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Your data matches 18 different statistics following compositions of up to 3 maps.
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Matching statistic: St001454
St001454: Plane partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> 0
[[1],[1]]
=> 0
[[2]]
=> 0
[[1,1]]
=> 0
[[1],[1],[1]]
=> 0
[[2],[1]]
=> 1
[[1,1],[1]]
=> 0
[[3]]
=> 0
[[2,1]]
=> 1
[[1,1,1]]
=> 0
[[1],[1],[1],[1]]
=> 0
[[2],[1],[1]]
=> 1
[[2],[2]]
=> 0
[[1,1],[1],[1]]
=> 0
[[1,1],[1,1]]
=> 0
[[3],[1]]
=> 2
[[2,1],[1]]
=> 1
[[1,1,1],[1]]
=> 0
[[4]]
=> 0
[[3,1]]
=> 2
[[2,2]]
=> 0
[[2,1,1]]
=> 1
[[1,1,1,1]]
=> 0
[[1],[1],[1],[1],[1]]
=> 0
[[2],[1],[1],[1]]
=> 1
[[2],[2],[1]]
=> 1
[[1,1],[1],[1],[1]]
=> 0
[[1,1],[1,1],[1]]
=> 0
[[3],[1],[1]]
=> 2
[[3],[2]]
=> 1
[[2,1],[1],[1]]
=> 1
[[2,1],[2]]
=> 1
[[2,1],[1,1]]
=> 1
[[1,1,1],[1],[1]]
=> 0
[[1,1,1],[1,1]]
=> 0
[[4],[1]]
=> 3
[[3,1],[1]]
=> 2
[[2,2],[1]]
=> 1
[[2,1,1],[1]]
=> 1
[[1,1,1,1],[1]]
=> 0
[[5]]
=> 0
[[4,1]]
=> 3
[[3,2]]
=> 1
[[3,1,1]]
=> 2
[[2,2,1]]
=> 1
[[2,1,1,1]]
=> 1
[[1,1,1,1,1]]
=> 0
[[1],[1],[1],[1],[1],[1]]
=> 0
[[2],[1],[1],[1],[1]]
=> 1
[[2],[2],[1],[1]]
=> 1
Description
The difference between the largest and smallest heights of a plane partition.
Matching statistic: St000225
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> [1]
=> 0
[[1],[1]]
=> [1,1]
=> [2]
=> 0
[[2]]
=> [2]
=> [1,1]
=> 0
[[1,1]]
=> [2]
=> [1,1]
=> 0
[[1],[1],[1]]
=> [1,1,1]
=> [3]
=> 0
[[2],[1]]
=> [2,1]
=> [2,1]
=> 1
[[1,1],[1]]
=> [2,1]
=> [2,1]
=> 1
[[3]]
=> [3]
=> [1,1,1]
=> 0
[[2,1]]
=> [3]
=> [1,1,1]
=> 0
[[1,1,1]]
=> [3]
=> [1,1,1]
=> 0
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [4]
=> 0
[[2],[1],[1]]
=> [2,1,1]
=> [3,1]
=> 2
[[2],[2]]
=> [2,2]
=> [2,2]
=> 0
[[1,1],[1],[1]]
=> [2,1,1]
=> [3,1]
=> 2
[[1,1],[1,1]]
=> [2,2]
=> [2,2]
=> 0
[[3],[1]]
=> [3,1]
=> [2,1,1]
=> 1
[[2,1],[1]]
=> [3,1]
=> [2,1,1]
=> 1
[[1,1,1],[1]]
=> [3,1]
=> [2,1,1]
=> 1
[[4]]
=> [4]
=> [1,1,1,1]
=> 0
[[3,1]]
=> [4]
=> [1,1,1,1]
=> 0
[[2,2]]
=> [4]
=> [1,1,1,1]
=> 0
[[2,1,1]]
=> [4]
=> [1,1,1,1]
=> 0
[[1,1,1,1]]
=> [4]
=> [1,1,1,1]
=> 0
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [5]
=> 0
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [4,1]
=> 3
[[2],[2],[1]]
=> [2,2,1]
=> [3,2]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [4,1]
=> 3
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [3,2]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [3,1,1]
=> 2
[[3],[2]]
=> [3,2]
=> [2,2,1]
=> 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [3,1,1]
=> 2
[[2,1],[2]]
=> [3,2]
=> [2,2,1]
=> 1
[[2,1],[1,1]]
=> [3,2]
=> [2,2,1]
=> 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [3,1,1]
=> 2
[[1,1,1],[1,1]]
=> [3,2]
=> [2,2,1]
=> 1
[[4],[1]]
=> [4,1]
=> [2,1,1,1]
=> 1
[[3,1],[1]]
=> [4,1]
=> [2,1,1,1]
=> 1
[[2,2],[1]]
=> [4,1]
=> [2,1,1,1]
=> 1
[[2,1,1],[1]]
=> [4,1]
=> [2,1,1,1]
=> 1
[[1,1,1,1],[1]]
=> [4,1]
=> [2,1,1,1]
=> 1
[[5]]
=> [5]
=> [1,1,1,1,1]
=> 0
[[4,1]]
=> [5]
=> [1,1,1,1,1]
=> 0
[[3,2]]
=> [5]
=> [1,1,1,1,1]
=> 0
[[3,1,1]]
=> [5]
=> [1,1,1,1,1]
=> 0
[[2,2,1]]
=> [5]
=> [1,1,1,1,1]
=> 0
[[2,1,1,1]]
=> [5]
=> [1,1,1,1,1]
=> 0
[[1,1,1,1,1]]
=> [5]
=> [1,1,1,1,1]
=> 0
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [6]
=> 0
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [5,1]
=> 4
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [4,2]
=> 2
Description
Difference between largest and smallest parts in a partition.
Matching statistic: St000460
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 89%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 89%
Values
[[1]]
=> [1]
=> []
=> ?
=> ? = 0
[[1],[1]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0}
[[2]]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0}
[[1,1]]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0}
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[2],[1]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1}
[[1,1],[1]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1}
[[3]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1}
