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Your data matches 28 different statistics following compositions of up to 3 maps.
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Matching statistic: St001768
St001768: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[-1] => 1
[1,2] => 1
[1,-2] => 1
[-1,2] => 1
[-1,-2] => 2
[2,1] => 1
[2,-1] => 1
[-2,1] => 1
[-2,-1] => 1
[1,2,3] => 1
[1,2,-3] => 1
[1,-2,3] => 1
[1,-2,-3] => 2
[-1,2,3] => 1
[-1,2,-3] => 4
[-1,-2,3] => 14
[-1,-2,-3] => 42
[1,3,2] => 1
[1,3,-2] => 1
[1,-3,2] => 1
[1,-3,-2] => 1
[-1,3,2] => 4
[-1,3,-2] => 9
[-1,-3,2] => 9
[-1,-3,-2] => 16
[2,1,3] => 1
[2,1,-3] => 2
[2,-1,3] => 1
[2,-1,-3] => 3
[-2,1,3] => 1
[-2,1,-3] => 3
[-2,-1,3] => 5
[-2,-1,-3] => 12
[2,3,1] => 1
[2,3,-1] => 1
[2,-3,1] => 2
[2,-3,-1] => 2
[-2,3,1] => 3
[-2,3,-1] => 5
[-2,-3,1] => 5
[-2,-3,-1] => 7
[3,1,2] => 1
[3,1,-2] => 2
[3,-1,2] => 3
[3,-1,-2] => 5
[-3,1,2] => 1
[-3,1,-2] => 2
[-3,-1,2] => 5
[-3,-1,-2] => 7
Description
The number of reduced words of a signed permutation.
This is the number of ways to write a permutation as a minimal length product of simple reflections.
Matching statistic: St000668
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 7%●distinct values known / distinct values provided: 3%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 7%●distinct values known / distinct values provided: 3%
Values
[1] => [1]
=> []
=> ? ∊ {1,1}
[-1] => []
=> ?
=> ? ∊ {1,1}
[1,2] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2}
[1,-2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[-1,2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[-1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2}
[2,1] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[2,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2}
[-2,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2}
[-2,-1] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-2,3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-2,-3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,2,3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,2,-3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-2,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-2,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,3,2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,3,-2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-3,2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-3,-2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,3,2] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,3,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-3,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-3,-2] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,1,-3] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-1,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-1,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,1,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,1,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-1,-3] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,3,1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,3,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-3,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-3,-1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,3,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,3,-1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-3,1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-3,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,1,2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,-1,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,-1,-2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,1,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,1,-2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,-1,2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,-1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,2,1] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,-4] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3,4] => [1,1,1]
=> [1,1]
=> 1
[1,-2,3,4] => [1,1,1]
=> [1,1]
=> 1
[-1,2,3,4] => [1,1,1]
=> [1,1]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,2,-4,-3] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[-2,-1,3,4] => [2,1,1]
=> [1,1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> 2
[2,1,-4,-3] => [2,2]
=> [2]
=> 2
[-2,-1,4,3] => [2,2]
=> [2]
=> 2
[-2,-1,-4,-3] => [2,2]
=> [2]
=> 2
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[-3,2,-1,4] => [2,1,1]
=> [1,1]
=> 1
[3,4,1,2] => [2,2]
=> [2]
=> 2
[3,-4,1,-2] => [2,2]
=> [2]
=> 2
[-3,4,-1,2] => [2,2]
=> [2]
=> 2
[-3,-4,-1,-2] => [2,2]
=> [2]
=> 2
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[-4,2,3,-1] => [2,1,1]
=> [1,1]
=> 1
[4,3,2,1] => [2,2]
=> [2]
=> 2
[4,-3,-2,1] => [2,2]
=> [2]
=> 2
[-4,3,2,-1] => [2,2]
=> [2]
=> 2
[-4,-3,-2,-1] => [2,2]
=> [2]
=> 2
Description
The least common multiple of the parts of the partition.
