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Your data matches 96 different statistics following compositions of up to 3 maps.
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Matching statistic: St000832
St000832: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 1
[2,1] => 1
[1,2,3] => 2
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 2
[1,2,3,4] => 3
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 3
[2,1,3,4] => 1
[2,1,4,3] => 1
[2,3,1,4] => 1
[2,3,4,1] => 3
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 1
[3,2,1,4] => 3
[3,2,4,1] => 1
[3,4,1,2] => 1
[3,4,2,1] => 1
[4,1,2,3] => 3
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 1
[4,3,1,2] => 1
[4,3,2,1] => 3
[1,2,3,4,5] => 4
[1,2,3,5,4] => 2
[1,2,4,3,5] => 1
[1,2,4,5,3] => 1
[1,2,5,3,4] => 1
[1,2,5,4,3] => 4
[1,3,2,4,5] => 1
[1,3,2,5,4] => 1
[1,3,4,2,5] => 1
[1,3,4,5,2] => 3
[1,3,5,2,4] => 1
[1,3,5,4,2] => 1
[1,4,2,3,5] => 1
[1,4,2,5,3] => 1
[1,4,3,2,5] => 4
[1,4,3,5,2] => 1
[1,4,5,2,3] => 1
[1,4,5,3,2] => 1
Description
The number of permutations obtained by reversing blocks of three consecutive numbers.
Matching statistic: St001289
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001289: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 82%●distinct values known / distinct values provided: 60%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001289: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 82%●distinct values known / distinct values provided: 60%
Values
[1,2] => [1,1]
=> [1]
=> [1,0]
=> 1
[2,1] => [2]
=> []
=> []
=> ? = 1
[1,2,3] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,3,2] => [2,1]
=> [1]
=> [1,0]
=> 1
[2,1,3] => [2,1]
=> [1]
=> [1,0]
=> 1
[2,3,1] => [3]
=> []
=> []
=> ? ∊ {2,2}
[3,1,2] => [3]
=> []
=> []
=> ? ∊ {2,2}
[3,2,1] => [2,1]
=> [1]
=> [1,0]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> [1,0]
=> 1
[1,4,2,3] => [3,1]
=> [1]
=> [1,0]
=> 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[2,3,1,4] => [3,1]
=> [1]
=> [1,0]
=> 1
[2,3,4,1] => [4]
=> []
=> []
=> ? ∊ {1,3,3,3,3,3}
[2,4,1,3] => [4]
=> []
=> []
=> ? ∊ {1,3,3,3,3,3}
[2,4,3,1] => [3,1]
=> [1]
=> [1,0]
=> 1
[3,1,2,4] => [3,1]
=> [1]
=> [1,0]
=> 1
[3,1,4,2] => [4]
=> []
=> []
=> ? ∊ {1,3,3,3,3,3}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> [1,0]
=> 1
[3,4,1,2] => [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[3,4,2,1] => [4]
=> []
=> []
=> ? ∊ {1,3,3,3,3,3}
[4,1,2,3] => [4]
=> []
=> []
=> ? ∊ {1,3,3,3,3,3}
[4,1,3,2] => [3,1]
=> [1]
=> [1,0]
=> 1
[4,2,1,3] => [3,1]
=> [1]
=> [1,0]
=> 1
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[4,3,1,2] => [4]
=> []
=> []
=> ? ∊ {1,3,3,3,3,3}
[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]
=> 3
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[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]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> [1,0]
=> 1
[1,3,5,2,4] => [4,1]
=> [1]
=> [1,0]
=> 1
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,4,2,5,3] => [4,1]
=> [1]
=> [1,0]
=> 1
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[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]
=> 1
[1,5,2,3,4] => [4,1]
=> [1]
=> [1,0]
=> 1
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[1,5,4,2,3] => [4,1]
=> [1]
=> [1,0]
=> 1
[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]
=> 3
[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
