Your data matches 9 different statistics following compositions of up to 3 maps.
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Matching statistic: St000115
(load all 3 compositions to match this statistic)
Mapping
St000115: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0]]
 => 0
[[1,0],[1]]
 => 1
[[2,0],[0]]
 => 0
[[2,0],[1]]
 => 1
[[2,0],[2]]
 => 2
[[1,1],[1]]
 => 1
[[1,0,0],[0,0],[0]]
 => 0
[[1,0,0],[1,0],[0]]
 => 0
[[1,0,0],[1,0],[1]]
 => 1
[[3,0],[0]]
 => 0
[[3,0],[1]]
 => 1
[[3,0],[2]]
 => 2
[[3,0],[3]]
 => 3
[[2,1],[1]]
 => 1
[[2,1],[2]]
 => 2
[[2,0,0],[0,0],[0]]
 => 0
[[2,0,0],[1,0],[0]]
 => 0
[[2,0,0],[1,0],[1]]
 => 1
[[2,0,0],[2,0],[0]]
 => 0
[[2,0,0],[2,0],[1]]
 => 1
[[2,0,0],[2,0],[2]]
 => 2
[[1,1,0],[1,0],[0]]
 => 0
[[1,1,0],[1,0],[1]]
 => 1
[[1,1,0],[1,1],[1]]
 => 1
[[1,0,0,0],[0,0,0],[0,0],[0]]
 => 0
[[1,0,0,0],[1,0,0],[0,0],[0]]
 => 0
[[1,0,0,0],[1,0,0],[1,0],[0]]
 => 0
[[1,0,0,0],[1,0,0],[1,0],[1]]
 => 1
[[4,0],[0]]
 => 0
[[4,0],[1]]
 => 1
[[4,0],[2]]
 => 2
[[4,0],[3]]
 => 3
[[4,0],[4]]
 => 4
[[3,1],[1]]
 => 1
[[3,1],[2]]
 => 2
[[3,1],[3]]
 => 3
[[2,2],[2]]
 => 2
[[3,0,0],[0,0],[0]]
 => 0
[[3,0,0],[1,0],[0]]
 => 0
[[3,0,0],[1,0],[1]]
 => 1
[[3,0,0],[2,0],[0]]
 => 0
[[3,0,0],[2,0],[1]]
 => 1
[[3,0,0],[2,0],[2]]
 => 2
[[3,0,0],[3,0],[0]]
 => 0
[[3,0,0],[3,0],[1]]
 => 1
[[3,0,0],[3,0],[2]]
 => 2
[[3,0,0],[3,0],[3]]
 => 3
[[2,1,0],[1,0],[0]]
 => 0
[[2,1,0],[1,0],[1]]
 => 1
[[2,1,0],[1,1],[1]]
 => 1
Description
The single entry in the last row.
Matching statistic: St000668
Mapping
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 43%
Values
[[1,0],[0]]
 => [[2]]
 => [1]
 => []
 => ? ∊ {0,1}
[[1,0],[1]]
 => [[1]]
 => [1]
 => []
 => ? ∊ {0,1}
[[2,0],[0]]
 => [[2,2]]
 => [2]
 => []
 => ? ∊ {0,1,1,2}
[[2,0],[1]]
 => [[1,2]]
 => [2]
 => []
 => ? ∊ {0,1,1,2}
[[2,0],[2]]
 => [[1,1]]
 => [2]
 => []
 => ? ∊ {0,1,1,2}
[[1,1],[1]]
 => [[1],[2]]
 => [1,1]
 => [1]
 => ? ∊ {0,1,1,2}
[[1,0,0],[0,0],[0]]
 => [[3]]
 => [1]
 => []
 => ? ∊ {0,0,1}
[[1,0,0],[1,0],[0]]
 => [[2]]
 => [1]
 => []
 => ? ∊ {0,0,1}
[[1,0,0],[1,0],[1]]
 => [[1]]
 => [1]
 => []
 => ? ∊ {0,0,1}
[[3,0],[0]]
 => [[2,2,2]]
 => [3]
 => []
 => ? ∊ {0,1,1,2,2,3}
[[3,0],[1]]
 => [[1,2,2]]
 => [3]
 => []
 => ? ∊ {0,1,1,2,2,3}
[[3,0],[2]]
 => [[1,1,2]]
 => [3]
 => []
 => ? ∊ {0,1,1,2,2,3}
[[3,0],[3]]
 => [[1,1,1]]
 => [3]
 => []
 => ? ∊ {0,1,1,2,2,3}
[[2,1],[1]]
 => [[1,2],[2]]
 => [2,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3}
[[2,1],[2]]
 => [[1,1],[2]]
 => [2,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3}
[[2,0,0],[0,0],[0]]
 => [[3,3]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[1,0],[0]]
 => [[2,3]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[1,0],[1]]
 => [[1,3]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[2,0],[0]]
 => [[2,2]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[2,0],[1]]
 => [[1,2]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[2,0],[2]]
 => [[1,1]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[1,1,0],[1,0],[0]]
 => [[2],[3]]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[1,1,0],[1,0],[1]]
 => [[1],[3]]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[1,1,0],[1,1],[1]]
 => [[1],[2]]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[1,0,0,0],[0,0,0],[0,0],[0]]
 => [[4]]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
[[1,0,0,0],[1,0,0],[0,0],[0]]
 => [[3]]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
[[1,0,0,0],[1,0,0],[1,0],[0]]
 => [[2]]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
[[1,0,0,0],[1,0,0],[1,0],[1]]
 => [[1]]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
[[4,0],[0]]
 => [[2,2,2,2]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[4,0],[1]]
 => [[1,2,2,2]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[4,0],[2]]
 => [[1,1,2,2]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[4,0],[3]]
 => [[1,1,1,2]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[4,0],[4]]
 => [[1,1,1,1]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[3,1],[1]]
 => [[1,2,2],[2]]
 => [3,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3,3,4}
[[3,1],[2]]
 => [[1,1,2],[2]]
 => [3,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3,3,4}
[[3,1],[3]]
 => [[1,1,1],[2]]
 => [3,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3,3,4}
[[2,2],[2]]
 => [[1,1],[2,2]]
 => [2,2]
 => [2]
 => 2
[[3,0,0],[0,0],[0]]
 => [[3,3,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[1,0],[0]]
 => [[2,3,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[1,0],[1]]
 => [[1,3,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[2,0],[0]]
 => [[2,2,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[2,0],[1]]
 => [[1,2,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[2,0],[2]]
 => [[1,1,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[3,0],[0]]
 => [[2,2,2]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[3,0],[1]]
 => [[1,2,2]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[3,0],[2]]
 => [[1,1,2]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[3,0],[3]]
 => [[1,1,1]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[1,0],[0]]
 => [[2,3],[3]]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[1,0],[1]]
 => [[1,3],[3]]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[1,1],[1]]
 => [[1,3],[2]]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[2,0],[0]]
 => [[2,2],[3]]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,1],[1,1],[1]]
 => [[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => 1
[[3,2],[2]]
 => [[1,1,2],[2,2]]
 => [3,2]
 => [2]
 => 2
[[3,2],[3]]
 => [[1,1,1],[2,2]]
 => [3,2]
 => [2]
 => 2
[[2,2,0],[2,0],[0]]
 => [[2,2],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,0],[1]]
 => [[1,2],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,0],[2]]
 => [[1,1],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,1],[1]]
 => [[1,2],[2,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,1],[2]]
 => [[1,1],[2,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,2],[2]]
 => [[1,1],[2,2]]
 => [2,2]
 => [2]
 => 2
[[2,1,1],[1,1],[1]]
 => [[1,3],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1],[2,1],[1]]
 => [[1,2],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1],[2,1],[2]]
 => [[1,1],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 1
[[1,1,1,0],[1,1,0],[1,0],[0]]
 => [[2],[3],[4]]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1,1,0],[1,1,0],[1,0],[1]]
 => [[1],[3],[4]]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1,1,0],[1,1,0],[1,1],[1]]
 => [[1],[2],[4]]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1,1,0],[1,1,1],[1,1],[1]]
 => [[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => 1
[[4,2],[2]]
 => [[1,1,2,2],[2,2]]
 => [4,2]
 => [2]
 => 2
[[4,2],[3]]
 => [[1,1,1,2],[2,2]]
 => [4,2]
 => [2]
 => 2
[[4,2],[4]]
 => [[1,1,1,1],[2,2]]
 => [4,2]
 => [2]
 => 2
[[3,3],[3]]
 => [[1,1,1],[2,2,2]]
 => [3,3]
 => [3]
 => 3
[[2,2,0,0],[2,0,0],[0,0],[0]]
 => [[3,3],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[1,0],[0]]
 => [[2,3],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[1,0],[1]]
 => [[1,3],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[2,0],[0]]
 => [[2,2],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[2,0],[1]]
 => [[1,2],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[2,0],[2]]
 => [[1,1],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[1,0],[0]]
 => [[2,3],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[1,0],[1]]
 => [[1,3],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[1,1],[1]]
 => [[1,3],[2,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,0],[0]]
 => [[2,2],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,0],[1]]
 => [[1,2],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,0],[2]]
 => [[1,1],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,1],[1]]
 => [[1,2],[2,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,1],[2]]
 => [[1,1],[2,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,0],[0]]
 => [[2,2],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,0],[1]]
 => [[1,2],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,0],[2]]
 => [[1,1],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,1],[1]]
 => [[1,2],[2,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,1],[2]]
 => [[1,1],[2,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,2],[2]]
 => [[1,1],[2,2]]
 => [2,2]
 => [2]
 => 2
[[2,1,1,0],[1,1,0],[1,0],[0]]
 => [[2,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[1,1,0],[1,0],[1]]
 => [[1,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[1,1,0],[1,1],[1]]
 => [[1,4],[2],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[1,1,1],[1,1],[1]]
 => [[1,4],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[1,0],[0]]
 => [[2,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[1,0],[1]]
 => [[1,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[1,1],[1]]
 => [[1,3],[2],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[2,0],[0]]
 => [[2,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[2,0],[1]]
 => [[1,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
Description
The least common multiple of the parts of the partition.
