Your data matches 28 different statistics following compositions of up to 3 maps.
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Matching statistic: St001675
(load all 18 compositions to match this statistic)
Mapping
St001675: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,1] => 2
[2] => 1
[1,1,1] => 3
[1,2] => 0
[2,1] => 0
[3] => 1
[1,1,1,1] => 4
[1,1,2] => 1
[1,2,1] => 3
[1,3] => 0
[2,1,1] => 1
[2,2] => 2
[3,1] => 0
[4] => 1
[1,1,1,1,1] => 5
[1,1,1,2] => 2
[1,1,2,1] => 2
[1,1,3] => 1
[1,2,1,1] => 2
[1,2,2] => 1
[1,3,1] => 3
[1,4] => 0
[2,1,1,1] => 2
[2,1,2] => 3
[2,2,1] => 1
[2,3] => 0
[3,1,1] => 1
[3,2] => 0
[4,1] => 0
[5] => 1
[1,1,1,1,1,1] => 6
[1,1,1,1,2] => 3
[1,1,1,2,1] => 3
[1,1,1,3] => 2
[1,1,2,1,1] => 5
[1,1,2,2] => 0
[1,1,3,1] => 2
[1,1,4] => 1
[1,2,1,1,1] => 3
[1,2,1,2] => 0
[1,2,2,1] => 4
[1,2,3] => 1
[1,3,1,1] => 2
[1,3,2] => 1
[1,4,1] => 3
[1,5] => 0
[2,1,1,1,1] => 3
[2,1,1,2] => 4
[2,1,2,1] => 0
Description
The number of parts equal to the part in the reversed composition.
Matching statistic: St001247
Mapping
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001247: Integer partitions ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 55%
Values
[1] => [1] => [[1],[]]
 => []
 => ? = 1
[1,1] => [2] => [[2],[]]
 => []
 => ? ∊ {1,2}
[2] => [1] => [[1],[]]
 => []
 => ? ∊ {1,2}
[1,1,1] => [3] => [[3],[]]
 => []
 => ? ∊ {0,0,1,3}
[1,2] => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,0,1,3}
[2,1] => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,0,1,3}
[3] => [1] => [[1],[]]
 => []
 => ? ∊ {0,0,1,3}
[1,1,1,1] => [4] => [[4],[]]
 => []
 => ? ∊ {0,0,1,1,2,3,4}
[1,1,2] => [2,1] => [[2,2],[1]]
 => [1]
 => 1
[1,2,1] => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,1,1,2,3,4}
[1,3] => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,0,1,1,2,3,4}
[2,1,1] => [1,2] => [[2,1],[]]
 => []
 => ? ∊ {0,0,1,1,2,3,4}
[2,2] => [2] => [[2],[]]
 => []
 => ? ∊ {0,0,1,1,2,3,4}
[3,1] => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,0,1,1,2,3,4}
[4] => [1] => [[1],[]]
 => []
 => ? ∊ {0,0,1,1,2,3,4}
[1,1,1,1,1] => [5] => [[5],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,5}
[1,1,1,2] => [3,1] => [[3,3],[2]]
 => [2]
 => 0
[1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 2
[1,1,3] => [2,1] => [[2,2],[1]]
 => [1]
 => 1
[1,2,1,1] => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,5}
[1,2,2] => [1,2] => [[2,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,5}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,5}
[1,4] => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,5}
[2,1,1,1] => [1,3] => [[3,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,5}
[2,1,2] => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,5}
[2,2,1] => [2,1] => [[2,2],[1]]
 => [1]
 => 1
[2,3] => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,5}
[3,1,1] => [1,2] => [[2,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,5}
[3,2] => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,5}
[4,1] => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,5}
[5] => [1] => [[1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,2,2,2,3,3,5}
[1,1,1,1,1,1] => [6] => [[6],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,1,1,2] => [4,1] => [[4,4],[3]]
 => [3]
 => 1
[1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 0
[1,1,1,3] => [3,1] => [[3,3],[2]]
 => [2]
 => 0
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 2
[1,1,2,2] => [2,2] => [[3,2],[1]]
 => [1]
 => 1
[1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 2
[1,1,4] => [2,1] => [[2,2],[1]]
 => [1]
 => 1
[1,2,1,1,1] => [1,1,3] => [[3,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,2,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,2,2,1] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 1
[1,2,3] => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,3,1,1] => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,3,2] => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,4,1] => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,5] => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,5,6}
[2,1,1,1,1] => [1,4] => [[4,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,5,6}
[2,1,1,2] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 1
[2,1,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,5,6}
[2,1,3] => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,5,6}
[2,2,1,1] => [2,2] => [[3,2],[1]]
 => [1]
 => 1
[2,2,2] => [3] => [[3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,5,6}
[2,3,1] => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,5,6}
[2,4] => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,5,6}
[3,1,1,1] => [1,3] => [[3,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,5,6}
[3,1,2] => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,5,6}
[3,2,1] => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,5,6}
[3,3] => [2] => [[2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,5,6}
[4,1,1] => [1,2] => [[2,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,5,6}
[4,2] => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,5,6}
[5,1] => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,5,6}
[6] => [1] => [[1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,1,1,1,1,1] => [7] => [[7],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,7}
[1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => [4]
 => 1
[1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 2
[1,1,1,1,3] => [4,1] => [[4,4],[3]]
 => [3]
 => 1
[1,1,1,2,1,1] => [3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 0
[1,1,1,2,2] => [3,2] => [[4,3],[2]]
 => [2]
 => 0
[1,1,1,3,1] => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 0
[1,1,1,4] => [3,1] => [[3,3],[2]]
 => [2]
 => 0
[1,1,2,1,1,1] => [2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => 2
[1,1,2,1,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 3
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1
[1,1,2,3] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 2
[1,1,3,1,1] => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 2
[1,1,3,2] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 2
[1,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 2
[1,1,5] => [2,1] => [[2,2],[1]]
 => [1]
 => 1
[1,2,1,1,1,1] => [1,1,4] => [[4,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,7}
[1,2,1,1,2] => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 1
[1,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => 1
[2,1,1,1,2] => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 0
[2,1,1,2,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 2
[2,1,1,3] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 1
[2,2,1,1,1] => [2,3] => [[4,2],[1]]
 => [1]
 => 1
[2,2,1,2] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 2
[2,2,2,1] => [3,1] => [[3,3],[2]]
 => [2]
 => 0
[2,2,3] => [2,1] => [[2,2],[1]]
 => [1]
 => 1
