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Your data matches 31 different statistics following compositions of up to 3 maps.
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Matching statistic: St001898
St001898: Ordered set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[{1}] => 0
[{1},{2}] => 0
[{2},{1}] => 0
[{1,2}] => 0
[{1},{2},{3}] => 0
[{1},{3},{2}] => 1
[{2},{1},{3}] => 0
[{3},{1},{2}] => 0
[{2},{3},{1}] => 0
[{3},{2},{1}] => 0
[{1},{2,3}] => 0
[{2},{1,3}] => 0
[{3},{1,2}] => 0
[{1,2},{3}] => 0
[{1,3},{2}] => 0
[{2,3},{1}] => 0
[{1,2,3}] => 0
[{1},{2},{3},{4}] => 0
[{1},{2},{4},{3}] => 2
[{1},{3},{2},{4}] => 1
[{1},{4},{2},{3}] => 2
[{1},{3},{4},{2}] => 2
[{1},{4},{3},{2}] => 3
[{2},{1},{3},{4}] => 0
[{2},{1},{4},{3}] => 2
[{3},{1},{2},{4}] => 0
[{4},{1},{2},{3}] => 0
[{3},{1},{4},{2}] => 1
[{4},{1},{3},{2}] => 1
[{2},{3},{1},{4}] => 0
[{2},{4},{1},{3}] => 1
[{3},{2},{1},{4}] => 0
[{4},{2},{1},{3}] => 0
[{3},{4},{1},{2}] => 0
[{4},{3},{1},{2}] => 0
[{2},{3},{4},{1}] => 0
[{2},{4},{3},{1}] => 1
[{3},{2},{4},{1}] => 0
[{4},{2},{3},{1}] => 0
[{3},{4},{2},{1}] => 0
[{4},{3},{2},{1}] => 0
[{1},{2},{3,4}] => 0
[{1},{3},{2,4}] => 1
[{1},{4},{2,3}] => 2
[{2},{1},{3,4}] => 0
[{3},{1},{2,4}] => 0
[{4},{1},{2,3}] => 0
[{2},{3},{1,4}] => 0
[{2},{4},{1,3}] => 1
[{3},{2},{1,4}] => 0
Description
The number of occurrences of an 132 pattern in an ordered set partition.
An occurrence of a pattern $\pi\in\mathfrak S_k$ ordered set partition with blocks $B_1|\dots|B_\ell$ is a sequence of elements $e_1,\dots,e_k$ with $e_i\in B_{j_i}$ and $j_1 < \dots < j_k$ order-isomorphic to $\pi$.
Matching statistic: St000319
Mp00287: Ordered set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 59%●distinct values known / distinct values provided: 43%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 59%●distinct values known / distinct values provided: 43%
Values
[{1}] => [1] => [[1],[]]
=> []
=> ? = 0
[{1},{2}] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0}
[{2},{1}] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0}
[{1,2}] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,0}
[{1},{2},{3}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1},{3},{2}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{2},{1},{3}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{3},{1},{2}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{2},{3},{1}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{3},{2},{1}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1},{2,3}] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{2},{1,3}] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{3},{1,2}] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1,2},{3}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{1,3},{2}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{2,3},{1}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{1,2,3}] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1},{2},{3},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{2},{4},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{3},{2},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{4},{2},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{3},{4},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{4},{3},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{1},{3},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{1},{4},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{1},{2},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{1},{2},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{1},{4},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{1},{3},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{3},{1},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{4},{1},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{2},{1},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{2},{1},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{4},{1},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{3},{1},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{3},{4},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{4},{3},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{2},{4},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{2},{3},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{4},{2},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{3},{2},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{2},{3,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{3},{2,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{4},{2,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{1},{3,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{1},{2,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{1},{2,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{3},{1,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{4},{1,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{2},{1,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{2},{1,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{4},{1,2}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{3},{1,2}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{2,3},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1},{2,4},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1},{3,4},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{1,3},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{1,4},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{1,2},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{1,2},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{1,4},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{1,3},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{3,4},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{2,4},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{2,3},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1,2},{3},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,2},{4},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,3},{2},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,4},{2},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,3},{4},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,4},{3},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,3},{1},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,4},{1},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{3,4},{1},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,3},{4},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,4},{3},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{3,4},{2},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,2},{3,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,3},{2,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,4},{2,3}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{2,3},{1,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{2,4},{1,3}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{3,4},{1,2}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,2,3},{4}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{1,2,4},{3}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{1,3,4},{2}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{2,3,4},{1}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{1},{2},{3,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{2},{3,5},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{2},{4,5},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{2,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{2,5},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{2,3},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{2,3},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{2,5},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{2,4},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{4,5},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{3,5},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{3,4},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{2},{1},{3,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
Mp00287: Ordered set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 59%●distinct values known / distinct values provided: 43%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 59%●distinct values known / distinct values provided: 43%
Values
[{1}] => [1] => [[1],[]]
=> []
=> ? = 0
[{1},{2}] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0}
[{2},{1}] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0}
[{1,2}] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,0}
[{1},{2},{3}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1},{3},{2}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{2},{1},{3}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{3},{1},{2}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{2},{3},{1}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{3},{2},{1}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1},{2,3}] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{2},{1,3}] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{3},{1,2}] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1,2},{3}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{1,3},{2}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{2,3},{1}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{1,2,3}] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1},{2},{3},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{2},{4},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{3},{2},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{4},{2},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{3},{4},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{4},{3},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{1},{3},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{1},{4},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{1},{2},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{1},{2},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{1},{4},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{1},{3},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{3},{1},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{4},{1},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{2},{1},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{2},{1},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{4},{1},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{3},{1},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{3},{4},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{4},{3},