Matching statistic: St000707
Mapping
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000707: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 43%
Values
[[1,0],[0]]
 => [[2]]
 => [1]
 => []
 => ? ∊ {0,1}
[[1,0],[1]]
 => [[1]]
 => [1]
 => []
 => ? ∊ {0,1}
[[2,0],[0]]
 => [[2,2]]
 => [2]
 => []
 => ? ∊ {0,1,1,2}
[[2,0],[1]]
 => [[1,2]]
 => [2]
 => []
 => ? ∊ {0,1,1,2}
[[2,0],[2]]
 => [[1,1]]
 => [2]
 => []
 => ? ∊ {0,1,1,2}
[[1,1],[1]]
 => [[1],[2]]
 => [1,1]
 => [1]
 => ? ∊ {0,1,1,2}
[[1,0,0],[0,0],[0]]
 => [[3]]
 => [1]
 => []
 => ? ∊ {0,0,1}
[[1,0,0],[1,0],[0]]
 => [[2]]
 => [1]
 => []
 => ? ∊ {0,0,1}
[[1,0,0],[1,0],[1]]
 => [[1]]
 => [1]
 => []
 => ? ∊ {0,0,1}
[[3,0],[0]]
 => [[2,2,2]]
 => [3]
 => []
 => ? ∊ {0,1,1,2,2,3}
[[3,0],[1]]
 => [[1,2,2]]
 => [3]
 => []
 => ? ∊ {0,1,1,2,2,3}
[[3,0],[2]]
 => [[1,1,2]]
 => [3]
 => []
 => ? ∊ {0,1,1,2,2,3}
[[3,0],[3]]
 => [[1,1,1]]
 => [3]
 => []
 => ? ∊ {0,1,1,2,2,3}
[[2,1],[1]]
 => [[1,2],[2]]
 => [2,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3}
[[2,1],[2]]
 => [[1,1],[2]]
 => [2,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3}
[[2,0,0],[0,0],[0]]
 => [[3,3]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[1,0],[0]]
 => [[2,3]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[1,0],[1]]
 => [[1,3]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[2,0],[0]]
 => [[2,2]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[2,0],[1]]
 => [[1,2]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[2,0],[2]]
 => [[1,1]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[1,1,0],[1,0],[0]]
 => [[2],[3]]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[1,1,0],[1,0],[1]]
 => [[1],[3]]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[1,1,0],[1,1],[1]]
 => [[1],[2]]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[1,0,0,0],[0,0,0],[0,0],[0]]
 => [[4]]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
[[1,0,0,0],[1,0,0],[0,0],[0]]
 => [[3]]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
[[1,0,0,0],[1,0,0],[1,0],[0]]
 => [[2]]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
[[1,0,0,0],[1,0,0],[1,0],[1]]
 => [[1]]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
[[4,0],[0]]
 => [[2,2,2,2]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[4,0],[1]]
 => [[1,2,2,2]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[4,0],[2]]
 => [[1,1,2,2]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[4,0],[3]]
 => [[1,1,1,2]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[4,0],[4]]
 => [[1,1,1,1]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[3,1],[1]]
 => [[1,2,2],[2]]
 => [3,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3,3,4}
[[3,1],[2]]
 => [[1,1,2],[2]]
 => [3,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3,3,4}
[[3,1],[3]]
 => [[1,1,1],[2]]
 => [3,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3,3,4}
[[2,2],[2]]
 => [[1,1],[2,2]]
 => [2,2]
 => [2]
 => 2
[[3,0,0],[0,0],[0]]
 => [[3,3,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[1,0],[0]]
 => [[2,3,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[1,0],[1]]
 => [[1,3,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[2,0],[0]]
 => [[2,2,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[2,0],[1]]
 => [[1,2,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[2,0],[2]]
 => [[1,1,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[3,0],[0]]
 => [[2,2,2]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[3,0],[1]]
 => [[1,2,2]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[3,0],[2]]
 => [[1,1,2]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[3,0],[3]]
 => [[1,1,1]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[1,0],[0]]
 => [[2,3],[3]]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[1,0],[1]]
 => [[1,3],[3]]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[1,1],[1]]
 => [[1,3],[2]]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[2,0],[0]]
 => [[2,2],[3]]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,1],[1,1],[1]]
 => [[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => 1
[[3,2],[2]]
 => [[1,1,2],[2,2]]
 => [3,2]
 => [2]
 => 2
[[3,2],[3]]
 => [[1,1,1],[2,2]]
 => [3,2]
 => [2]
 => 2
[[2,2,0],[2,0],[0]]
 => [[2,2],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,0],[1]]
 => [[1,2],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,0],[2]]
 => [[1,1],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,1],[1]]
 => [[1,2],[2,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,1],[2]]
 => [[1,1],[2,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,2],[2]]
 => [[1,1],[2,2]]
 => [2,2]
 => [2]
 => 2
[[2,1,1],[1,1],[1]]
 => [[1,3],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1],[2,1],[1]]
 => [[1,2],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1],[2,1],[2]]
 => [[1,1],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 1
[[1,1,1,0],[1,1,0],[1,0],[0]]
 => [[2],[3],[4]]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1,1,0],[1,1,0],[1,0],[1]]
 => [[1],[3],[4]]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1,1,0],[1,1,0],[1,1],[1]]
 => [[1],[2],[4]]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1,1,0],[1,1,1],[1,1],[1]]
 => [[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => 1
[[4,2],[2]]
 => [[1,1,2,2],[2,2]]
 => [4,2]
 => [2]
 => 2
[[4,2],[3]]
 => [[1,1,1,2],[2,2]]
 => [4,2]
 => [2]
 => 2
[[4,2],[4]]
 => [[1,1,1,1],[2,2]]
 => [4,2]
 => [2]
 => 2
[[3,3],[3]]
 => [[1,1,1],[2,2,2]]
 => [3,3]
 => [3]
 => 6
[[2,2,0,0],[2,0,0],[0,0],[0]]
 => [[3,3],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[1,0],[0]]
 => [[2,3],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[1,0],[1]]
 => [[1,3],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[2,0],[0]]
 => [[2,2],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[2,0],[1]]
 => [[1,2],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[2,0],[2]]
 => [[1,1],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[1,0],[0]]
 => [[2,3],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[1,0],[1]]
 => [[1,3],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[1,1],[1]]
 => [[1,3],[2,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,0],[0]]
 => [[2,2],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,0],[1]]
 => [[1,2],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,0],[2]]
 => [[1,1],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,1],[1]]
 => [[1,2],[2,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,1],[2]]
 => [[1,1],[2,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,0],[0]]
 => [[2,2],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,0],[1]]
 => [[1,2],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,0],[2]]
 => [[1,1],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,1],[1]]
 => [[1,2],[2,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,1],[2]]
 => [[1,1],[2,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,2],[2]]
 => [[1,1],[2,2]]
 => [2,2]
 => [2]
 => 2
[[2,1,1,0],[1,1,0],[1,0],[0]]
 => [[2,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[1,1,0],[1,0],[1]]
 => [[1,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[1,1,0],[1,1],[1]]
 => [[1,4],[2],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[1,1,1],[1,1],[1]]
 => [[1,4],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[1,0],[0]]
 => [[2,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[1,0],[1]]
 => [[1,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[1,1],[1]]
 => [[1,3],[2],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[2,0],[0]]
 => [[2,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[2,0],[1]]
 => [[1,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
Description
The product of the factorials of the parts.