Matching statistic: St000706
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000706: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 7%●distinct values known / distinct values provided: 4%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000706: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 7%●distinct values known / distinct values provided: 4%
Values
[1] => [1]
=> []
=> ? ∊ {1,1}
[-1] => []
=> ?
=> ? ∊ {1,1}
[1,2] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2}
[1,-2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[-1,2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[-1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2}
[2,1] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[2,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2}
[-2,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2}
[-2,-1] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[1,2,3] => [1,1,1]
=> [1,1]
=> 2
[1,2,-3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-2,3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-2,-3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,2,3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,2,-3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-2,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-2,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,3,2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,3,-2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-3,2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-3,-2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,3,2] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,3,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-3,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-3,-2] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,1,-3] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-1,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-1,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,1,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,1,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-1,-3] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,3,1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,3,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-3,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-3,-1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,3,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,3,-1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-3,1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-3,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,1,2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,-1,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,-1,-2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,1,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,1,-2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,-1,2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,-1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,2,1] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 6
[1,2,3,-4] => [1,1,1]
=> [1,1]
=> 2
[1,2,-3,4] => [1,1,1]
=> [1,1]
=> 2
[1,-2,3,4] => [1,1,1]
=> [1,1]
=> 2
[-1,2,3,4] => [1,1,1]
=> [1,1]
=> 2
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 2
[1,2,-4,-3] => [2,1,1]
=> [1,1]
=> 2
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 2
[1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 2
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 2
[1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 2
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 2
[-2,-1,3,4] => [2,1,1]
=> [1,1]
=> 2
[2,1,4,3] => [2,2]
=> [2]
=> 1
[2,1,-4,-3] => [2,2]
=> [2]
=> 1
[-2,-1,4,3] => [2,2]
=> [2]
=> 1
[-2,-1,-4,-3] => [2,2]
=> [2]
=> 1
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 2
[-3,2,-1,4] => [2,1,1]
=> [1,1]
=> 2
[3,4,1,2] => [2,2]
=> [2]
=> 1
[3,-4,1,-2] => [2,2]
=> [2]
=> 1
[-3,4,-1,2] => [2,2]
=> [2]
=> 1
[-3,-4,-1,-2] => [2,2]
=> [2]
=> 1
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 2
[-4,2,3,-1] => [2,1,1]
=> [1,1]
=> 2
[4,3,2,1] => [2,2]
=> [2]
=> 1
[4,-3,-2,1] => [2,2]
=> [2]
=> 1
[-4,3,2,-1] => [2,2]
=> [2]
=> 1
[-4,-3,-2,-1] => [2,2]
=> [2]
=> 1
Description
The product of the factorials of the multiplicities of an integer partition.
Matching statistic: St000707
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000707: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 7%●distinct values known / distinct values provided: 3%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000707: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 7%●distinct values known / distinct values provided: 3%
Values
[1] => [1]
=> []
=> ? ∊ {1,1}
[-1] => []
=> ?
=> ? ∊ {1,1}
[1,2] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2}
[1,-2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[-1,2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[-1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2}
[2,1] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[2,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2}
[-2,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2}
[-2,-1] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-2,3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-2,-3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,2,3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,2,-3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-2,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-2,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,3,2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,3,-2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-3,2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-3,-2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,3,2] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,3,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-3,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-3,-2] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,1,-3] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-1,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-1,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,1,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,1,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-1,-3] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,3,1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,3,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-3,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-3,-1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,3,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,3,-1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-3,1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-3,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,1,2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,-1,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,-1,-2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,1,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,1,-2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,-1,2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,-1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,2,1] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,-4] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3,4] => [1,1,1]
=> [1,1]
=> 1
[1,-2,3,4] => [1,1,1]
=> [1,1]
=> 1
[-1,2,3,4] => [1,1,1]
=> [1,1]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,2,-4,-3] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[-2,-1,3,4] => [2,1,1]
=> [1,1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> 2
[2,1,-4,-3] => [2,2]
=> [2]
=> 2
[-2,-1,4,3] => [2,2]
=> [2]
=> 2
[-2,-1,-4,-3] => [2,2]
=> [2]
=> 2
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[-3,2,-1,4] => [2,1,1]
=> [1,1]
=> 1
[3,4,1,2] => [2,2]
=> [2]
=> 2
[3,-4,1,-2] => [2,2]
=> [2]
=> 2
[-3,4,-1,2] => [2,2]
=> [2]
=> 2
[-3,-4,-1,-2] => [2,2]
=> [2]
=> 2
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[-4,2,3,-1] => [2,1,1]
=> [1,1]
=> 1
[4,3,2,1] => [2,2]
=> [2]
=> 2
[4,-3,-2,1] => [2,2]
=> [2]
=> 2
[-4,3,2,-1] => [2,2]
=> [2]
=> 2
[-4,-3,-2,-1] => [2,2]
=> [2]
=> 2
Description
The product of the factorials of the parts.