[2,3,4,5,1] => [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[2,3,5,1,4] => [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[2,4,1,5,3] => [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[2,4,5,3,1] => [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[2,5,1,3,4] => [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[2,5,4,1,3] => [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[3,1,4,5,2] => [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[3,1,5,2,4] => [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[3,4,2,5,1] => [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[3,4,5,1,2] => [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[3,5,2,1,4] => [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[3,5,4,2,1] => [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[4,1,2,5,3] => [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[4,1,5,3,2] => [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[4,3,1,5,2] => [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[4,3,5,2,1] => [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[4,5,1,2,3] => [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[4,5,2,3,1] => [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[5,1,2,3,4] => [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[5,1,4,2,3] => [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[5,3,1,2,4] => [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[5,3,4,1,2] => [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[5,4,1,3,2] => [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[5,4,2,1,3] => [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[2,3,4,5,6,1] => [6]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[2,3,4,6,1,5] => [6]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[2,3,5,1,6,4] => [6]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[2,3,5,6,4,1] => [6]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[2,3,6,1,4,5] => [6]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[2,3,6,5,1,4] => [6]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[2,4,1,5,6,3] => [6]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[2,4,1,6,3,5] => [6]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[2,4,5,3,6,1] => [6]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[2,4,5,6,1,3] => [6]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[2,4,6,3,1,5] => [6]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[2,4,6,5,3,1] => [6]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[2,5,1,3,6,4] => [6]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[2,5,1,6,4,3] => [6]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[2,5,4,1,6,3] => [6]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[2,5,4,6,3,1] => [6]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[2,5,6,1,3,4] => [6]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
Description
The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero.
This n-fold tensor product seems to be always injective.
Matching statistic: St001924
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St001924: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 82%●distinct values known / distinct values provided: 60%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St001924: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 82%●distinct values known / distinct values provided: 60%
Values
[1,2] => [1,1]
=> [1]
=> [1]
=> 1
[2,1] => [2]
=> []
=> ?
=> ? = 1
[1,2,3] => [1,1,1]
=> [1,1]
=> [2]
=> 1
[1,3,2] => [2,1]
=> [1]
=> [1]
=> 1
[2,1,3] => [2,1]
=> [1]
=> [1]
=> 1
[2,3,1] => [3]
=> []
=> ?
=> ? ∊ {2,2}
[3,1,2] => [3]
=> []
=> ?
=> ? ∊ {2,2}
[3,2,1] => [2,1]
=> [1]
=> [1]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> [1]
=> 1
[1,4,2,3] => [3,1]
=> [1]
=> [1]
=> 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> [1,1]
=> 1
[2,3,1,4] => [3,1]
=> [1]
=> [1]
=> 1
[2,3,4,1] => [4]
=> []
=> ?
=> ? ∊ {1,3,3,3,3,3}
[2,4,1,3] => [4]
=> []
=> ?
=> ? ∊ {1,3,3,3,3,3}
[2,4,3,1] => [3,1]
=> [1]
=> [1]
=> 1
[3,1,2,4] => [3,1]
=> [1]
=> [1]
=> 1
[3,1,4,2] => [4]
=> []
=> ?
=> ? ∊ {1,3,3,3,3,3}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> [1]
=> 1
[3,4,1,2] => [2,2]
=> [2]
=> [1,1]
=> 1
[3,4,2,1] => [4]
=> []
=> ?
=> ? ∊ {1,3,3,3,3,3}
[4,1,2,3] => [4]
=> []
=> ?
=> ? ∊ {1,3,3,3,3,3}
[4,1,3,2] => [3,1]
=> [1]
=> [1]
=> 1
[4,2,1,3] => [3,1]
=> [1]
=> [1]
=> 1
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[4,3,1,2] => [4]
=> []
=> ?