Matching statistic: St000708
Mapping
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 43%
Values
[[1,0],[0]]
 => [[2]]
 => [1]
 => []
 => ? ∊ {0,1}
[[1,0],[1]]
 => [[1]]
 => [1]
 => []
 => ? ∊ {0,1}
[[2,0],[0]]
 => [[2,2]]
 => [2]
 => []
 => ? ∊ {0,1,1,2}
[[2,0],[1]]
 => [[1,2]]
 => [2]
 => []
 => ? ∊ {0,1,1,2}
[[2,0],[2]]
 => [[1,1]]
 => [2]
 => []
 => ? ∊ {0,1,1,2}
[[1,1],[1]]
 => [[1],[2]]
 => [1,1]
 => [1]
 => ? ∊ {0,1,1,2}
[[1,0,0],[0,0],[0]]
 => [[3]]
 => [1]
 => []
 => ? ∊ {0,0,1}
[[1,0,0],[1,0],[0]]
 => [[2]]
 => [1]
 => []
 => ? ∊ {0,0,1}
[[1,0,0],[1,0],[1]]
 => [[1]]
 => [1]
 => []
 => ? ∊ {0,0,1}
[[3,0],[0]]
 => [[2,2,2]]
 => [3]
 => []
 => ? ∊ {0,1,1,2,2,3}
[[3,0],[1]]
 => [[1,2,2]]
 => [3]
 => []
 => ? ∊ {0,1,1,2,2,3}
[[3,0],[2]]
 => [[1,1,2]]
 => [3]
 => []
 => ? ∊ {0,1,1,2,2,3}
[[3,0],[3]]
 => [[1,1,1]]
 => [3]
 => []
 => ? ∊ {0,1,1,2,2,3}
[[2,1],[1]]
 => [[1,2],[2]]
 => [2,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3}
[[2,1],[2]]
 => [[1,1],[2]]
 => [2,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3}
[[2,0,0],[0,0],[0]]
 => [[3,3]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[1,0],[0]]
 => [[2,3]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[1,0],[1]]
 => [[1,3]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[2,0],[0]]
 => [[2,2]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[2,0],[1]]
 => [[1,2]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[2,0],[2]]
 => [[1,1]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[1,1,0],[1,0],[0]]
 => [[2],[3]]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[1,1,0],[1,0],[1]]
 => [[1],[3]]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[1,1,0],[1,1],[1]]
 => [[1],[2]]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[1,0,0,0],[0,0,0],[0,0],[0]]
 => [[4]]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
[[1,0,0,0],[1,0,0],[0,0],[0]]
 => [[3]]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
[[1,0,0,0],[1,0,0],[1,0],[0]]
 => [[2]]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
[[1,0,0,0],[1,0,0],[1,0],[1]]
 => [[1]]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
[[4,0],[0]]
 => [[2,2,2,2]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[4,0],[1]]
 => [[1,2,2,2]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[4,0],[2]]
 => [[1,1,2,2]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[4,0],[3]]
 => [[1,1,1,2]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[4,0],[4]]
 => [[1,1,1,1]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[3,1],[1]]
 => [[1,2,2],[2]]
 => [3,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3,3,4}
[[3,1],[2]]
 => [[1,1,2],[2]]
 => [3,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3,3,4}
[[3,1],[3]]
 => [[1,1,1],[2]]
 => [3,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3,3,4}
[[2,2],[2]]
 => [[1,1],[2,2]]
 => [2,2]
 => [2]
 => 2
[[3,0,0],[0,0],[0]]
 => [[3,3,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[1,0],[0]]
 => [[2,3,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[1,0],[1]]
 => [[1,3,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[2,0],[0]]
 => [[2,2,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[2,0],[1]]
 => [[1,2,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[2,0],[2]]
 => [[1,1,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[3,0],[0]]
 => [[2,2,2]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[3,0],[1]]
 => [[1,2,2]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[3,0],[2]]
 => [[1,1,2]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[3,0],[3]]
 => [[1,1,1]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[1,0],[0]]
 => [[2,3],[3]]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[1,0],[1]]
 => [[1,3],[3]]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[1,1],[1]]
 => [[1,3],[2]]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[2,0],[0]]
 => [[2,2],[3]]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,1],[1,1],[1]]
 => [[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => 1
[[3,2],[2]]
 => [[1,1,2],[2,2]]
 => [3,2]
 => [2]
 => 2
[[3,2],[3]]
 => [[1,1,1],[2,2]]
 => [3,2]
 => [2]
 => 2
[[2,2,0],[2,0],[0]]
 => [[2,2],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,0],[1]]
 => [[1,2],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,0],[2]]
 => [[1,1],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,1],[1]]
 => [[1,2],[2,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,1],[2]]
 => [[1,1],[2,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,2],[2]]
 => [[1,1],[2,2]]
 => [2,2]
 => [2]
 => 2
[[2,1,1],[1,1],[1]]
 => [[1,3],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1],[2,1],[1]]
 => [[1,2],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1],[2,1],[2]]
 => [[1,1],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 1
[[1,1,1,0],[1,1,0],[1,0],[0]]
 => [[2],[3],[4]]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1,1,0],[1,1,0],[1,0],[1]]
 => [[1],[3],[4]]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1,1,0],[1,1,0],[1,1],[1]]
 => [[1],[2],[4]]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1,1,0],[1,1,1],[1,1],[1]]
 => [[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => 1
[[4,2],[2]]
 => [[1,1,2,2],[2,2]]
 => [4,2]
 => [2]
 => 2
[[4,2],[3]]
 => [[1,1,1,2],[2,2]]
 => [4,2]
 => [2]
 => 2
[[4,2],[4]]
 => [[1,1,1,1],[2,2]]
 => [4,2]
 => [2]
 => 2
[[3,3],[3]]
 => [[1,1,1],[2,2,2]]
 => [3,3]
 => [3]
 => 3
[[2,2,0,0],[2,0,0],[0,0],[0]]
 => [[3,3],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[1,0],[0]]
 => [[2,3],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[1,0],[1]]
 => [[1,3],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[2,0],[0]]
 => [[2,2],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[2,0],[1]]
 => [[1,2],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[2,0],[2]]
 => [[1,1],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[1,0],[0]]
 => [[2,3],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[1,0],[1]]
 => [[1,3],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[1,1],[1]]
 => [[1,3],[2,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,0],[0]]
 => [[2,2],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,0],[1]]
 => [[1,2],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,0],[2]]
 => [[1,1],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,1],[1]]
 => [[1,2],[2,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,1],[2]]
 => [[1,1],[2,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,0],[0]]
 => [[2,2],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,0],[1]]
 => [[1,2],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,0],[2]]
 => [[1,1],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,1],[1]]
 => [[1,2],[2,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,1],[2]]
 => [[1,1],[2,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,2],[2]]
 => [[1,1],[2,2]]
 => [2,2]
 => [2]
 => 2
[[2,1,1,0],[1,1,0],[1,0],[0]]
 => [[2,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[1,1,0],[1,0],[1]]
 => [[1,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[1,1,0],[1,1],[1]]
 => [[1,4],[2],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[1,1,1],[1,1],[1]]
 => [[1,4],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[1,0],[0]]
 => [[2,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[1,0],[1]]
 => [[1,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[1,1],[1]]
 => [[1,3],[2],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[2,0],[0]]
 => [[2,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[2,0],[1]]
 => [[1,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
Description
The product of the parts of an integer partition.
