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Your data matches 129 different statistics following compositions of up to 3 maps.
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Matching statistic: St000357
St000357: 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] => 1
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 3
[1,2,4,3] => 2
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 1
[2,1,4,3] => 0
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 0
[2,4,3,1] => 0
[3,1,2,4] => 1
[3,1,4,2] => 0
[3,2,1,4] => 0
[3,2,4,1] => 0
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 1
[4,1,3,2] => 0
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 6
[1,2,3,5,4] => 5
[1,2,4,3,5] => 4
[1,2,4,5,3] => 4
[1,2,5,3,4] => 3
[1,2,5,4,3] => 3
[1,3,2,4,5] => 3
[1,3,2,5,4] => 2
[1,3,4,2,5] => 3
[1,3,4,5,2] => 3
[1,3,5,2,4] => 2
[1,3,5,4,2] => 2
[1,4,2,3,5] => 2
[1,4,2,5,3] => 1
[1,4,3,2,5] => 1
[1,4,3,5,2] => 1
[1,4,5,2,3] => 1
Description
The number of occurrences of the pattern 12-3.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $12\!\!-\!\!3$.
Matching statistic: St000360
St000360: 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] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 0
[1,4,2,3] => 0
[1,4,3,2] => 1
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 0
[2,4,3,1] => 1
[3,1,2,4] => 0
[3,1,4,2] => 0
[3,2,1,4] => 1
[3,2,4,1] => 1
[3,4,1,2] => 0
[3,4,2,1] => 1
[4,1,2,3] => 0
[4,1,3,2] => 0
[4,2,1,3] => 1
[4,2,3,1] => 1
[4,3,1,2] => 2
[4,3,2,1] => 3
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 0
[1,2,5,3,4] => 0
[1,2,5,4,3] => 1
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 0
[1,3,4,5,2] => 0
[1,3,5,2,4] => 0
[1,3,5,4,2] => 1
[1,4,2,3,5] => 0
[1,4,2,5,3] => 0
[1,4,3,2,5] => 1
[1,4,3,5,2] => 1
[1,4,5,2,3] => 0
Description
The number of occurrences of the pattern 32-1.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $32\!\!-\!\!1$.
Matching statistic: St000319
(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
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 65%●distinct values known / distinct values provided: 45%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 65%●distinct values known / distinct values provided: 45%
Values
[1] => [1,0]
=> []
=> ?
=> ? = 0
[1,2] => [1,0,1,0]
=> [1]
=> []
=> ? ∊ {0,0}
[2,1] => [1,1,0,0]
=> []
=> ?
=> ? ∊ {0,0}
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> [1]
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> [1]
=> 0
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> []
=> ? ∊ {0,0,0,1}
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,1}
[3,1,2] => [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,1}
[3,2,1] => [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,1}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 0
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 0
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 0
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 0
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> 2
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,1]
=> 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1]
=> 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [2,2,1]
=> 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,1,1]
=> 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,1,1]
=> 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,1,1]
=> 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [2,1,1]
=> 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> 0
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> 0
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> 0
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> 0
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,2]
=> 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2]
=> 1
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [2,2]
=> 1
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2]
=> 1
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2]
=> 1
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,1]
=> 2
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,1]
=> 2
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,1]
=> 1
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [2,1]
=> 1
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,1]
=> 1
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,1]
=> 1
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
(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
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 65%●distinct values known / distinct values provided: 45%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 65%●distinct values known / distinct values provided: 45%
Values
[1] => [1,0]
=> []
=> ?
=> ? = 0
[1,2] => [1,0,1,0]
=> [1]
=> []
=> ? ∊ {0,0}
[2,1] => [1,1,0,0]
=> []
=> ?
=> ? ∊ {0,0}
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> [1]
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> [1]
=> 0
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> []
=> ? ∊ {0,0,0,1}
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,1}
[3,1,2] => [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,1}
[3,2,1] => [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,1}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 0
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 0
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 0
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 0
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> 2
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,1]
=> 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1]
=> 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [2,2,1]
=> 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,1,1]
=> 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,1,1]
=> 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,1,1]
=> 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [2,1,1]
=> 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> 0
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> 0
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> 0
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> 0
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,2]
=> 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2]
=> 1
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [2,2]
=> 1
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2]
=> 1
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2]
=> 1
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,1]
=> 2
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,1]
=> 2
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,1]
=> 1
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [2,1]
=> 1
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,1]
=> 1
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,1]
=> 1
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
Description
The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St001280
(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
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 36% ●values known / values provided: 65%●distinct values known / distinct values provided: 36%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 36% ●values known / values provided: 65%●distinct values known / distinct values provided: 36%
Values
[1] => [1,0]
=> []
=> ?
