Your data matches 11 different statistics following compositions of up to 3 maps.
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Matching statistic: St001455
Mapping
St001455: Plane partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => 0
[[1],[1]]
 => 1
[[2]]
 => 0
[[1,1]]
 => 1
[[1],[1],[1]]
 => 1
[[2],[1]]
 => 0
[[1,1],[1]]
 => 1
[[3]]
 => 0
[[2,1]]
 => 0
[[1,1,1]]
 => 1
[[1],[1],[1],[1]]
 => 1
[[2],[1],[1]]
 => 1
[[2],[2]]
 => 2
[[1,1],[1],[1]]
 => 1
[[1,1],[1,1]]
 => 1
[[3],[1]]
 => 0
[[2,1],[1]]
 => 1
[[1,1,1],[1]]
 => 1
[[4]]
 => 0
[[3,1]]
 => 0
[[2,2]]
 => 2
[[2,1,1]]
 => 1
[[1,1,1,1]]
 => 1
[[1],[1],[1],[1],[1]]
 => 1
[[2],[1],[1],[1]]
 => 1
[[2],[2],[1]]
 => 2
[[1,1],[1],[1],[1]]
 => 1
[[1,1],[1,1],[1]]
 => 1
[[3],[1],[1]]
 => 1
[[3],[2]]
 => 0
[[2,1],[1],[1]]
 => 1
[[2,1],[2]]
 => 2
[[2,1],[1,1]]
 => 1
[[1,1,1],[1],[1]]
 => 1
[[1,1,1],[1,1]]
 => 1
[[4],[1]]
 => 0
[[3,1],[1]]
 => 1
[[2,2],[1]]
 => 2
[[2,1,1],[1]]
 => 1
[[1,1,1,1],[1]]
 => 1
[[5]]
 => 0
[[4,1]]
 => 0
[[3,2]]
 => 0
[[3,1,1]]
 => 1
[[2,2,1]]
 => 2
[[2,1,1,1]]
 => 1
[[1,1,1,1,1]]
 => 1
[[1],[1],[1],[1],[1],[1]]
 => 1
[[2],[1],[1],[1],[1]]
 => 1
[[2],[2],[1],[1]]
 => 2
Description
Largest repeated part of a plane partition, and zero if no part is repeated.
Matching statistic: St000143
Mapping
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000143: 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]
 => 1
[[1,1]]
 => [2]
 => [1,1]
 => 1
[[1],[1],[1]]
 => [1,1,1]
 => [3]
 => 0
[[2],[1]]
 => [2,1]
 => [2,1]
 => 0
[[1,1],[1]]
 => [2,1]
 => [2,1]
 => 0
[[3]]
 => [3]
 => [1,1,1]
 => 1
[[2,1]]
 => [3]
 => [1,1,1]
 => 1
[[1,1,1]]
 => [3]
 => [1,1,1]
 => 1
[[1],[1],[1],[1]]
 => [1,1,1,1]
 => [4]
 => 0
[[2],[1],[1]]
 => [2,1,1]
 => [3,1]
 => 0
[[2],[2]]
 => [2,2]
 => [2,2]
 => 2
[[1,1],[1],[1]]
 => [2,1,1]
 => [3,1]
 => 0
[[1,1],[1,1]]
 => [2,2]
 => [2,2]
 => 2
[[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]
 => 1
[[3,1]]
 => [4]
 => [1,1,1,1]
 => 1
[[2,2]]
 => [4]
 => [1,1,1,1]
 => 1
[[2,1,1]]
 => [4]
 => [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]
 => [5]
 => 0
[[2],[1],[1],[1]]
 => [2,1,1,1]
 => [4,1]
 => 0
[[2],[2],[1]]
 => [2,2,1]
 => [3,2]
 => 0
[[1,1],[1],[1],[1]]
 => [2,1,1,1]
 => [4,1]
 => 0
[[1,1],[1,1],[1]]
 => [2,2,1]
 => [3,2]
 => 0
[[3],[1],[1]]
 => [3,1,1]
 => [3,1,1]
 => 1
[[3],[2]]
 => [3,2]
 => [2,2,1]
 => 2
[[2,1],[1],[1]]
 => [3,1,1]
 => [3,1,1]
 => 1
[[2,1],[2]]
 => [3,2]
 => [2,2,1]
 => 2
[[2,1],[1,1]]
 => [3,2]
 => [2,2,1]
 => 2
[[1,1,1],[1],[1]]
 => [3,1,1]
 => [3,1,1]
 => 1
[[1,1,1],[1,1]]
 => [3,2]
 => [2,2,1]
 => 2
[[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]
 => 1
[[4,1]]
 => [5]
 => [1,1,1,1,1]
 => 1
[[3,2]]
 => [5]
 => [1,1,1,1,1]
 => 1
[[3,1,1]]
 => [5]
 => [1,1,1,1,1]
 => 1
[[2,2,1]]
 => [5]
 => [1,1,1,1,1]
 => 1
[[2,1,1,1]]
 => [5]
 => [1,1,1,1,1]
 => 1
[[1,1,1,1,1]]
 => [5]
 => [1,1,1,1,1]
 => 1
[[1],[1],[1],[1],[1],[1]]
 => [1,1,1,1,1,1]
 => [6]
 => 0
[[2],[1],[1],[1],[1]]
 => [2,1,1,1,1]
 => [5,1]
 => 0
[[2],[2],[1],[1]]
 => [2,2,1,1]
 => [4,2]
 => 0
Description
The largest repeated part of a partition. If the parts of the partition are all distinct, the value of the statistic is defined to be zero.
