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Your data matches 214 different statistics following compositions of up to 3 maps.
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Matching statistic: St001594
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
St001594: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1
[1,0,1,0]
=> 0
[1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> 0
[1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> 0
[1,1,1,0,0,0]
=> 2
[1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> 0
Description
The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied.
See the link for the definition.
Matching statistic: St001222
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001222: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 84%●distinct values known / distinct values provided: 75%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001222: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 84%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> []
=> ?
=> ?
=> ? = 1
[1,0,1,0]
=> [1]
=> []
=> []
=> ? ∊ {0,1}
[1,1,0,0]
=> []
=> ?
=> ?
=> ? ∊ {0,1}
[1,0,1,0,1,0]
=> [2,1]
=> [1]
=> [1,0]
=> 0
[1,0,1,1,0,0]
=> [1,1]
=> [1]
=> [1,0]
=> 0
[1,1,0,0,1,0]
=> [2]
=> []
=> []
=> ? ∊ {0,1,2}
[1,1,0,1,0,0]
=> [1]
=> []
=> []
=> ? ∊ {0,1,2}
[1,1,1,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {0,1,2}
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> [1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> [1,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> [1,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [3]
=> []
=> []
=> ? ∊ {0,1,2,2}
[1,1,1,0,0,1,0,0]
=> [2]
=> []
=> []
=> ? ∊ {0,1,2,2}
[1,1,1,0,1,0,0,0]
=> [1]
=> []
=> []
=> ? ∊ {0,1,2,2}
[1,1,1,1,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {0,1,2,2}
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [3]
=> [1,0,1,0,1,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [3]
=> [1,0,1,0,1,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2]
=> [1,0,1,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1]
=> [1,0]
=> 0
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1]
=> [1,0]
=> 0
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1]
=> [1,0]
=> 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1]
=> [1,0]
=> 0
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> []
=> ? ∊ {0,0,1,1,3}
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> []
=> ? ∊ {0,0,1,1,3}
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> []
=> ? ∊ {0,0,1,1,3}
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> []
=> ? ∊ {0,0,1,1,3}
[1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,1,3}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,3,3,3,3}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,3,3,3,3}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,3,3,3,3}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,3,3,3,3}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,3,3,3,3}
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,3,3,3,3}
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,3,3,3,3}
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,3,3,3,3}
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,3,3,3,3}
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,3,3,3,3}
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,3,3,3,3}
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,3,3,3,3}
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,3,3,3,3}
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,3,3,3,3}
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,3,3,3,3}
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,3,3,3,3}
Description
Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module.
Matching statistic: St001714
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001714: Integer partitions ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 100%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001714: Integer partitions ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [[1],[]]
=> []
=> ? = 1
[1,0,1,0]
=> [1,0,1,0]
=> [[1,1],[]]
=> []
=> ? ∊ {0,1}
[1,1,0,0]
=> [1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {0,1}
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {0,0,1,2}
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {0,0,1,2}
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 0
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {0,0,1,2}
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {0,0,1,2}
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 0
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 0
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 0
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 0
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 0
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 0
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 0
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 0
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 0
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 0
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,3}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,3}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,3}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,3}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,3}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,3}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,3}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,3}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 0
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,3}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,3}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,3}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,3}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,3}
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,3}
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,3}
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,3}
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,3}
Description
The number of subpartitions of an integer partition that do not dominate the conjugate subpartition.
In particular, partitions with statistic $0$ are wide partitions.
