Your data matches 100 different statistics following compositions of up to 3 maps.
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Matching statistic: St000752
Mp00060: Permutations Robinson-Schensted tableau shapeInteger partitions
St000752: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 0
[1,2] => [2]
=> 0
[2,1] => [1,1]
=> 0
[1,2,3] => [3]
=> 1
[1,3,2] => [2,1]
=> 0
[2,1,3] => [2,1]
=> 0
[2,3,1] => [2,1]
=> 0
[3,1,2] => [2,1]
=> 0
[3,2,1] => [1,1,1]
=> 0
[1,2,3,4] => [4]
=> 2
[1,2,4,3] => [3,1]
=> 1
[1,3,2,4] => [3,1]
=> 1
[1,3,4,2] => [3,1]
=> 1
[1,4,2,3] => [3,1]
=> 1
[1,4,3,2] => [2,1,1]
=> 0
[2,1,3,4] => [3,1]
=> 1
[2,1,4,3] => [2,2]
=> 0
[2,3,1,4] => [3,1]
=> 1
[2,3,4,1] => [3,1]
=> 1
[2,4,1,3] => [2,2]
=> 0
[2,4,3,1] => [2,1,1]
=> 0
[3,1,2,4] => [3,1]
=> 1
[3,1,4,2] => [2,2]
=> 0
[3,2,1,4] => [2,1,1]
=> 0
[3,2,4,1] => [2,1,1]
=> 0
[3,4,1,2] => [2,2]
=> 0
[3,4,2,1] => [2,1,1]
=> 0
[4,1,2,3] => [3,1]
=> 1
[4,1,3,2] => [2,1,1]
=> 0
[4,2,1,3] => [2,1,1]
=> 0
[4,2,3,1] => [2,1,1]
=> 0
[4,3,1,2] => [2,1,1]
=> 0
[4,3,2,1] => [1,1,1,1]
=> 0
[1,2,3,4,5] => [5]
=> 0
[1,2,3,5,4] => [4,1]
=> 2
[1,2,4,3,5] => [4,1]
=> 2
[1,2,4,5,3] => [4,1]
=> 2
[1,2,5,3,4] => [4,1]
=> 2
[1,2,5,4,3] => [3,1,1]
=> 1
[1,3,2,4,5] => [4,1]
=> 2
[1,3,2,5,4] => [3,2]
=> 1
[1,3,4,2,5] => [4,1]
=> 2
[1,3,4,5,2] => [4,1]
=> 2
[1,3,5,2,4] => [3,2]
=> 1
[1,3,5,4,2] => [3,1,1]
=> 1
[1,4,2,3,5] => [4,1]
=> 2
[1,4,2,5,3] => [3,2]
=> 1
[1,4,3,2,5] => [3,1,1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> 1
[1,4,5,2,3] => [3,2]
=> 1
Description
The Grundy value for the game 'Couples are forever' on an integer partition. Two players alternately choose a part of the partition greater than two, and split it into two parts. The player facing a partition with all parts at most two looses.
Mp00252: Permutations restrictionPermutations
Mp00108: Permutations cycle typeInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St000698: Integer partitions ⟶ ℤResult quality: 62% values known / values provided: 62%distinct values known / distinct values provided: 75%
Values
[1] => [] => []
=> ?