Matching statistic: St001587
Mapping
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St001587: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1]
 => [1]
 => [1]
 => 0
[[1],[1]]
 => [1,1]
 => [2]
 => [1,1]
 => 0
[[2]]
 => [2]
 => [1,1]
 => [2]
 => 1
[[1,1]]
 => [2]
 => [1,1]
 => [2]
 => 1
[[1],[1],[1]]
 => [1,1,1]
 => [3]
 => [3]
 => 0
[[2],[1]]
 => [2,1]
 => [2,1]
 => [1,1,1]
 => 0
[[1,1],[1]]
 => [2,1]
 => [2,1]
 => [1,1,1]
 => 0
[[3]]
 => [3]
 => [1,1,1]
 => [2,1]
 => 1
[[2,1]]
 => [3]
 => [1,1,1]
 => [2,1]
 => 1
[[1,1,1]]
 => [3]
 => [1,1,1]
 => [2,1]
 => 1
[[1],[1],[1],[1]]
 => [1,1,1,1]
 => [4]
 => [1,1,1,1]
 => 0
[[2],[1],[1]]
 => [2,1,1]
 => [3,1]
 => [3,1]
 => 0
[[2],[2]]
 => [2,2]
 => [2,2]
 => [4]
 => 2
[[1,1],[1],[1]]
 => [2,1,1]
 => [3,1]
 => [3,1]
 => 0
[[1,1],[1,1]]
 => [2,2]
 => [2,2]
 => [4]
 => 2
[[3],[1]]
 => [3,1]
 => [2,1,1]
 => [2,1,1]
 => 1
[[2,1],[1]]
 => [3,1]
 => [2,1,1]
 => [2,1,1]
 => 1
[[1,1,1],[1]]
 => [3,1]
 => [2,1,1]
 => [2,1,1]
 => 1
[[4]]
 => [4]
 => [1,1,1,1]
 => [2,2]
 => 1
[[3,1]]
 => [4]
 => [1,1,1,1]
 => [2,2]
 => 1
[[2,2]]
 => [4]
 => [1,1,1,1]
 => [2,2]
 => 1
[[2,1,1]]
 => [4]
 => [1,1,1,1]
 => [2,2]
 => 1
[[1,1,1,1]]
 => [4]
 => [1,1,1,1]
 => [2,2]
 => 1
[[1],[1],[1],[1],[1]]
 => [1,1,1,1,1]
 => [5]
 => [5]
 => 0
[[2],[1],[1],[1]]
 => [2,1,1,1]
 => [4,1]
 => [1,1,1,1,1]
 => 0
[[2],[2],[1]]
 => [2,2,1]
 => [3,2]
 => [3,1,1]
 => 0
[[1,1],[1],[1],[1]]
 => [2,1,1,1]
 => [4,1]
 => [1,1,1,1,1]
 => 0
[[1,1],[1,1],[1]]
 => [2,2,1]
 => [3,2]
 => [3,1,1]
 => 0
[[3],[1],[1]]
 => [3,1,1]
 => [3,1,1]
 => [3,2]
 => 1
[[3],[2]]
 => [3,2]
 => [2,2,1]
 => [4,1]
 => 2
[[2,1],[1],[1]]
 => [3,1,1]
 => [3,1,1]
 => [3,2]
 => 1
[[2,1],[2]]
 => [3,2]
 => [2,2,1]
 => [4,1]
 => 2
[[2,1],[1,1]]
 => [3,2]
 => [2,2,1]
 => [4,1]
 => 2
[[1,1,1],[1],[1]]
 => [3,1,1]
 => [3,1,1]
 => [3,2]
 => 1
[[1,1,1],[1,1]]
 => [3,2]
 => [2,2,1]
 => [4,1]
 => 2
[[4],[1]]
 => [4,1]
 => [2,1,1,1]
 => [2,1,1,1]
 => 1
[[3,1],[1]]
 => [4,1]
 => [2,1,1,1]
 => [2,1,1,1]
 => 1
[[2,2],[1]]
 => [4,1]
 => [2,1,1,1]
 => [2,1,1,1]
 => 1
[[2,1,1],[1]]
 => [4,1]
 => [2,1,1,1]
 => [2,1,1,1]
 => 1
[[1,1,1,1],[1]]
 => [4,1]
 => [2,1,1,1]
 => [2,1,1,1]
 => 1
[[5]]
 => [5]
 => [1,1,1,1,1]
 => [2,2,1]
 => 1
[[4,1]]
 => [5]
 => [1,1,1,1,1]
 => [2,2,1]
 => 1
[[3,2]]
 => [5]
 => [1,1,1,1,1]
 => [2,2,1]
 => 1
[[3,1,1]]
 => [5]
 => [1,1,1,1,1]
 => [2,2,1]
 => 1
[[2,2,1]]
 => [5]
 => [1,1,1,1,1]
 => [2,2,1]
 => 1
[[2,1,1,1]]
 => [5]
 => [1,1,1,1,1]
 => [2,2,1]
 => 1
[[1,1,1,1,1]]
 => [5]
 => [1,1,1,1,1]
 => [2,2,1]
 => 1
[[1],[1],[1],[1],[1],[1]]
 => [1,1,1,1,1,1]
 => [6]
 => [3,3]
 => 0
[[2],[1],[1],[1],[1]]
 => [2,1,1,1,1]
 => [5,1]
 => [5,1]
 => 0
[[2],[2],[1],[1]]
 => [2,2,1,1]
 => [4,2]
 => [1,1,1,1,1,1]
 => 0
Description
Half of the largest even part of an integer partition. The largest even part is recorded by [[St000995]].
Matching statistic: St000208
Mapping
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 67%
Values
[[1]]
 => [1]
 => []
 => ?
 => ? = 0
[[1],[1]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,1,1}
[[2]]
 => [2]
 => []
 => ?
 => ? ∊ {0,1,1}
[[1,1]]
 => [2]
 => []
 => ?
 => ? ∊ {0,1,1}
[[1],[1],[1]]
 => [1,1,1]
 => [1,1]
 => [1]
 => 1
[[2],[1]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1}
[[1,1],[1]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1}
[[3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,1,1}
[[2,1]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,1,1}
[[1,1,1]]
 => [3]
 => []
 => ?
 => ? ∊ {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]
 => 1
[[2],[2]]
 => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2}
[[1,1],[1],[1]]
 => [2,1,1]
 => [1,1]
 => [1]
 => 1
[[1,1],[1,1]]
 => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2}
[[3],[1]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2}
[[2,1],[1]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2}