Matching statistic: St000708
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 7%●distinct values known / distinct values provided: 3%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 7%●distinct values known / distinct values provided: 3%
Values
[1] => [1]
=> []
=> ? ∊ {1,1}
[-1] => []
=> ?
=> ? ∊ {1,1}
[1,2] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2}
[1,-2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[-1,2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[-1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2}
[2,1] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[2,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2}
[-2,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2}
[-2,-1] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-2,3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-2,-3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,2,3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,2,-3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-2,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-2,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,3,2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,3,-2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-3,2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-3,-2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,3,2] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,3,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-3,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-3,-2] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,1,-3] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-1,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-1,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,1,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,1,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-1,-3] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,3,1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,3,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-3,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-3,-1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,3,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,3,-1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-3,1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-3,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,1,2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,-1,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,-1,-2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,1,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,1,-2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,-1,2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,-1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,2,1] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,-4] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3,4] => [1,1,1]
=> [1,1]
=> 1
[1,-2,3,4] => [1,1,1]
=> [1,1]
=> 1
[-1,2,3,4] => [1,1,1]
=> [1,1]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,2,-4,-3] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[-2,-1,3,4] => [2,1,1]
=> [1,1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> 2
[2,1,-4,-3] => [2,2]
=> [2]
=> 2
[-2,-1,4,3] => [2,2]
=> [2]
=> 2
[-2,-1,-4,-3] => [2,2]
=> [2]
=> 2
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[-3,2,-1,4] => [2,1,1]
=> [1,1]
=> 1
[3,4,1,2] => [2,2]
=> [2]
=> 2
[3,-4,1,-2] => [2,2]
=> [2]
=> 2
[-3,4,-1,2] => [2,2]
=> [2]
=> 2
[-3,-4,-1,-2] => [2,2]
=> [2]
=> 2
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[-4,2,3,-1] => [2,1,1]
=> [1,1]
=> 1
[4,3,2,1] => [2,2]
=> [2]
=> 2
[4,-3,-2,1] => [2,2]
=> [2]
=> 2
[-4,3,2,-1] => [2,2]
=> [2]
=> 2
[-4,-3,-2,-1] => [2,2]
=> [2]
=> 2
Description
The product of the parts of an integer partition.
Matching statistic: St000770
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000770: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 7%●distinct values known / distinct values provided: 3%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000770: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 7%●distinct values known / distinct values provided: 3%
Values
[1] => [1]
=> []
=> ? ∊ {1,1}
[-1] => []
=> ?
