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Matching statistic: St000480
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Mp00079: Set partitions —shape⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1]
=> 0
{{1,2}}
=> [2]
=> 1
{{1},{2}}
=> [1,1]
=> 0
{{1,2,3}}
=> [3]
=> 1
{{1,2},{3}}
=> [2,1]
=> 1
{{1,3},{2}}
=> [2,1]
=> 1
{{1},{2,3}}
=> [2,1]
=> 1
{{1},{2},{3}}
=> [1,1,1]
=> 0
{{1,2,3,4}}
=> [4]
=> 1
{{1,2,3},{4}}
=> [3,1]
=> 1
{{1,2,4},{3}}
=> [3,1]
=> 1
{{1,2},{3,4}}
=> [2,2]
=> 1
{{1,2},{3},{4}}
=> [2,1,1]
=> 1
{{1,3,4},{2}}
=> [3,1]
=> 1
{{1,3},{2,4}}
=> [2,2]
=> 1
{{1,3},{2},{4}}
=> [2,1,1]
=> 1
{{1,4},{2,3}}
=> [2,2]
=> 1
{{1},{2,3,4}}
=> [3,1]
=> 1
{{1},{2,3},{4}}
=> [2,1,1]
=> 1
{{1,4},{2},{3}}
=> [2,1,1]
=> 1
{{1},{2,4},{3}}
=> [2,1,1]
=> 1
{{1},{2},{3,4}}
=> [2,1,1]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> 0
{{1,2,3,4,5}}
=> [5]
=> 1
{{1,2,3,4},{5}}
=> [4,1]
=> 1
{{1,2,3,5},{4}}
=> [4,1]
=> 1
{{1,2,3},{4,5}}
=> [3,2]
=> 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> 1
{{1,2,4,5},{3}}
=> [4,1]
=> 1
{{1,2,4},{3,5}}
=> [3,2]
=> 1
{{1,2,4},{3},{5}}
=> [3,1,1]
=> 1
{{1,2,5},{3,4}}
=> [3,2]
=> 1
{{1,2},{3,4,5}}
=> [3,2]
=> 1
{{1,2},{3,4},{5}}
=> [2,2,1]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> 1
{{1,3,4},{2,5}}
=> [3,2]
=> 1
{{1,3,4},{2},{5}}
=> [3,1,1]
=> 1
{{1,3,5},{2,4}}
=> [3,2]
=> 1
{{1,3},{2,4,5}}
=> [3,2]
=> 1
{{1,3},{2,4},{5}}
=> [2,2,1]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> 1
{{1,4},{2,3,5}}
=> [3,2]
=> 1
Description
The number of lower covers of a partition in dominance order.
According to [1], Corollary 2.4, the maximum number of elements one element (apparently for $n\neq 2$) can cover is
$$
\frac{1}{2}(\sqrt{1+8n}-3)
$$
and an element which covers this number of elements is given by $(c+i,c,c-1,\dots,3,2,1)$, where $1\leq i\leq c+2$.
