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Your data matches 33 different statistics following compositions of up to 3 maps.
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Matching statistic: St001940
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(load all 2 compositions to match this statistic)
St001940: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 0
[1,1]
=> 0
[3]
=> 0
[2,1]
=> 1
[1,1,1]
=> 0
[4]
=> 0
[3,1]
=> 1
[2,2]
=> 1
[2,1,1]
=> 0
[1,1,1,1]
=> 0
[5]
=> 0
[4,1]
=> 1
[3,2]
=> 0
[3,1,1]
=> 0
[2,2,1]
=> 2
[2,1,1,1]
=> 0
[1,1,1,1,1]
=> 0
[6]
=> 0
[5,1]
=> 1
[4,2]
=> 0
[4,1,1]
=> 0
[3,3]
=> 0
[3,2,1]
=> 1
[3,1,1,1]
=> 0
[2,2,2]
=> 0
[2,2,1,1]
=> 1
[2,1,1,1,1]
=> 0
[1,1,1,1,1,1]
=> 0
[7]
=> 0
[6,1]
=> 1
[5,2]
=> 0
[5,1,1]
=> 0
[4,3]
=> 0
[4,2,1]
=> 1
[4,1,1,1]
=> 0
[3,3,1]
=> 1
[3,2,2]
=> 1
[3,2,1,1]
=> 0
[3,1,1,1,1]
=> 0
[2,2,2,1]
=> 1
[2,2,1,1,1]
=> 1
[2,1,1,1,1,1]
=> 0
[1,1,1,1,1,1,1]
=> 0
[8]
=> 0
[7,1]
=> 1
[6,2]
=> 0
[6,1,1]
=> 0
[5,3]
=> 0
[5,2,1]
=> 1
Description
The number of distinct parts that are equal to their multiplicity in the integer partition.
Matching statistic: St001525
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001525: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 78%●distinct values known / distinct values provided: 75%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001525: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 78%●distinct values known / distinct values provided: 75%
Values
[1]
=> []
=> ?
=> ?
=> ? = 1
[2]
=> []
=> ?
=> ?
=> ? ∊ {0,0}
[1,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0}
[3]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1}
[2,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,1}
[1,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,1}
[4]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,1}
[3,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,0,1}
[2,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0,1}
[2,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1}
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[5]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,2}
[4,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,0,0,2}
[3,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0,0,2}
[3,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,2}
[2,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,2}
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,1}
[5,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1}
[4,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1}
[4,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1}
[3,3]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1}
[3,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1}
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[2,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1}
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[7]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1}
[6,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1}
[5,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1}
[5,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1}
[4,3]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1}
[4,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1}
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[3,3,1]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1}
[3,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1}
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[8]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,2}
[7,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,2}
[6,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,2}
[6,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,2}
[5,3]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,2}
[5,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,2}
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[4,4]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,2}
[4,3,1]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,2}
[4,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,2}
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,3,2]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,2}
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[9]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2}
[8,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2}
[7,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2}
[7,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2}
[6,3]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2}
[6,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2}
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[5,4]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2}
[5,3,1]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2}
[5,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2}
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,4,1]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2}
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[3,3,2,1]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[3,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[2,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 0
[7,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[6,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[5,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
Description
The number of symmetric hooks on the diagonal of a partition.
Matching statistic: St001593
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001593: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 78%●distinct values known / distinct values provided: 75%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001593: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 78%●distinct values known / distinct values provided: 75%
Values
[1]
=> []
=> ?
=> ?
=> ? = 1
[2]
=> []
=> ?
=> ?