[3,1,1,2] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 1
[3,3,1] => [2,1] => [[2,2],[1]]
 => [1]
 => 1
[1,1,1,1,1,1,2] => [6,1] => [[6,6],[5]]
 => [5]
 => 0
[1,1,1,1,1,2,1] => [5,1,1] => [[5,5,5],[4,4]]
 => [4,4]
 => 2
[1,1,1,1,1,3] => [5,1] => [[5,5],[4]]
 => [4]
 => 1
[1,1,1,1,2,1,1] => [4,1,2] => [[5,4,4],[3,3]]
 => [3,3]
 => 2
[1,1,1,1,2,2] => [4,2] => [[5,4],[3]]
 => [3]
 => 1
[1,1,1,1,3,1] => [4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 2
[1,1,1,1,4] => [4,1] => [[4,4],[3]]
 => [3]
 => 1
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
 => [2,2]
 => 0
[1,1,1,2,1,2] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => 0
Description
The number of parts of a partition that are not congruent 2 modulo 3.
Matching statistic: St001384
Mapping
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001384: Integer partitions ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 82%
Values
[1] => [1] => [[1],[]]
 => []
 => ? = 1
[1,1] => [2] => [[2],[]]
 => []
 => ? ∊ {1,2}
[2] => [1] => [[1],[]]
 => []
 => ? ∊ {1,2}
[1,1,1] => [3] => [[3],[]]
 => []
 => ? ∊ {0,0,1,3}
[1,2] => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,0,1,3}
[2,1] => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,0,1,3}
[3] => [1] => [[1],[]]
 => []
 => ? ∊ {0,0,1,3}
[1,1,1,1] => [4] => [[4],[]]
 => []
 => ? ∊ {0,1,1,1,2,3,4}
[1,1,2] => [2,1] => [[2,2],[1]]
 => [1]
 => 0
[1,2,1] => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,1,1,1,2,3,4}
[1,3] => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,1,1,1,2,3,4}
[2,1,1] => [1,2] => [[2,1],[]]
 => []
 => ? ∊ {0,1,1,1,2,3,4}
[2,2] => [2] => [[2],[]]
 => []
 => ? ∊ {0,1,1,1,2,3,4}
[3,1] => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,1,1,1,2,3,4}
[4] => [1] => [[1],[]]
 => []
 => ? ∊ {0,1,1,1,2,3,4}
[1,1,1,1,1] => [5] => [[5],[]]
 => []
 => ? ∊ {0,0,1,1,1,2,2,2,2,3,3,5}
[1,1,1,2] => [3,1] => [[3,3],[2]]
 => [2]
 => 1
[1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 1
[1,1,3] => [2,1] => [[2,2],[1]]
 => [1]
 => 0
[1,2,1,1] => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {0,0,1,1,1,2,2,2,2,3,3,5}
[1,2,2] => [1,2] => [[2,1],[]]
 => []
 => ? ∊ {0,0,1,1,1,2,2,2,2,3,3,5}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,1,1,1,2,2,2,2,3,3,5}
[1,4] => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,0,1,1,1,2,2,2,2,3,3,5}
[2,1,1,1] => [1,3] => [[3,1],[]]
 => []
 => ? ∊ {0,0,1,1,1,2,2,2,2,3,3,5}
[2,1,2] => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,1,1,1,2,2,2,2,3,3,5}
[2,2,1] => [2,1] => [[2,2],[1]]
 => [1]
 => 0
[2,3] => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,0,1,1,1,2,2,2,2,3,3,5}
[3,1,1] => [1,2] => [[2,1],[]]
 => []
 => ? ∊ {0,0,1,1,1,2,2,2,2,3,3,5}
[3,2] => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,0,1,1,1,2,2,2,2,3,3,5}
[4,1] => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,0,1,1,1,2,2,2,2,3,3,5}
[5] => [1] => [[1],[]]
 => []
 => ? ∊ {0,0,1,1,1,2,2,2,2,3,3,5}
[1,1,1,1,1,1] => [6] => [[6],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,1,1,2] => [4,1] => [[4,4],[3]]
 => [3]
 => 2
[1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 1
[1,1,1,3] => [3,1] => [[3,3],[2]]
 => [2]
 => 1
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 1
[1,1,2,2] => [2,2] => [[3,2],[1]]
 => [1]
 => 0
[1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 1
[1,1,4] => [2,1] => [[2,2],[1]]
 => [1]
 => 0
[1,2,1,1,1] => [1,1,3] => [[3,1,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,2,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,2,2,1] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[1,2,3] => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,3,1,1] => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,3,2] => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,4,1] => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,5] => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[2,1,1,1,1] => [1,4] => [[4,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[2,1,1,2] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[2,1,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[2,1,3] => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[2,2,1,1] => [2,2] => [[3,2],[1]]
 => [1]
 => 0
[2,2,2] => [3] => [[3],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[2,3,1] => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[2,4] => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[3,1,1,1] => [1,3] => [[3,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[3,1,2] => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[3,2,1] => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[3,3] => [2] => [[2],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[4,1,1] => [1,2] => [[2,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[4,2] => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[5,1] => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[6] => [1] => [[1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,1,1,1,1,1] => [7] => [[7],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,7}
[1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
 => [4]
 => 3
[1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 3
[1,1,1,1,3] => [4,1] => [[4,4],[3]]
 => [3]
 => 2
[1,1,1,2,1,1] => [3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 1
[1,1,1,2,2] => [3,2] => [[4,3],[2]]
 => [2]
 => 1
[1,1,1,3,1] => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 1
[1,1,1,4] => [3,1] => [[3,3],[2]]
 => [2]
 => 1
[1,1,2,1,1,1] => [2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => 1
[1,1,2,1,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 2
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 0
[1,1,2,3] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 1
[1,1,3,1,1] => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 1
[1,1,3,2] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 1
[1,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 1
[1,1,5] => [2,1] => [[2,2],[1]]
 => [1]
 => 0
[1,2,1,1,1,1] => [1,1,4] => [[4,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,7}
[1,2,1,1,2] => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 0
[1,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => 0
[2,1,1,1,2] => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 1
[2,1,1,2,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 1
[2,1,1,3] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[2,2,1,1,1] => [2,3] => [[4,2],[1]]
 => [1]
 => 0
[2,2,1,2] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 1
[2,2,2,1] => [3,1] => [[3,3],[2]]
 => [2]
 => 1
[2,2,3] => [2,1] => [[2,2],[1]]
 => [1]
 => 0
[3,1,1,2] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[3,3,1] => [2,1] => [[2,2],[1]]
 => [1]
 => 0
[1,1,1,1,1,1,2] => [6,1] => [[6,6],[5]]
 => [5]
 => 4
[1,1,1,1,1,2,1] => [5,1,1] => [[5,5,5],[4,4]]
 => [4,4]
 => 5
[1,1,1,1,1,3] => [5,1] => [[5,5],[4]]
 => [4]
 => 3
