivalently, if $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ is an integer partition, then the statistic is
$$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$$
The following 46 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001964The interval resolution global dimension of a poset. St001570The minimal number of edges to add to make a graph Hamiltonian. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001875The number of simple modules with projective dimension at most 1. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of 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. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000264The girth of a graph, which is not a tree.
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