=> ? ∊ {1,3,3,3,3,3}
[4,3,2,1] => [2,2]
=> [2]
=> [1,1]
=> 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> [1]
=> 1
[1,3,5,2,4] => [4,1]
=> [1]
=> [1]
=> 1
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,4,2,5,3] => [4,1]
=> [1]
=> [1]
=> 1
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 1
[1,4,5,3,2] => [4,1]
=> [1]
=> [1]
=> 1
[1,5,2,3,4] => [4,1]
=> [1]
=> [1]
=> 1
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3
[1,5,4,2,3] => [4,1]
=> [1]
=> [1]
=> 1
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 1
[2,3,4,5,1] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[2,3,5,1,4] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[2,4,1,5,3] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[2,4,5,3,1] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[2,5,1,3,4] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[2,5,4,1,3] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[3,1,4,5,2] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[3,1,5,2,4] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[3,4,2,5,1] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[3,4,5,1,2] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[3,5,2,1,4] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[3,5,4,2,1] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[4,1,2,5,3] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[4,1,5,3,2] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[4,3,1,5,2] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[4,3,5,2,1] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[4,5,1,2,3] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[4,5,2,3,1] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[5,1,2,3,4] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[5,1,4,2,3] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[5,3,1,2,4] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[5,3,4,1,2] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[5,4,1,3,2] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[5,4,2,1,3] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,4}
[2,3,4,5,6,1] => [6]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[2,3,4,6,1,5] => [6]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[2,3,5,1,6,4] => [6]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[2,3,5,6,4,1] => [6]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[2,3,6,1,4,5] => [6]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[2,3,6,5,1,4] => [6]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[2,4,1,5,6,3] => [6]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[2,4,1,6,3,5] => [6]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[2,4,5,3,6,1] => [6]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[2,4,5,6,1,3] => [6]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[2,4,6,3,1,5] => [6]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[2,4,6,5,3,1] => [6]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[2,5,1,3,6,4] => [6]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[2,5,1,6,4,3] => [6]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[2,5,4,1,6,3] => [6]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[2,5,4,6,3,1] => [6]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[2,5,6,1,3,4] => [6]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
Description
The number of cells in an integer partition whose arm and leg length coincide.
Matching statistic: St001198
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 81%●distinct values known / distinct values provided: 40%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 81%●distinct values known / distinct values provided: 40%
Values
[1,2] => [1,2] => [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[2,1] => [1,2] => [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[3,1,2] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {2,2} + 1
[3,2,1] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {2,2} + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,4,2,3] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {3,3,3,3,3,3} + 1
[1,4,3,2] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {3,3,3,3,3,3} + 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {3,3,3,3,3,3} + 1
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {3,3,3,3,3,3} + 1
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[3,1,4,2] => [1,3,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[3,2,4,1] => [1,3,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[3,4,1,2] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[3,4,2,1] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[4,1,2,3] => [1,4,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {3,3,3,3,3,3} + 1
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[4,2,1,3] => [1,4,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {3,3,3,3,3,3} + 1
[4,2,3,1] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[4,3,1,2] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[4,3,2,1] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,2,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,5,2,4] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,3,5,4,2] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,4,2,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,2,4,5,3] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,4,3,5,2] => [1,2,4,5,3] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,4,5,3,2] => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,5,2,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,5,2,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,5,3,2,4] => [1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,5,3,4,2] => [1,2,5,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,5,4,2,3] => [1,2,5,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,5,4,3,2] => [1,2,5,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[2,1,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[2,3,5,1,4] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[2,5,1,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[2,5,3,1,4] => [1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[3,1,5,2,4] => [1,3,5,4,2] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[3,2,5,1,4] => [1,3,5,4,2] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[3,5,1,2,4] => [1,3,2,5,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[3,5,1,4,2] => [1,3,2,5,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[3,5,2,1,4] => [1,3,2,5,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[3,5,2,4,1] => [1,3,2,5,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[5,1,2,3,4] => [1,5,4,3,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[5,1,4,2,3] => [1,5,3,4,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[5,2,1,3,4] => [1,5,4,3,