[1,1,1,1,2,1,1] => [4,1,2] => [[5,4,4],[3,3]]
 => [3,3]
 => 3
[1,1,1,1,2,2] => [4,2] => [[5,4],[3]]
 => [3]
 => 2
[1,1,1,1,3,1] => [4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 3
[1,1,1,1,4] => [4,1] => [[4,4],[3]]
 => [3]
 => 2
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
 => [2,2]
 => 1
[1,1,1,2,1,2] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => 3
Description
The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains.
Matching statistic: St001630
Mapping
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 18%
Values
[1] => [1] => [[1],[]]
 => ([],1)
 => ? = 1
[1,1] => [2] => [[2],[]]
 => ([],1)
 => ? ∊ {1,2}
[2] => [1] => [[1],[]]
 => ([],1)
 => ? ∊ {1,2}
[1,1,1] => [3] => [[3],[]]
 => ([],1)
 => ? ∊ {0,0,1,3}
[1,2] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,3}
[2,1] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,3}
[3] => [1] => [[1],[]]
 => ([],1)
 => ? ∊ {0,0,1,3}
[1,1,1,1] => [4] => [[4],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[1,1,2] => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[1,2,1] => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[1,3] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[2,1,1] => [1,2] => [[2,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[2,2] => [2] => [[2],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[3,1] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[4] => [1] => [[1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[1,1,1,1,1] => [5] => [[5],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,1,1,2] => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,1,3] => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,2,1,1] => [1,1,2] => [[2,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,2,2] => [1,2] => [[2,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,4] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[2,1,1,1] => [1,3] => [[3,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[2,1,2] => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[2,2,1] => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[2,3] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[3,1,1] => [1,2] => [[2,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[3,2] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[4,1] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[5] => [1] => [[1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,1,1,1,1,1] => [6] => [[6],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,1,1,2] => [4,1] => [[4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,1,3] => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,2,2] => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,4] => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,2,1,1,1] => [1,1,3] => [[3,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,2,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,2,2,1] => [1,2,1] => [[2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,2,3] => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,3,1,1] => [1,1,2] => [[2,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,3,2] => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,4,1] => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,5] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[2,1,1,1,1] => [1,4] => [[4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[2,1,1,2] => [1,2,1] => [[2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[2,1,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,1,2,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,2,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,1,2,2,1,1] => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[1,2,1,1,2,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,2,2,1,1,1] => [1,2,3] => [[4,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[2,1,1,2,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,1,2,2] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[2,2,1,1,2] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,1,1,2,1,1,1] => [4,1,3] => [[6,4,4],[3,3]]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,1,1,2,2,1] => [4,2,1] => [[5,5,4],[4,3]]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,1,2,1,1,2] => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,1,1,2,2,1,1] => [3,2,2] => [[5,4,3],[3,2]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
[1,1,1,2,2,2] => [3,3] => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,1,3,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,2,1,1,1,2] => [2,1,3,1] => [[4,4,2,2],[3,1,1]]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 2
[1,1,2,1,1,2,1] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2
[1,1,2,1,1,3] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,1,2,2,1,1,1] => [2,2,3] => [[5,3,2],[2,1]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
[1,1,2,2,1,2] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,2,2,2,1] => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,1,2,2,3] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,3,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,1,3,3,1] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,2,1,1,1,2,1] => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
 => ([(0,2),(2,1)],3)
 => 1
[1,2,1,1,2,1,1] => [1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2
[1,2,1,1,2,2] => [1,1,2,2] => [[3,2,1,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,2,1,1,3,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,2,2,1,1,1,1] => [1,2,4] => [[5,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,2,2,1,1,2] => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2
[1,2,2,2,1,1] => [1,3,2] => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,1,1,2,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,3,3,1,1] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[2,1,1,1,2,1,1] => [1,3,1,2] => [[4,3,3,1],[2,2]]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 2
[2,1,1,1,2,2] => [1,3,2] => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,1,2,1,1,1] => [1,2,1,3] => [[4,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,1,2,2,1] => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2
[2,1,1,3,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,2,2,1,1] => [1,1,2,2] => [[3,2,1,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[2,2,1,1,1,2] => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,2,1,1,2,1] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 1
[2,2,1,1,3] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1
[2,2,2,1,1,1] => [3,3] => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 1
[3,1,1,2,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,1,1,2,2] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[3,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,1,1,2,1,1,2] => [4,1,2,1] => [[5,5,4,4],[4,3,3]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,1,1,1,2,2,1,1] => [4,2,2] => [[6,5,4],[4,3]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St001875