=> ? ∊ {1,1}
[1,2] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2}
[1,-2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[-1,2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[-1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2}
[2,1] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[2,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2}
[-2,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2}
[-2,-1] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-2,3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-2,-3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,2,3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,2,-3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-2,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-2,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,3,2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,3,-2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-3,2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-3,-2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,3,2] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,3,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-3,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-3,-2] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,1,-3] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-1,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-1,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,1,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,1,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-1,-3] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,3,1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,3,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-3,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-3,-1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,3,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,3,-1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-3,1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-3,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,1,2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,-1,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,-1,-2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,1,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,1,-2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,-1,2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,-1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,2,1] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,-4] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3,4] => [1,1,1]
=> [1,1]
=> 1
[1,-2,3,4] => [1,1,1]
=> [1,1]
=> 1
[-1,2,3,4] => [1,1,1]
=> [1,1]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,2,-4,-3] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[-2,-1,3,4] => [2,1,1]
=> [1,1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> 2
[2,1,-4,-3] => [2,2]
=> [2]
=> 2
[-2,-1,4,3] => [2,2]
=> [2]
=> 2
[-2,-1,-4,-3] => [2,2]
=> [2]
=> 2
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[-3,2,-1,4] => [2,1,1]
=> [1,1]
=> 1
[3,4,1,2] => [2,2]
=> [2]
=> 2
[3,-4,1,-2] => [2,2]
=> [2]
=> 2
[-3,4,-1,2] => [2,2]
=> [2]
=> 2
[-3,-4,-1,-2] => [2,2]
=> [2]
=> 2
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[-4,2,3,-1] => [2,1,1]
=> [1,1]
=> 1
[4,3,2,1] => [2,2]
=> [2]
=> 2
[4,-3,-2,1] => [2,2]
=> [2]
=> 2
[-4,3,2,-1] => [2,2]
=> [2]
=> 2
[-4,-3,-2,-1] => [2,2]
=> [2]
=> 2
Description
The major index of an integer partition when read from bottom to top.
This is the sum of the positions of the corners of the shape of an integer partition when reading from bottom to top.
For example, the partition $\lambda = (8,6,6,4,3,3)$ has corners at positions 3,6,9, and 13, giving a major index of 31.
Matching statistic: St000813
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000813: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 7%●distinct values known / distinct values provided: 4%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000813: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 7%●distinct values known / distinct values provided: 4%
Values
[1] => [1]
=> []
=> ? ∊ {1,1}
[-1] => []
=> ?
=> ? ∊ {1,1}
[1,2] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2}
[1,-2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[-1,2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[-1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2}
[2,1] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[2,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2}
[-2,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2}
[-2,-1] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[1,2,3] => [1,1,1]
=> [1,1]
=> 3
[1,2,-3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-2,3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-2,-3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,2,3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,2,-3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-2,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-2,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,3,2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,3,-2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-3,2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-3,-2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,3,2] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,3,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-3,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-3,-2] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,1,-3] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-1,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-1,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,1,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,1,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-1,-3] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,3,1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,3,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-3,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-3,-1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,3,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,3,-1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-3,1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-3,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,1,2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,-1,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,-1,-2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,1,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,1,-2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,-1,2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,-1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,2,1] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 10
[1,2,3,-4] => [1,1,1]
=> [1,1]
=> 3
[1,2,-3,4] => [1,1,1]
=> [1,1]
=> 3
[1,-2,3,4] => [1,1,1]
=> [1,1]
=> 3
[-1,2,3,4] => [1,1,1]
=> [1,1]
=> 3
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 3
[1,2,-4,-3] => [2,1,1]
=> [1,1]
=> 3
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 3
[1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 3
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 3
[1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 3
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 3
[-2,-1,3,4] => [2,1,1]
=> [1,1]
=> 3
[2,1,4,3] => [2,2]
=> [2]
=> 1
[2,1,-4,-3] => [2,2]
=> [2]
=> 1
[-2,-1,4,3] => [2,2]
=> [2]
=> 1
[-2,-1,-4,-3] => [2,2]
=> [2]
=> 1
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 3
[-3,2,-1,4] => [2,1,1]
=> [1,1]
=> 3
[3,4,1,2] => [2,2]
=> [2]
=> 1
[3,-4,1,-2] => [2,2]
=> [2]
=> 1
[-3,4,-1,2] => [2,2]
=> [2]
=> 1
[-3,-4,-1,-2] => [2,2]
=> [2]
=> 1
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 3
[-4,2,3,-1] => [2,1,1]
=> [1,1]
=> 3
[4,3,2,1] => [2,2]
=> [2]
=> 1
[4,-3,-2,1] => [2,2]
=> [2]
=> 1
[-4,3,2,-1] => [2,2]
=> [2]
=> 1
[-4,-3,-2,-1] => [2,2]
=> [2]
=> 1
Description
The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition.
This is also the sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to monomial symmetric functions.