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[5,2,4,1,3] => [1,5,3,4,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[5,4,1,3,2] => [1,5,2,4,3] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[5,4,2,3,1] => [1,5,2,4,3] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[5,4,3,1,2] => [1,5,2,4,3] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[5,4,3,2,1] => [1,5,2,4,3] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,2,3,6,4,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,2,3,6,5,4] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,2,4,6,3,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,2,4,6,5,3] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,2,6,3,4,5] => [1,2,3,6,5,4] => [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,2,6,4,3,5] => [1,2,3,6,5,4] => [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,3,2,6,4,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,3,2,6,5,4] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,3,4,6,2,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,3,4,6,5,2] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,3,6,2,4,5] => [1,2,3,6,5,4] => [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,3,6,4,2,5] => [1,2,3,6,5,4] => [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,4,2,6,3,5] => [1,2,4,6,5,3] => [6,4,1,2,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,4,3,6,2,5] => [1,2,4,6,5,3] => [6,4,1,2,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,4,6,2,3,5] => [1,2,4,3,6,5] => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,4,6,2,5,3] => [1,2,4,3,6,5] => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6} + 1
Description
The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001206
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 81%●distinct values known / distinct values provided: 40%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 81%●distinct values known / distinct values provided: 40%
Values
[1,2] => [1,2] => [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[2,1] => [1,2] => [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[3,1,2] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {2,2} + 1
[3,2,1] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {2,2} + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,4,2,3] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {3,3,3,3,3,3} + 1
[1,4,3,2] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {3,3,3,3,3,3} + 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {3,3,3,3,3,3} + 1
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {3,3,3,3,3,3} + 1
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[3,1,4,2] => [1,3,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[3,2,4,1] => [1,3,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[3,4,1,2] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[3,4,2,1] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[4,1,2,3] => [1,4,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {3,3,3,3,3,3} + 1
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[4,2,1,3] => [1,4,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {3,3,3,3,3,3} + 1
[4,2,3,1] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[4,3,1,2] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[4,3,2,1] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,2,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,5,2,4] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,3,5,4,2] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,4,2,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,2,4,5,3] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,4,3,5,2] => [1,2,4,5,3] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,4,5,3,2] => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,5,2,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,5,2,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,5,3,2,4] => [1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,5,3,4,2] => [1,2,5,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,5,4,2,3] => [1,2,5,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,5,4,3,2] => [1,2,5,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[2,1,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[2,3,5,1,4] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[2,5,1,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[2,5,3,1,4] => [1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[3,1,5,2,4] => [1,3,5,4,2] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[3,2,5,1,4] => [1,3,5,4,2] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[3,5,1,2,4] => [1,3,2,5,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[3,5,1,4,2] => [1,3,2,5,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[3,5,2,1,4] => [1,3,2,5,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[3,5,2,4,1] => [1,3,2,5,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[5,1,2,3,4] => [1,5,4,3,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[5,1,4,2,3] => [1,5,3,4,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[5,2,1,3,4] => [1,5,4,3,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[5,2,4,1,3] => [1,5,3,4,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[5,4,1,3,2] => [1,5,2,4,3] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[5,4,2,3,1] => [1,5,2,4,3] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[5,4,3,1,2] => [1,5,2,4,3] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[5,4,3,2,1] => [1,5,2,4,3] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} + 1
[1,2,3,6,4,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,2,3,6,5,4] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,2,4,6,3,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,2,4,6,5,3] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,2,6,3,4,5] => [1,2,3,6,5,4] => [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,2,6,4,3,5] => [1,2,3,6,5,4] => [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,3,2,6,4,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,3,2,6,5,4] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,3,4,6,2,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,3,4,6,5,2] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,3,6,2,4,5] => [1,2,3,6,5,4] => [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,3,6,4,2,5] => [1,2,3,6,5,4] => [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,4,2,6,3,5] => [1,2,4,6,5,3] => [6,4,1,2,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,4,3,6,2,5] => [1,2,4,6,5,3] => [6,4,1,2,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,4,6,2,3,5] => [1,2,4,3,6,5] => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[1,4,6,2,5,3] => [1,2,4,3,6,5] => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6} + 1
Description
The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.