Mapping
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 36%
Values
[1] => [1] => [[1],[]]
 => ([],1)
 => ? = 1
[1,1] => [2] => [[2],[]]
 => ([],1)
 => ? ∊ {1,2}
[2] => [1] => [[1],[]]
 => ([],1)
 => ? ∊ {1,2}
[1,1,1] => [3] => [[3],[]]
 => ([],1)
 => ? ∊ {0,0,1,3}
[1,2] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,3}
[2,1] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,3}
[3] => [1] => [[1],[]]
 => ([],1)
 => ? ∊ {0,0,1,3}
[1,1,1,1] => [4] => [[4],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[1,1,2] => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[1,2,1] => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[1,3] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[2,1,1] => [1,2] => [[2,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[2,2] => [2] => [[2],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[3,1] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[4] => [1] => [[1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[1,1,1,1,1] => [5] => [[5],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,1,1,2] => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,1,3] => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,2,1,1] => [1,1,2] => [[2,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,2,2] => [1,2] => [[2,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,4] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[2,1,1,1] => [1,3] => [[3,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[2,1,2] => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[2,2,1] => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[2,3] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[3,1,1] => [1,2] => [[2,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[3,2] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[4,1] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[5] => [1] => [[1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,1,1,1,1,1] => [6] => [[6],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,1,1,2] => [4,1] => [[4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,1,3] => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,2,2] => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,4] => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,2,1,1,1] => [1,1,3] => [[3,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,2,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,2,2,1] => [1,2,1] => [[2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,2,3] => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,3,1,1] => [1,1,2] => [[2,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,3,2] => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,4,1] => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,5] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[2,1,1,1,1] => [1,4] => [[4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[2,1,1,2] => [1,2,1] => [[2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[2,1,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 3
[1,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,2,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 3
[1,1,2,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
[1,1,2,2,1,1] => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 4
[1,2,1,1,2,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => 3
[1,2,2,1,1,1] => [1,2,3] => [[4,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 3
[2,1,1,2,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
[2,1,1,2,2] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 3
[2,2,1,1,2] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,1,2,1,1,1] => [4,1,3] => [[6,4,4],[3,3]]
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,1,2,2,1] => [4,2,1] => [[5,5,4],[4,3]]
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,2,1,1,2] => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
[1,1,1,2,2,1,1] => [3,2,2] => [[5,4,3],[3,2]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 5
[1,1,1,2,2,2] => [3,3] => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,3,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
 => ([(0,2),(2,1)],3)
 => 3
[1,1,2,1,1,1,2] => [2,1,3,1] => [[4,4,2,2],[3,1,1]]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 4
[1,1,2,1,1,2,1] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 4
[1,1,2,1,1,3] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
[1,1,2,2,1,1,1] => [2,2,3] => [[5,3,2],[2,1]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 5
[1,1,2,2,1,2] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 3
[1,1,2,2,2,1] => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
[1,1,2,2,3] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 3
[1,1,3,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
[1,1,3,3,1] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 3
[1,2,1,1,1,2,1] => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
 => ([(0,2),(2,1)],3)
 => 3
[1,2,1,1,2,1,1] => [1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 4
[1,2,1,1,2,2] => [1,1,2,2] => [[3,2,1,1],[1]]
 => ([(0,2),(2,1)],3)
 => 3
[1,2,1,1,3,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => 3
[1,2,2,1,1,1,1] => [1,2,4] => [[5,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 3
[1,2,2,1,1,2] => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 4
[1,2,2,2,1,1] => [1,3,2] => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
[1,3,1,1,2,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => 3
[1,3,3,1,1] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 3
[2,1,1,1,2,1,1] => [1,3,1,2] => [[4,3,3,1],[2,2]]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 4
[2,1,1,1,2,2] => [1,3,2] => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
[2,1,1,2,1,1,1] => [1,2,1,3] => [[4,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
[2,1,1,2,2,1] => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 4
[2,1,1,3,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
[2,1,2,2,1,1] => [1,1,2,2] => [[3,2,1,1],[1]]
 => ([(0,2),(2,1)],3)
 => 3
[2,2,1,1,1,2] => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
[2,2,1,1,2,1] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 3
[2,2,1,1,3] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 3
[2,2,2,1,1,1] => [3,3] => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 3
[3,1,1,2,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
[3,1,1,2,2] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 3
[3,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,1,2,1,1,2] => [4,1,2,1] => [[5,5,4,4],[4,3,3]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
[1,1,1,1,2,2,1,1] => [4,2,2] => [[6,5,4],[4,3]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 5
Description
The number of simple modules with projective dimension at most 1.