[[2,1]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1}
[[1,1,1]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1}
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[2],[2]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,2}
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1],[1,1]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,2}
[[3],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,2}
[[2,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,2}
[[1,1,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,2}
[[4]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,2}
[[3,1]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,2}
[[2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,2}
[[2,1,1]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,2}
[[1,1,1,1]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,2}
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[3],[2]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[2,1],[2]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[[2,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[[4],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[[3,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[[2,2],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[[2,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[[1,1,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[[5]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[[4,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[[3,2]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[[3,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[[2,2,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[[2,1,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[[1,1,1,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[3],[3]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,4}
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[2,1],[2,1]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,4}
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[1,1,1],[1,1,1]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,4}
[[4],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[4],[2]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,4}
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[3,1],[2]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,4}
[[3,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,4}
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[2,2],[2]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,4}
[[2,2],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,4}
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[2,1,1],[2]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,4}
[[2,1,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,4}
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,1,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,4}
[[5],[1]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,4}
[[4,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,4}
[[3,2],[1]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,4}
[[3,1,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,4}
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 5
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 3
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 3
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 3
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 3
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[3],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[3],[3],[1]]
=> [3,3,1]
=> [3,1]
=> [1]
=> 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[2,1],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [3,1]
=> [1]
=> 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
Description
The hook length of the last cell along the main diagonal of an integer partition.
Matching statistic: St001250
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001250: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001250: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> []
=> ?