=> ? = 0
[1,2] => [1,0,1,0]
=> [1]
=> []
=> ? ∊ {0,0}
[2,1] => [1,1,0,0]
=> []
=> ?
=> ? ∊ {0,0}
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> [1]
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> [1]
=> 0
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> []
=> ? ∊ {0,0,0,1}
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,1}
[3,1,2] => [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,1}
[3,2,1] => [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,1}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 0
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 0
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 0
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 0
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,3}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> 2
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,1]
=> 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1]
=> 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [2,2,1]
=> 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,1,1]
=> 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,1,1]
=> 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,1,1]
=> 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [2,1,1]
=> 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> 0
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> 0
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> 0
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> 0
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,2]
=> 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2]
=> 2
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [2,2]
=> 2
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2]
=> 2
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2]
=> 2
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,1]
=> 1
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,1]
=> 1
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,1]
=> 1
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [2,1]
=> 1
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,1]
=> 1
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,1]
=> 1
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000771
(load all 37 compositions to match this statistic)
(load all 37 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,2] => ([],2)
=> ([],2)
=> ? = 0 + 1
[2,1] => ([(0,1)],2)
=> ([],1)
=> 1 = 0 + 1
[1,2,3] => ([],3)
=> ([],3)
=> ? ∊ {0,0,1} + 1
[1,3,2] => ([(1,2)],3)
=> ([],2)
=> ? ∊ {0,0,1} + 1
[2,1,3] => ([(1,2)],3)
=> ([],2)
=> ? ∊ {0,0,1} + 1
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1 = 0 + 1
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1 = 0 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> 1 = 0 + 1
[1,2,3,4] => ([],4)
=> ([],4)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,3} + 1
[1,2,4,3] => ([(2,3)],4)
=> ([],3)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,3} + 1
[1,3,2,4] => ([(2,3)],4)
=> ([],3)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,3} + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,3} + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,3} + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,3} + 1
[2,1,3,4] => ([(2,3)],4)
=> ([],3)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,3} + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([],2)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,3} + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,3} + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 0 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,3} + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,3} + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 0 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 0 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 0 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 0 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 0 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 0 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> 1 = 0 + 1
[1,2,3,4,5] => ([],5)
=> ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[1,2,3,5,4] => ([(3,4)],5)
=> ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[1,2,4,3,5] => ([(3,4)],5)
=> ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[1,3,2,4,5] => ([(3,4)],5)
=> ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[2,1,3,4,5] => ([(3,4)],5)
=> ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 0 + 1
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 0 + 1
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> 2 = 1 + 1
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 0 + 1
[4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 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: St001604
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 36% ●values known / values provided: 62%●distinct values known / distinct values provided: 36%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 36% ●values known / values provided: 62%●distinct values known / distinct values provided: 36%
Values
[1] => [1] => [1,0]
=> []
=> ? = 0
[1,2] => [1,2] => [1,0,1,0]
=> [1]
=> ? ∊ {0,0}
[2,1] => [2,1] => [1,1,0,0]
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> ? ∊ {0,0,0,0,0,1}
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> ? ∊ {0,0,0,0,0,1}
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [2]
=> ? ∊ {0,0,0,0,0,1}
[2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,0,0,0,0,1}
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,1}
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,1}
[1,2,3,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[1,2,4,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[1,3,2,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[2,1,3,4] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
[2,3,1,4] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,3}
[2,3,4,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,3}
[2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,3}
[2,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,3}
[3,1,2,4] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,3}
[3,1,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,3}
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,3}
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,3}
[3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,3}
[4,1,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,3}
[4,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,3}
[4,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,3}
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,3}
[4,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,3}
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,3}
[1,2,3,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,2,3,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,2,4,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,2,4,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,2,5,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,2,5,4,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,3,2,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,3,2,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,3,4,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,3,4,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,3,5,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,4,2,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,4,2,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,4,3,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,4,3,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,4,5,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,5,2,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,5,2,4,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,5,3,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,5,3,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,5,4,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[2,1,3,4,5] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 2
[2,1,3,5,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 2
[2,1,4,3,5] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 2
[2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 2
[2,1,5,3,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 2
[2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 2
[2,3,1,4,5] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 0
[2,3,1,5,4] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 0
[2,3,4,1,5] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 0
[2,3,4,5,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 0