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{2},{4},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{2},{3},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{4},{2},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{3},{2},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{2},{3,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{3},{2,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{4},{2,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{1},{3,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{1},{2,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{1},{2,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{3},{1,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{4},{1,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{2},{1,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{2},{1,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{4},{1,2}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{3},{1,2}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{2,3},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1},{2,4},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1},{3,4},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{1,3},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{1,4},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{1,2},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{1,2},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{1,4},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{1,3},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{3,4},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{2,4},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{2,3},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1,2},{3},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,2},{4},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,3},{2},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,4},{2},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,3},{4},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,4},{3},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,3},{1},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,4},{1},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{3,4},{1},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,3},{4},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,4},{3},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{3,4},{2},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,2},{3,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,3},{2,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,4},{2,3}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{2,3},{1,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{2,4},{1,3}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{3,4},{1,2}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,2,3},{4}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{1,2,4},{3}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{1,3,4},{2}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{2,3,4},{1}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{1},{2},{3,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{2},{3,5},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{2},{4,5},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{2,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{2,5},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{2,3},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{2,3},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{2,5},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{2,4},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{4,5},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{3,5},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{3,4},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{2},{1},{3,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
Description
The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St000944
Mp00287: Ordered set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000944: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 59%●distinct values known / distinct values provided: 29%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000944: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 59%●distinct values known / distinct values provided: 29%
Values
[{1}] => [1] => [[1],[]]
=> []
=> ? = 0
[{1},{2}] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0}
[{2},{1}] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0}
[{1,2}] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,0}
[{1},{2},{3}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1},{3},{2}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{2},{1},{3}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{3},{1},{2}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{2},{3},{1}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{3},{2},{1}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1},{2,3}] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{2},{1,3}] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{3},{1,2}] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1,2},{3}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{1,3},{2}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{2,3},{1}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{1,2,3}] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1},{2},{3},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{2},{4},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{3},{2},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{4},{2},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{3},{4},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{4},{3},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{1},{3},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{1},{4},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{1},{2},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{1},{2},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{1},{4},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{1},{3},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{3},{1},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{4},{1},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{2},{1},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{2},{1},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{4},{1},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{3},{1},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{3},{4},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{4},{3},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{2},{4},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{2},{3},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{4},{2},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{3},{2},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{2},{3,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{3},{2,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{4},{2,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{1},{3,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{1},{2,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{1},{2,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{3},{1,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{4},{1,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{2},{1,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{2},{1,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{4},{1,2}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{3},{1,2}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{2,3},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1},{2,4},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1},{3,4},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{1,3},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{1,4},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{1,2},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{1,2},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{1,4},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{1,3},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{3,4},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{2,4},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{2,3},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1,2},{3},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,2},{4},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,3},{2},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,4},{2},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,3},{4},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,4},{3},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,3},{1},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,4},{1},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{3,4},{1},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,3},{4},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,4},{3},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{3,4},{2},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,2},{3,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,3},{2,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,4},{2,3}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{2,3},{1,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{2,4},{1,3}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{3,4},{1,2}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,2,3},{4}] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[{1,2,4},{3}] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[{1,3,4},{2}] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[{2,3,4},{1}] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[{1},{2},{3,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{2},{3,5},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{2},{4,5},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{2,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{2,5},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{2,3},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{2,3},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{2,5},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{2,4},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{4,5},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{3,5},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{3,4},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{2},{1},{3,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
Description
The 3-degree of an integer partition.