Matching statistic: St000481
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1]
=> [1]
=> 0
{{1,2}}
=> [2]
=> [1,1]
=> 1
{{1},{2}}
=> [1,1]
=> [2]
=> 0
{{1,2,3}}
=> [3]
=> [1,1,1]
=> 1
{{1,2},{3}}
=> [2,1]
=> [2,1]
=> 1
{{1,3},{2}}
=> [2,1]
=> [2,1]
=> 1
{{1},{2,3}}
=> [2,1]
=> [2,1]
=> 1
{{1},{2},{3}}
=> [1,1,1]
=> [3]
=> 0
{{1,2,3,4}}
=> [4]
=> [1,1,1,1]
=> 1
{{1,2,3},{4}}
=> [3,1]
=> [2,1,1]
=> 1
{{1,2,4},{3}}
=> [3,1]
=> [2,1,1]
=> 1
{{1,2},{3,4}}
=> [2,2]
=> [2,2]
=> 1
{{1,2},{3},{4}}
=> [2,1,1]
=> [3,1]
=> 1
{{1,3,4},{2}}
=> [3,1]
=> [2,1,1]
=> 1
{{1,3},{2,4}}
=> [2,2]
=> [2,2]
=> 1
{{1,3},{2},{4}}
=> [2,1,1]
=> [3,1]
=> 1
{{1,4},{2,3}}
=> [2,2]
=> [2,2]
=> 1
{{1},{2,3,4}}
=> [3,1]
=> [2,1,1]
=> 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [3,1]
=> 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [3,1]
=> 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [3,1]
=> 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [3,1]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [4]
=> 0
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1]
=> 1
{{1,2,3,4},{5}}
=> [4,1]
=> [2,1,1,1]
=> 1
{{1,2,3,5},{4}}
=> [4,1]
=> [2,1,1,1]
=> 1
{{1,2,3},{4,5}}
=> [3,2]
=> [2,2,1]
=> 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [3,1,1]
=> 1
{{1,2,4,5},{3}}
=> [4,1]
=> [2,1,1,1]
=> 1
{{1,2,4},{3,5}}
=> [3,2]
=> [2,2,1]
=> 1
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [3,1,1]
=> 1
{{1,2,5},{3,4}}
=> [3,2]
=> [2,2,1]
=> 1
{{1,2},{3,4,5}}
=> [3,2]
=> [2,2,1]
=> 1
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [3,2]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [3,1,1]
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [3,2]
=> 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [3,2]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> [2,1,1,1]
=> 1
{{1,3,4},{2,5}}
=> [3,2]
=> [2,2,1]
=> 1
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [3,1,1]
=> 1
{{1,3,5},{2,4}}
=> [3,2]
=> [2,2,1]
=> 1
{{1,3},{2,4,5}}
=> [3,2]
=> [2,2,1]
=> 1
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [3,2]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [3,1,1]
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [3,2]
=> 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [3,2]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> [2,2,1]
=> 1
{{1,4},{2,3,5}}
=> [3,2]
=> [2,2,1]
=> 1
Description
The number of upper covers of a partition in dominance order.
Matching statistic: St001165
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001165: Dyck paths ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001165: Dyck paths ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
{{1,2}}
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
{{1},{2}}
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
{{1,2,3}}
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
{{1,2},{3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
{{1},{2,3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
{{1,2,3,4}}
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
{{1,2,3},{4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
{{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
{{1,2},{3,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
{{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
{{1,3},{2,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
{{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
{{1},{2,3,4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2,3,4,5}}
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
{{1,2,3,4},{5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
{{1,2,3,5},{4}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
{{1,2,3},{4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
{{1,2,4,5},{3}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
{{1,2,4},{3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
{{1,2,5},{3,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2},{3,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
{{1,3,4},{2,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
{{1,3,5},{2,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,3},{2,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
{{1,4,5},{2,3}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,4},{2,3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2,3,4,5,6,7}}
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,2,3,4,5,6},{7}}
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,2,3,4,5,7},{6}}
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,2,3,4,5},{6},{7}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,2,3,4,6,7},{5}}
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,2,3,4,6},{5},{7}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,2,3,4,7},{5},{6}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,2,3,4},{5},{6},{7}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,2,3,5,6,7},{4}}
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,2,3,5,6},{4},{7}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,2,3,5,7},{4},{6}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,2,3,5},{4},{6},{7}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,2,3,6,7},{4},{5}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,2,3,6},{4},{5},{7}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,2,3,7},{4},{5},{6}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,2,3},{4},{5},{6},{7}}
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,2,4,5,6,7},{3}}
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,2,4,5,6},{3},{7}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,2,4,5,7},{3},{6}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,2,4,5},{3},{6},{7}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,2,4,6,7},{3},{5}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,2,4,6},{3},{5},{7}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,2,4,7},{3},{5},{6}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,2,4},{3},{5},{6},{7}}