=> ? = 0
[1,2] => [1] => [1]
=> []
=> ? ∊ {0,0}
[2,1] => [1] => [1]
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1}
[1,3,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1}
[2,1,3] => [2,1] => [2]
=> []
=> ? ∊ {0,0,0,0,0,1}
[2,3,1] => [2,1] => [2]
=> []
=> ? ∊ {0,0,0,0,0,1}
[3,1,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1}
[3,2,1] => [2,1] => [2]
=> []
=> ? ∊ {0,0,0,0,0,1}
[1,2,3,4] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,2,4,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,3,2,4] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[1,3,4,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[1,4,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,4,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[2,1,3,4] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[2,1,4,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[2,3,1,4] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[2,3,4,1] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[2,4,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[2,4,3,1] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[3,1,2,4] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[3,1,4,2] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[3,2,1,4] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[3,2,4,1] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[3,4,1,2] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[3,4,2,1] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[4,1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[4,1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[4,2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[4,2,3,1] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[4,3,1,2] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[4,3,2,1] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[1,2,3,4,5] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,5,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,4,3,5] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,2,4,5,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,2,5,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,5,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,4,5] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,5,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,3,4,2,5] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,3,5,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,3,5] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,4,3,5,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,4,5,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,5,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,5,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,5,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,5,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,4,5] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,5,4] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[2,1,4,3,5] => [2,1,4,3] => [2,2]
=> [2]
=> 1
[2,1,4,5,3] => [2,1,4,3] => [2,2]
=> [2]
=> 1
[2,1,5,3,4] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[2,1,5,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 1
[2,3,1,4,5] => [2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,1,5,4] => [2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,1,5] => [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,5,1] => [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,4,1] => [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,3,5] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,3,1,5] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,3,5,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,1,3] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,3,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,3,4] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[2,5,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 1
[2,5,3,1,4] => [2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,1,4,5] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,2,1,5,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,2,5,1,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,4,1,2,5] => [3,4,1,2] => [2,2]
=> [2]
=> 1
[3,4,1,5,2] => [3,4,1,2] => [2,2]
=> [2]
=> 1
[3,4,5,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 1
[3,5,2,1,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,5,4,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 1
[4,2,3,1,5] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[4,2,3,5,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[4,2,5,3,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[4,3,2,1,5] => [4,3,2,1] => [2,2]
=> [2]
=> 1
[4,3,2,5,1] => [4,3,2,1] => [2,2]
=> [2]
=> 1
[4,3,5,2,1] => [4,3,2,1] => [2,2]
=> [2]
=> 1
[4,5,2,3,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[4,5,3,2,1] => [4,3,2,1] => [2,2]
=> [2]
=> 1
[5,1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[5,1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[5,1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[5,1,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[5,2,1,3,4] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[5,2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 1
Description
The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. For any positive integer $k$, one associates a $k$-core to a partition by repeatedly removing all rim hooks of size $k$. This statistic counts the $2$-rim hooks that are removed in this process to obtain a $2$-core.
Matching statistic: St001199
Mp00108: Permutations cycle typeInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
St001199: Dyck paths ⟶ ℤResult quality: 50% values known / values provided: 54%distinct values known / distinct values provided: 50%
Values
[1] => [1]
=> []
=> []
=> ? = 0
[1,2] => [1,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0}
[2,1] => [2]
=> []
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,0,0,0,0,1}
[1,3,2] => [2,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,1}
[2,1,3] => [2,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,1}
[2,3,1] => [3]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1}
[3,1,2] => [3]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1}
[3,2,1] => [2,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,1}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[1,3,4,2] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[1,4,2,3] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[2,1,4,3] => [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[2,3,1,4] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[2,3,4,1] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[2,4,1,3] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[2,4,3,1] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[3,1,2,4] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[3,1,4,2] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[3,2,4,1] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[3,4,1,2] => [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[3,4,2,1] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[4,1,2,3] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[4,1,3,2] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[4,2,1,3] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[4,3,1,2] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[4,3,2,1] => [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,4,5,3,2] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,3,4] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,5,4,2,3] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,4,5,3] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[2,1,5,3,4] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,1,5,4] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[2,3,4,1,5] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,5,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,4,1] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,3,5] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,1,3] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[2,5,4,3,1] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[3,1,2,5,4] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[3,4,1,5,2] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[3,4,5,2,1] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[3,5,1,2,4] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[3,5,4,1,2] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[4,1,5,2,3] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[4,2,3,1,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[4,2,5,1,3] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[4,3,2,1,5] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[4,3,2,5,1] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[4,3,5,1,2] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[4,5,1,3,2] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[4,5,2,1,3] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[4,5,3,1,2] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[5,1,4,3,2] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[5,2,3,4,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[5,2,4,3,1] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[5,3,2,1,4] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[5,3,2,4,1] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[5,3,4,2,1] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[5,4,1,2,3] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[5,4,2,3,1] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[5,4,3,2,1] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Mp00159: Permutations Demazure product with inversePermutations