[[1,1,1],[1]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2}
[[4]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1,1,1,2,2}
[[3,1]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1,1,1,2,2}
[[2,2]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1,1,1,2,2}
[[2,1,1]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1,1,1,2,2}
[[1,1,1,1]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1,1,1,2,2}
[[1],[1],[1],[1],[1]]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[[2],[1],[1],[1]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2],[2],[1]]
 => [2,2,1]
 => [2,1]
 => [1]
 => 1
[[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]
 => 1
[[3],[1],[1]]
 => [3,1,1]
 => [1,1]
 => [1]
 => 1
[[3],[2]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[2,1],[1],[1]]
 => [3,1,1]
 => [1,1]
 => [1]
 => 1
[[2,1],[2]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[2,1],[1,1]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[1,1,1],[1],[1]]
 => [3,1,1]
 => [1,1]
 => [1]
 => 1
[[1,1,1],[1,1]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[4],[1]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[3,1],[1]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[2,2],[1]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[2,1,1],[1]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[1,1,1,1],[1]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[5]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[4,1]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[3,2]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[3,1,1]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[2,2,1]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[2,1,1,1]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[1,1,1,1,1]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[1],[1],[1],[1],[1],[1]]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
[[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]
 => 1
[[3],[3]]
 => [3,3]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[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]
 => 1
[[2,1],[1,1],[1]]
 => [3,2,1]
 => [2,1]
 => [1]
 => 1
[[2,1],[2,1]]
 => [3,3]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[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]
 => 1
[[1,1,1],[1,1,1]]
 => [3,3]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[4],[1],[1]]
 => [4,1,1]
 => [1,1]
 => [1]
 => 1
[[4],[2]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[3,1],[1],[1]]
 => [4,1,1]
 => [1,1]
 => [1]
 => 1
[[3,1],[2]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[3,1],[1,1]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[2,2],[1],[1]]
 => [4,1,1]
 => [1,1]
 => [1]
 => 1
[[2,2],[2]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[2,2],[1,1]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[2,1,1],[1],[1]]
 => [4,1,1]
 => [1,1]
 => [1]
 => 1
[[2,1,1],[2]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[2,1,1],[1,1]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[1,1,1,1],[1],[1]]
 => [4,1,1]
 => [1,1]
 => [1]
 => 1
[[1,1,1,1],[1,1]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[5],[1]]
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[4,1],[1]]
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[3,2],[1]]
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[3,1,1],[1]]
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[1],[1],[1],[1],[1],[1],[1]]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 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
[[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]
 => 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
[[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]
 => 1
[[1,1,1],[1,1],[1],[1]]
 => [3,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that $P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has only integer lattice points as vertices. See also [[St000205]], [[St000206]] and [[St000207]].