=> ? = 0
[[1],[1]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0}
[[2]]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0}
[[1,1]]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0}
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[2],[1]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1}
[[1,1],[1]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1}
[[3]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1}
[[2,1]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1}
[[1,1,1]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1}
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[2],[2]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,2}
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1],[1,1]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,2}
[[3],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,2}
[[2,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,2}
[[1,1,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,2}
[[4]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,2}
[[3,1]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,2}
[[2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,2}
[[2,1,1]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,2}
[[1,1,1,1]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,2}
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[3],[2]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[2,1],[2]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[[2,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[[4],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[[3,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[[2,2],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[[2,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[[1,1,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[[5]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[[4,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[[3,2]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[[3,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[[2,2,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[[2,1,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[[1,1,1,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,3}
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[3],[3]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,4}
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[2,1],[2,1]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,4}
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[1,1,1],[1,1,1]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,4}
[[4],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[4],[2]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,4}
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[3,1],[2]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,4}
[[3,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,4}
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[2,2],[2]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,4}
[[2,2],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,4}
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[2,1,1],[2]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,4}
[[2,1,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,4}
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,1,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,4}
[[5],[1]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,4}
[[4,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,4}
[[3,2],[1]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,4}
[[3,1,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,4}
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 5
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 3
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 2
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 3
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 2
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[3],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[3],[3],[1]]
=> [3,3,1]
=> [3,1]
=> [1]
=> 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[2,1],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [3,1]
=> [1]
=> 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
Description
The number of parts of a partition that are not congruent 0 modulo 3.
Matching statistic: St000668
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 39%●distinct values known / distinct values provided: 33%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 39%●distinct values known / distinct values provided: 33%
Values
[[1]]
=> [1]
=> []
=> ?
=> ? = 0
[[1],[1]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0}
[[2]]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0}
[[1,1]]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0}
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1}
[[2],[1]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1}
[[1,1],[1]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1}
[[3]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1}
[[2,1]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1}
[[1,1,1]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1}
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[2],[2]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[1,1],[1,1]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[3],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[2,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[1,1,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[4]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[3,1]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[2,1,1]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[1,1,1,1]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[3],[2]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,1],[2]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[1,1,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[4],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[3,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,2],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[1,1,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[5]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[4,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[3,2]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[3,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,2,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,1,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[1,1,1,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[3],[3]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[2,1],[2,1]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[1,1,1],[1,1,1]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 2
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 2
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[3],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[4],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[3,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2,2],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1
[[2],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[[2],[2],[1],[1],[1],[1]]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[[2],[2],[2],[1],[1]]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 2
[[2],[2],[2],[2]]
=> [2,2,2,2]
=> [2,2,2]
=> [2,2]
=> 2
[[1,1],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[[1,1],[1,1],[1],[1],[1],[1]]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[[1,1],[1,1],[1,1],[1],[1]]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 2
[[1,1],[1,1],[1,1],[1,1]]
=> [2,2,2,2]
=> [2,2,2]
=> [2,2]
=> 2
[[3],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[3],[2],[1],[1],[1]]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[3],[2],[2],[1]]
=> [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 2
[[3],[3],[1],[1]]
=> [3,3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
Description
The least common multiple of the parts of the partition.
Matching statistic: St000681
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 89%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 89%
Values
[[1]]
=> [1]
=> []
=> ?
=> ? = 0
[[1],[1]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0}
[[2]]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0}
[[1,1]]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0}
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1}
[[2],[1]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1}
[[1,1],[1]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1}
[[3]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1}
[[2,1]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1}
[[1,1,1]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1}
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[2],[2]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[1,1],[1,1]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[3],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[2,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[1,1,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[4]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[3,1]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[2,1,1]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[1,1,1,1]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3}
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3}
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3}
[[3],[2]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3}
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3}
[[2,1],[2]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3}
[[2,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3}
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3}
[[1,1,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3}
[[4],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3}
[[3,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3}
[[2,2],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3}
[[2,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3}
[[1,1,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3}
[[5]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3}
[[4,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3}
[[3,2]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3}
[[3,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3}
[[2,2,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3}
[[2,1,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3}
[[1,1,1,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3}
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,4,4}
[[3],[3]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,4,4}
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,4,4}
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,4,4}
[[2,1],[2,1]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,4,4}
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,4,4}
[[1,1,1],[1,1,1]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,4,4}
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 4
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[3],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[4],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[3,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2,2],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 5
[[2],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 4
[[2],[2],[1],[1],[1],[1]]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 3
[[2],[2],[2],[1],[1]]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 3
[[2],[2],[2],[2]]
=> [2,2,2,2]
=> [2,2,2]
=> [2,2]
=> 2
[[1,1],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 4
[[1,1],[1,1],[1],[1],[1],[1]]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 3
[[1,1],[1,1],[1,1],[1],[1]]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 3
[[1,1],[1,1],[1,1],[1,1]]
=> [2,2,2,2]
=> [2,2,2]
=> [2,2]
=> 2
[[3],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
[[3],[2],[1],[1],[1]]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[[3],[2],[2],[1]]
=> [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
[[3],[3],[1],[1]]
=> [3,3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
Description
The Grundy value of Chomp on Ferrers diagrams.