Matching statistic: St000815
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000815: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 7%●distinct values known / distinct values provided: 3%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000815: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 7%●distinct values known / distinct values provided: 3%
Values
[1] => [1]
=> []
=> ? ∊ {1,1}
[-1] => []
=> ?
=> ? ∊ {1,1}
[1,2] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2}
[1,-2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[-1,2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[-1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2}
[2,1] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[2,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2}
[-2,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2}
[-2,-1] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-2,3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-2,-3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,2,3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,2,-3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-2,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-2,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,3,2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,3,-2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-3,2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-3,-2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,3,2] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,3,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-3,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-3,-2] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,1,-3] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-1,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-1,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,1,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,1,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-1,-3] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,3,1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,3,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-3,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-3,-1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,3,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,3,-1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-3,1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-3,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,1,2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,-1,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,-1,-2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,1,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,1,-2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,-1,2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,-1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,2,1] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,-4] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3,4] => [1,1,1]
=> [1,1]
=> 1
[1,-2,3,4] => [1,1,1]
=> [1,1]
=> 1
[-1,2,3,4] => [1,1,1]
=> [1,1]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,2,-4,-3] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[-2,-1,3,4] => [2,1,1]
=> [1,1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> 2
[2,1,-4,-3] => [2,2]
=> [2]
=> 2
[-2,-1,4,3] => [2,2]
=> [2]
=> 2
[-2,-1,-4,-3] => [2,2]
=> [2]
=> 2
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[-3,2,-1,4] => [2,1,1]
=> [1,1]
=> 1
[3,4,1,2] => [2,2]
=> [2]
=> 2
[3,-4,1,-2] => [2,2]
=> [2]
=> 2
[-3,4,-1,2] => [2,2]
=> [2]
=> 2
[-3,-4,-1,-2] => [2,2]
=> [2]
=> 2
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[-4,2,3,-1] => [2,1,1]
=> [1,1]
=> 1
[4,3,2,1] => [2,2]
=> [2]
=> 2
[4,-3,-2,1] => [2,2]
=> [2]
=> 2
[-4,3,2,-1] => [2,2]
=> [2]
=> 2
[-4,-3,-2,-1] => [2,2]
=> [2]
=> 2
Description
The number of semistandard Young tableaux of partition weight of given shape.
The weight of a semistandard Young tableaux is the sequence $(m_1, m_2,\dots)$, where $m_i$ is the number of occurrences of the number $i$ in the tableau. This statistic counts those tableaux whose weight is a weakly decreasing sequence.
Alternatively, this is the sum of the entries in the column specified by the partition of the change of basis matrix from Schur functions to monomial symmetric functions.
Matching statistic: St000933
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000933: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 7%●distinct values known / distinct values provided: 3%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000933: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 7%●distinct values known / distinct values provided: 3%
Values
[1] => [1]
=> []
=> ? ∊ {1,1}
[-1] => []
=> ?
=> ? ∊ {1,1}
[1,2] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2}
[1,-2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[-1,2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[-1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2}
[2,1] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[2,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2}
[-2,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2}
[-2,-1] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-2,3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-2,-3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,2,3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,2,-3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-2,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-2,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,3,2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,3,-2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-3,2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-3,-2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,3,2] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,3,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-3,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-3,-2] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,1,-3] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-1,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-1,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,1,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,1,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-1,-3] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,3,1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,3,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-3,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-3,-1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,3,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,3,-1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-3,1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-3,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,1,2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,-1,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,-1,-2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,1,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,1,-2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,-1,2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,-1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,2,1] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,-4] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3,4] => [1,1,1]
=> [1,1]
=> 1
[1,-2,3,4] => [1,1,1]
=> [1,1]
=> 1
[-1,2,3,4] => [1,1,1]
=> [1,1]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,2,-4,-3] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[-2,-1,3,4] => [2,1,1]
=> [1,1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> 2
[2,1,-4,-3] => [2,2]
=> [2]
=> 2
[-2,-1,4,3] => [2,2]
=> [2]
=> 2
[-2,-1,-4,-3] => [2,2]
=> [2]
=> 2
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[-3,2,-1,4] => [2,1,1]
=> [1,1]
=> 1
[3,4,1,2] => [2,2]
=> [2]
=> 2
[3,-4,1,-2] => [2,2]
=> [2]
=> 2
[-3,4,-1,2] => [2,2]
=> [2]
=> 2
[-3,-4,-1,-2] => [2,2]
=> [2]
=> 2
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[-4,2,3,-1] => [2,1,1]
=> [1,1]
=> 1
[4,3,2,1] => [2,2]
=> [2]
=> 2
[4,-3,-2,1] => [2,2]
=> [2]
=> 2
[-4,3,2,-1] => [2,2]
=> [2]
=> 2
[-4,-3,-2,-1] => [2,2]
=> [2]
=> 2
Description
The number of multipartitions of sizes given by an integer partition.