Matching statistic: St000770
Mapping
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000770: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 43%
Values
[[1,0],[0]]
 => [[2]]
 => [1]
 => []
 => ? ∊ {0,1}
[[1,0],[1]]
 => [[1]]
 => [1]
 => []
 => ? ∊ {0,1}
[[2,0],[0]]
 => [[2,2]]
 => [2]
 => []
 => ? ∊ {0,1,1,2}
[[2,0],[1]]
 => [[1,2]]
 => [2]
 => []
 => ? ∊ {0,1,1,2}
[[2,0],[2]]
 => [[1,1]]
 => [2]
 => []
 => ? ∊ {0,1,1,2}
[[1,1],[1]]
 => [[1],[2]]
 => [1,1]
 => [1]
 => ? ∊ {0,1,1,2}
[[1,0,0],[0,0],[0]]
 => [[3]]
 => [1]
 => []
 => ? ∊ {0,0,1}
[[1,0,0],[1,0],[0]]
 => [[2]]
 => [1]
 => []
 => ? ∊ {0,0,1}
[[1,0,0],[1,0],[1]]
 => [[1]]
 => [1]
 => []
 => ? ∊ {0,0,1}
[[3,0],[0]]
 => [[2,2,2]]
 => [3]
 => []
 => ? ∊ {0,1,1,2,2,3}
[[3,0],[1]]
 => [[1,2,2]]
 => [3]
 => []
 => ? ∊ {0,1,1,2,2,3}
[[3,0],[2]]
 => [[1,1,2]]
 => [3]
 => []
 => ? ∊ {0,1,1,2,2,3}
[[3,0],[3]]
 => [[1,1,1]]
 => [3]
 => []
 => ? ∊ {0,1,1,2,2,3}
[[2,1],[1]]
 => [[1,2],[2]]
 => [2,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3}
[[2,1],[2]]
 => [[1,1],[2]]
 => [2,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3}
[[2,0,0],[0,0],[0]]
 => [[3,3]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[1,0],[0]]
 => [[2,3]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[1,0],[1]]
 => [[1,3]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[2,0],[0]]
 => [[2,2]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[2,0],[1]]
 => [[1,2]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[2,0],[2]]
 => [[1,1]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[1,1,0],[1,0],[0]]
 => [[2],[3]]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[1,1,0],[1,0],[1]]
 => [[1],[3]]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[1,1,0],[1,1],[1]]
 => [[1],[2]]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[1,0,0,0],[0,0,0],[0,0],[0]]
 => [[4]]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
[[1,0,0,0],[1,0,0],[0,0],[0]]
 => [[3]]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
[[1,0,0,0],[1,0,0],[1,0],[0]]
 => [[2]]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
[[1,0,0,0],[1,0,0],[1,0],[1]]
 => [[1]]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
[[4,0],[0]]
 => [[2,2,2,2]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[4,0],[1]]
 => [[1,2,2,2]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[4,0],[2]]
 => [[1,1,2,2]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[4,0],[3]]
 => [[1,1,1,2]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[4,0],[4]]
 => [[1,1,1,1]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[3,1],[1]]
 => [[1,2,2],[2]]
 => [3,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3,3,4}
[[3,1],[2]]
 => [[1,1,2],[2]]
 => [3,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3,3,4}
[[3,1],[3]]
 => [[1,1,1],[2]]
 => [3,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3,3,4}
[[2,2],[2]]
 => [[1,1],[2,2]]
 => [2,2]
 => [2]
 => 2
[[3,0,0],[0,0],[0]]
 => [[3,3,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[1,0],[0]]
 => [[2,3,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[1,0],[1]]
 => [[1,3,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[2,0],[0]]
 => [[2,2,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[2,0],[1]]
 => [[1,2,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[2,0],[2]]
 => [[1,1,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[3,0],[0]]
 => [[2,2,2]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[3,0],[1]]
 => [[1,2,2]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[3,0],[2]]
 => [[1,1,2]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[3,0],[3]]
 => [[1,1,1]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[1,0],[0]]
 => [[2,3],[3]]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[1,0],[1]]
 => [[1,3],[3]]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[1,1],[1]]
 => [[1,3],[2]]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[2,0],[0]]
 => [[2,2],[3]]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,1],[1,1],[1]]
 => [[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => 1
[[3,2],[2]]
 => [[1,1,2],[2,2]]
 => [3,2]
 => [2]
 => 2
[[3,2],[3]]
 => [[1,1,1],[2,2]]
 => [3,2]
 => [2]
 => 2
[[2,2,0],[2,0],[0]]
 => [[2,2],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,0],[1]]
 => [[1,2],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,0],[2]]
 => [[1,1],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,1],[1]]
 => [[1,2],[2,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,1],[2]]
 => [[1,1],[2,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,2],[2]]
 => [[1,1],[2,2]]
 => [2,2]
 => [2]
 => 2
[[2,1,1],[1,1],[1]]
 => [[1,3],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1],[2,1],[1]]
 => [[1,2],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1],[2,1],[2]]
 => [[1,1],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 1
[[1,1,1,0],[1,1,0],[1,0],[0]]
 => [[2],[3],[4]]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1,1,0],[1,1,0],[1,0],[1]]
 => [[1],[3],[4]]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1,1,0],[1,1,0],[1,1],[1]]
 => [[1],[2],[4]]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1,1,0],[1,1,1],[1,1],[1]]
 => [[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => 1
[[4,2],[2]]
 => [[1,1,2,2],[2,2]]
 => [4,2]
 => [2]
 => 2
[[4,2],[3]]
 => [[1,1,1,2],[2,2]]
 => [4,2]
 => [2]
 => 2
[[4,2],[4]]
 => [[1,1,1,1],[2,2]]
 => [4,2]
 => [2]
 => 2
[[3,3],[3]]
 => [[1,1,1],[2,2,2]]
 => [3,3]
 => [3]
 => 3
[[2,2,0,0],[2,0,0],[0,0],[0]]
 => [[3,3],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[1,0],[0]]
 => [[2,3],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[1,0],[1]]
 => [[1,3],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[2,0],[0]]
 => [[2,2],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[2,0],[1]]
 => [[1,2],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[2,0],[2]]
 => [[1,1],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[1,0],[0]]
 => [[2,3],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[1,0],[1]]
 => [[1,3],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[1,1],[1]]
 => [[1,3],[2,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,0],[0]]
 => [[2,2],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,0],[1]]
 => [[1,2],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,0],[2]]
 => [[1,1],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,1],[1]]
 => [[1,2],[2,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,1],[2]]
 => [[1,1],[2,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,0],[0]]
 => [[2,2],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,0],[1]]
 => [[1,2],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,0],[2]]
 => [[1,1],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,1],[1]]
 => [[1,2],[2,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,1],[2]]
 => [[1,1],[2,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,2],[2]]
 => [[1,1],[2,2]]
 => [2,2]
 => [2]
 => 2
[[2,1,1,0],[1,1,0],[1,0],[0]]
 => [[2,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[1,1,0],[1,0],[1]]
 => [[1,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[1,1,0],[1,1],[1]]
 => [[1,4],[2],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[1,1,1],[1,1],[1]]
 => [[1,4],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[1,0],[0]]
 => [[2,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[1,0],[1]]
 => [[1,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[1,1],[1]]
 => [[1,3],[2],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[2,0],[0]]
 => [[2,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[2,0],[1]]
 => [[1,2],[3],[4]]
 => [2,1,1]
 => [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.