Mp00065: Permutations permutation posetPosets
Mp00205: Posets maximal antichainsLattices
St001878: Lattices ⟶ ℤResult quality: 50% values known / values provided: 52%distinct values known / distinct values provided: 50%
Values
[1] => [1] => ([],1)
=> ([],1)
=> ? = 0
[1,2] => [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0}
[2,1] => [2,1] => ([],2)
=> ([],1)
=> ? ∊ {0,0}
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0}
[2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0}
[2,3,1] => [3,2,1] => ([],3)
=> ([],1)
=> ? ∊ {0,0,0,0,0}
[3,1,2] => [3,2,1] => ([],3)
=> ([],1)
=> ? ∊ {0,0,0,0,0}
[3,2,1] => [3,2,1] => ([],3)
=> ([],1)
=> ? ∊ {0,0,0,0,0}
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,4,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,4,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 1
[2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,3,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,3,4,1] => [4,2,3,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,4,1,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,4,3,1] => [4,3,2,1] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,1,2,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,1,4,2] => [4,2,3,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,2,4,1] => [4,2,3,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,4,1,2] => [4,3,2,1] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,4,2,1] => [4,3,2,1] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[4,1,2,3] => [4,2,3,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[4,1,3,2] => [4,2,3,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[4,2,1,3] => [4,3,2,1] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[4,2,3,1] => [4,3,2,1] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[4,3,1,2] => [4,3,2,1] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[4,3,2,1] => [4,3,2,1] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,5,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,2,5,3,4] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,2,5,4,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,4,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,4,5,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,5,2,4] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,3,5,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,4,2,3,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,4,2,5,3] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,4,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,4,3,5,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,4,5,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,4,5,3,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,5,2,3,4] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,5,2,4,3] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,5,3,2,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,5,3,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,5,4,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,5,4,3,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,1,3,4,5] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,1,3,5,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,1,4,3,5] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,1,4,5,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,1,5,3,4] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,1,5,4,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,3,1,4,5] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,3,1,5,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,3,4,1,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,3,4,5,1] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,3,5,1,4] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3,5,4,1] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,4,1,3,5] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[2,4,1,5,3] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,4,3,1,5] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,4,3,5,1] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,4,5,1,3] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,4,5,3,1] => [5,4,3,2,1] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,5,1,3,4] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,5,1,4,3] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,5,3,1,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,5,3,4,1] => [5,4,3,2,1] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,5,4,1,3] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,5,4,3,1] => [5,4,3,2,1] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,1,2,4,5] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,1,2,5,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,1,4,2,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,1,4,5,2] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,1,5,2,4] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,1,5,4,2] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,2,1,4,5] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,2,1,5,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,2,4,1,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,2,4,5,1] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,2,5,1,4] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,2,5,4,1] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,4,1,2,5] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,4,1,5,2] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,5,1,2,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,5,2,1,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[4,1,2,3,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[4,1,2,5,3] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[4,1,3,2,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Mp00108: Permutations cycle typeInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St000137: Integer partitions ⟶ ℤResult quality: 50% values known / values provided: 50%distinct values known / distinct values provided: 50%
Values
[1] => [1]
=> []
=> ?
=> ? = 0
[1,2] => [1,1]
=> [1]
=> []
=> ? ∊ {0,0}
[2,1] => [2]
=> []
=> ?
=> ? ∊ {0,0}
[1,2,3] => [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[2,3,1] => [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0}
[3,1,2] => [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0}
[3,2,1] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[2,3,4,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[2,4,1,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[2,4,3,1] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[3,1,4,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[3,4,2,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[4,1,2,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[4,1,3,2] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[4,2,1,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,3,1,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[4,3,2,1] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,2,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[1,4,5,3,2] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,4,2,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,1,4,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,5,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,3,1,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,1,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,5,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,4,1] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,4,3,5,1] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,1,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,3,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,3,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,4,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,3,1,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,5,4,1,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,4,3,1] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,2,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,1,2,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,4,2,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,5,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,1,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,1,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,3,1,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,2,3,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,5,1,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,3,2,1,5] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,5,3,1,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[5,1,3,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[5,2,1,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[5,2,3,1,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
Description
The Grundy value of an integer partition. Consider the two-player game on an integer partition. In each move, a player removes either a box, or a 2x2-configuration of boxes such that the resulting diagram is still a partition. The first player that cannot move lose. This happens exactly when the empty partition is reached. The grundy value of an integer partition is defined as the grundy value of this two-player game as defined in [1]. This game was described to me during Norcom 2013, by Urban Larsson, and it seems to be quite difficult to give a good description of the partitions with Grundy value 0.