Matching statistic: St000704
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000704: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 80%●distinct values known / distinct values provided: 60%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000704: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 80%●distinct values known / distinct values provided: 60%
Values
[1,2] => [2]
=> [2]
=> []
=> ? ∊ {1,1}
[2,1] => [1,1]
=> [1,1]
=> [1]
=> ? ∊ {1,1}
[1,2,3] => [3]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,2,2}
[1,3,2] => [2,1]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2}
[2,1,3] => [2,1]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2}
[2,3,1] => [2,1]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2}
[3,1,2] => [2,1]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2}
[3,2,1] => [1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,3,4] => [4]
=> [2,2]
=> [2]
=> 1
[1,2,4,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,3,2,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,3,4,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,4,2,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3}
[2,1,3,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[2,1,4,3] => [2,2]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3}
[2,3,1,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[2,3,4,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[2,4,1,3] => [2,2]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3}
[2,4,3,1] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3}
[3,1,2,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[3,1,4,2] => [2,2]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3}
[3,2,1,4] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3}
[3,2,4,1] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3}
[3,4,1,2] => [2,2]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3}
[3,4,2,1] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3}
[4,1,2,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[4,1,3,2] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3}
[4,2,1,3] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3}
[4,2,3,1] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3}
[4,3,1,2] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3}
[4,3,2,1] => [1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,4,5] => [5]
=> [2,2,1]
=> [2,1]
=> 2
[1,2,3,5,4] => [4,1]
=> [3,2]
=> [2]
=> 1
[1,2,4,3,5] => [4,1]
=> [3,2]
=> [2]
=> 1
[1,2,4,5,3] => [4,1]
=> [3,2]
=> [2]
=> 1
[1,2,5,3,4] => [4,1]
=> [3,2]
=> [2]
=> 1
[1,2,5,4,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,3,2,4,5] => [4,1]
=> [3,2]
=> [2]
=> 1
[1,3,2,5,4] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,3,4,2,5] => [4,1]
=> [3,2]
=> [2]
=> 1
[1,3,4,5,2] => [4,1]
=> [3,2]
=> [2]
=> 1
[1,3,5,2,4] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,3,5,4,2] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,4,2,3,5] => [4,1]
=> [3,2]
=> [2]
=> 1
[1,4,2,5,3] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,4,3,2,5] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,4,5,2,3] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,4,5,3,2] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,5,2,3,4] => [4,1]
=> [3,2]
=> [2]
=> 1
[1,5,2,4,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,5,3,4,2] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,5,4,2,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,5,4,3,2] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[2,1,3,4,5] => [4,1]
=> [3,2]
=> [2]
=> 1
[2,1,3,5,4] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,1,4,3,5] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,1,4,5,3] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,1,5,3,4] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,1,5,4,3] => [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,3,1,4,5] => [4,1]
=> [3,2]
=> [2]
=> 1
[2,3,1,5,4] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,3,4,1,5] => [4,1]
=> [3,2]
=> [2]
=> 1
[2,3,4,5,1] => [4,1]
=> [3,2]
=> [2]
=> 1
[2,3,5,1,4] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,3,5,4,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[2,4,1,3,5] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,4,1,5,3] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,4,3,1,5] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[2,4,3,5,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[2,4,5,1,3] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,4,5,3,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[2,5,1,3,4] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,5,1,4,3] => [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,5,3,1,4] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[2,5,3,4,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[2,5,4,1,3] => [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,5,4,3,1] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[3,1,2,4,5] => [4,1]
=> [3,2]
=> [2]
=> 1
[3,1,2,5,4] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[3,1,4,2,5] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[3,1,4,5,2] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[3,1,5,2,4] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[3,1,5,4,2] => [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[3,2,1,4,5] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[3,2,1,5,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[3,2,4,1,5] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[3,2,4,5,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[3,2,5,1,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[3,2,5,4,1] => [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[3,4,1,2,5] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[3,4,1,5,2] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[3,4,2,1,5] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[3,4,2,5,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[3,4,5,1,2] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[3,4,5,2,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[3,5,1,2,4] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[3,5,1,4,2] => [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
Description
The number of semistandard tableaux on a given integer partition with minimal maximal entry.