Matching statistic: St001877
Mapping
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 36%
Values
[1] => [1] => [[1],[]]
 => ([],1)
 => ? = 1
[1,1] => [2] => [[2],[]]
 => ([],1)
 => ? ∊ {1,2}
[2] => [1] => [[1],[]]
 => ([],1)
 => ? ∊ {1,2}
[1,1,1] => [3] => [[3],[]]
 => ([],1)
 => ? ∊ {0,0,1,3}
[1,2] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,3}
[2,1] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,3}
[3] => [1] => [[1],[]]
 => ([],1)
 => ? ∊ {0,0,1,3}
[1,1,1,1] => [4] => [[4],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[1,1,2] => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[1,2,1] => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[1,3] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[2,1,1] => [1,2] => [[2,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[2,2] => [2] => [[2],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[3,1] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[4] => [1] => [[1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[1,1,1,1,1] => [5] => [[5],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,1,1,2] => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,1,3] => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,2,1,1] => [1,1,2] => [[2,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,2,2] => [1,2] => [[2,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,4] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[2,1,1,1] => [1,3] => [[3,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[2,1,2] => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[2,2,1] => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[2,3] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[3,1,1] => [1,2] => [[2,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[3,2] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[4,1] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[5] => [1] => [[1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,1,1,1,1,1] => [6] => [[6],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,1,1,2] => [4,1] => [[4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,1,3] => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,2,2] => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,4] => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,2,1,1,1] => [1,1,3] => [[3,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,2,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,2,2,1] => [1,2,1] => [[2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,2,3] => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,3,1,1] => [1,1,2] => [[2,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,3,2] => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,4,1] => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,5] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[2,1,1,1,1] => [1,4] => [[4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[2,1,1,2] => [1,2,1] => [[2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[2,1,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,1,2,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,2,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,2,2,1,1] => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
[1,2,1,1,2,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,2,2,1,1,1] => [1,2,3] => [[4,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[2,1,1,2,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,1,1,2,2] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[2,2,1,1,2] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,1,1,2,1,1,1] => [4,1,3] => [[6,4,4],[3,3]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,1,1,2,2,1] => [4,2,1] => [[5,5,4],[4,3]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,1,2,1,1,2] => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,1,2,2,1,1] => [3,2,2] => [[5,4,3],[3,2]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 1
[1,1,1,2,2,2] => [3,3] => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,1,3,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,2,1,1,1,2] => [2,1,3,1] => [[4,4,2,2],[3,1,1]]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 2
[1,1,2,1,1,2,1] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[1,1,2,1,1,3] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,2,2,1,1,1] => [2,2,3] => [[5,3,2],[2,1]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 1
[1,1,2,2,1,2] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,2,2,2,1] => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,2,2,3] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,3,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,3,3,1] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,2,1,1,1,2,1] => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
 => ([(0,2),(2,1)],3)
 => 0
[1,2,1,1,2,1,1] => [1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[1,2,1,1,2,2] => [1,1,2,2] => [[3,2,1,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,2,1,1,3,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,2,2,1,1,1,1] => [1,2,4] => [[5,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,2,2,1,1,2] => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[1,2,2,2,1,1] => [1,3,2] => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,3,1,1,2,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,3,3,1,1] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[2,1,1,1,2,1,1] => [1,3,1,2] => [[4,3,3,1],[2,2]]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 2
[2,1,1,1,2,2] => [1,3,2] => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,1,1,2,1,1,1] => [1,2,1,3] => [[4,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,1,1,2,2,1] => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[2,1,1,3,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,1,2,2,1,1] => [1,1,2,2] => [[3,2,1,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[2,2,1,1,1,2] => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,2,1,1,2,1] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[2,2,1,1,3] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[2,2,2,1,1,1] => [3,3] => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 0
[3,1,1,2,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[3,1,1,2,2] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[3,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,1,1,2,1,1,2] => [4,1,2,1] => [[5,5,4,4],[4,3,3]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,1,1,2,2,1,1] => [4,2,2] => [[6,5,4],[4,3]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 1
Description
Number of indecomposable injective modules with projective dimension 2.