Matching statistic: St000506
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000506: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 79%●distinct values known / distinct values provided: 50%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000506: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 79%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1] => [1]
=> []
=> ? = 1
[1,0,1,0]
=> [1,2] => [1,1]
=> [1]
=> 0
[1,1,0,0]
=> [2,1] => [2]
=> []
=> ? = 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [1]
=> 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [1]
=> 0
[1,1,0,1,0,0]
=> [2,3,1] => [3]
=> []
=> ? ∊ {0,2}
[1,1,1,0,0,0]
=> [3,1,2] => [3]
=> []
=> ? ∊ {0,2}
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1]
=> [1]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [3,1]
=> [1]
=> 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1]
=> [1]
=> 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4]
=> []
=> ? ∊ {1,1,2,2}
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [4]
=> []
=> ? ∊ {1,1,2,2}
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,1]
=> [1]
=> 0
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [4]
=> []
=> ? ∊ {1,1,2,2}
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,2]
=> [2]
=> 0
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {1,1,2,2}
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1]
=> [1]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [4,1]
=> [1]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [4,1]
=> [1]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,2,1]
=> [2,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [4,1]
=> [1]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2]
=> [2]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [3,2]
=> [2]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> [2]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1]
=> [1]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3}
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3}
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [4,1]
=> [1]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3}
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [3,2]
=> [2]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3}
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [3,2]
=> [2]
=> 0
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [4,1]
=> [1]
=> 0
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3}
[1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3}
[1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [2,2,1]
=> [2,1]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => [3,2]
=> [2]
=> 0
[1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3}
[1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => [3,2]
=> [2]
=> 0
[1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [4,1]
=> [1]
=> 0
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3}
[1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [3,2]
=> [2]
=> 0
[1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3}
[1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [3,1,1,1]
=> [1,1,1]
=> 0
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => [6]
=> []
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,1,5] => [6]
=> []
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,1,6,4] => [6]
=> []
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,1,4,5] => [6]
=> []
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,6,3] => [6]
=> []
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,1,6,3,5] => [6]
=> []
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,5,6,1,3] => [6]
=> []
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,1,3,6,4] => [6]
=> []
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,5,6,1,3,4] => [6]
=> []
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,1,3,4,5] => [6]
=> []
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,2] => [6]
=> []
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,1,4,6,2,5] => [6]
=> []
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,6,4] => [6]
=> []
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,1,6,2,4,5] => [6]
=> []
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,1,6,2] => [6]
=> []
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,6,1,2,5] => [6]
=> []
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,1,2,5,6,3] => [6]
=> []
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,1,2,6,3,5] => [6]
=> []
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [4,1,5,6,2,3] => [6]
=> []
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,5,1,2,6,3] => [6]
=> []
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [4,6,1,2,3,5] => [6]
=> []
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,6,4] => [6]
=> []
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,1,6,2,3,4] => [6]
=> []
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,1,2,3,4,5] => [6]
=> []
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
Description
The number of standard desarrangement tableaux of shape equal to the given partition.
A '''standard desarrangement tableau''' is a standard tableau whose first ascent is even. Here, an ascent of a standard tableau is an entry $i$ such that $i+1$ appears to the right or above $i$ in the tableau (with respect to English tableau notation).
This is also the nullity of the random-to-random operator (and the random-to-top) operator acting on the simple module of the symmetric group indexed by the given partition. See also:
* [[St000046]]: The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition
* [[St000500]]: Eigenvalues of the random-to-random operator acting on the regular representation.
Matching statistic: St001525
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001525: Integer partitions ⟶ ℤResult quality: 74% ●values known / values provided: 74%●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: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> []
=> ?
=> ?
=> ? = 1
[1,0,1,0]
=> [1]
=> []
=> ?
=> ? ∊ {0,1}
[1,1,0,0]
=> []
=> ?
=> ?
=> ? ∊ {0,1}
[1,0,1,0,1,0]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,2}
[1,0,1,1,0,0]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,2}
[1,1,0,0,1,0]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0,1,2}
[1,1,0,1,0,0]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,0,1,2}
[1,1,1,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,1,2}
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,2}
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,2}
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,2}
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,2}
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,2}
[1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,2,2}
[1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,2,2}
[1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,2,2}
[1,1,1,1,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,2,2}
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> [2,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,1]
=> [2,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> [2]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,2]
=> [2]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [2,2]
=> [2]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,1]
=> [1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,1]
=> [1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,1]
=> [1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3}
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3}
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3}
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3}
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3}
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3}
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3}
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3}
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3}
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3}
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3}
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3}
[1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [4,3,2,1]
=> [3,2,1]
=> 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [4,3,2,1]
=> [3,2,1]
=> 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [3,3,2,1]
=> [3,2,1]
=> 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [3,3,2,1]
=> [3,2,1]
=> 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [3,3,2,1]
=> [3,2,1]
=> 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [4,2,2,1]
=> [2,2,1]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [4,2,2,1]
=> [2,2,1]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [3,2,2,1]
=> [2,2,1]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [3,2,2,1]
=> [2,2,1]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [3,2,2,1]
=> [2,2,1]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [4,3,1,1]
=> [3,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [4,3,1,1]
=> [3,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [3,3,1,1]
=> [3,1,1]
=> 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3}
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3}
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3}
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3}
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3}
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3}
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3}
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3}
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3}
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3}
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3}
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3}
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3}
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3}
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3}
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3}
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3}
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3}
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3}
Description
The number of symmetric hooks on the diagonal of a partition.