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,2,5,6,7},{3},{4}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,2,5,6},{3},{4},{7}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,2,5,7},{3},{4},{6}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,2,5},{3},{4},{6},{7}}
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,2,6,7},{3},{4},{5}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,2,6},{3},{4},{5},{7}}
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,2,7},{3},{4},{5},{6}}
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,2},{3},{4},{5},{6},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,3,4,5,6,7},{2}}
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,3,4,5,6},{2},{7}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,3,4,5,7},{2},{6}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,3,4,5},{2},{6},{7}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,3,4,6,7},{2},{5}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,3,4,6},{2},{5},{7}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,3,4,7},{2},{5},{6}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,3,4},{2},{5},{6},{7}}
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,3,5,6,7},{2},{4}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,3,5,6},{2},{4},{7}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,3,5,7},{2},{4},{6}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,3,5},{2},{4},{6},{7}}
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,3,6,7},{2},{4},{5}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,3,6},{2},{4},{5},{7}}
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,3,7},{2},{4},{5},{6}}
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1,3},{2},{4},{5},{6},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1},{2,3,4,5,6,7}}
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
{{1},{2,3,4,5,6},{7}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
Description
Number of simple modules with even projective dimension in the corresponding Nakayama algebra.
Matching statistic: St001199
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 82%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 82%●distinct values known / distinct values provided: 67%
Values
{{1}}
=> [1]
=> []
=> []
=> ? = 0
{{1,2}}
=> [2]
=> []
=> []
=> ? ∊ {0,1}
{{1},{2}}
=> [1,1]
=> [1]
=> [1,0]
=> ? ∊ {0,1}
{{1,2,3}}
=> [3]
=> []
=> []
=> ? ∊ {0,1,1,1,1}
{{1,2},{3}}
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {0,1,1,1,1}
{{1,3},{2}}
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {0,1,1,1,1}
{{1},{2,3}}
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {0,1,1,1,1}
{{1},{2},{3}}
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,1,1,1,1}
{{1,2,3,4}}
=> [4]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1}
{{1,2,3},{4}}
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1}
{{1,2,4},{3}}
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3,4}}
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1}
{{1,3,4},{2}}
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2,4}}
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1}
{{1,4},{2,3}}
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
{{1},{2,3,4}}
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1}
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1}
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1}
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1}
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1}
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
{{1,2,3,4,5}}
=> [5]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,3,4},{5}}
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,3,5},{4}}
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,3},{4,5}}
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,4,5},{3}}
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,4},{3,5}}
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,5},{3,4}}
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
{{1,2},{3,4,5}}
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,4},{2,5}}
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,5},{2,4}}
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
{{1,3},{2,4,5}}
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
{{1,4},{2,3,5}}
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,5},{2,3,4}}
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
{{1},{2,3,4,5}}
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,4},{2},{3,5}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2,4},{3,5}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
{{1,5},{2},{3,4}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1},{2,5},{3,4}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
{{1,2,3,4,5,6}}
=> [6]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,3,4},{5,6}}
=> [4,2]
=> [2]
=> [1,0,1,0]
=> 1
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,3,5},{4,6}}
=> [4,2]
=> [2]
=> [1,0,1,0]
=> 1
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,3,6},{4,5}}
=> [4,2]
=> [2]
=> [1,0,1,0]
=> 1
{{1,2,3},{4,5,6}}
=> [3,3]
=> [3]
=> [1,0,1,0,1,0]
=> 2
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,4,5},{3,6}}
=> [4,2]
=> [2]
=> [1,0,1,0]
=> 1
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,4,6},{3,5}}
=> [4,2]
=> [2]
=> [1,0,1,0]
=> 1
{{1,2,4,6},{3},{5}}
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,5,6},{3},{4}}
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St000668
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 67%
Values
{{1}}
=> [1]
=> []
=> ?
=> ? = 0
{{1,2}}
=> [2]
=> []
=> ?
=> ? ∊ {0,1}
{{1},{2}}
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1}
{{1,2,3}}
=> [3]
=> []
=> ?