Matching statistic: St001878
Mapping
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 18%
Values
[1] => [1] => [[1],[]]
 => ([],1)
 => ? = 1
[1,1] => [2] => [[2],[]]
 => ([],1)
 => ? ∊ {1,2}
[2] => [1] => [[1],[]]
 => ([],1)
 => ? ∊ {1,2}
[1,1,1] => [3] => [[3],[]]
 => ([],1)
 => ? ∊ {0,0,1,3}
[1,2] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,3}
[2,1] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,3}
[3] => [1] => [[1],[]]
 => ([],1)
 => ? ∊ {0,0,1,3}
[1,1,1,1] => [4] => [[4],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[1,1,2] => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[1,2,1] => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[1,3] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[2,1,1] => [1,2] => [[2,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[2,2] => [2] => [[2],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[3,1] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[4] => [1] => [[1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[1,1,1,1,1] => [5] => [[5],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,1,1,2] => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,1,3] => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,2,1,1] => [1,1,2] => [[2,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,2,2] => [1,2] => [[2,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,4] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[2,1,1,1] => [1,3] => [[3,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[2,1,2] => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[2,2,1] => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[2,3] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[3,1,1] => [1,2] => [[2,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[3,2] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[4,1] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[5] => [1] => [[1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,1,1,1,1,1] => [6] => [[6],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,1,1,2] => [4,1] => [[4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,1,3] => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,2,2] => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,4] => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,2,1,1,1] => [1,1,3] => [[3,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,2,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,2,2,1] => [1,2,1] => [[2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,2,3] => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,3,1,1] => [1,1,2] => [[2,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,3,2] => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,4,1] => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,5] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[2,1,1,1,1] => [1,4] => [[4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[2,1,1,2] => [1,2,1] => [[2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[2,1,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,1,2,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,2,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,1,2,2,1,1] => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[1,2,1,1,2,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,2,2,1,1,1] => [1,2,3] => [[4,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[2,1,1,2,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,1,2,2] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[2,2,1,1,2] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,1,1,2,1,1,1] => [4,1,3] => [[6,4,4],[3,3]]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,1,1,2,2,1] => [4,2,1] => [[5,5,4],[4,3]]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,1,2,1,1,2] => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,1,1,2,2,1,1] => [3,2,2] => [[5,4,3],[3,2]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
[1,1,1,2,2,2] => [3,3] => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,1,3,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,2,1,1,1,2] => [2,1,3,1] => [[4,4,2,2],[3,1,1]]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 2
[1,1,2,1,1,2,1] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[1,1,2,1,1,3] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,1,2,2,1,1,1] => [2,2,3] => [[5,3,2],[2,1]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
[1,1,2,2,1,2] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,2,2,2,1] => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,1,2,2,3] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,3,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,1,3,3,1] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,2,1,1,1,2,1] => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
 => ([(0,2),(2,1)],3)
 => 1
[1,2,1,1,2,1,1] => [1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[1,2,1,1,2,2] => [1,1,2,2] => [[3,2,1,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,2,1,1,3,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,2,2,1,1,1,1] => [1,2,4] => [[5,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,2,2,1,1,2] => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[1,2,2,2,1,1] => [1,3,2] => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,1,1,2,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,3,3,1,1] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[2,1,1,1,2,1,1] => [1,3,1,2] => [[4,3,3,1],[2,2]]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 2
[2,1,1,1,2,2] => [1,3,2] => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,1,2,1,1,1] => [1,2,1,3] => [[4,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,1,2,2,1] => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[2,1,1,3,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,2,2,1,1] => [1,1,2,2] => [[3,2,1,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[2,2,1,1,1,2] => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,2,1,1,2,1] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 1
[2,2,1,1,3] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1
[2,2,2,1,1,1] => [3,3] => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 1
[3,1,1,2,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,1,1,2,2] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[3,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,1,1,2,1,1,2] => [4,1,2,1] => [[5,5,4,4],[4,3,3]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,1,1,1,2,2,1,1] => [4,2,2] => [[6,5,4],[4,3]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St001876
Mapping
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 27%
Values
[1] => [1] => [[1],[]]
 => ([],1)
 => ? = 1
[1,1] => [2] => [[2],[]]
 => ([],1)
 => ? ∊ {1,2}
[2] => [1] => [[1],[]]
 => ([],1)
 => ? ∊ {1,2}
[1,1,1] => [3] => [[3],[]]
 => ([],1)
 => ? ∊ {0,0,1,3}
[1,2] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,3}
[2,1] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,3}
[3] => [1] => [[1],[]]
 => ([],1)
 => ? ∊ {0,0,1,3}
[1,1,1,1] => [4] => [[4],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[1,1,2] => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[1,2,1] => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[1,3] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[2,1,1] => [1,2] => [[2,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[2,2] => [2] => [[2],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[3,1] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[4] => [1] => [[1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,2,3,4}