Matching statistic: St000815
Mapping
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000815: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 43%
Values
[[1,0],[0]]
 => [[2]]
 => [1]
 => []
 => ? ∊ {0,1}
[[1,0],[1]]
 => [[1]]
 => [1]
 => []
 => ? ∊ {0,1}
[[2,0],[0]]
 => [[2,2]]
 => [2]
 => []
 => ? ∊ {0,1,1,2}
[[2,0],[1]]
 => [[1,2]]
 => [2]
 => []
 => ? ∊ {0,1,1,2}
[[2,0],[2]]
 => [[1,1]]
 => [2]
 => []
 => ? ∊ {0,1,1,2}
[[1,1],[1]]
 => [[1],[2]]
 => [1,1]
 => [1]
 => ? ∊ {0,1,1,2}
[[1,0,0],[0,0],[0]]
 => [[3]]
 => [1]
 => []
 => ? ∊ {0,0,1}
[[1,0,0],[1,0],[0]]
 => [[2]]
 => [1]
 => []
 => ? ∊ {0,0,1}
[[1,0,0],[1,0],[1]]
 => [[1]]
 => [1]
 => []
 => ? ∊ {0,0,1}
[[3,0],[0]]
 => [[2,2,2]]
 => [3]
 => []
 => ? ∊ {0,1,1,2,2,3}
[[3,0],[1]]
 => [[1,2,2]]
 => [3]
 => []
 => ? ∊ {0,1,1,2,2,3}
[[3,0],[2]]
 => [[1,1,2]]
 => [3]
 => []
 => ? ∊ {0,1,1,2,2,3}
[[3,0],[3]]
 => [[1,1,1]]
 => [3]
 => []
 => ? ∊ {0,1,1,2,2,3}
[[2,1],[1]]
 => [[1,2],[2]]
 => [2,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3}
[[2,1],[2]]
 => [[1,1],[2]]
 => [2,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3}
[[2,0,0],[0,0],[0]]
 => [[3,3]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[1,0],[0]]
 => [[2,3]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[1,0],[1]]
 => [[1,3]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[2,0],[0]]
 => [[2,2]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[2,0],[1]]
 => [[1,2]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[2,0],[2]]
 => [[1,1]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[1,1,0],[1,0],[0]]
 => [[2],[3]]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[1,1,0],[1,0],[1]]
 => [[1],[3]]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[1,1,0],[1,1],[1]]
 => [[1],[2]]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[1,0,0,0],[0,0,0],[0,0],[0]]
 => [[4]]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
[[1,0,0,0],[1,0,0],[0,0],[0]]
 => [[3]]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
[[1,0,0,0],[1,0,0],[1,0],[0]]
 => [[2]]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
[[1,0,0,0],[1,0,0],[1,0],[1]]
 => [[1]]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
[[4,0],[0]]
 => [[2,2,2,2]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[4,0],[1]]
 => [[1,2,2,2]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[4,0],[2]]
 => [[1,1,2,2]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[4,0],[3]]
 => [[1,1,1,2]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[4,0],[4]]
 => [[1,1,1,1]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[3,1],[1]]
 => [[1,2,2],[2]]
 => [3,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3,3,4}
[[3,1],[2]]
 => [[1,1,2],[2]]
 => [3,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3,3,4}
[[3,1],[3]]
 => [[1,1,1],[2]]
 => [3,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3,3,4}
[[2,2],[2]]
 => [[1,1],[2,2]]
 => [2,2]
 => [2]
 => 2
[[3,0,0],[0,0],[0]]
 => [[3,3,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[1,0],[0]]
 => [[2,3,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[1,0],[1]]
 => [[1,3,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[2,0],[0]]
 => [[2,2,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[2,0],[1]]
 => [[1,2,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[2,0],[2]]
 => [[1,1,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[3,0],[0]]
 => [[2,2,2]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[3,0],[1]]
 => [[1,2,2]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[3,0],[2]]
 => [[1,1,2]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[3,0],[3]]
 => [[1,1,1]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[1,0],[0]]
 => [[2,3],[3]]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[1,0],[1]]
 => [[1,3],[3]]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[1,1],[1]]
 => [[1,3],[2]]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[2,0],[0]]
 => [[2,2],[3]]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,1],[1,1],[1]]
 => [[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => 1
[[3,2],[2]]
 => [[1,1,2],[2,2]]
 => [3,2]
 => [2]
 => 2
[[3,2],[3]]
 => [[1,1,1],[2,2]]
 => [3,2]
 => [2]
 => 2
[[2,2,0],[2,0],[0]]
 => [[2,2],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,0],[1]]
 => [[1,2],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,0],[2]]
 => [[1,1],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,1],[1]]
 => [[1,2],[2,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,1],[2]]
 => [[1,1],[2,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,2],[2]]
 => [[1,1],[2,2]]
 => [2,2]
 => [2]
 => 2
[[2,1,1],[1,1],[1]]
 => [[1,3],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1],[2,1],[1]]
 => [[1,2],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1],[2,1],[2]]
 => [[1,1],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 1
[[1,1,1,0],[1,1,0],[1,0],[0]]
 => [[2],[3],[4]]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1,1,0],[1,1,0],[1,0],[1]]
 => [[1],[3],[4]]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1,1,0],[1,1,0],[1,1],[1]]
 => [[1],[2],[4]]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1,1,0],[1,1,1],[1,1],[1]]
 => [[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => 1
[[4,2],[2]]
 => [[1,1,2,2],[2,2]]
 => [4,2]
 => [2]
 => 2
[[4,2],[3]]
 => [[1,1,1,2],[2,2]]
 => [4,2]
 => [2]
 => 2
[[4,2],[4]]
 => [[1,1,1,1],[2,2]]
 => [4,2]
 => [2]
 => 2
[[3,3],[3]]
 => [[1,1,1],[2,2,2]]
 => [3,3]
 => [3]
 => 3
[[2,2,0,0],[2,0,0],[0,0],[0]]
 => [[3,3],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[1,0],[0]]
 => [[2,3],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[1,0],[1]]
 => [[1,3],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[2,0],[0]]
 => [[2,2],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[2,0],[1]]
 => [[1,2],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[2,0],[2]]
 => [[1,1],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[1,0],[0]]
 => [[2,3],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[1,0],[1]]
 => [[1,3],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[1,1],[1]]
 => [[1,3],[2,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,0],[0]]
 => [[2,2],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,0],[1]]
 => [[1,2],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,0],[2]]
 => [[1,1],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,1],[1]]
 => [[1,2],[2,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,1],[2]]
 => [[1,1],[2,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,0],[0]]
 => [[2,2],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,0],[1]]
 => [[1,2],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,0],[2]]
 => [[1,1],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,1],[1]]
 => [[1,2],[2,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,1],[2]]
 => [[1,1],[2,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,2],[2]]
 => [[1,1],[2,2]]
 => [2,2]
 => [2]
 => 2
[[2,1,1,0],[1,1,0],[1,0],[0]]
 => [[2,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[1,1,0],[1,0],[1]]
 => [[1,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[1,1,0],[1,1],[1]]
 => [[1,4],[2],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[1,1,1],[1,1],[1]]
 => [[1,4],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[1,0],[0]]
 => [[2,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[1,0],[1]]
 => [[1,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[1,1],[1]]
 => [[1,3],[2],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[2,0],[0]]
 => [[2,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[2,0],[1]]
 => [[1,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
Description
The number of semistandard Young tableaux of partition weight of given shape. The weight of a semistandard Young tableaux is the sequence $(m_1, m_2,\dots)$, where $m_i$ is the number of occurrences of the number $i$ in the tableau. This statistic counts those tableaux whose weight is a weakly decreasing sequence. Alternatively, this is the sum of the entries in the column specified by the partition of the change of basis matrix from Schur functions to monomial symmetric functions.