This is, for an integer partition $\lambda = (\lambda_1 > \cdots > \lambda_k > 0)$, the number of [[SemistandardTableaux|semistandard tableaux]] of shape $\lambda$ with maximal entry $k$.
Equivalently, this is the evaluation $s_\lambda(1,\ldots,1)$ of the Schur function $s_\lambda$ in $k$ variables, or, explicitly,
$$ \prod_{(i,j) \in L} \frac{k + j - i}{ \operatorname{hook}(i,j) }$$
where the product is over all cells $(i,j) \in L$ and $\operatorname{hook}(i,j)$ is the hook length of a cell.
See [Theorem 6.3, 1] for details.
Matching statistic: St001128
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001128: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 80%●distinct values known / distinct values provided: 40%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001128: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 80%●distinct values known / distinct values provided: 40%
Values
[1,2] => [2]
=> [2]
=> []
=> ? ∊ {1,1}
[2,1] => [1,1]
=> [1,1]
=> [1]
=> ? ∊ {1,1}
[1,2,3] => [3]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,2,2}
[1,3,2] => [2,1]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2}
[2,1,3] => [2,1]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2}
[2,3,1] => [2,1]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2}
[3,1,2] => [2,1]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2}
[3,2,1] => [1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,3,4] => [4]
=> [2,2]
=> [2]
=> 1
[1,2,4,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,3,2,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,3,4,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,4,2,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3}
[2,1,3,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[2,1,4,3] => [2,2]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3}
[2,3,1,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[2,3,4,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[2,4,1,3] => [2,2]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3}
[2,4,3,1] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3}
[3,1,2,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[3,1,4,2] => [2,2]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3}
[3,2,1,4] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3}
[3,2,4,1] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3}
[3,4,1,2] => [2,2]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3}
[3,4,2,1] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3}
[4,1,2,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[4,1,3,2] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3}
[4,2,1,3] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3}
[4,2,3,1] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3}
[4,3,1,2] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3}
[4,3,2,1] => [1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,4,5] => [5]
=> [2,2,1]
=> [2,1]
=> 2
[1,2,3,5,4] => [4,1]
=> [3,2]
=> [2]
=> 1
[1,2,4,3,5] => [4,1]
=> [3,2]
=> [2]
=> 1
[1,2,4,5,3] => [4,1]
=> [3,2]
=> [2]
=> 1
[1,2,5,3,4] => [4,1]
=> [3,2]
=> [2]
=> 1
[1,2,5,4,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,3,2,4,5] => [4,1]
=> [3,2]
=> [2]
=> 1
[1,3,2,5,4] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,3,4,2,5] => [4,1]
=> [3,2]
=> [2]
=> 1
[1,3,4,5,2] => [4,1]
=> [3,2]
=> [2]
=> 1
[1,3,5,2,4] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,3,5,4,2] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,4,2,3,5] => [4,1]
=> [3,2]
=> [2]
=> 1
[1,4,2,5,3] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,4,3,2,5] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,4,5,2,3] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,4,5,3,2] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,5,2,3,4] => [4,1]
=> [3,2]
=> [2]
=> 1
[1,5,2,4,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,5,3,4,2] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,5,4,2,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,5,4,3,2] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[2,1,3,4,5] => [4,1]
=> [3,2]
=> [2]
=> 1
[2,1,3,5,4] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,1,4,3,5] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,1,4,5,3] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,1,5,3,4] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,1,5,4,3] => [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,3,1,4,5] => [4,1]
=> [3,2]
=> [2]
=> 1
[2,3,1,5,4] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,3,4,1,5] => [4,1]
=> [3,2]
=> [2]
=> 1
[2,3,4,5,1] => [4,1]
=> [3,2]
=> [2]
=> 1
[2,3,5,1,4] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,3,5,4,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[2,4,1,3,5] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,4,1,5,3] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,4,3,1,5] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[2,4,3,5,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[2,4,5,1,3] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,4,5,3,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[2,5,1,3,4] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,5,1,4,3] => [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,5,3,1,4] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[2,5,3,4,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[2,5,4,1,3] => [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,5,4,3,1] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[3,1,2,4,5] => [4,1]
=> [3,2]
=> [2]
=> 1
[3,1,2,5,4] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[3,1,4,2,5] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[3,1,4,5,2] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[3,1,5,2,4] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[3,1,5,4,2] => [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[3,2,1,4,5] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[3,2,1,5,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[3,2,4,1,5] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[3,2,4,5,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[3,2,5,1,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[3,2,5,4,1] => [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[3,4,1,2,5] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[3,4,1,5,2] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[3,4,2,1,5] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[3,4,2,5,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[3,4,5,1,2] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[3,4,5,2,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[3,5,1,2,4] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[3,5,1,4,2] => [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
Description
The exponens consonantiae of a partition.