[2,3,5,1,4] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 0
[2,3,5,4,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 0
[2,4,1,3,5] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 0
[2,4,1,5,3] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 0
[2,4,3,1,5] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 0
[2,4,3,5,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 0
[2,4,5,1,3] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 0
[3,4,1,2,5] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[3,4,1,5,2] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[3,4,2,1,5] => [3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[3,4,2,5,1] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[3,4,5,1,2] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[3,4,5,2,1] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[3,5,1,2,4] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[3,5,1,4,2] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[3,5,2,1,4] => [3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[3,5,2,4,1] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[3,5,4,1,2] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[3,5,4,2,1] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,1,2,3,5] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,1,2,5,3] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,1,3,2,5] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,1,3,5,2] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,1,5,2,3] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,1,5,3,2] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,2,3,1,5] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,2,3,5,1] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,2,5,1,3] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,2,5,3,1] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,3,5,1,2] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,3,5,2,1] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,5,1,2,3] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[4,5,1,3,2] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St000567
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000567: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 62%●distinct values known / distinct values provided: 45%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000567: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 62%●distinct values known / distinct values provided: 45%
Values
[1] => [1] => [1]
=> []
=> ? = 0
[1,2] => [1,2] => [2]
=> []
=> ? ∊ {0,0}
[2,1] => [1,2] => [2]
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,1}
[1,3,2] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,1}
[2,1,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,1}
[2,3,1] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,1}
[3,1,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1}
[3,2,1] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1}
[1,2,3,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[1,2,4,3] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[1,3,2,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[1,3,4,2] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[1,4,2,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[1,4,3,2] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[2,1,3,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[2,1,4,3] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[2,3,1,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[2,3,4,1] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[2,4,1,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[2,4,3,1] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[3,1,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[3,1,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[3,2,1,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[3,2,4,1] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[3,4,1,2] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[3,4,2,1] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[4,1,2,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[4,1,3,2] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[4,2,1,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[4,2,3,1] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[4,3,1,2] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[4,3,2,1] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[1,2,3,4,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[1,2,3,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[1,2,4,3,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[1,2,4,5,3] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[1,2,5,3,4] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[1,2,5,4,3] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[1,3,2,4,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[1,3,2,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[1,3,4,2,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[1,3,4,5,2] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[1,3,5,2,4] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[1,3,5,4,2] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[1,4,2,3,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[1,4,2,5,3] => [1,2,4,5,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[1,4,3,2,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[1,4,3,5,2] => [1,2,4,5,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[1,4,5,2,3] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[1,4,5,3,2] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[1,5,2,3,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,5,2,4,3] => [1,2,5,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[1,5,3,2,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,1,3,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,3,1,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1
[3,1,5,2,4] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,1,5,4,2] => [1,3,5,2,4] => [3,2]
=> [2]
=> 0
[3,2,5,1,4] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,2,5,4,1] => [1,3,5,2,4] => [3,2]
=> [2]
=> 0
[3,4,5,1,2] => [1,3,5,2,4] => [3,2]
=> [2]
=> 0
[3,4,5,2,1] => [1,3,5,2,4] => [3,2]
=> [2]
=> 0
[3,5,1,2,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 0
[3,5,1,4,2] => [1,3,2,5,4] => [3,2]
=> [2]
=> 0
[3,5,2,1,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 0
[3,5,2,4,1] => [1,3,2,5,4] => [3,2]
=> [2]
=> 0
[4,1,2,3,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1
[4,1,2,5,3] => [1,4,5,3,2] => [3,1,1]
=> [1,1]
=> 1
[4,1,3,5,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> 0
[4,1,5,3,2] => [1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 1
[4,2,1,3,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1
[4,2,1,5,3] => [1,4,5,3,2] => [3,1,1]
=> [1,1]
=> 1
[4,2,3,5,1] => [1,4,5,2,3] => [3,2]
=> [2]
=> 0
[4,2,5,3,1] => [1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 1
[4,3,1,5,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> 0
[4,3,2,5,1] => [1,4,5,2,3] => [3,2]
=> [2]
=> 0
[4,5,1,2,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 0
[4,5,1,3,2] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1
[4,5,2,1,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 0
[4,5,2,3,1] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1
[4,5,3,1,2] => [1,4,2,5,3] => [3,2]
=> [2]
=> 0
[4,5,3,2,1] => [1,4,2,5,3] => [3,2]
=> [2]
=> 0
[5,1,2,3,4] => [1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 3
[5,1,2,4,3] => [1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 1
[5,1,3,2,4] => [1,5,4,2,3] => [3,1,1]
=> [1,1]
=> 1
[5,1,4,2,3] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 1
[5,2,1,3,4] => [1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 3
[5,2,1,4,3] => [1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 1
[5,2,3,1,4] => [1,5,4,2,3] => [3,1,1]
=> [1,1]
=> 1
[5,2,4,1,3] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 1
[5,3,1,2,4] => [1,5,4,2,3] => [3,1,1]
=> [1,1]
=> 1
[5,3,2,1,4] => [1,5,4,2,3] => [3,1,1]
=> [1,1]
=> 1
[5,4,1,2,3] => [1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 1
[5,4,1,3,2] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[5,4,2,1,3] => [1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 1
[5,4,2,3,1] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[5,4,3,1,2] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[5,4,3,2,1] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,2,6,3,4,5] => [1,2,3,6,5,4] => [4,1,1]
=> [1,1]
=> 1
[1,2,6,4,3,5] => [1,2,3,6,5,4] => [4,1,1]
=> [1,1]
=> 1
Description
The sum of the products of all pairs of parts.