Players take turns and choose a cell of the diagram, cutting off all cells below and to the right of this cell in English notation. The player who is left with the single cell partition looses. The traditional version is played on chocolate bars, see [1].
This statistic is the Grundy value of the partition, that is, the smallest non-negative integer which does not occur as value of a partition obtained by a single move.
Matching statistic: St000698
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 56%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 56%
Values
[[1]]
=> [1]
=> []
=> ?
=> ? = 0
[[1],[1]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0}
[[2]]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0}
[[1,1]]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0}
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1}
[[2],[1]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1}
[[1,1],[1]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1}
[[3]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1}
[[2,1]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1}
[[1,1,1]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1}
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[2],[2]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[1,1],[1,1]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[3],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[2,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[1,1,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[4]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[3,1]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[2,1,1]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[1,1,1,1]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[3],[2]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,1],[2]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[1,1,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[4],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[3,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,2],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[1,1,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[5]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[4,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[3,2]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[3,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,2,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,1,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[1,1,1,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[3],[3]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[2,1],[2,1]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[1,1,1],[1,1,1]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 2
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[3],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[4],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[3,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2,2],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 3
[[2],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 2
[[2],[2],[1],[1],[1],[1]]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 2
[[2],[2],[2],[1],[1]]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 2
[[2],[2],[2],[2]]
=> [2,2,2,2]
=> [2,2,2]
=> [2,2]
=> 2
[[1,1],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 2
[[1,1],[1,1],[1],[1],[1],[1]]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 2
[[1,1],[1,1],[1,1],[1],[1]]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 2
[[1,1],[1,1],[1,1],[1,1]]
=> [2,2,2,2]
=> [2,2,2]
=> [2,2]
=> 2
[[3],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2
[[3],[2],[1],[1],[1]]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[3],[2],[2],[1]]
=> [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
[[3],[3],[1],[1]]
=> [3,3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
Description
The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core.
For any positive integer $k$, one associates a $k$-core to a partition by repeatedly removing all rim hooks of size $k$.
This statistic counts the $2$-rim hooks that are removed in this process to obtain a $2$-core.
Matching statistic: St000707
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000707: Integer partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 56%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000707: Integer partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 56%
Values
[[1]]
=> [1]
=> []
=> ?
=> ? = 0
[[1],[1]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0}
[[2]]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0}
[[1,1]]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0}
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1}
[[2],[1]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1}
[[1,1],[1]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1}
[[3]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1}
[[2,1]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1}
[[1,1,1]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1}
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[2],[2]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[1,1],[1,1]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[3],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[2,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[1,1,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[4]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[3,1]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[2,1,1]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[1,1,1,1]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[3],[2]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,1],[2]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[1,1,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[4],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[3,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,2],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[1,1,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[5]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[4,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[3,2]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[3,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,2,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,1,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[1,1,1,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[3],[3]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[2,1],[2,1]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[1,1,1],[1,1,1]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 2
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 2
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[3],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[4],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[3,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2,2],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1
[[2],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[[2],[2],[1],[1],[1],[1]]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[[2],[2],[2],[1],[1]]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 2
[[2],[2],[2],[2]]
=> [2,2,2,2]
=> [2,2,2]
=> [2,2]
=> 4
[[1,1],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[[1,1],[1,1],[1],[1],[1],[1]]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[[1,1],[1,1],[1,1],[1],[1]]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 2
[[1,1],[1,1],[1,1],[1,1]]
=> [2,2,2,2]
=> [2,2,2]
=> [2,2]
=> 4
[[3],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[3],[2],[1],[1],[1]]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[3],[2],[2],[1]]
=> [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 2
[[3],[3],[1],[1]]
=> [3,3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
Description
The product of the factorials of the parts.