This is the quotient of the least common multiple and the greatest common divior of the parts of the partiton. See [1, Caput sextum, §19-§22].
Matching statistic: St001199
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 79%●distinct values known / distinct values provided: 40%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 79%●distinct values known / distinct values provided: 40%
Values
[1,2] => [1] => [1,0]
=> [1,1,0,0]
=> ? ∊ {1,1}
[2,1] => [1] => [1,0]
=> [1,1,0,0]
=> ? ∊ {1,1}
[1,2,3] => [1,2] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,3,2] => [1,2] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[2,1,3] => [2,1] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> ? ∊ {1,2,2}
[2,3,1] => [2,1] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> ? ∊ {1,2,2}
[3,1,2] => [1,2] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[3,2,1] => [2,1] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> ? ∊ {1,2,2}
[1,2,3,4] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,4,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,3,2,4] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,3,4,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,4,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,4,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[2,1,3,4] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,1,4,3] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,3,1,4] => [2,3,1] => [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,3,4,1] => [2,3,1] => [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,4,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,4,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[3,1,2,4] => [3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,3,3,3,3,3,3}
[3,1,4,2] => [3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,3,3,3,3,3,3}
[3,2,1,4] => [3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,3,3,3,3,3,3}
[3,2,4,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,3,3,3,3,3,3}
[3,4,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,3,3,3,3,3,3}
[3,4,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,3,3,3,3,3,3}
[4,1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[4,2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[4,2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[4,3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,3,3,3,3,3,3}
[4,3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,3,3,3,3,3,3}
[1,2,3,4,5] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,3,5,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,4,3,5] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,2,4,5,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,2,5,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,5,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,3,2,4,5] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,3,2,5,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,3,4,2,5] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,3,4,5,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,3,5,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,3,5,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,4,2,3,5] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,4,2,5,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,4,3,2,5] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,4,3,5,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,4,5,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,4,5,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,5,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,5,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,5,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,5,3,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,5,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,5,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[2,1,3,4,5] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[2,1,3,5,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[2,1,4,3,5] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[2,1,4,5,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[2,1,5,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[2,1,5,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[2,3,1,4,5] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[4,1,2,3,5] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[4,1,2,5,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[4,1,3,2,5] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[4,1,3,5,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[4,1,5,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[4,1,5,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[4,2,1,3,5] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[4,2,1,5,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[4,2,3,1,5] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[4,2,3,5,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[4,2,5,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[4,2,5,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[4,3,1,2,5] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[4,3,1,5,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[4,3,2,1,5] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[4,3,2,5,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[4,3,5,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[4,3,5,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[4,5,1,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[4,5,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[4,5,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[4,5,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[4,5,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[4,5,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[5,4,1,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[5,4,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[5,4,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[5,4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[5,4,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[5,4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[5,1,2,3,4,6] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[5,1,2,3,6,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[5,1,2,4,3,6] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[5,1,2,4,6,3] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[5,1,2,6,3,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[5,1,2,6,4,3] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
[5,1,3,2,4,6] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6}
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St000207
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000207: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 78%●distinct values known / distinct values provided: 40%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000207: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 78%●distinct values known / distinct values provided: 40%
Values
[1,2] => [2]
=> []
=> ?