For an integer partition $\lambda$, this is given by the exponent of 3 in the Gram determinant of the integal Specht module of the symmetric group indexed by $\lambda$.
This stupid comment should not be accepted as an edit!
Matching statistic: St001248
Mp00287: Ordered set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001248: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 59%●distinct values known / distinct values provided: 43%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001248: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 59%●distinct values known / distinct values provided: 43%
Values
[{1}] => [1] => [[1],[]]
=> []
=> ? = 0
[{1},{2}] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0}
[{2},{1}] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0}
[{1,2}] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,0}
[{1},{2},{3}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1},{3},{2}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{2},{1},{3}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{3},{1},{2}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{2},{3},{1}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{3},{2},{1}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1},{2,3}] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{2},{1,3}] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{3},{1,2}] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1,2},{3}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{1,3},{2}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{2,3},{1}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{1,2,3}] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1},{2},{3},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{1},{2},{4},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{1},{3},{2},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{1},{4},{2},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{1},{3},{4},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{1},{4},{3},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{2},{1},{3},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{2},{1},{4},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{3},{1},{2},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{4},{1},{2},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{3},{1},{4},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{4},{1},{3},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{2},{3},{1},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{2},{4},{1},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{3},{2},{1},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{4},{2},{1},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{3},{4},{1},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{4},{3},{1},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{2},{3},{4},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{2},{4},{3},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{3},{2},{4},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{4},{2},{3},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{3},{4},{2},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{4},{3},{2},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{1},{2},{3,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{1},{3},{2,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{1},{4},{2,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{2},{1},{3,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{3},{1},{2,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{4},{1},{2,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{2},{3},{1,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{2},{4},{1,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{3},{2},{1,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{4},{2},{1,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{3},{4},{1,2}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{4},{3},{1,2}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{1},{2,3},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1},{2,4},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1},{3,4},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{1,3},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{1,4},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{1,2},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{1,2},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{1,4},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{1,3},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{3,4},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{2,4},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{2,3},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1,2},{3},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,2},{4},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,3},{2},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,4},{2},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,3},{4},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,4},{3},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,3},{1},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,4},{1},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{3,4},{1},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,3},{4},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,4},{3},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{3,4},{2},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,2},{3,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,3},{2,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,4},{2,3}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{2,3},{1,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{2,4},{1,3}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{3,4},{1,2}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,2,3},{4}] => [3,1] => [[3,3],[2]]
=> [2]
=> 2
[{1,2,4},{3}] => [3,1] => [[3,3],[2]]
=> [2]
=> 2
[{1,3,4},{2}] => [3,1] => [[3,3],[2]]
=> [2]
=> 2
[{2,3,4},{1}] => [3,1] => [[3,3],[2]]
=> [2]
=> 2
[{1},{2},{3,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{2},{3,5},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{2},{4,5},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{2,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{2,5},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{2,3},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{2,3},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{2,5},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{2,4},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{4,5},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{3,5},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{3,4},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{2},{1},{3,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
Description
Sum of the even parts of a partition.