This is the evaluation of the second elementary symmetric polynomial which is equal to
$$e_2(\lambda) = \binom{n+1}{2} - \sum_{i=1}^\ell\binom{\lambda_i+1}{2}$$
for a partition $\lambda = (\lambda_1,\dots,\lambda_\ell) \vdash n$, see [1].
This is the maximal number of inversions a permutation with the given shape can have, see [2, cor.2.4].
Matching statistic: St000512
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000512: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 61%●distinct values known / distinct values provided: 45%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000512: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 61%●distinct values known / distinct values provided: 45%
Values
[1] => [1]
=> []
=> ? = 0
[1,2] => [1,1]
=> [1]
=> ? ∊ {0,0}
[2,1] => [2]
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1}
[2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1}
[2,3,1] => [3]
=> []
=> ? ∊ {0,0,0,0,1}
[3,1,2] => [3]
=> []
=> ? ∊ {0,0,0,0,1}
[3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 0
[2,1,4,3] => [2,2]
=> [2]
=> 0
[2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[3,1,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,2,4,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[3,4,1,2] => [2,2]
=> [2]
=> 0
[3,4,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[4,1,2,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[4,1,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[4,2,1,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,3,1,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[4,3,2,1] => [2,2]
=> [2]
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 4
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> 0
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> 0
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> 0
[1,3,4,5,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,5,6}
[1,3,5,2,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,5,6}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 0
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> 0
[1,4,2,5,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,5,6}
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 0
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> 1
[1,4,5,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,5,6}
[1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,5,6}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 0
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,5,4,2,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,5,6}
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 1
[2,1,4,5,3] => [3,2]
=> [2]
=> 0
[2,1,5,3,4] => [3,2]
=> [2]
=> 0
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> 0
[2,3,1,5,4] => [3,2]
=> [2]
=> 0
[2,3,4,1,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,5,6}
[2,3,4,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,5,6}
[2,3,5,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,5,6}
[2,3,5,4,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,5,6}
[2,4,1,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,5,6}
[2,4,1,5,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,5,6}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> 0
[2,4,3,5,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,5,6}
[2,4,5,1,3] => [3,2]
=> [2]
=> 0
[2,4,5,3,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,5,6}
[2,5,1,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,5,6}
[2,5,1,4,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,5,6}
[2,5,3,1,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,5,6}
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> 0
[2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,5,6}
[2,5,4,3,1] => [3,2]
=> [2]
=> 0
[3,1,2,4,5] => [3,1,1]
=> [1,1]
=> 0
[3,1,2,5,4] => [3,2]
=> [2]
=> 0
[3,1,4,2,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,5,6}
[3,1,4,5,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,5,6}
[3,1,5,2,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,5,6}
[3,1,5,4,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,5,6}
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 1
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> 0
[3,2,4,5,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,5,6}
[3,2,5,1,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,5,6}
[3,2,5,4,1] => [3,1,1]
=> [1,1]
=> 0
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> 1
[3,4,1,5,2] => [3,2]
=> [2]
=> 0
[3,4,2,1,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,5,6}
[3,4,2,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,5,6}
[3,4,5,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,5,6}
[3,4,5,2,1] => [3,2]
=> [2]
=> 0
[3,5,2,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,5,6}
Description
The number of invariant subsets of size 3 when acting with a permutation of given cycle type.