=> ? ∊ {0,1,1,1,1}
{{1,2},{3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1}
{{1,3},{2}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1}
{{1},{2,3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1}
{{1},{2},{3}}
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1}
{{1,2,3,4}}
=> [4]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,3},{4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,4},{3}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,4},{2}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,4},{2,3}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2,3,4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2,3,4,5}}
=> [5]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,3,4},{5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,3,5},{4}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,3},{4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,4,5},{3}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,4},{3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,5},{3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,4},{2,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,5},{2,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,4},{2,3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,5},{2,3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2,3,4,5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3},{4},{5,6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,3},{2},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3},{2,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2},{4},{5,6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,4},{2,3},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,5},{2,3},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1},{2,3,5},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,6},{2,3},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1},{2,3},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,6},{2,3},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1},{2,3,6},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2,3},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
Description
The least common multiple of the parts of the partition.
Matching statistic: St000704
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000704: Integer partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000704: Integer partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 67%
Values
{{1}}
=> [1]
=> []
=> ?
=> ? = 0
{{1,2}}
=> [2]
=> []
=> ?
=> ? ∊ {0,1}
{{1},{2}}
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1}
{{1,2,3}}
=> [3]
=> []
=> ?
=> ? ∊ {0,1,1,1,1}
{{1,2},{3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1}
{{1,3},{2}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1}
{{1},{2,3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1}
{{1},{2},{3}}
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1}
{{1,2,3,4}}
=> [4]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,3},{4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,4},{3}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,4},{2}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,4},{2,3}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2,3,4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2,3,4,5}}
=> [5]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,3,4},{5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,3,5},{4}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,3},{4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,4,5},{3}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,4},{3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,5},{3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,4},{2,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,5},{2,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,4},{2,3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,5},{2,3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2,3,4,5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3},{4},{5,6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,3},{2},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3},{2,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2},{4},{5,6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,4},{2,3},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,5},{2,3},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1},{2,3,5},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,6},{2,3},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1},{2,3},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,6},{2,3},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1},{2,3,6},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2,3},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
Description
The number of semistandard tableaux on a given integer partition with minimal maximal entry.
This is, for an integer partition $\lambda = (\lambda_1 > \cdots > \lambda_k > 0)$, the number of [[SemistandardTableaux|semistandard tableaux]] of shape $\lambda$ with maximal entry $k$.
Equivalently, this is the evaluation $s_\lambda(1,\ldots,1)$ of the Schur function $s_\lambda$ in $k$ variables, or, explicitly,
$$ \prod_{(i,j) \in L} \frac{k + j - i}{ \operatorname{hook}(i,j) }$$
where the product is over all cells $(i,j) \in L$ and $\operatorname{hook}(i,j)$ is the hook length of a cell.
See [Theorem 6.3, 1] for details.
Matching statistic: St000707
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000707: Integer partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000707: Integer partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 67%
Values
{{1}}
=> [1]
=> []
=> ?
=> ? = 0
{{1,2}}
=> [2]
=> []
=> ?
=> ? ∊ {0,1}
{{1},{2}}
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1}
{{1,2,3}}
=> [3]
=> []
=> ?
=> ? ∊ {0,1,1,1,1}
{{1,2},{3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1}
{{1,3},{2}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1}
{{1},{2,3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1}
{{1},{2},{3}}
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1}
{{1,2,3,4}}
=> [4]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,3},{4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,4},{3}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,4},{2}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,4},{2,3}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2,3,4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2,3,4,5}}
=> [5]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,3,4},{5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,3,5},{4}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,3},{4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,4,5},{3}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,4},{3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,5},{3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,4},{2,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,5},{2,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,4},{2,3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,5},{2,3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2,3,4,5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3},{4},{5,6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,3},{2},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3},{2,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2},{4},{5,6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,4},{2,3},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,5},{2,3},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1},{2,3,5},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,6},{2,3},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1},{2,3},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,6},{2,3},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1},{2,3,6},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2,3},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
Description
The product of the factorials of the parts.