This is, for $\lambda = (\lambda_1,\ldots,\lambda_n)$, this is the number of $n$-tuples $(\lambda^{(1)},\ldots,\lambda^{(n)})$ of partitions $\lambda^{(i)}$ such that $\lambda^{(i)} \vdash \lambda_i$.
Matching statistic: St000937
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000937: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 7%●distinct values known / distinct values provided: 3%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000937: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 7%●distinct values known / distinct values provided: 3%
Values
[1] => [1]
=> []
=> ? ∊ {1,1}
[-1] => []
=> ?
=> ? ∊ {1,1}
[1,2] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2}
[1,-2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[-1,2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[-1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2}
[2,1] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[2,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2}
[-2,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2}
[-2,-1] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2}
[1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-2,3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-2,-3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,2,3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,2,-3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-2,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-2,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,3,2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,3,-2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-3,2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,-3,-2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,3,2] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,3,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-3,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-1,-3,-2] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,1,-3] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-1,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-1,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,1,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,1,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-1,-3] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,3,1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,3,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-3,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[2,-3,-1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,3,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,3,-1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-3,1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-2,-3,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,1,2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,-1,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,-1,-2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,1,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,1,-2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,-1,2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[-3,-1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[3,2,1] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,5,7,7,9,9,12,14,16,42}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 2
[1,2,3,-4] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3,4] => [1,1,1]
=> [1,1]
=> 1
[1,-2,3,4] => [1,1,1]
=> [1,1]
=> 1
[-1,2,3,4] => [1,1,1]
=> [1,1]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,2,-4,-3] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[-2,-1,3,4] => [2,1,1]
=> [1,1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> 2
[2,1,-4,-3] => [2,2]
=> [2]
=> 2
[-2,-1,4,3] => [2,2]
=> [2]
=> 2
[-2,-1,-4,-3] => [2,2]
=> [2]
=> 2
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[-3,2,-1,4] => [2,1,1]
=> [1,1]
=> 1
[3,4,1,2] => [2,2]
=> [2]
=> 2
[3,-4,1,-2] => [2,2]
=> [2]
=> 2
[-3,4,-1,2] => [2,2]
=> [2]
=> 2
[-3,-4,-1,-2] => [2,2]
=> [2]
=> 2
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[-4,2,3,-1] => [2,1,1]
=> [1,1]
=> 1
[4,3,2,1] => [2,2]
=> [2]
=> 2
[4,-3,-2,1] => [2,2]
=> [2]
=> 2
[-4,3,2,-1] => [2,2]
=> [2]
=> 2
[-4,-3,-2,-1] => [2,2]
=> [2]
=> 2
Description
The number of positive values of the symmetric group character corresponding to the partition.
For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugacy class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $2$.
The following 18 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St001568The smallest positive integer that does not appear twice in the partition. St000418The number of Dyck paths that are weakly below a Dyck path. St000420The number of Dyck paths that are weakly above a Dyck path. St000438The position of the last up step in a Dyck path. St000444The length of the maximal rise of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000981The length of the longest zigzag subpath. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001531Number of partial orders contained in the poset determined by the Dyck path. St001808The box weight or horizontal decoration of a Dyck path. St001959The product of the heights of the peaks of a Dyck path.
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