Matching statistic: St000933
Mapping
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000933: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 43%
Values
[[1,0],[0]]
 => [[2]]
 => [1]
 => []
 => ? ∊ {0,1}
[[1,0],[1]]
 => [[1]]
 => [1]
 => []
 => ? ∊ {0,1}
[[2,0],[0]]
 => [[2,2]]
 => [2]
 => []
 => ? ∊ {0,1,1,2}
[[2,0],[1]]
 => [[1,2]]
 => [2]
 => []
 => ? ∊ {0,1,1,2}
[[2,0],[2]]
 => [[1,1]]
 => [2]
 => []
 => ? ∊ {0,1,1,2}
[[1,1],[1]]
 => [[1],[2]]
 => [1,1]
 => [1]
 => ? ∊ {0,1,1,2}
[[1,0,0],[0,0],[0]]
 => [[3]]
 => [1]
 => []
 => ? ∊ {0,0,1}
[[1,0,0],[1,0],[0]]
 => [[2]]
 => [1]
 => []
 => ? ∊ {0,0,1}
[[1,0,0],[1,0],[1]]
 => [[1]]
 => [1]
 => []
 => ? ∊ {0,0,1}
[[3,0],[0]]
 => [[2,2,2]]
 => [3]
 => []
 => ? ∊ {0,1,1,2,2,3}
[[3,0],[1]]
 => [[1,2,2]]
 => [3]
 => []
 => ? ∊ {0,1,1,2,2,3}
[[3,0],[2]]
 => [[1,1,2]]
 => [3]
 => []
 => ? ∊ {0,1,1,2,2,3}
[[3,0],[3]]
 => [[1,1,1]]
 => [3]
 => []
 => ? ∊ {0,1,1,2,2,3}
[[2,1],[1]]
 => [[1,2],[2]]
 => [2,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3}
[[2,1],[2]]
 => [[1,1],[2]]
 => [2,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3}
[[2,0,0],[0,0],[0]]
 => [[3,3]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[1,0],[0]]
 => [[2,3]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[1,0],[1]]
 => [[1,3]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[2,0],[0]]
 => [[2,2]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[2,0],[1]]
 => [[1,2]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[2,0],[2]]
 => [[1,1]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[1,1,0],[1,0],[0]]
 => [[2],[3]]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[1,1,0],[1,0],[1]]
 => [[1],[3]]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[1,1,0],[1,1],[1]]
 => [[1],[2]]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[1,0,0,0],[0,0,0],[0,0],[0]]
 => [[4]]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
[[1,0,0,0],[1,0,0],[0,0],[0]]
 => [[3]]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
[[1,0,0,0],[1,0,0],[1,0],[0]]
 => [[2]]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
[[1,0,0,0],[1,0,0],[1,0],[1]]
 => [[1]]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
[[4,0],[0]]
 => [[2,2,2,2]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[4,0],[1]]
 => [[1,2,2,2]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[4,0],[2]]
 => [[1,1,2,2]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[4,0],[3]]
 => [[1,1,1,2]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[4,0],[4]]
 => [[1,1,1,1]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[3,1],[1]]
 => [[1,2,2],[2]]
 => [3,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3,3,4}
[[3,1],[2]]
 => [[1,1,2],[2]]
 => [3,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3,3,4}
[[3,1],[3]]
 => [[1,1,1],[2]]
 => [3,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3,3,4}
[[2,2],[2]]
 => [[1,1],[2,2]]
 => [2,2]
 => [2]
 => 2
[[3,0,0],[0,0],[0]]
 => [[3,3,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[1,0],[0]]
 => [[2,3,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[1,0],[1]]
 => [[1,3,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[2,0],[0]]
 => [[2,2,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[2,0],[1]]
 => [[1,2,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[2,0],[2]]
 => [[1,1,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[3,0],[0]]
 => [[2,2,2]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[3,0],[1]]
 => [[1,2,2]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[3,0],[2]]
 => [[1,1,2]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[3,0],[3]]
 => [[1,1,1]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[1,0],[0]]
 => [[2,3],[3]]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[1,0],[1]]
 => [[1,3],[3]]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[1,1],[1]]
 => [[1,3],[2]]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[2,0],[0]]
 => [[2,2],[3]]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,1],[1,1],[1]]
 => [[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => 1
[[3,2],[2]]
 => [[1,1,2],[2,2]]
 => [3,2]
 => [2]
 => 2
[[3,2],[3]]
 => [[1,1,1],[2,2]]
 => [3,2]
 => [2]
 => 2
[[2,2,0],[2,0],[0]]
 => [[2,2],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,0],[1]]
 => [[1,2],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,0],[2]]
 => [[1,1],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,1],[1]]
 => [[1,2],[2,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,1],[2]]
 => [[1,1],[2,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,2],[2]]
 => [[1,1],[2,2]]
 => [2,2]
 => [2]
 => 2
[[2,1,1],[1,1],[1]]
 => [[1,3],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1],[2,1],[1]]
 => [[1,2],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1],[2,1],[2]]
 => [[1,1],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 1
[[1,1,1,0],[1,1,0],[1,0],[0]]
 => [[2],[3],[4]]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1,1,0],[1,1,0],[1,0],[1]]
 => [[1],[3],[4]]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1,1,0],[1,1,0],[1,1],[1]]
 => [[1],[2],[4]]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1,1,0],[1,1,1],[1,1],[1]]
 => [[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => 1
[[4,2],[2]]
 => [[1,1,2,2],[2,2]]
 => [4,2]
 => [2]
 => 2
[[4,2],[3]]
 => [[1,1,1,2],[2,2]]
 => [4,2]
 => [2]
 => 2
[[4,2],[4]]
 => [[1,1,1,1],[2,2]]
 => [4,2]
 => [2]
 => 2
[[3,3],[3]]
 => [[1,1,1],[2,2,2]]
 => [3,3]
 => [3]
 => 3
[[2,2,0,0],[2,0,0],[0,0],[0]]
 => [[3,3],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[1,0],[0]]
 => [[2,3],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[1,0],[1]]
 => [[1,3],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[2,0],[0]]
 => [[2,2],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[2,0],[1]]
 => [[1,2],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[2,0],[2]]
 => [[1,1],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[1,0],[0]]
 => [[2,3],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[1,0],[1]]
 => [[1,3],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[1,1],[1]]
 => [[1,3],[2,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,0],[0]]
 => [[2,2],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,0],[1]]
 => [[1,2],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,0],[2]]
 => [[1,1],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,1],[1]]
 => [[1,2],[2,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,1],[2]]
 => [[1,1],[2,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,0],[0]]
 => [[2,2],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,0],[1]]
 => [[1,2],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,0],[2]]
 => [[1,1],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,1],[1]]
 => [[1,2],[2,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,1],[2]]
 => [[1,1],[2,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,2],[2]]
 => [[1,1],[2,2]]
 => [2,2]
 => [2]
 => 2
[[2,1,1,0],[1,1,0],[1,0],[0]]
 => [[2,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[1,1,0],[1,0],[1]]
 => [[1,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[1,1,0],[1,1],[1]]
 => [[1,4],[2],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[1,1,1],[1,1],[1]]
 => [[1,4],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[1,0],[0]]
 => [[2,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[1,0],[1]]
 => [[1,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[1,1],[1]]
 => [[1,3],[2],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[2,0],[0]]
 => [[2,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[2,0],[1]]
 => [[1,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
Description
The number of multipartitions of sizes given by an integer partition. This is, for $\lambda = (\lambda_1,\ldots,\lambda_n)$, this is the number of $n$-tuples $(\lambda^{(1)},\ldots,\lambda^{(n)})$ of partitions $\lambda^{(i)}$ such that $\lambda^{(i)} \vdash \lambda_i$.