Matching statistic: St000755
Mapping
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000755: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 57%●distinct values known / distinct values provided: 33%
Values
[[1]]
 => [1]
 => []
 => ?
 => ? = 0
[[1],[1]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,1,1}
[[2]]
 => [2]
 => []
 => ?
 => ? ∊ {0,1,1}
[[1,1]]
 => [2]
 => []
 => ?
 => ? ∊ {0,1,1}
[[1],[1],[1]]
 => [1,1,1]
 => [1,1]
 => [1]
 => 1
[[2],[1]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1}
[[1,1],[1]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1}
[[3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,1,1}
[[2,1]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,1,1}
[[1,1,1]]
 => [3]
 => []
 => ?
 => ? ∊ {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]
 => 1
[[2],[2]]
 => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2}
[[1,1],[1],[1]]
 => [2,1,1]
 => [1,1]
 => [1]
 => 1
[[1,1],[1,1]]
 => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2}
[[3],[1]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2}
[[2,1],[1]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2}
[[1,1,1],[1]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2}
[[4]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1,1,1,2,2}
[[3,1]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1,1,1,2,2}
[[2,2]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1,1,1,2,2}
[[2,1,1]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1,1,1,2,2}
[[1,1,1,1]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1,1,1,2,2}
[[1],[1],[1],[1],[1]]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[[2],[1],[1],[1]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2],[2],[1]]
 => [2,2,1]
 => [2,1]
 => [1]
 => 1
[[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]
 => 1
[[3],[1],[1]]
 => [3,1,1]
 => [1,1]
 => [1]
 => 1
[[3],[2]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[2,1],[1],[1]]
 => [3,1,1]
 => [1,1]
 => [1]
 => 1
[[2,1],[2]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[2,1],[1,1]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[1,1,1],[1],[1]]
 => [3,1,1]
 => [1,1]
 => [1]
 => 1
[[1,1,1],[1,1]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[4],[1]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[3,1],[1]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[2,2],[1]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[2,1,1],[1]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[1,1,1,1],[1]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[5]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[4,1]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[3,2]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[3,1,1]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[2,2,1]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[2,1,1,1]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[1,1,1,1,1]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[1],[1],[1],[1],[1],[1]]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
[[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]
 => 1
[[3],[3]]
 => [3,3]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[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]
 => 1
[[2,1],[1,1],[1]]
 => [3,2,1]
 => [2,1]
 => [1]
 => 1
[[2,1],[2,1]]
 => [3,3]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[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]
 => 1
[[1,1,1],[1,1,1]]
 => [3,3]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[4],[1],[1]]
 => [4,1,1]
 => [1,1]
 => [1]
 => 1
[[4],[2]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[3,1],[1],[1]]
 => [4,1,1]
 => [1,1]
 => [1]
 => 1
[[3,1],[2]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[3,1],[1,1]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[2,2],[1],[1]]
 => [4,1,1]
 => [1,1]
 => [1]
 => 1
[[2,2],[2]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[2,2],[1,1]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[2,1,1],[1],[1]]
 => [4,1,1]
 => [1,1]
 => [1]
 => 1
[[2,1,1],[2]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[2,1,1],[1,1]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[1,1,1,1],[1],[1]]
 => [4,1,1]
 => [1,1]
 => [1]
 => 1
[[1,1,1,1],[1,1]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[5],[1]]
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[4,1],[1]]
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[3,2],[1]]
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[3,1,1],[1]]
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[1],[1],[1],[1],[1],[1],[1]]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 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
[[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]
 => 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
[[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]
 => 1
[[1,1,1],[1,1],[1],[1]]
 => [3,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
Description
The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. Consider the recurrence $$f(n)=\sum_{p\in\lambda} f(n-p).$$ This statistic returns the number of distinct real roots of the associated characteristic polynomial. For example, the partition $(2,1)$ corresponds to the recurrence $f(n)=f(n-1)+f(n-2)$ with associated characteristic polynomial $x^2-x-1$, which has two real roots.
Matching statistic: St001389
Mapping
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 67%
Values
[[1]]
 => [1]
 => []
 => ?
 => ? = 0
[[1],[1]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,1,1}
[[2]]
 => [2]
 => []
 => ?
 => ? ∊ {0,1,1}
[[1,1]]
 => [2]
 => []
 => ?