Matching statistic: St000708
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 67%
Values
{{1}}
=> [1]
=> []
=> ?
=> ? = 0
{{1,2}}
=> [2]
=> []
=> ?
=> ? ∊ {0,1}
{{1},{2}}
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1}
{{1,2,3}}
=> [3]
=> []
=> ?
=> ? ∊ {0,1,1,1,1}
{{1,2},{3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1}
{{1,3},{2}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1}
{{1},{2,3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1}
{{1},{2},{3}}
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1}
{{1,2,3,4}}
=> [4]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,3},{4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,4},{3}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,4},{2}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,4},{2,3}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2,3,4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2,3,4,5}}
=> [5]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,3,4},{5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,3,5},{4}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,3},{4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,4,5},{3}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,4},{3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,5},{3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,4},{2,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,5},{2,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,4},{2,3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,5},{2,3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2,3,4,5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3},{4},{5,6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,3},{2},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3},{2,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2},{4},{5,6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,4},{2,3},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,5},{2,3},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1},{2,3,5},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,6},{2,3},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1},{2,3},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,6},{2,3},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1},{2,3,6},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2,3},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
Description
The product of the parts of an integer partition.
Matching statistic: St000933
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000933: Integer partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000933: Integer partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 67%
Values
{{1}}
=> [1]
=> []
=> ?
=> ? = 0
{{1,2}}
=> [2]
=> []
=> ?
=> ? ∊ {0,1}
{{1},{2}}
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1}
{{1,2,3}}
=> [3]
=> []
=> ?
=> ? ∊ {0,1,1,1,1}
{{1,2},{3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1}
{{1,3},{2}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1}
{{1},{2,3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1}
{{1},{2},{3}}
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1}
{{1,2,3,4}}
=> [4]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,3},{4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,4},{3}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,4},{2}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,4},{2,3}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2,3,4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2,3,4,5}}
=> [5]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,3,4},{5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,3,5},{4}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,3},{4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,4,5},{3}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,4},{3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,5},{3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,4},{2,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,5},{2,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,4},{2,3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,5},{2,3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2,3,4,5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3},{4},{5,6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,3},{2},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3},{2,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2},{4},{5,6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,4},{2,3},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,5},{2,3},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1},{2,3,5},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,6},{2,3},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1},{2,3},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,6},{2,3},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1},{2,3,6},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2,3},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
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: St001128
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001128: Integer partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001128: Integer partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 67%
Values
{{1}}
=> [1]
=> []
=> ?
=> ? = 0
{{1,2}}
=> [2]
=> []
=> ?
=> ? ∊ {0,1}
{{1},{2}}
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1}
{{1,2,3}}
=> [3]
=> []
=> ?
=> ? ∊ {0,1,1,1,1}
{{1,2},{3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1}
{{1,3},{2}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1}
{{1},{2,3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1}
{{1},{2},{3}}
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1}
{{1,2,3,4}}
=> [4]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,3},{4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,4},{3}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,4},{2}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,4},{2,3}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2,3,4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2,3,4,5}}
=> [5]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,3,4},{5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,3,5},{4}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,3},{4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,4,5},{3}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,4},{3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,5},{3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,4},{2,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,5},{2,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,4},{2,3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,5},{2,3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2,3,4,5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3},{4},{5,6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,3},{2},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3},{2,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2},{4},{5,6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,4},{2,3},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,5},{2,3},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1},{2,3,5},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,6},{2,3},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1},{2,3},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,6},{2,3},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1},{2,3,6},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2,3},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
Description
The exponens consonantiae of a partition.
This is the quotient of the least common multiple and the greatest common divior of the parts of the partiton. See [1, Caput sextum, §19-§22].
The following 10 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000260The radius of a connected graph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000862The number of parts of the shifted shape of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001734The lettericity of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001555The order of a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001624The breadth of a lattice.
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