Matching statistic: St001122
Mp00108: Permutations cycle typeInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St001122: Integer partitions ⟶ ℤResult quality: 50% values known / values provided: 50%distinct values known / distinct values provided: 50%
Values
[1] => [1]
=> []
=> ?
=> ? = 0
[1,2] => [1,1]
=> [1]
=> []
=> ? ∊ {0,0}
[2,1] => [2]
=> []
=> ?
=> ? ∊ {0,0}
[1,2,3] => [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[2,3,1] => [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0}
[3,1,2] => [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0}
[3,2,1] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[2,3,4,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[2,4,1,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[2,4,3,1] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[3,1,4,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[3,4,2,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[4,1,2,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[4,1,3,2] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[4,2,1,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,3,1,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[4,3,2,1] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,2,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[1,4,5,3,2] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,4,2,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,1,4,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,5,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,3,1,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,1,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,5,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,4,1] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,4,3,5,1] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,1,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,3,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,3,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,4,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,3,1,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,5,4,1,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,4,3,1] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,2,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,1,2,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,4,2,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,5,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,1,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,1,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,3,1,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,2,3,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,5,1,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,3,2,1,5] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,5,3,1,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[5,1,3,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[5,2,1,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[5,2,3,1,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
Description
The multiplicity of the sign representation in the Kronecker square corresponding to a partition. The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$: $$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$ This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{1^n}$, for $\lambda\vdash n$. It equals $1$ if and only if $\lambda$ is self-conjugate.
Matching statistic: St001442
Mp00108: Permutations cycle typeInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St001442: Integer partitions ⟶ ℤResult quality: 50% values known / values provided: 50%distinct values known / distinct values provided: 50%
Values
[1] => [1]
=> []
=> ?
=> ? = 0
[1,2] => [1,1]
=> [1]
=> []
=> ? ∊ {0,0}
[2,1] => [2]
=> []
=> ?
=> ? ∊ {0,0}
[1,2,3] => [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[2,3,1] => [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0}
[3,1,2] => [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0}
[3,2,1] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[2,3,4,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[2,4,1,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[2,4,3,1] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[3,1,4,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[3,4,2,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[4,1,2,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[4,1,3,2] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[4,2,1,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,3,1,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[4,3,2,1] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,2,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[1,4,5,3,2] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,4,2,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,1,4,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,5,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,3,1,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,1,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,5,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,4,1] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,4,3,5,1] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,1,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,3,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,3,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,4,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,3,1,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,5,4,1,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,4,3,1] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,2,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,1,2,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,4,2,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,5,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,1,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,1,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,3,1,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,2,3,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,5,1,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,3,2,1,5] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,5,3,1,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[5,1,3,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[5,2,1,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[5,2,3,1,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
Description
The number of standard Young tableaux whose major index is divisible by the size of a given integer partition.
Matching statistic: St001525
Mp00108: Permutations cycle typeInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St001525: Integer partitions ⟶ ℤResult quality: 50% values known / values provided: 50%distinct values known / distinct values provided: 50%
Values
[1] => [1]
=> []
=> ?
=> ? = 0
[1,2] => [1,1]
=> [1]
=> []
=> ? ∊ {0,0}
[2,1] => [2]
=> []
=> ?
=> ? ∊ {0,0}
[1,2,3] => [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[2,3,1] => [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0}
[3,1,2] => [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0}
[3,2,1] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[2,3,4,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[2,4,1,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[2,4,3,1] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[3,1,4,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[3,4,2,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[4,1,2,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[4,1,3,2] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[4,2,1,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,3,1,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[4,3,2,1] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,2,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[1,4,5,3,2] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,4,2,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,1,4,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,5,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,3,1,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,1,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,5,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,4,1] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,4,3,5,1] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,1,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,3,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,3,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,4,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,3,1,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,5,4,1,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,4,3,1] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,2,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,1,2,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,4,2,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,5,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,1,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,1,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,3,1,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,2,3,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,5,1,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,3,2,1,5] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,5,3,1,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[5,1,3,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[5,2,1,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[5,2,3,1,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
Description
The number of symmetric hooks on the diagonal of a partition.