 => ? ∊ {0,1,1}
[[1],[1],[1]]
 => [1,1,1]
 => [1,1]
 => [1]
 => 1
[[2],[1]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1}
[[1,1],[1]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1}
[[3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,1,1}
[[2,1]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,1,1}
[[1,1,1]]
 => [3]
 => []
 => ?
 => ? ∊ {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]
 => 1
[[2],[2]]
 => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2}
[[1,1],[1],[1]]
 => [2,1,1]
 => [1,1]
 => [1]
 => 1
[[1,1],[1,1]]
 => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2}
[[3],[1]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2}
[[2,1],[1]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2}
[[1,1,1],[1]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,2,2}
[[4]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1,1,1,2,2}
[[3,1]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1,1,1,2,2}
[[2,2]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1,1,1,2,2}
[[2,1,1]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1,1,1,2,2}
[[1,1,1,1]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1,1,1,2,2}
[[1],[1],[1],[1],[1]]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[[2],[1],[1],[1]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2],[2],[1]]
 => [2,2,1]
 => [2,1]
 => [1]
 => 1
[[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]
 => 1
[[3],[1],[1]]
 => [3,1,1]
 => [1,1]
 => [1]
 => 1
[[3],[2]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[2,1],[1],[1]]
 => [3,1,1]
 => [1,1]
 => [1]
 => 1
[[2,1],[2]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[2,1],[1,1]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[1,1,1],[1],[1]]
 => [3,1,1]
 => [1,1]
 => [1]
 => 1
[[1,1,1],[1,1]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[4],[1]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[3,1],[1]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[2,2],[1]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[2,1,1],[1]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[1,1,1,1],[1]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[5]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[4,1]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[3,2]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[3,1,1]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[2,2,1]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[2,1,1,1]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[1,1,1,1,1]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2}
[[1],[1],[1],[1],[1],[1]]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
[[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]
 => 1
[[3],[3]]
 => [3,3]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[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]
 => 1
[[2,1],[1,1],[1]]
 => [3,2,1]
 => [2,1]
 => [1]
 => 1
[[2,1],[2,1]]
 => [3,3]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[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]
 => 1
[[1,1,1],[1,1,1]]
 => [3,3]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[4],[1],[1]]
 => [4,1,1]
 => [1,1]
 => [1]
 => 1
[[4],[2]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[3,1],[1],[1]]
 => [4,1,1]
 => [1,1]
 => [1]
 => 1
[[3,1],[2]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[3,1],[1,1]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[2,2],[1],[1]]
 => [4,1,1]
 => [1,1]
 => [1]
 => 1
[[2,2],[2]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[2,2],[1,1]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[2,1,1],[1],[1]]
 => [4,1,1]
 => [1,1]
 => [1]
 => 1
[[2,1,1],[2]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[2,1,1],[1,1]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[1,1,1,1],[1],[1]]
 => [4,1,1]
 => [1,1]
 => [1]
 => 1
[[1,1,1,1],[1,1]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[5],[1]]
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[4,1],[1]]
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[3,2],[1]]
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[3,1,1],[1]]
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[1],[1],[1],[1],[1],[1],[1]]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 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
[[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]
 => 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
[[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]
 => 1
[[1,1,1],[1,1],[1],[1]]
 => [3,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
Description
The number of partitions of the same length below the given integer partition. For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is $$ \det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$
Matching statistic: St001491
Mapping
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 50%
Values
[[1]]
 => [1]
 => []
 => => ? = 0
[[1],[1]]
 => [1,1]
 => [1]
 => 10 => 1
[[2]]
 => [2]
 => []
 => => ? ∊ {0,1}
[[1,1]]
 => [2]
 => []
 => => ? ∊ {0,1}
[[1],[1],[1]]
 => [1,1,1]
 => [1,1]
 => 110 => 1
[[2],[1]]
 => [2,1]
 => [1]
 => 10 => 1
[[1,1],[1]]
 => [2,1]
 => [1]
 => 10 => 1
[[3]]
 => [3]
 => []
 => => ? ∊ {0,0,0}
[[2,1]]
 => [3]
 => []
 => => ? ∊ {0,0,0}
[[1,1,1]]
 => [3]
 => []