=> ? ∊ {0,0}
[1,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0}
[3]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1}
[2,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,1}
[1,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,1}
[4]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,1}
[3,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,0,1}
[2,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0,1}
[2,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1}
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[5]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,2}
[4,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,0,0,2}
[3,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0,0,2}
[3,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,2}
[2,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,2}
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,1}
[5,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1}
[4,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1}
[4,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1}
[3,3]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1}
[3,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1}
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[2,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1}
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[7]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1}
[6,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1}
[5,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1}
[5,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1}
[4,3]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1}
[4,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1}
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[3,3,1]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1}
[3,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1}
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[8]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,2}
[7,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,2}
[6,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,2}
[6,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2}
[5,3]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,2}
[5,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2}
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[4,4]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,2}
[4,3,1]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2}
[4,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2}
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,3,2]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2}
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[9]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2}
[8,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2}
[7,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2}
[7,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2}
[6,3]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2}
[6,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2}
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[5,4]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2}
[5,3,1]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2}
[5,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2}
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,4,1]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2}
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[3,3,2,1]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[3,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[2,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 0
[7,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[6,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[5,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
Description
This is the number of standard Young tableaux of the given shifted shape.
For an integer partition $\lambda = (\lambda_1,\dots,\lambda_k)$, the shifted diagram is obtained by moving the $i$-th row in the diagram $i-1$ boxes to the right, i.e.,
$$\lambda^∗ = \{(i, j) | 1 \leq i \leq k, i \leq j \leq \lambda_i + i − 1 \}.$$
In particular, this statistic is zero if and only if $\lambda_{i+1} = \lambda_i$ for some $i$.
Matching statistic: St000225
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 77%●distinct values known / distinct values provided: 75%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 77%●distinct values known / distinct values provided: 75%
Values
[1]
=> []
=> ?
=> ?
=> ? = 1
[2]
=> []
=> ?
=> ?
=> ? ∊ {0,0}
[1,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0}
[3]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1}
[2,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,1}
[1,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,1}
[4]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,1}
[3,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,1,1}
[2,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,1,1}
[2,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1}
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[5]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,1,2}
[4,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,0,1,2}
[3,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0,1,2}
[3,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,2}
[2,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,2}
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,1,1,1}
[5,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1}
[4,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1}
[4,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[3,3]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1}
[3,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[2,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[7]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1}
[6,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1}
[5,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1}
[5,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1}
[4,3]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1}
[4,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1}
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[3,3,1]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1}
[3,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1}
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[8]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,2}
[7,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,2}
[6,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,2}
[6,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,2}
[5,3]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,2}
[5,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,2}
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[4,4]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,2}
[4,3,1]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,2}
[4,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,2}
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,3,2]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,2}
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[9]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2}
[8,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2}
[7,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2}
[7,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2}
[6,3]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2}
[6,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2}
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[5,4]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2}
[5,3,1]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2}
[5,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2}
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,4,1]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2}
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[3,3,2,1]
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[3,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[2,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 0
[7,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[6,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
[5,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
Description
Difference between largest and smallest parts in a partition.
Matching statistic: St000661
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000661: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 75%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000661: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 75%●distinct values known / distinct values provided: 50%
Values
[1]
=> []
=> ?
=> ?
=> ? = 1
[2]
=> []
=> ?
=> ?
=> ? ∊ {0,0}
[1,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0}
[3]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1}
[2,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,1}
[1,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,1}
[4]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,1}
[3,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,1,1}
[2,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,1,1}
[2,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,1,1}
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[5]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,1,2}
[4,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,1,2}
[3,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,0,1,2}
[3,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,1,2}
[2,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,1,2}
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[6]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,1,1,1}
[5,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1}
[4,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1}
[4,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,1,1,1}
[3,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1}
[3,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,1,1,1}
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[7]
=> []
=> ?
=> ?
=> ? ∊ {0,1,1,1,1,1,1}
[6,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1}
[5,2]
=> [2]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1}
[5,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> ? ∊ {0,1,1,1,1,1,1}
[4,3]
=> [3]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1}
[4,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {0,1,1,1,1,1,1}
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[3,3,1]
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,1,1,1,1,1,1}
[3,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[8]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,1,1,1,1,2}
[7,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,2}
[6,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,2}
[6,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,1,1,1,1,2}
[5,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,2}
[5,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,1,1,1,1,2}
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[4,4]
=> [4]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,2}
[4,3,1]
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,1,1,1,1,2}
[4,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[3,3,2]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 0
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[9]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,2}
[8,1]
=> [1]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2}
[7,2]
=> [2]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2}
[7,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2}
[6,3]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2}
[6,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2}
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[5,4]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2}
[5,3,1]
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2}
[5,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[4,4,1]
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2}
[4,3,2]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 0
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[3,3,3]
=> [3,3]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[3,3,2,1]
=> [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[3,2,2,2]
=> [2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[3,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[2,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[10]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2}
[9,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2}
[8,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2}
[8,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2}
[7,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2}
Description
The number of rises of length 3 of a Dyck path.