Matching statistic: St000708
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 56%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 56%
Values
[[1]]
=> [1]
=> []
=> ?
=> ? = 0
[[1],[1]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0}
[[2]]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0}
[[1,1]]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0}
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1}
[[2],[1]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1}
[[1,1],[1]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1}
[[3]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1}
[[2,1]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1}
[[1,1,1]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1}
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[2],[2]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[1,1],[1,1]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[3],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[2,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[1,1,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[4]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[3,1]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[2,1,1]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[1,1,1,1]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[3],[2]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,1],[2]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[1,1,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[4],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[3,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,2],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[1,1,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[5]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[4,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[3,2]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[3,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,2,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,1,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[1,1,1,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[3],[3]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[2,1],[2,1]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[1,1,1],[1,1,1]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 2
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 2
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[3],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[4],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[3,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2,2],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1
[[2],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[[2],[2],[1],[1],[1],[1]]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[[2],[2],[2],[1],[1]]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 2
[[2],[2],[2],[2]]
=> [2,2,2,2]
=> [2,2,2]
=> [2,2]
=> 4
[[1,1],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[[1,1],[1,1],[1],[1],[1],[1]]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[[1,1],[1,1],[1,1],[1],[1]]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 2
[[1,1],[1,1],[1,1],[1,1]]
=> [2,2,2,2]
=> [2,2,2]
=> [2,2]
=> 4
[[3],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[3],[2],[1],[1],[1]]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[3],[2],[2],[1]]
=> [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 2
[[3],[3],[1],[1]]
=> [3,3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
Description
The product of the parts of an integer partition.
Matching statistic: St000770
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000770: Integer partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 78%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000770: Integer partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 78%
Values
[[1]]
=> [1]
=> []
=> ?
=> ? = 0
[[1],[1]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0}
[[2]]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0}
[[1,1]]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0}
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1}
[[2],[1]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1}
[[1,1],[1]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1}
[[3]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1}
[[2,1]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1}
[[1,1,1]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1}
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[2],[2]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[1,1],[1,1]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[3],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[2,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[1,1,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[4]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[3,1]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[2,1,1]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[1,1,1,1]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2}
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[3],[2]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,1],[2]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[1,1,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[4],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[3,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,2],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[1,1,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[5]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[4,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[3,2]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[3,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,2,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[2,1,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[1,1,1,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3}
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[3],[3]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[2,1],[2,1]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[1,1,1],[1,1,1]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,4,4}
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 4
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 4
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[3],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[4],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[3,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2,2],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1
[[2],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[[2],[2],[1],[1],[1],[1]]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[[2],[2],[2],[1],[1]]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 5
[[2],[2],[2],[2]]
=> [2,2,2,2]
=> [2,2,2]
=> [2,2]
=> 2
[[1,1],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[[1,1],[1,1],[1],[1],[1],[1]]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[[1,1],[1,1],[1,1],[1],[1]]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 5
[[1,1],[1,1],[1,1],[1,1]]
=> [2,2,2,2]
=> [2,2,2]
=> [2,2]
=> 2
[[3],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[3],[2],[1],[1],[1]]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[3],[2],[2],[1]]
=> [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 4
[[3],[3],[1],[1]]
=> [3,3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
Description
The major index of an integer partition when read from bottom to top.
This is the sum of the positions of the corners of the shape of an integer partition when reading from bottom to top.
For example, the partition $\lambda = (8,6,6,4,3,3)$ has corners at positions 3,6,9, and 13, giving a major index of 31.
The following 8 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000933The number of multipartitions of sizes given by an integer partition. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path.
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