 => => ? ∊ {0,0,0}
[[1],[1],[1],[1]]
 => [1,1,1,1]
 => [1,1,1]
 => 1110 => 2
[[2],[1],[1]]
 => [2,1,1]
 => [1,1]
 => 110 => 1
[[2],[2]]
 => [2,2]
 => [2]
 => 100 => 1
[[1,1],[1],[1]]
 => [2,1,1]
 => [1,1]
 => 110 => 1
[[1,1],[1,1]]
 => [2,2]
 => [2]
 => 100 => 1
[[3],[1]]
 => [3,1]
 => [1]
 => 10 => 1
[[2,1],[1]]
 => [3,1]
 => [1]
 => 10 => 1
[[1,1,1],[1]]
 => [3,1]
 => [1]
 => 10 => 1
[[4]]
 => [4]
 => []
 => => ? ∊ {0,0,0,1,2}
[[3,1]]
 => [4]
 => []
 => => ? ∊ {0,0,0,1,2}
[[2,2]]
 => [4]
 => []
 => => ? ∊ {0,0,0,1,2}
[[2,1,1]]
 => [4]
 => []
 => => ? ∊ {0,0,0,1,2}
[[1,1,1,1]]
 => [4]
 => []
 => => ? ∊ {0,0,0,1,2}
[[1],[1],[1],[1],[1]]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 11110 => ? ∊ {0,0,0,1,1,1,2,2}
[[2],[1],[1],[1]]
 => [2,1,1,1]
 => [1,1,1]
 => 1110 => 2
[[2],[2],[1]]
 => [2,2,1]
 => [2,1]
 => 1010 => 0
[[1,1],[1],[1],[1]]
 => [2,1,1,1]
 => [1,1,1]
 => 1110 => 2
[[1,1],[1,1],[1]]
 => [2,2,1]
 => [2,1]
 => 1010 => 0
[[3],[1],[1]]
 => [3,1,1]
 => [1,1]
 => 110 => 1
[[3],[2]]
 => [3,2]
 => [2]
 => 100 => 1
[[2,1],[1],[1]]
 => [3,1,1]
 => [1,1]
 => 110 => 1
[[2,1],[2]]
 => [3,2]
 => [2]
 => 100 => 1
[[2,1],[1,1]]
 => [3,2]
 => [2]
 => 100 => 1
[[1,1,1],[1],[1]]
 => [3,1,1]
 => [1,1]
 => 110 => 1
[[1,1,1],[1,1]]
 => [3,2]
 => [2]
 => 100 => 1
[[4],[1]]
 => [4,1]
 => [1]
 => 10 => 1
[[3,1],[1]]
 => [4,1]
 => [1]
 => 10 => 1
[[2,2],[1]]
 => [4,1]
 => [1]
 => 10 => 1
[[2,1,1],[1]]
 => [4,1]
 => [1]
 => 10 => 1
[[1,1,1,1],[1]]
 => [4,1]
 => [1]
 => 10 => 1
[[5]]
 => [5]
 => []
 => => ? ∊ {0,0,0,1,1,1,2,2}
[[4,1]]
 => [5]
 => []
 => => ? ∊ {0,0,0,1,1,1,2,2}
[[3,2]]
 => [5]
 => []
 => => ? ∊ {0,0,0,1,1,1,2,2}
[[3,1,1]]
 => [5]
 => []
 => => ? ∊ {0,0,0,1,1,1,2,2}
[[2,2,1]]
 => [5]
 => []
 => => ? ∊ {0,0,0,1,1,1,2,2}
[[2,1,1,1]]
 => [5]
 => []
 => => ? ∊ {0,0,0,1,1,1,2,2}
[[1,1,1,1,1]]
 => [5]
 => []
 => => ? ∊ {0,0,0,1,1,1,2,2}
[[1],[1],[1],[1],[1],[1]]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 111110 => ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3}
[[2],[1],[1],[1],[1]]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 11110 => ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3}
[[2],[2],[1],[1]]
 => [2,2,1,1]
 => [2,1,1]
 => 10110 => ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3}
[[2],[2],[2]]
 => [2,2,2]
 => [2,2]
 => 1100 => 1
[[1,1],[1],[1],[1],[1]]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 11110 => ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3}
[[1,1],[1,1],[1],[1]]
 => [2,2,1,1]
 => [2,1,1]
 => 10110 => ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3}
[[1,1],[1,1],[1,1]]
 => [2,2,2]
 => [2,2]
 => 1100 => 1
[[3],[1],[1],[1]]
 => [3,1,1,1]
 => [1,1,1]
 => 1110 => 2
[[3],[2],[1]]
 => [3,2,1]
 => [2,1]
 => 1010 => 0
[[3],[3]]
 => [3,3]
 => [3]
 => 1000 => 1
[[2,1],[1],[1],[1]]
 => [3,1,1,1]
 => [1,1,1]
 => 1110 => 2
[[2,1],[2],[1]]
 => [3,2,1]
 => [2,1]
 => 1010 => 0
[[2,1],[1,1],[1]]
 => [3,2,1]
 => [2,1]
 => 1010 => 0
[[2,1],[2,1]]
 => [3,3]
 => [3]
 => 1000 => 1
[[1,1,1],[1],[1],[1]]
 => [3,1,1,1]
 => [1,1,1]
 => 1110 => 2
[[1,1,1],[1,1],[1]]
 => [3,2,1]
 => [2,1]
 => 1010 => 0
[[1,1,1],[1,1,1]]
 => [3,3]
 => [3]
 => 1000 => 1
[[4],[1],[1]]
 => [4,1,1]
 => [1,1]
 => 110 => 1
[[4],[2]]
 => [4,2]
 => [2]
 => 100 => 1
[[3,1],[1],[1]]
 => [4,1,1]
 => [1,1]
 => 110 => 1
[[3,1],[2]]
 => [4,2]
 => [2]
 => 100 => 1
[[3,1],[1,1]]
 => [4,2]
 => [2]
 => 100 => 1
[[2,2],[1],[1]]
 => [4,1,1]
 => [1,1]
 => 110 => 1
[[2,2],[2]]
 => [4,2]
 => [2]
 => 100 => 1
[[2,2],[1,1]]
 => [4,2]
 => [2]
 => 100 => 1
[[2,1,1],[1],[1]]
 => [4,1,1]
 => [1,1]
 => 110 => 1
[[2,1,1],[2]]
 => [4,2]
 => [2]
 => 100 => 1
[[6]]
 => [6]
 => []
 => => ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3}
[[5,1]]
 => [6]
 => []
 => => ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3}
[[4,2]]
 => [6]
 => []
 => => ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3}
[[4,1,1]]
 => [6]
 => []
 => => ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3}
[[3,3]]
 => [6]
 => []
 => => ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3}
[[3,2,1]]
 => [6]
 => []
 => => ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3}
[[3,1,1,1]]
 => [6]
 => []
 => => ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3}
[[2,2,2]]
 => [6]
 => []
 => => ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3}
[[2,2,1,1]]
 => [6]
 => []
 => => ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3}
[[2,1,1,1,1]]
 => [6]
 => []
 => => ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3}
[[1,1,1,1,1,1]]
 => [6]
 => []
 => => ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,3,3}
[[1],[1],[1],[1],[1],[1],[1]]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1111110 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[[2],[1],[1],[1],[1],[1]]
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => 111110 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[[2],[2],[1],[1],[1]]
 => [2,2,1,1,1]
 => [2,1,1,1]
 => 101110 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[[2],[2],[2],[1]]
 => [2,2,2,1]
 => [2,2,1]
 => 11010 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[[1,1],[1],[1],[1],[1],[1]]
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => 111110 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[[1,1],[1,1],[1],[1],[1]]