Matching statistic: St001279
Mp00287: Ordered set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001279: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 59%●distinct values known / distinct values provided: 57%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001279: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 59%●distinct values known / distinct values provided: 57%
Values
[{1}] => [1] => [[1],[]]
=> []
=> ? = 0
[{1},{2}] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0}
[{2},{1}] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0}
[{1,2}] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,0}
[{1},{2},{3}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1},{3},{2}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{2},{1},{3}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{3},{1},{2}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{2},{3},{1}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{3},{2},{1}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1},{2,3}] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{2},{1,3}] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{3},{1,2}] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1,2},{3}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{1,3},{2}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{2,3},{1}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{1,2,3}] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1},{2},{3},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{1},{2},{4},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{1},{3},{2},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{1},{4},{2},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{1},{3},{4},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{1},{4},{3},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{2},{1},{3},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{2},{1},{4},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{3},{1},{2},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{4},{1},{2},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{3},{1},{4},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{4},{1},{3},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{2},{3},{1},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{2},{4},{1},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{3},{2},{1},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{4},{2},{1},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{3},{4},{1},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{4},{3},{1},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{2},{3},{4},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{2},{4},{3},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{3},{2},{4},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{4},{2},{3},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{3},{4},{2},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{4},{3},{2},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{1},{2},{3,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{1},{3},{2,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{1},{4},{2,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{2},{1},{3,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{3},{1},{2,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{4},{1},{2,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{2},{3},{1,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{2},{4},{1,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{3},{2},{1,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{4},{2},{1,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{3},{4},{1,2}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{4},{3},{1,2}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
[{1},{2,3},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1},{2,4},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1},{3,4},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{1,3},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{1,4},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{1,2},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{1,2},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{1,4},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{1,3},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{3,4},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{2,4},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{2,3},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1,2},{3},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,2},{4},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,3},{2},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,4},{2},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,3},{4},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,4},{3},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,3},{1},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,4},{1},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{3,4},{1},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,3},{4},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,4},{3},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{3,4},{2},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,2},{3,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,3},{2,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,4},{2,3}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{2,3},{1,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{2,4},{1,3}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{3,4},{1,2}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,2,3},{4}] => [3,1] => [[3,3],[2]]
=> [2]
=> 2
[{1,2,4},{3}] => [3,1] => [[3,3],[2]]
=> [2]
=> 2
[{1,3,4},{2}] => [3,1] => [[3,3],[2]]
=> [2]
=> 2
[{2,3,4},{1}] => [3,1] => [[3,3],[2]]
=> [2]
=> 2
[{1},{2},{3,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{2},{3,5},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{2},{4,5},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{2,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{2,5},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{2,3},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{2,3},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{2,5},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{2,4},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{4,5},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{3,5},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{3,4},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{2},{1},{3,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
Description
The sum of the parts of an integer partition that are at least two.