Matching statistic: St000937
Mapping
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000937: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 43%
Values
[[1,0],[0]]
 => [[2]]
 => [1]
 => []
 => ? ∊ {0,1}
[[1,0],[1]]
 => [[1]]
 => [1]
 => []
 => ? ∊ {0,1}
[[2,0],[0]]
 => [[2,2]]
 => [2]
 => []
 => ? ∊ {0,1,1,2}
[[2,0],[1]]
 => [[1,2]]
 => [2]
 => []
 => ? ∊ {0,1,1,2}
[[2,0],[2]]
 => [[1,1]]
 => [2]
 => []
 => ? ∊ {0,1,1,2}
[[1,1],[1]]
 => [[1],[2]]
 => [1,1]
 => [1]
 => ? ∊ {0,1,1,2}
[[1,0,0],[0,0],[0]]
 => [[3]]
 => [1]
 => []
 => ? ∊ {0,0,1}
[[1,0,0],[1,0],[0]]
 => [[2]]
 => [1]
 => []
 => ? ∊ {0,0,1}
[[1,0,0],[1,0],[1]]
 => [[1]]
 => [1]
 => []
 => ? ∊ {0,0,1}
[[3,0],[0]]
 => [[2,2,2]]
 => [3]
 => []
 => ? ∊ {0,1,1,2,2,3}
[[3,0],[1]]
 => [[1,2,2]]
 => [3]
 => []
 => ? ∊ {0,1,1,2,2,3}
[[3,0],[2]]
 => [[1,1,2]]
 => [3]
 => []
 => ? ∊ {0,1,1,2,2,3}
[[3,0],[3]]
 => [[1,1,1]]
 => [3]
 => []
 => ? ∊ {0,1,1,2,2,3}
[[2,1],[1]]
 => [[1,2],[2]]
 => [2,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3}
[[2,1],[2]]
 => [[1,1],[2]]
 => [2,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3}
[[2,0,0],[0,0],[0]]
 => [[3,3]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[1,0],[0]]
 => [[2,3]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[1,0],[1]]
 => [[1,3]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[2,0],[0]]
 => [[2,2]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[2,0],[1]]
 => [[1,2]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[2,0,0],[2,0],[2]]
 => [[1,1]]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[1,1,0],[1,0],[0]]
 => [[2],[3]]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[1,1,0],[1,0],[1]]
 => [[1],[3]]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[1,1,0],[1,1],[1]]
 => [[1],[2]]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2}
[[1,0,0,0],[0,0,0],[0,0],[0]]
 => [[4]]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
[[1,0,0,0],[1,0,0],[0,0],[0]]
 => [[3]]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
[[1,0,0,0],[1,0,0],[1,0],[0]]
 => [[2]]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
[[1,0,0,0],[1,0,0],[1,0],[1]]
 => [[1]]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
[[4,0],[0]]
 => [[2,2,2,2]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[4,0],[1]]
 => [[1,2,2,2]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[4,0],[2]]
 => [[1,1,2,2]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[4,0],[3]]
 => [[1,1,1,2]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[4,0],[4]]
 => [[1,1,1,1]]
 => [4]
 => []
 => ? ∊ {0,1,1,2,2,3,3,4}
[[3,1],[1]]
 => [[1,2,2],[2]]
 => [3,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3,3,4}
[[3,1],[2]]
 => [[1,1,2],[2]]
 => [3,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3,3,4}
[[3,1],[3]]
 => [[1,1,1],[2]]
 => [3,1]
 => [1]
 => ? ∊ {0,1,1,2,2,3,3,4}
[[2,2],[2]]
 => [[1,1],[2,2]]
 => [2,2]
 => [2]
 => 2
[[3,0,0],[0,0],[0]]
 => [[3,3,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[1,0],[0]]
 => [[2,3,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[1,0],[1]]
 => [[1,3,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[2,0],[0]]
 => [[2,2,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[2,0],[1]]
 => [[1,2,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[2,0],[2]]
 => [[1,1,3]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[3,0],[0]]
 => [[2,2,2]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[3,0],[1]]
 => [[1,2,2]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[3,0],[2]]
 => [[1,1,2]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[3,0],[3]]
 => [[1,1,1]]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[1,0],[0]]
 => [[2,3],[3]]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[1,0],[1]]
 => [[1,3],[3]]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[1,1],[1]]
 => [[1,3],[2]]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[2,0],[0]]
 => [[2,2],[3]]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,1],[1,1],[1]]
 => [[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => 1
[[3,2],[2]]
 => [[1,1,2],[2,2]]
 => [3,2]
 => [2]
 => 2
[[3,2],[3]]
 => [[1,1,1],[2,2]]
 => [3,2]
 => [2]
 => 2
[[2,2,0],[2,0],[0]]
 => [[2,2],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,0],[1]]
 => [[1,2],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,0],[2]]
 => [[1,1],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,1],[1]]
 => [[1,2],[2,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,1],[2]]
 => [[1,1],[2,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0],[2,2],[2]]
 => [[1,1],[2,2]]
 => [2,2]
 => [2]
 => 2
[[2,1,1],[1,1],[1]]
 => [[1,3],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1],[2,1],[1]]
 => [[1,2],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1],[2,1],[2]]
 => [[1,1],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 1
[[1,1,1,0],[1,1,0],[1,0],[0]]
 => [[2],[3],[4]]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1,1,0],[1,1,0],[1,0],[1]]
 => [[1],[3],[4]]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1,1,0],[1,1,0],[1,1],[1]]
 => [[1],[2],[4]]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1,1,0],[1,1,1],[1,1],[1]]
 => [[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => 1
[[4,2],[2]]
 => [[1,1,2,2],[2,2]]
 => [4,2]
 => [2]
 => 2
[[4,2],[3]]
 => [[1,1,1,2],[2,2]]
 => [4,2]
 => [2]
 => 2
[[4,2],[4]]
 => [[1,1,1,1],[2,2]]
 => [4,2]
 => [2]
 => 2
[[3,3],[3]]
 => [[1,1,1],[2,2,2]]
 => [3,3]
 => [3]
 => 3
[[2,2,0,0],[2,0,0],[0,0],[0]]
 => [[3,3],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[1,0],[0]]
 => [[2,3],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[1,0],[1]]
 => [[1,3],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[2,0],[0]]
 => [[2,2],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[2,0],[1]]
 => [[1,2],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,0,0],[2,0],[2]]
 => [[1,1],[4,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[1,0],[0]]
 => [[2,3],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[1,0],[1]]
 => [[1,3],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[1,1],[1]]
 => [[1,3],[2,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,0],[0]]
 => [[2,2],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,0],[1]]
 => [[1,2],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,0],[2]]
 => [[1,1],[3,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,1],[1]]
 => [[1,2],[2,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,1,0],[2,1],[2]]
 => [[1,1],[2,4]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,0],[0]]
 => [[2,2],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,0],[1]]
 => [[1,2],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,0],[2]]
 => [[1,1],[3,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,1],[1]]
 => [[1,2],[2,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,1],[2]]
 => [[1,1],[2,3]]
 => [2,2]
 => [2]
 => 2
[[2,2,0,0],[2,2,0],[2,2],[2]]
 => [[1,1],[2,2]]
 => [2,2]
 => [2]
 => 2
[[2,1,1,0],[1,1,0],[1,0],[0]]
 => [[2,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[1,1,0],[1,0],[1]]
 => [[1,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[1,1,0],[1,1],[1]]
 => [[1,4],[2],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[1,1,1],[1,1],[1]]
 => [[1,4],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[1,0],[0]]
 => [[2,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[1,0],[1]]
 => [[1,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[1,1],[1]]
 => [[1,3],[2],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[2,0],[0]]
 => [[2,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
[[2,1,1,0],[2,1,0],[2,0],[1]]
 => [[1,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1
Description
The number of positive values of the symmetric group character corresponding to the partition. For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugacy class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $2$.