[1,1,1,1,1] => [5] => [[5],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,1,1,2] => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,1,3] => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,2,1,1] => [1,1,2] => [[2,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,2,2] => [1,2] => [[2,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,4] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[2,1,1,1] => [1,3] => [[3,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[2,1,2] => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[2,2,1] => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[2,3] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[3,1,1] => [1,2] => [[2,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[3,2] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[4,1] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[5] => [1] => [[1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,5}
[1,1,1,1,1,1] => [6] => [[6],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,1,1,2] => [4,1] => [[4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,1,3] => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,2,2] => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,4] => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,2,1,1,1] => [1,1,3] => [[3,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,2,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,2,2,1] => [1,2,1] => [[2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,2,3] => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,3,1,1] => [1,1,2] => [[2,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,3,2] => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,4,1] => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,5] => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[2,1,1,1,1] => [1,4] => [[4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[2,1,1,2] => [1,2,1] => [[2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[2,1,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,1,2,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,2,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,2,2,1,1] => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
[1,2,1,1,2,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,2,2,1,1,1] => [1,2,3] => [[4,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[2,1,1,2,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,1,1,2,2] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[2,2,1,1,2] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,1,1,2,1,1,1] => [4,1,3] => [[6,4,4],[3,3]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,1,1,2,2,1] => [4,2,1] => [[5,5,4],[4,3]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,1,2,1,1,2] => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,1,2,2,1,1] => [3,2,2] => [[5,4,3],[3,2]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 1
[1,1,1,2,2,2] => [3,3] => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,1,3,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,2,1,1,2,1] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[1,1,2,1,1,3] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,2,2,1,1,1] => [2,2,3] => [[5,3,2],[2,1]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 1
[1,1,2,2,1,2] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,2,2,2,1] => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,2,2,3] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,3,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,3,3,1] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,2,1,1,1,2,1] => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
 => ([(0,2),(2,1)],3)
 => 0
[1,2,1,1,2,1,1] => [1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[1,2,1,1,2,2] => [1,1,2,2] => [[3,2,1,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,2,1,1,3,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,2,2,1,1,1,1] => [1,2,4] => [[5,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,2,2,1,1,2] => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[1,2,2,2,1,1] => [1,3,2] => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,3,1,1,2,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,3,3,1,1] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[2,1,1,1,2,2] => [1,3,2] => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,1,1,2,1,1,1] => [1,2,1,3] => [[4,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,1,1,2,2,1] => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[2,1,1,3,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,1,2,2,1,1] => [1,1,2,2] => [[3,2,1,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[2,2,1,1,1,2] => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,2,1,1,2,1] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[2,2,1,1,3] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[2,2,2,1,1,1] => [3,3] => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 0
[3,1,1,2,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[3,1,1,2,2] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[3,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,1,1,2,1,1,2] => [4,1,2,1] => [[5,5,4,4],[4,3,3]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,1,1,2,2,1,1] => [4,2,2] => [[6,5,4],[4,3]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 1
[1,1,1,1,2,2,2] => [4,3] => [[6,4],[3]]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,1,1,3,1,1,1] => [4,1,3] => [[6,4,4],[3,3]]
 => ([(0,2),(2,1)],3)
 => 0
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Matching statistic: St000714
(load all 2 compositions to match this statistic)
Mapping
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000714: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 82%
Values
[1] => [[1],[]]
 => []
 => ?
 => ? = 1
[1,1] => [[1,1],[]]
 => []
 => ?
 => ? ∊ {1,2}
[2] => [[2],[]]
 => []
 => ?
 => ? ∊ {1,2}
[1,1,1] => [[1,1,1],[]]
 => []
 => ?
 => ? ∊ {0,0,1,3}
[1,2] => [[2,1],[]]
 => []
 => ?
 => ? ∊ {0,0,1,3}
[2,1] => [[2,2],[1]]
 => [1]
 => []
 => ? ∊ {0,0,1,3}
[3] => [[3],[]]
 => []
 => ?
 => ? ∊ {0,0,1,3}
[1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ?
 => ? ∊ {0,0,1,1,1,2,3,4}
[1,1,2] => [[2,1,1],[]]
 => []
 => ?
 => ? ∊ {0,0,1,1,1,2,3,4}
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => []
 => ? ∊ {0,0,1,1,1,2,3,4}
[1,3] => [[3,1],[]]
 => []
 => ?
 => ? ∊ {0,0,1,1,1,2,3,4}
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => [1]
 => ? ∊ {0,0,1,1,1,2,3,4}
[2,2] => [[3,2],[1]]
 => [1]
 => []
 => ? ∊ {0,0,1,1,1,2,3,4}
[3,1] => [[3,3],[2]]
 => [2]
 => []
 => ? ∊ {0,0,1,1,1,2,3,4}
[4] => [[4],[]]
 => []
 => ?
 => ? ∊ {0,0,1,1,1,2,3,4}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ?
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,5}
[1,1,1,2] => [[2,1,1,1],[]]
 => []
 => ?
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,5}
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,5}
[1,1,3] => [[3,1,1],[]]
 => []
 => ?
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,5}
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,5}
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,5}
[1,3,1] => [[3,3,1],[2]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,5}
[1,4] => [[4,1],[]]
 => []
 => ?
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,5}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,5}
[2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,5}
[2,3] => [[4,2],[1]]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,5}
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => [2]
 => 3
[3,2] => [[4,3],[2]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,5}
[4,1] => [[4,4],[3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,5}
[5] => [[5],[]]
 => []
 => ?