Matching statistic: St001280
Mp00287: Ordered set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 59%●distinct values known / distinct values provided: 43%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 59%●distinct values known / distinct values provided: 43%
Values
[{1}] => [1] => [[1],[]]
=> []
=> ? = 0
[{1},{2}] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0}
[{2},{1}] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0}
[{1,2}] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,0}
[{1},{2},{3}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1},{3},{2}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{2},{1},{3}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{3},{1},{2}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{2},{3},{1}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{3},{2},{1}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1},{2,3}] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{2},{1,3}] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{3},{1,2}] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1,2},{3}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{1,3},{2}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{2,3},{1}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{1,2,3}] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1},{2},{3},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{2},{4},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{3},{2},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{4},{2},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{3},{4},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{4},{3},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{1},{3},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{1},{4},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{1},{2},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{1},{2},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{1},{4},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{1},{3},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{3},{1},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{4},{1},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{2},{1},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{2},{1},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{4},{1},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{3},{1},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{3},{4},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{4},{3},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{2},{4},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{2},{3},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{4},{2},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{3},{2},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{2},{3,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{3},{2,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{4},{2,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{1},{3,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{1},{2,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{1},{2,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{3},{1,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{4},{1,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{2},{1,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{2},{1,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{4},{1,2}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{3},{1,2}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{2,3},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1},{2,4},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1},{3,4},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{1,3},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{1,4},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{1,2},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{1,2},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{1,4},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{1,3},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{3,4},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{2,4},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{2,3},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1,2},{3},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,2},{4},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,3},{2},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,4},{2},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,3},{4},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,4},{3},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,3},{1},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,4},{1},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{3,4},{1},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,3},{4},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,4},{3},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{3,4},{2},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,2},{3,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,3},{2,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,4},{2,3}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{2,3},{1,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{2,4},{1,3}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{3,4},{1,2}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,2,3},{4}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{1,2,4},{3}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{1,3,4},{2}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{2,3,4},{1}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{1},{2},{3,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{2},{3,5},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{2},{4,5},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{2,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{2,5},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{2,3},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{2,3},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{2,5},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{2,4},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{4,5},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{3,5},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{3,4},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{2},{1},{3,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St001392
Mp00287: Ordered set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001392: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 59%●distinct values known / distinct values provided: 43%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001392: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 59%●distinct values known / distinct values provided: 43%
Values
[{1}] => [1] => [[1],[]]
=> []
=> ? = 0
[{1},{2}] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0}
[{2},{1}] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0}
[{1,2}] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,0}
[{1},{2},{3}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1},{3},{2}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{2},{1},{3}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{3},{1},{2}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{2},{3},{1}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{3},{2},{1}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1},{2,3}] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{2},{1,3}] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{3},{1,2}] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1,2},{3}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{1,3},{2}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{2,3},{1}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{1,2,3}] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1},{2},{3},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{2},{4},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{3},{2},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{4},{2},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{3},{4},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{4},{3},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{1},{3},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{1},{4},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{1},{2},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{1},{2},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{1},{4},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{1},{3},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{3},{1},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{4},{1},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{2},{1},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{2},{1},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{4},{1},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{3},{1},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{3},{4},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{4},{3},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{2},{4},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{2},{3},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{4},{2},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{3},{2},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{2},{3,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{3},{2,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{4},{2,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{1},{3,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{1},{2,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{1},{2,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{3},{1,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{4},{1,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{2},{1,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{2},{1,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{4},{1,2}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{3},{1,2}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{2,3},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1},{2,4},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1},{3,4},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{1,3},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{1,4},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{1,2},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{1,2},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{1,4},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{1,3},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{3,4},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{2,4},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{2,3},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1,2},{3},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,2},{4},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,3},{2},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,4},{2},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,3},{4},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,4},{3},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,3},{1},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,4},{1},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{3,4},{1},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,3},{4},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,4},{3},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{3,4},{2},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,2},{3,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,3},{2,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,4},{2,3}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{2,3},{1,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{2,4},{1,3}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{3,4},{1,2}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,2,3},{4}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{1,2,4},{3}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{1,3,4},{2}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{2,3,4},{1}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{1},{2},{3,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{2},{3,5},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{2},{4,5},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{2,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{2,5},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{2,3},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{2,3},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{2,5},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{2,4},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{4,5},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{3,5},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{3,4},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{2},{1},{3,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
Description
The largest nonnegative integer which is not a part and is smaller than the largest part of the partition.