Matching statistic: St000319
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[1],[]]
=> []
=> ? = 1
[1,0,1,0]
=> [[1,1],[]]
=> []
=> ? ∊ {0,1}
[1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {0,1}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {0,0,1,2}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {0,0,1,2}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 0
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {0,0,1,2}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {0,0,1,2}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 0
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 0
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 1
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 0
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 0
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 0
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 0
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> 0
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 0
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> 0
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> 0
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[1],[]]
=> []
=> ? = 1
[1,0,1,0]
=> [[1,1],[]]
=> []
=> ? ∊ {0,1}
[1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {0,1}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {0,0,1,2}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {0,0,1,2}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 0
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {0,0,1,2}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {0,0,1,2}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 0
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 0
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 1
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 0
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 0
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 0
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 0
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> 0
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 0
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> 0
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> 0
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
Description
The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St001122
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001122: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 68%●distinct values known / distinct values provided: 50%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001122: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 68%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [[1],[]]
=> []
=> ? = 1
[1,0,1,0]
=> [[1,1],[]]
=> []
=> ? ∊ {0,1}
[1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {0,1}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,2}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {0,0,0,2}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {0,0,0,2}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {0,0,0,2}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,2,2}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,2,2}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,2,2}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,2,2}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 0
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,2,2}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,2,2}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,2,2}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,2,2}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,3}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,3}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,3}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,3}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,3}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,3}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,3}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,3}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,3}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 0
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,3}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,3}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,3}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,3}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 0
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,3}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,3}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,3}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> 0
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> 0
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> 0
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> 0
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> 0
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
Description
The multiplicity of the sign representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{1^n}$, for $\lambda\vdash n$. It equals $1$ if and only if $\lambda$ is self-conjugate.
Matching statistic: St001280
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[1],[]]
=> []
=> ? = 1
[1,0,1,0]
=> [[1,1],[]]
=> []
=> ? ∊ {0,1}
[1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {0,1}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {0,0,1,2}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {0,0,1,2}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 0
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {0,0,1,2}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {0,0,1,2}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 0
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 0
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 1
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 0
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 0
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 0
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 0
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> 0
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 0
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> 0
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> 0
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3}
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St001384
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001384: Integer partitions ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001384: Integer partitions ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[1],[]]
=> []
=> ? = 1
[1,0,1,0]
=> [[1,1],[]]
=> []
=> ? ∊ {0,1}
[1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {0,1}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {0,0,1,2}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {0,0,1,2}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 0
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {0,0,1,2}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {0,0,1,2}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,2,2}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,2,2}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 0
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,2,2}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,2,2}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 0
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 1
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,2,2}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 0
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 0
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,2,2}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,2,2}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,2,2}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 0
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 0
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> 0
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 0
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 0
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> 0
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> 0
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
Description
The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains.
The following 204 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000010The length of the partition. St000053The number of valleys of the Dyck path. St000120The number of left tunnels of a Dyck path. St000147The largest part of an integer partition. St000148The number of odd parts of a partition. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000183The side length of the Durfee square of an integer partition. St000306The bounce count of a Dyck path. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000369The dinv deficit of a Dyck path. St000378The diagonal inversion number of an integer partition. St000442The maximal area to the right of an up step of a Dyck path. St000475The number of parts equal to 1 in a partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000548The number of different non-empty partial sums of an integer partition. St000549The number of odd partial sums of an integer partition. St000658The number of rises of length 2 of a Dyck path. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000783The side length of the largest staircase partition fitting into a partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000885The number of critical steps in the Catalan decomposition of a binary word. St000897The number of different multiplicities of parts of an integer partition. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000992The alternating sum of the parts of an integer partition. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001484The number of singletons of an integer partition. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001955The number of natural descents for set-valued two row standard Young tableaux. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000659The number of rises of length at least 2 of a Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St000934The 2-degree of an integer partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000663The number of right floats of a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000732The number of double deficiencies of a permutation. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000137The Grundy value of an integer partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St001175The size of a partition minus the hook length of the base cell. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001593This is the number of standard Young tableaux of the given shifted shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000929The constant term of the character polynomial of an integer partition. St001651The Frankl number of a lattice. St000376The bounce deficit of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001568The smallest positive integer that does not appear twice in the partition. St000646The number of big ascents of a permutation. St001435The number of missing boxes in the first row. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001557The number of inversions of the second entry of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001570The minimal number of edges to add to make a graph Hamiltonian. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000650The number of 3-rises of a permutation. St000837The number of ascents of distance 2 of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. 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. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000944The 3-degree of an integer partition. St001176The size of a partition minus its first part. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001389The number of partitions of the same length below the given integer partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001571The Cartan determinant of the integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. 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. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001961The sum of the greatest common divisors of all pairs of parts. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000478Another weight of a partition according to Alladi. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000993The multiplicity of the largest part of an integer partition. St001383The BG-rank of an integer partition. St000455The second largest eigenvalue of a graph if it is integral. St000284The Plancherel distribution on integer partitions. St000567The sum of the products of all pairs of parts. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001128The exponens consonantiae of a partition. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001811The Castelnuovo-Mumford regularity of a permutation. St001948The number of augmented double ascents of a permutation. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. 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. St000648The number of 2-excedences of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001816Eigenvalues of the top-to-random operator acting on a simple module. 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. St000668The least common multiple of the parts of the partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000326The position of the first one in a binary word after appending a 1 at the end. St000454The largest eigenvalue of a graph if it is integral. St000215The number of adjacencies of a permutation, zero appended. St000546The number of global descents of a permutation.
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