Matching statistic: St000455
Mapping
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 14%
Values
[[1,0],[0]]
 => [[2]]
 => ([(0,1)],2)
 => ([],2)
 => ? ∊ {0,1}
[[1,0],[1]]
 => [[1]]
 => ([],1)
 => ([],1)
 => ? ∊ {0,1}
[[2,0],[0]]
 => [[2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? ∊ {0,1,1,2}
[[2,0],[1]]
 => [[1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ? ∊ {0,1,1,2}
[[2,0],[2]]
 => [[1,1]]
 => ([],1)
 => ([],1)
 => ? ∊ {0,1,1,2}
[[1,1],[1]]
 => [[1],[2]]
 => ([],1)
 => ([],1)
 => ? ∊ {0,1,1,2}
[[1,0,0],[0,0],[0]]
 => [[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? ∊ {0,0,1}
[[1,0,0],[1,0],[0]]
 => [[2]]
 => ([(0,1)],2)
 => ([],2)
 => ? ∊ {0,0,1}
[[1,0,0],[1,0],[1]]
 => [[1]]
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,1}
[[3,0],[0]]
 => [[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? ∊ {0,1,1,2,2,3}
[[3,0],[1]]
 => [[1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? ∊ {0,1,1,2,2,3}
[[3,0],[2]]
 => [[1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ? ∊ {0,1,1,2,2,3}
[[3,0],[3]]
 => [[1,1,1]]
 => ([],1)
 => ([],1)
 => ? ∊ {0,1,1,2,2,3}
[[2,1],[1]]
 => [[1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ? ∊ {0,1,1,2,2,3}
[[2,1],[2]]
 => [[1,1],[2]]
 => ([],1)
 => ([],1)
 => ? ∊ {0,1,1,2,2,3}
[[2,0,0],[0,0],[0]]
 => [[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[2,0,0],[1,0],[0]]
 => [[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[2,0,0],[1,0],[1]]
 => [[1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? ∊ {0,0,1,1,1,1,2}
[[2,0,0],[2,0],[0]]
 => [[2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? ∊ {0,0,1,1,1,1,2}
[[2,0,0],[2,0],[1]]
 => [[1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ? ∊ {0,0,1,1,1,1,2}
[[2,0,0],[2,0],[2]]
 => [[1,1]]
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,2}
[[1,1,0],[1,0],[0]]
 => [[2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? ∊ {0,0,1,1,1,1,2}
[[1,1,0],[1,0],[1]]
 => [[1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => ? ∊ {0,0,1,1,1,1,2}
[[1,1,0],[1,1],[1]]
 => [[1],[2]]
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,2}
[[1,0,0,0],[0,0,0],[0,0],[0]]
 => [[4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? ∊ {0,0,0,1}
[[1,0,0,0],[1,0,0],[0,0],[0]]
 => [[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? ∊ {0,0,0,1}
[[1,0,0,0],[1,0,0],[1,0],[0]]
 => [[2]]
 => ([(0,1)],2)
 => ([],2)
 => ? ∊ {0,0,0,1}
[[1,0,0,0],[1,0,0],[1,0],[1]]
 => [[1]]
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,1}
[[4,0],[0]]
 => [[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? ∊ {0,1,1,2,2,2,3,3,4}
[[4,0],[1]]
 => [[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? ∊ {0,1,1,2,2,2,3,3,4}
[[4,0],[2]]
 => [[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? ∊ {0,1,1,2,2,2,3,3,4}
[[4,0],[3]]
 => [[1,1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ? ∊ {0,1,1,2,2,2,3,3,4}
[[4,0],[4]]
 => [[1,1,1,1]]
 => ([],1)
 => ([],1)
 => ? ∊ {0,1,1,2,2,2,3,3,4}
[[3,1],[1]]
 => [[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? ∊ {0,1,1,2,2,2,3,3,4}
[[3,1],[2]]
 => [[1,1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ? ∊ {0,1,1,2,2,2,3,3,4}
[[3,1],[3]]
 => [[1,1,1],[2]]
 => ([],1)
 => ([],1)
 => ? ∊ {0,1,1,2,2,2,3,3,4}
[[2,2],[2]]
 => [[1,1],[2,2]]
 => ([],1)
 => ([],1)
 => ? ∊ {0,1,1,2,2,2,3,3,4}
[[3,0,0],[0,0],[0]]
 => [[3,3,3]]
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[1,0],[0]]
 => [[2,3,3]]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[1,0],[1]]
 => [[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[3,0,0],[2,0],[0]]
 => [[2,2,3]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[2,0],[1]]
 => [[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[3,0,0],[2,0],[2]]
 => [[1,1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[3,0],[0]]
 => [[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[3,0],[1]]
 => [[1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[3,0],[2]]
 => [[1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,0,0],[3,0],[3]]
 => [[1,1,1]]
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[1,0],[0]]
 => [[2,3],[3]]
 => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
 => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[1,0],[1]]
 => [[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[1,1],[1]]
 => [[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[2,0],[0]]
 => [[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[2,0],[1]]
 => [[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[2,0],[2]]
 => [[1,1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,1,0],[2,1],[1]]
 => [[1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,0,0,0],[2,0,0],[0,0],[0]]
 => [[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[2,0,0,0],[2,0,0],[1,0],[0]]
 => [[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[1,1,0,0],[1,0,0],[0,0],[0]]
 => [[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[1,1,0,0],[1,0,0],[1,0],[0]]
 => [[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[4,0,0],[2,0],[2]]
 => [[1,1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[4,0,0],[3,0],[2]]
 => [[1,1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[3,1,0],[1,0],[1]]
 => [[1,3,3],[3]]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(5,6)],7)
 => 0
[[3,1,0],[1,1],[1]]
 => [[1,3,3],[2]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[3,1,0],[2,1],[1]]
 => [[1,2,3],[2]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[2,2,0],[2,0],[0]]
 => [[2,2],[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[2,2,0],[2,0],[1]]
 => [[1,2],[3,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[3,0,0,0],[3,0,0],[1,0],[1]]
 => [[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[3,0,0,0],[3,0,0],[2,0],[1]]
 => [[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[2,1,0,0],[1,0,0],[1,0],[1]]
 => [[1,4],[4]]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(5,6)],7)
 => 0
[[2,1,0,0],[1,1,0],[1,0],[1]]
 => [[1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => 0
[[2,1,0,0],[2,0,0],[1,0],[1]]
 => [[1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[2,1,0,0],[2,0,0],[2,0],[1]]
 => [[1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[0,0],[0]]
 => [[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[1,0],[0]]
 => [[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[1,1,0,0,0],[1,1,0,0],[1,0,0],[0,0],[0]]
 => [[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[1,1,0,0,0],[1,1,0,0],[1,0,0],[1,0],[0]]
 => [[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[4,0,0,0],[4,0,0],[2,0],[2]]
 => [[1,1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[4,0,0,0],[4,0,0],[3,0],[2]]
 => [[1,1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[3,1,0,0],[2,0,0],[2,0],[2]]
 => [[1,1,4],[4]]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(5,6)],7)
 => 0
[[3,1,0,0],[2,1,0],[2,0],[2]]
 => [[1,1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => 0
[[3,1,0,0],[3,0,0],[2,0],[2]]
 => [[1,1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[3,1,0,0],[3,0,0],[3,0],[2]]
 => [[1,1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[3,1,0,0],[3,1,0],[1,0],[1]]
 => [[1,3,3],[3]]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(5,6)],7)
 => 0
[[3,1,0,0],[3,1,0],[1,1],[1]]
 => [[1,3,3],[2]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[3,1,0,0],[3,1,0],[2,1],[1]]
 => [[1,2,3],[2]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[2,2,0,0],[2,0,0],[2,0],[2]]
 => [[1,1],[4,4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[2,2,0,0],[2,1,0],[1,0],[1]]
 => [[1,3],[3,4]]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(5,6)],7)
 => 0
[[2,2,0,0],[2,1,0],[1,1],[1]]
 => [[1,3],[2,4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[2,2,0,0],[2,1,0],[2,0],[2]]
 => [[1,1],[3,4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[2,2,0,0],[2,1,0],[2,1],[1]]
 => [[1,2],[2,4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[2,2,0,0],[2,2,0],[2,0],[0]]
 => [[2,2],[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 0
[[2,2,0,0],[2,2,0],[2,0],[1]]
 => [[1,2],[3,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 0
[[2,1,1,0],[2,1,0],[1,0],[1]]
 => [[1,3],[3],[4]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => 0
[[2,1,1,0],[2,1,0],[1,1],[1]]
 => [[1,3],[2],[4]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => 0
[[2,1,1,0],[2,1,0],[2,1],[1]]
 => [[1,2],[2],[4]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 0
Description
The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.