Matching statistic: St000566
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 55% ●values known / values provided: 61%●distinct values known / distinct values provided: 55%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 55% ●values known / values provided: 61%●distinct values known / distinct values provided: 55%
Values
[1] => [1]
=> []
=> []
=> ? = 0
[1,2] => [1,1]
=> [1]
=> [1]
=> ? ∊ {0,0}
[2,1] => [2]
=> []
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,1,1]
=> [1,1]
=> [2]
=> 1
[1,3,2] => [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0}
[2,1,3] => [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0}
[2,3,1] => [3]
=> []
=> []
=> ? ∊ {0,0,0,0,0}
[3,1,2] => [3]
=> []
=> []
=> ? ∊ {0,0,0,0,0}
[3,2,1] => [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[1,4,2,3] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> [1,1]
=> 0
[2,3,1,4] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[2,3,4,1] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[2,4,1,3] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[2,4,3,1] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[3,1,2,4] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[3,1,4,2] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[3,4,1,2] => [2,2]
=> [2]
=> [1,1]
=> 0
[3,4,2,1] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[4,1,2,3] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[4,1,3,2] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[4,2,1,3] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[4,3,1,2] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[4,3,2,1] => [2,2]
=> [2]
=> [1,1]
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 6
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [2,1]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,5}
[1,3,5,2,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,5}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,4,2,5,3] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,5}
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [2,1]
=> 1
[1,4,5,3,2] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,5}
[1,5,2,3,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,5}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[1,5,4,2,3] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,5}
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [2,1]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [2,1]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [2,1]
=> 1
[2,1,4,5,3] => [3,2]
=> [2]
=> [1,1]
=> 0
[2,1,5,3,4] => [3,2]
=> [2]
=> [1,1]
=> 0
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [2,1]
=> 1
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[2,3,1,5,4] => [3,2]
=> [2]
=> [1,1]
=> 0
[2,3,4,1,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,5}
[2,3,4,5,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,5}
[2,3,5,1,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,5}
[2,3,5,4,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,5}
[2,4,1,3,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,5}
[2,4,1,5,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,5}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[2,4,3,5,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,5}
[2,4,5,1,3] => [3,2]
=> [2]
=> [1,1]
=> 0
[2,4,5,3,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,5}
[2,5,1,3,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,5}
[2,5,1,4,3] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,5}
[2,5,3,1,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,5}
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[2,5,4,1,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,5}
[2,5,4,3,1] => [3,2]
=> [2]
=> [1,1]
=> 0
[3,1,2,4,5] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[3,1,2,5,4] => [3,2]
=> [2]
=> [1,1]
=> 0
[3,1,4,2,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,5}
[3,1,4,5,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,5}
[3,1,5,2,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,5}
[3,1,5,4,2] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,5}
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [2,1]
=> 1
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[3,2,4,5,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,5}
[3,2,5,1,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,5}
[3,2,5,4,1] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> [2,1]
=> 1
[3,4,1,5,2] => [3,2]
=> [2]
=> [1,1]
=> 0
[3,4,2,1,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,5}
[3,4,2,5,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,5}
[3,4,5,1,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,5}
[3,4,5,2,1] => [3,2]
=> [2]
=> [1,1]
=> 0
[3,5,2,1,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,5}
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 119 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000693The modular (standard) major index of a standard tableau. St000984The number of boxes below precisely one peak. St001480The number of simple summands of the module J^2/J^3. St001176The size of a partition minus its first part. St001961The sum of the greatest common divisors of all pairs of parts. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001525The number of symmetric hooks on the diagonal of a partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. 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. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000941The number of characters of the symmetric group whose value on the partition is even. St001964The interval resolution global dimension of a poset. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001914The size of the orbit of an integer partition in Bulgarian solitaire. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. 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. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001279The sum of the parts of an integer partition that are at least two. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001541The Gini index of an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001651The Frankl number of a lattice. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000454The largest eigenvalue of a graph if it is integral. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St000456The monochromatic index of a connected graph. St000668The least common multiple of the parts of the partition. 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. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000699The toughness times the least common multiple of 1,. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000934The 2-degree of an integer partition. St001846The number of elements which do not have a complement in the lattice. St000260The radius of a connected graph. St000284The Plancherel distribution on integer partitions. St000455The second largest eigenvalue of a graph if it is integral. St000478Another weight of a partition according to Alladi. St000901The cube of the number of standard Young tableaux with shape given by the partition. 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. St001435The number of missing boxes in the first row. St001845The number of join irreducibles minus the rank of a lattice. St001487The number of inner corners of a skew partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001060The distinguishing index of a graph. St000909The number of maximal chains of maximal size in a poset. St001875The number of simple modules with projective dimension at most 1. St000527The width of the poset. St000259The diameter 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. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000632The jump number of the poset. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001902The number of potential covers of a poset. St000181The number of connected components of the Hasse diagram for the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000908The length of the shortest maximal antichain in a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001472The permanent of the Coxeter matrix of the poset. St001510The number of self-evacuating linear extensions of a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001779The order of promotion on the set of linear extensions of a poset. 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. St001438The number of missing boxes of a skew partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type.
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