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,5}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,4,5,6}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,4,5,6}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,4,5,6}
[1,1,1,3] => [[3,1,1,1],[]]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,4,5,6}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,4,5,6}
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,4,5,6}
[1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,4,5,6}
[1,1,4] => [[4,1,1],[]]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,4,5,6}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,4,5,6}
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,4,5,6}
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,4,5,6}
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => [2]
 => 3
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,4,5,6}
[1,4,1] => [[4,4,1],[3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,4,5,6}
[1,5] => [[5,1],[]]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,4,5,6}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 1
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,4,5,6}
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 2
[2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,4,5,6}
[2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,4,5,6}
[2,4] => [[5,2],[1]]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,4,5,6}
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => [2,2]
 => 1
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => [2]
 => 3
[3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => [2]
 => 3
[3,3] => [[5,3],[2]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,4,5,6}
[4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => [3]
 => 4
[4,2] => [[5,4],[3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,4,5,6}
[5,1] => [[5,5],[4]]
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,4,5,6}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
 => [2,2]
 => [2]
 => 3
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 1
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 2
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
 => [2,2,2]
 => [2,2]
 => 1
[1,3,1,2] => [[4,3,3,1],[2,2]]
 => [2,2]
 => [2]
 => 3
[1,3,2,1] => [[4,4,3,1],[3,2]]
 => [3,2]
 => [2]
 => 3
[1,4,1,1] => [[4,4,4,1],[3,3]]
 => [3,3]
 => [3]
 => 4
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[2,1,1,3] => [[4,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
 => [2,2,1,1]
 => [2,1,1]
 => 0
[2,1,2,2] => [[4,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 1
[2,1,3,1] => [[4,4,2,2],[3,1,1]]
 => [3,1,1]
 => [1,1]
 => 1
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
 => [2,2,2,1]
 => [2,2,1]
 => 0
[2,2,1,2] => [[4,3,3,2],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 2
[2,2,2,1] => [[4,4,3,2],[3,2,1]]
 => [3,2,1]
 => [2,1]
 => 2
[2,3,1,1] => [[4,4,4,2],[3,3,1]]
 => [3,3,1]
 => [3,1]
 => 3
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]]
 => [2,2,2,2]
 => [2,2,2]
 => 0
[3,1,1,2] => [[4,3,3,3],[2,2,2]]
 => [2,2,2]
 => [2,2]
 => 1
[3,1,2,1] => [[4,4,3,3],[3,2,2]]
 => [3,2,2]
 => [2,2]
 => 1
[3,1,3] => [[5,3,3],[2,2]]
 => [2,2]
 => [2]
 => 3
[3,2,1,1] => [[4,4,4,3],[3,3,2]]
 => [3,3,2]
 => [3,2]
 => 2
[3,2,2] => [[5,4,3],[3,2]]
 => [3,2]
 => [2]
 => 3
[3,3,1] => [[5,5,3],[4,2]]
 => [4,2]
 => [2]
 => 3
[4,1,1,1] => [[4,4,4,4],[3,3,3]]
 => [3,3,3]
 => [3,3]
 => 1
[4,1,2] => [[5,4,4],[3,3]]
 => [3,3]
 => [3]
 => 4
[4,2,1] => [[5,5,4],[4,3]]
 => [4,3]
 => [3]
 => 4
[5,1,1] => [[5,5,5],[4,4]]
 => [4,4]
 => [4]
 => 5
[1,1,1,2,1,1,1] => [[2,2,2,2,1,1,1],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1
[1,1,2,1,1,1,1] => [[2,2,2,2,2,1,1],[1,1,1,1]]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,1,2,1,1,2] => [[3,2,2,2,1,1],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1
[1,1,2,1,2,1] => [[3,3,2,2,1,1],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 1
[1,1,2,2,1,1] => [[3,3,3,2,1,1],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 2
[1,1,3,1,2] => [[4,3,3,1,1],[2,2]]
 => [2,2]
 => [2]
 => 3
Description
The number of semistandard Young tableau of given shape, with entries at most 2. This is also the dimension of the corresponding irreducible representation of $GL_2$.
Matching statistic: St000284
Mapping
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000284: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 9%
Values
[1] => ([],1)
 => [1]
 => []
 => ? = 1
[1,1] => ([(0,1)],2)
 => [2]
 => []
 => ? ∊ {1,2}
[2] => ([],2)
 => [1,1]
 => [1]
 => ? ∊ {1,2}
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ? ∊ {0,0,3}
[1,2] => ([(1,2)],3)
 => [2,1]
 => [1]
 => ? ∊ {0,0,3}
[2,1] => ([(0,2),(1,2)],3)
 => [3]
 => []
 => ? ∊ {0,0,3}
[3] => ([],3)
 => [1,1,1]
 => [1,1]
 => 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,1,2,3,4}
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => ? ∊ {0,0,1,2,3,4}
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,1,2,3,4}
[1,3] => ([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,1,2,3,4}
[2,2] => ([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => ? ∊ {0,0,1,2,3,4}
[3,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,1,2,3,4}
[4] => ([],4)
 => [1,1,1,1]
 => [1,1,1]
 => 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,3,5}
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,3,5}
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,3,5}
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,3,5}
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,3,5}
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,3,5}
[1,4] => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,3,5}
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,3,5}
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,3,5}
[2,3] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,3,5}
[3,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,3,5}
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,3,5}
[5] => ([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 1
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[1,5] => ([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 1
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[2,4] => ([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 1
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[3,3] => ([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,6}
[6] => ([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
[1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,7}
[1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,7}
[1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => 1
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 1
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => 1
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => 1
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => 1
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 1
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => 1
[1,6] => ([(5,6)],7)
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => 1
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 1
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => 1
[2,5] => ([(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => 1
[3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => 1
[3,4] => ([(3,6),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 1
[4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => 1
[7] => ([],7)
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
[1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,1,1,1,1]
 => [1,1,1,1]
 => 1
[1,1,3,3] => ([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [1,1]
 => 1
[1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [1,1]
 => 1
[1,3,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [1,1]
 => 1
[1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [1,1]
 => 1
[2,1,2,3] => ([(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [1,1]
 => 1
[2,2,1,3] => ([(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [1,1]
 => 1
[2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [1,1]
 => 1
[3,1,1,3] => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [1,1]
 => 1
[3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [1,1]
 => 1
[4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [1,1]
 => 1
[4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
 => [5,1,1,1]
 => [1,1,1]
 => 1
[8] => ([],8)
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => 1
Description
The Plancherel distribution on integer partitions. This is defined as the distribution induced by the RSK shape of the uniform distribution on permutations. In other words, this is the size of the preimage of the map 'Robinson-Schensted tableau shape' from permutations to integer partitions. Equivalently, this is given by the square of the number of standard Young tableaux of the given shape.
The following 18 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. 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. 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. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000941The number of characters of the symmetric group whose value on the partition is even. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000456The monochromatic index of a connected graph. St001118The acyclic chromatic index of a graph.