Matching statistic: St001593
Mp00108: Permutations cycle typeInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St001593: Integer partitions ⟶ ℤResult quality: 50% values known / values provided: 50%distinct values known / distinct values provided: 50%
Values
[1] => [1]
=> []
=> ?
=> ? = 0
[1,2] => [1,1]
=> [1]
=> []
=> ? ∊ {0,0}
[2,1] => [2]
=> []
=> ?
=> ? ∊ {0,0}
[1,2,3] => [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[2,3,1] => [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0}
[3,1,2] => [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0}
[3,2,1] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[2,3,4,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[2,4,1,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[2,4,3,1] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[3,1,4,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[3,4,2,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[4,1,2,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[4,1,3,2] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[4,2,1,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,3,1,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[4,3,2,1] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,2,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[1,4,5,3,2] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,4,2,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,1,4,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,5,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,3,1,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,1,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,5,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,4,1] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,4,3,5,1] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,1,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,3,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,3,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,4,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,3,1,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,5,4,1,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,4,3,1] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,2,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,1,2,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,4,2,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,5,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,1,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,1,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,3,1,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,2,3,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,5,1,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,3,2,1,5] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,5,3,1,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[5,1,3,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[5,2,1,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[5,2,3,1,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
Description
This is the number of standard Young tableaux of the given shifted shape. For an integer partition $\lambda = (\lambda_1,\dots,\lambda_k)$, the shifted diagram is obtained by moving the $i$-th row in the diagram $i-1$ boxes to the right, i.e., $$\lambda^∗ = \{(i, j) | 1 \leq i \leq k, i \leq j \leq \lambda_i + i − 1 \}.$$ In particular, this statistic is zero if and only if $\lambda_{i+1} = \lambda_i$ for some $i$.
Matching statistic: St001601
Mp00108: Permutations cycle typeInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St001601: Integer partitions ⟶ ℤResult quality: 50% values known / values provided: 50%distinct values known / distinct values provided: 50%
Values
[1] => [1]
=> []
=> ?
=> ? = 0
[1,2] => [1,1]
=> [1]
=> []
=> ? ∊ {0,0}
[2,1] => [2]
=> []
=> ?
=> ? ∊ {0,0}
[1,2,3] => [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[2,3,1] => [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0}
[3,1,2] => [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0}
[3,2,1] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[2,3,4,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[2,4,1,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[2,4,3,1] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[3,1,4,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[3,4,2,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[4,1,2,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[4,1,3,2] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[4,2,1,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,3,1,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[4,3,2,1] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2}
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,2,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[1,4,5,3,2] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,4,2,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,1,4,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,5,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,3,1,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,1,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,5,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,4,1] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,4,3,5,1] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,1,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,3,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,3,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,4,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,3,1,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,5,4,1,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,4,3,1] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,2,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,1,2,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,4,2,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,5,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,1,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,1,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,3,1,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,2,3,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,5,1,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,3,2,1,5] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,5,3,1,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[5,1,3,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[5,2,1,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[5,2,3,1,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees.
The following 90 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000260The radius of a connected graph. St000929The constant term of the character polynomial of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001568The smallest positive integer that does not appear twice in the partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000454The largest eigenvalue of a graph if it is integral. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. 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. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001280The number of parts of an integer partition that are at least two. St000481The number of upper covers of a partition in dominance order. St000480The number of lower covers of a partition in dominance order. St001651The Frankl number of a lattice. St000455The second largest eigenvalue of a graph if it is integral. St001964The interval resolution global dimension of a poset. St001845The number of join irreducibles minus the rank of a lattice. St001820The size of the image of the pop stack sorting operator. St000259The diameter of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001487The number of inner corners of a skew partition. St000944The 3-degree of an integer partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001846The number of elements which do not have a complement in the lattice. St001868The number of alignments of type NE of a signed permutation. St001875The number of simple modules with projective dimension at most 1. St000527The width of the poset. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001435The number of missing boxes in the first row. St000632The jump number of the poset. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001902The number of potential covers of a poset. St000181The number of connected components of the Hasse diagram for the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000908The length of the shortest maximal antichain in a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001472The permanent of the Coxeter matrix of the poset. St001510The number of self-evacuating linear extensions of a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001779The order of promotion on the set of linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001438The number of missing boxes of a skew partition.