Matching statistic: St001230
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001230: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 75%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001230: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 75%
Values
[1]
=> []
=> ?
=> ?
=> ? = 1
[2]
=> []
=> ?
=> ?
=> ? ∊ {0,0}
[1,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0}
[3]
=> []
=> ?
=> ?
=> ? ∊ {0,1}
[2,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1}
[1,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 0
[4]
=> []
=> ?
=> ?
=> ? ∊ {0,1,1}
[3,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1,1}
[2,2]
=> [2]
=> []
=> []
=> ? ∊ {0,1,1}
[2,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[5]
=> []
=> ?
=> ?
=> ? ∊ {0,1,2}
[4,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1,2}
[3,2]
=> [2]
=> []
=> []
=> ? ∊ {0,1,2}
[3,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 0
[2,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[6]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,1}
[5,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,1}
[4,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,0,1}
[4,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 0
[3,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,0,1}
[3,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[7]
=> []
=> ?
=> ?
=> ? ∊ {0,1,1,1}
[6,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1,1,1}
[5,2]
=> [2]
=> []
=> []
=> ? ∊ {0,1,1,1}
[5,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 0
[4,3]
=> [3]
=> []
=> []
=> ? ∊ {0,1,1,1}
[4,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[3,3,1]
=> [3,1]
=> [1]
=> [1,0]
=> 0
[3,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[8]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,2}
[7,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,0,2}
[6,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,0,0,2}
[6,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 0
[5,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,0,0,2}
[5,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[4,4]
=> [4]
=> []
=> []
=> ? ∊ {0,0,0,0,2}
[4,3,1]
=> [3,1]
=> [1]
=> [1,0]
=> 0
[4,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[3,3,2]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[9]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,1,1,2}
[8,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,1,1,1,2}
[7,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,1,1,1,2}
[7,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 0
[6,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,1,1,1,2}
[6,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[5,4]
=> [4]
=> []
=> []
=> ? ∊ {0,0,1,1,1,2}
[5,3,1]
=> [3,1]
=> [1]
=> [1,0]
=> 0
[5,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[4,4,1]
=> [4,1]
=> [1]
=> [1,0]
=> 0
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,1,1,1,2}
[10]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,2,2}
[9,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1,2,2}
[8,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1,2,2}
[7,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1,2,2}
[6,4]
=> [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1,2,2}
[5,5]
=> [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1,2,2}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,1,2,2}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,1,2,2}
[11]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2}
[10,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2}
[9,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2}
[8,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2}
[7,4]
=> [4]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2}
[6,5]
=> [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2}
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2}
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2}
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2}
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2}
[12]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2}
Description
The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property.
Matching statistic: St001123
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001123: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 75%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001123: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 75%
Values
[1]
=> []
=> ?
=> ?
=> ? = 1
[2]
=> []
=> ?
=> ?
=> ? ∊ {0,0}
[1,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0}
[3]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1}
[2,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,1}
[1,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,1}
[4]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,1,1}
[3,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,0,1,1}
[2,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0,1,1}
[2,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1}
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,1,1}
[5]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,2}
[4,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,2}
[3,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,2}
[3,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,2}
[2,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,2}
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,2}
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[6]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1}
[5,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1}
[4,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1}
[4,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1}
[3,3]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1}
[3,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1}
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1}
[2,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1}
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1}
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[7]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1}
[6,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1}
[5,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1}
[5,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1}
[4,3]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1}
[4,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1}
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1}
[3,3,1]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1}
[3,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1}
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1}
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1}
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[8]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[7,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[6,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[6,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[5,3]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[5,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[4,4]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[4,3,1]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[4,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,3,2]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[3,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[2,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 0
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[5,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[4,3,1,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[4,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[4,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[3,3,2,2]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[3,3,2,1,1]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[3,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[3,2,2,1,1,1]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[2,2,2,2,2]
=> [2,2,2,2]
=> [2,2,2]
=> [2,2]
=> 0
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 1
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 0
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> 0
[7,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
Description
The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{21^{n-2}}$, for $\lambda\vdash n$.