 => [2,2,1,1,1]
 => [2,1,1,1]
 => 101110 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[[1,1],[1,1],[1,1],[1]]
 => [2,2,2,1]
 => [2,2,1]
 => 11010 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[[3],[1],[1],[1],[1]]
 => [3,1,1,1,1]
 => [1,1,1,1]
 => 11110 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[[3],[2],[1],[1]]
 => [3,2,1,1]
 => [2,1,1]
 => 10110 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[[3],[3],[1]]
 => [3,3,1]
 => [3,1]
 => 10010 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[[2,1],[1],[1],[1],[1]]
 => [3,1,1,1,1]
 => [1,1,1,1]
 => 11110 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[[2,1],[2],[1],[1]]
 => [3,2,1,1]
 => [2,1,1]
 => 10110 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[[2,1],[1,1],[1],[1]]
 => [3,2,1,1]
 => [2,1,1]
 => 10110 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[[2,1],[2,1],[1]]
 => [3,3,1]
 => [3,1]
 => 10010 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[[1,1,1],[1],[1],[1],[1]]
 => [3,1,1,1,1]
 => [1,1,1,1]
 => 11110 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset. Let $A_n=K[x]/(x^n)$. We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
Matching statistic: St000668
Mapping
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: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 50%
Values
[[1]]
 => [1]
 => []
 => ?
 => ? = 0
[[1],[1]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,1,1}
[[2]]
 => [2]
 => []
 => ?
 => ? ∊ {0,1,1}
[[1,1]]
 => [2]
 => []
 => ?
 => ? ∊ {0,1,1}
[[1],[1],[1]]
 => [1,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,1,1,1}
[[2],[1]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1,1}
[[1,1],[1]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1,1}
[[3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1}
[[2,1]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1}
[[1,1,1]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1}
[[1],[1],[1],[1]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2],[1],[1]]
 => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[2],[2]]
 => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[1,1],[1],[1]]
 => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[1,1],[1,1]]
 => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[3],[1]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[2,1],[1]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[1,1,1],[1]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[4]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[3,1]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[2,2]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[2,1,1]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[1,1,1,1]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[1],[1],[1],[1],[1]]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[[2],[1],[1],[1]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2],[2],[1]]
 => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[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]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[3],[1],[1]]
 => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[3],[2]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[2,1],[1],[1]]
 => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[2,1],[2]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[2,1],[1,1]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[1,1,1],[1],[1]]
 => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[1,1,1],[1,1]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[4],[1]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[3,1],[1]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[2,2],[1]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[2,1,1],[1]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[1,1,1,1],[1]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[5]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[4,1]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[3,2]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[3,1,1]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[2,2,1]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[2,1,1,1]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[1,1,1,1,1]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[1],[1],[1],[1],[1],[1]]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
[[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]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[3],[3]]
 => [3,3]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[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]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[2,1],[1,1],[1]]
 => [3,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[2,1],[2,1]]
 => [3,3]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[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]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[1,1,1],[1,1,1]]
 => [3,3]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3}
[[1],[1],[1],[1],[1],[1],[1]]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 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: St000698
Mapping
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: 83%
Values
[[1]]
 => [1]
 => []
 => ?