Matching statistic: St001541
Mp00287: Ordered set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001541: Integer partitions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 71%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001541: Integer partitions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 71%
Values
[{1}] => [1] => [[1],[]]
=> []
=> ? = 0
[{1},{2}] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0}
[{2},{1}] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0}
[{1,2}] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,0}
[{1},{2},{3}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1},{3},{2}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{2},{1},{3}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{3},{1},{2}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{2},{3},{1}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{3},{2},{1}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1},{2,3}] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{2},{1,3}] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{3},{1,2}] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1,2},{3}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{1,3},{2}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{2,3},{1}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{1,2,3}] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1},{2},{3},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{2},{4},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{3},{2},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{4},{2},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{3},{4},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{4},{3},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{1},{3},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{1},{4},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{1},{2},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{1},{2},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{1},{4},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{1},{3},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{3},{1},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{4},{1},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{2},{1},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{2},{1},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{4},{1},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{3},{1},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{3},{4},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{4},{3},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{2},{4},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{2},{3},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{4},{2},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{3},{2},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{2},{3,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{3},{2,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{4},{2,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{1},{3,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{1},{2,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{1},{2,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{3},{1,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{4},{1,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{2},{1,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{2},{1,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{4},{1,2}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{3},{1,2}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{2,3},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1},{2,4},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1},{3,4},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{1,3},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{1,4},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{1,2},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{1,2},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{1,4},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{1,3},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{3,4},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{2,4},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{2,3},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1,2},{3},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,2},{4},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,3},{2},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,4},{2},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,3},{4},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,4},{3},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,3},{1},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,4},{1},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{3,4},{1},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,3},{4},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,4},{3},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{3,4},{2},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,2},{3,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,3},{2,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,4},{2,3}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{2,3},{1,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{2,4},{1,3}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{3,4},{1,2}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,2,3},{4}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{1,2,4},{3}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{1,3,4},{2}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{2,3,4},{1}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{1},{2},{3,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{2},{3,5},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{2},{4,5},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{2,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{2,5},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{2,3},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{2,3},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{2,5},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{2,4},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{4,5},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{3,5},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{3,4},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{2},{1},{3,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
Description
The Gini index of an integer partition.
As discussed in [1], this statistic is equal to [[St000567]] applied to the conjugate partition.
Matching statistic: St001587
Mp00287: Ordered set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001587: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 59%●distinct values known / distinct values provided: 29%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001587: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 59%●distinct values known / distinct values provided: 29%
Values
[{1}] => [1] => [[1],[]]
=> []
=> ? = 0
[{1},{2}] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0}
[{2},{1}] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0}
[{1,2}] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,0}
[{1},{2},{3}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1},{3},{2}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{2},{1},{3}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{3},{1},{2}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{2},{3},{1}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{3},{2},{1}] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1},{2,3}] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{2},{1,3}] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{3},{1,2}] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1,2},{3}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{1,3},{2}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{2,3},{1}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{1,2,3}] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[{1},{2},{3},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{2},{4},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{3},{2},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{4},{2},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{3},{4},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{4},{3},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{1},{3},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{1},{4},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{1},{2},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{1},{2},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{1},{4},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{1},{3},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{3},{1},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{4},{1},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{2},{1},{4}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{2},{1},{3}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{4},{1},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{3},{1},{2}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{3},{4},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{4},{3},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{2},{4},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{2},{3},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{4},{2},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{3},{2},{1}] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{2},{3,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{3},{2,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{4},{2,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{1},{3,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{1},{2,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{1},{2,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{3},{1,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{2},{4},{1,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{2},{1,4}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{2},{1,3}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{3},{4},{1,2}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{4},{3},{1,2}] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[{1},{2,3},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1},{2,4},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1},{3,4},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{1,3},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{1,4},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{1,2},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{1,2},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{1,4},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{1,3},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{3,4},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{2,4},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{2,3},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1,2},{3},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,2},{4},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,3},{2},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,4},{2},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,3},{4},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,4},{3},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,3},{1},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,4},{1},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{3,4},{1},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,3},{4},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,4},{3},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{3,4},{2},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,2},{3,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,3},{2,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,4},{2,3}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{2,3},{1,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{2,4},{1,3}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{3,4},{1,2}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,2,3},{4}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{1,2,4},{3}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{1,3,4},{2}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{2,3,4},{1}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{1},{2},{3,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{2},{3,5},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{2},{4,5},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{2,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{2,5},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{2,3},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{2,3},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{2,5},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{2,4},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{4,5},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{3,5},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{3,4},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{2},{1},{3,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
Description
Half of the largest even part of an integer partition.
The largest even part is recorded by [[St000995]].
The following 21 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001657The number of twos in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000478Another weight of a partition according to Alladi. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001651The Frankl number of a lattice. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
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