Matching statistic: St001124
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 75%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 75%
Values
[1]
=> []
=> ?
=> ?
=> ? = 1
[2]
=> []
=> ?
=> ?
=> ? ∊ {0,0}
[1,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0}
[3]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1}
[2,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,1}
[1,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,1}
[4]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,1,1}
[3,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,0,1,1}
[2,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0,1,1}
[2,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1}
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,1,1}
[5]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,1,2}
[4,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,2}
[3,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,2}
[3,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,2}
[2,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,2}
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,2}
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,1}
[5,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,1}
[4,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,1}
[4,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1}
[3,3]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,1}
[3,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1}
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1}
[2,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1}
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1}
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[7]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1}
[6,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1}
[5,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1}
[5,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1}
[4,3]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1}
[4,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1}
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1}
[3,3,1]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1}
[3,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1}
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1}
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1}
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[8]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[7,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[6,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[6,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[5,3]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[5,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[4,4]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[4,3,1]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[4,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,3,2]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[3,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[2,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 0
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[4,3,1,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[4,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[3,3,2,2]
=> [3,2,2]
=> [2,2]
=> [2]
=> 0
[3,3,2,1,1]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[3,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[3,2,2,1,1,1]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[2,2,2,2,2]
=> [2,2,2,2]
=> [2,2,2]
=> [2,2]
=> 0
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 1
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 0
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> 0
[7,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
Description
The multiplicity of the standard representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{(n-1)1}$, for $\lambda\vdash n > 1$. For $n\leq1$ the statistic is undefined.
It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.
Matching statistic: St000650
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000650: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 75%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000650: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 75%
Values
[1]
=> []
=> []
=> [] => ? = 1
[2]
=> []
=> []
=> [] => ? = 0
[1,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[3]
=> []
=> []
=> [] => ? = 1
[2,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[4]
=> []
=> []
=> [] => ? = 1
[3,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[5]
=> []
=> []
=> [] => ? = 2
[4,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 0
[6]
=> []
=> []
=> [] => ? = 0
[5,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 0
[7]
=> []
=> []
=> [] => ? ∊ {0,1}
[6,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 0
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? ∊ {0,1}
[8]
=> []
=> []
=> [] => ? ∊ {0,0,2}
[7,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 0
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 0
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? ∊ {0,0,2}
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => ? ∊ {0,0,2}
[9]
=> []
=> []
=> [] => ? ∊ {0,0,1,1,2}
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? ∊ {0,0,1,1,2}
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? ∊ {0,0,1,1,2}
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => ? ∊ {0,0,1,1,2}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,8,7,6,5,4,3,2] => ? ∊ {0,0,1,1,2}
[10]
=> []
=> []
=> [] => ? ∊ {0,0,0,0,0,1,2,2}
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? ∊ {0,0,0,0,0,1,2,2}
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? ∊ {0,0,0,0,0,1,2,2}
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => ? ∊ {0,0,0,0,0,1,2,2}
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => ? ∊ {0,0,0,0,0,1,2,2}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7,8,6,5,4,3,2] => ? ∊ {0,0,0,0,0,1,2,2}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,8,7,6,5,4,3,2] => ? ∊ {0,0,0,0,0,1,2,2}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10,9,8,7,6,5,4,3,2] => ? ∊ {0,0,0,0,0,1,2,2}
[11]
=> []
=> []
=> [] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2}
[5,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2}
[4,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2}
[4,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2}
[3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,5,7,4,3,2] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2}
[3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2}
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7,8,6,5,4,3,2] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2}
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,8,7,6,5,4,3,2] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2}
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2}
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,6,8,7,5,4,3,2] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2}
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,8,9,7,6,5,4,3,2] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2}
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10,9,8,7,6,5,4,3,2] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,10,9,8,7,6,5,4,3,2] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2}
[12]
=> []
=> []
=> [] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2}
[6,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2}
[6,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2}
[5,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2}
[5,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2}
[4,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,5,7,4,3,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2}
[4,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2}
[4,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7,8,6,5,4,3,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2}
[4,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,8,7,6,5,4,3,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2}
[3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,5,6,7,4,3,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2}
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,7,6,8,5,4,3,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2}
[3,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2}
[3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,6,8,7,5,4,3,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2}
Description
The number of 3-rises of a permutation.