 => ? = 0
[[1],[1]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,1,1}
[[2]]
 => [2]
 => []
 => ?
 => ? ∊ {0,1,1}
[[1,1]]
 => [2]
 => []
 => ?
 => ? ∊ {0,1,1}
[[1],[1],[1]]
 => [1,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,1,1,1}
[[2],[1]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1,1}
[[1,1],[1]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1,1}
[[3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1}
[[2,1]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1}
[[1,1,1]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1}
[[1],[1],[1],[1]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2],[1],[1]]
 => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[2],[2]]
 => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[1,1],[1],[1]]
 => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[1,1],[1,1]]
 => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[3],[1]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[2,1],[1]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[1,1,1],[1]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[4]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[3,1]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[2,2]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[2,1,1]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[1,1,1,1]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[1],[1],[1],[1],[1]]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[[2],[1],[1],[1]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2],[2],[1]]
 => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[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]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[3],[1],[1]]
 => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[3],[2]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[2,1],[1],[1]]
 => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[2,1],[2]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[2,1],[1,1]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[1,1,1],[1],[1]]
 => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[1,1,1],[1,1]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[4],[1]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[3,1],[1]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[2,2],[1]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[2,1,1],[1]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[1,1,1,1],[1]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[5]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[4,1]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[3,2]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[3,1,1]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[2,2,1]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[2,1,1,1]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[1,1,1,1,1]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[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]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3}
[[3],[3]]
 => [3,3]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3}
[[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]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3}
[[2,1],[1,1],[1]]
 => [3,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3}
[[2,1],[2,1]]
 => [3,3]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3}
[[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]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3}
[[1,1,1],[1,1,1]]
 => [3,3]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3}
[[1],[1],[1],[1],[1],[1],[1]]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 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: St001128
Mapping
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001128: Integer partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 50%
Values
[[1]]
 => [1]
 => []
 => ?
 => ? = 0
[[1],[1]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,1,1}
[[2]]
 => [2]
 => []
 => ?
 => ? ∊ {0,1,1}
[[1,1]]
 => [2]
 => []
 => ?
 => ? ∊ {0,1,1}
[[1],[1],[1]]
 => [1,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,1,1,1}
[[2],[1]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1,1}
[[1,1],[1]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1,1}
[[3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1}
[[2,1]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1}
[[1,1,1]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1}
[[1],[1],[1],[1]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2],[1],[1]]
 => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[2],[2]]
 => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[1,1],[1],[1]]
 => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[1,1],[1,1]]
 => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[3],[1]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[2,1],[1]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[1,1,1],[1]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[4]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[3,1]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[2,2]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[2,1,1]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[1,1,1,1]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2}
[[1],[1],[1],[1],[1]]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[[2],[1],[1],[1]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2],[2],[1]]
 => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[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]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[3],[1],[1]]
 => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[3],[2]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[2,1],[1],[1]]
 => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[2,1],[2]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[2,1],[1,1]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[1,1,1],[1],[1]]
 => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[1,1,1],[1,1]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[4],[1]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[3,1],[1]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[2,2],[1]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[2,1,1],[1]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[1,1,1,1],[1]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[5]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[4,1]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[3,2]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[3,1,1]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[2,2,1]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[2,1,1,1]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[1,1,1,1,1]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[[1],[1],[1],[1],[1],[1]]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
[[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]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3}
[[3],[3]]
 => [3,3]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3}
[[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]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3}
[[2,1],[1,1],[1]]
 => [3,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3}
[[2,1],[2,1]]
 => [3,3]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3}
[[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]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3}
[[1,1,1],[1,1,1]]
 => [3,3]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3}
[[1],[1],[1],[1],[1],[1],[1]]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 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]
 => 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]
 => 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]
 => 1
[[1,1],[1],[1],[1],[1],[1],[1]]
 => [2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 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]
 => 1
[[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 exponens consonantiae of a partition. This is the quotient of the least common multiple and the greatest common divior of the parts of the partiton. See [1, Caput sextum, §19-§22].
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St001568The smallest positive integer that does not appear twice in the partition.