=> ? ∊ {1,1}
[2,1] => [1,1]
=> [1]
=> []
=> ? ∊ {1,1}
[1,2,3] => [3]
=> []
=> ?
=> ? ∊ {1,1,1,2,2}
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,2,2}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,2,2}
[2,3,1] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,2,2}
[3,1,2] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,2,2}
[3,2,1] => [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,2,3,4] => [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,3,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[2,3,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[2,4,1,3] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[3,1,4,2] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,1,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,3,4,5] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,2,4,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,2,5,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,2,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,3,5,2,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,4,2,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,5,2,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[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] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,4,2,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,3,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,1,3,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,1,4,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,1,4,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,1,5,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,3,1,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,3,1,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,3,4,1,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,3,4,5,1] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,3,5,1,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,3,5,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,4,1,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,4,1,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,4,3,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,4,5,1,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,4,5,3,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,5,1,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,5,1,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,5,3,1,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,5,4,1,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,1,2,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[3,1,5,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,1,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,4,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,5,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,5,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,4,2,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,4,2,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,4,5,2,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,5,2,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,5,2,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,5,4,1,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,5,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,1,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,1,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,1,5,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,2,1,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has all vertices in integer lattice points.
Matching statistic: St000208
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 78%●distinct values known / distinct values provided: 40%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 78%●distinct values known / distinct values provided: 40%
Values
[1,2] => [2]
=> []
=> ?
=> ? ∊ {1,1}
[2,1] => [1,1]
=> [1]
=> []
=> ? ∊ {1,1}
[1,2,3] => [3]
=> []
=> ?
=> ? ∊ {1,1,1,2,2}
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,2,2}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,2,2}
[2,3,1] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,2,2}
[3,1,2] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,2,2}
[3,2,1] => [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,2,3,4] => [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,3,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[2,3,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[2,4,1,3] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[3,1,4,2] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,1,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,3,4,5] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,2,4,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,2,5,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,2,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,3,5,2,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,4,2,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,5,2,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[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] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,4,2,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,3,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,1,3,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,1,4,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,1,4,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,1,5,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,3,1,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,3,1,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,3,4,1,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,3,4,5,1] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,3,5,1,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,3,5,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,4,1,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,4,1,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,4,3,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,4,5,1,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,4,5,3,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,5,1,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[2,5,1,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,5,3,1,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,5,4,1,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,1,2,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[3,1,5,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,1,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,4,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,5,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,5,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,4,2,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,4,2,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,4,5,2,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,5,2,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,5,2,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,5,4,1,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,5,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,1,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,1,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,1,5,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,2,1,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has only integer lattice points as vertices.
See also [[St000205]], [[St000206]] and [[St000207]].
The following 86 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000260The radius of a connected graph. St001568The smallest positive integer that does not appear twice in the partition. St001256Number of simple reflexive modules that are 2-stable reflexive. St000781The number of proper colouring schemes of a Ferrers diagram. St000914The sum of the values of the Möbius function of a poset. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by 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. St000259The diameter of a connected graph. St000735The last entry on the main diagonal of a standard tableau. St001890The maximum magnitude of the Möbius function of a poset. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000454The largest eigenvalue of a graph if it is integral. St000284The Plancherel distribution on integer partitions. St000668The least common multiple of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St000618The number of self-evacuating tableaux of given shape. St001432The order dimension of the partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000456The monochromatic index of a connected graph. St001964The interval resolution global dimension of a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000741The Colin de Verdière graph invariant. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000937The number of positive values of the symmetric group character corresponding to the partition. St000850The number of 1/2-balanced pairs in a poset. St000455The second largest eigenvalue of a graph if it is integral. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000633The size of the automorphism group of a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000451The length of the longest pattern of the form k 1 2. St000534The number of 2-rises of a permutation. St000842The breadth of a permutation. St001330The hat guessing number of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number 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. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000553The number of blocks of a graph. St000552The number of cut vertices of a graph. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice.
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