A 3-rise of a permutation $\pi$ is an index $i$ such that $\pi(i)+3 = \pi(i+1)$.
For 1-rises, or successions, see [[St000441]], for 2-rises see [[St000534]].
Matching statistic: St000929
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 50%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 50%
Values
[1]
=> []
=> []
=> []
=> ? = 1
[2]
=> []
=> []
=> []
=> ? ∊ {0,0}
[1,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,0}
[3]
=> []
=> []
=> []
=> ? ∊ {0,0,1}
[2,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,0,1}
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> ? ∊ {0,0,1}
[4]
=> []
=> []
=> []
=> ? ∊ {0,0,0,1,1}
[3,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,0,0,1,1}
[2,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> ? ∊ {0,0,0,1,1}
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> ? ∊ {0,0,0,1,1}
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,0,0,1,1}
[5]
=> []
=> []
=> []
=> ? ∊ {0,0,0,0,2}
[4,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,0,0,0,2}
[3,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> ? ∊ {0,0,0,0,2}
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> ? ∊ {0,0,0,0,2}
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,0,0,0,2}
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 0
[6]
=> []
=> []
=> []
=> ? ∊ {0,0,0,0,1,1}
[5,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,0,0,0,1,1}
[4,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> ? ∊ {0,0,0,0,1,1}
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1}
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 0
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,0,0,0,1,1}
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1}
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 0
[7]
=> []
=> []
=> []
=> ? ∊ {0,1,1,1,1,1}
[6,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,1,1,1,1,1}
[5,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> ? ∊ {0,1,1,1,1,1}
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> ? ∊ {0,1,1,1,1,1}
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 0
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,1,1,1,1,1}
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 0
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,1,1,1,1}
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> 0
[8]
=> []
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,2}
[7,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2}
[6,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,2}
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2}
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 0
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,2}
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 0
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 0
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2}
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 0
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2}
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1]
=> ? ∊ {0,0,0,0,1,1,1,2}
[9]
=> []
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2}
[8,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2}
[7,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2}
[7,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2}
[6,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 0
[6,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1
[6,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2}
[5,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 0
[5,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 0
[5,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2}
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 0
[4,4,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 0
[4,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 0
[4,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 0
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 0
[3,3,3]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2}
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 0
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> 0
[3,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2}
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 0
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> 0
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> 0
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 0
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> 0
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2}
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2}
[10]
=> []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
[9,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
[8,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
[7,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 0
Description
The constant term of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1]. Indeed, this constant term is $0$ for partitions $\lambda \neq 1^n$ and $1$ for $\lambda = 1^n$.
The following 23 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000649The number of 3-excedences of a permutation. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001520The number of strict 3-descents. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001498The normalised height of a Nakayama algebra with magnitude 1. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000214The number of adjacencies of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000873The aix statistic of a permutation. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St000181The number of connected components of the Hasse diagram for the poset. St000640The rank of the largest boolean interval in a poset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000850The number of 1/2-balanced pairs in a poset. St001615The number of join prime elements of a lattice. St000633The size of the automorphism group of a poset. St000455The second largest eigenvalue of a graph if it is integral.
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