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Your data matches 81 different statistics following compositions of up to 3 maps.
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Matching statistic: St000737
St000737: Semistandard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> 1
[[2,2]]
=> 2
[[1],[2]]
=> 1
[[1,3]]
=> 1
[[2,3]]
=> 2
[[3,3]]
=> 3
[[1],[3]]
=> 1
[[2],[3]]
=> 2
[[1,1,2]]
=> 1
[[1,2,2]]
=> 1
[[2,2,2]]
=> 2
[[1,1],[2]]
=> 1
[[1,2],[2]]
=> 1
[[1,4]]
=> 1
[[2,4]]
=> 2
[[3,4]]
=> 3
[[4,4]]
=> 4
[[1],[4]]
=> 1
[[2],[4]]
=> 2
[[3],[4]]
=> 3
[[1,1,3]]
=> 1
[[1,2,3]]
=> 1
[[1,3,3]]
=> 1
[[2,2,3]]
=> 2
[[2,3,3]]
=> 2
[[3,3,3]]
=> 3
[[1,1],[3]]
=> 1
[[1,2],[3]]
=> 1
[[1,3],[2]]
=> 1
[[1,3],[3]]
=> 1
[[2,2],[3]]
=> 2
[[2,3],[3]]
=> 2
[[1],[2],[3]]
=> 1
[[1,1,1,2]]
=> 1
[[1,1,2,2]]
=> 1
[[1,2,2,2]]
=> 1
[[2,2,2,2]]
=> 2
[[1,1,1],[2]]
=> 1
[[1,1,2],[2]]
=> 1
[[1,2,2],[2]]
=> 1
[[1,1],[2,2]]
=> 2
[[1,5]]
=> 1
[[2,5]]
=> 2
[[3,5]]
=> 3
[[4,5]]
=> 4
[[5,5]]
=> 5
[[1],[5]]
=> 1
[[2],[5]]
=> 2
[[3],[5]]
=> 3
[[4],[5]]
=> 4
Description
The last entry on the main diagonal of a semistandard tableau.
Matching statistic: St000939
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000939: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000939: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 67%
Values
[[1,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[2,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[1],[2]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2}
[[1,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[2,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[3,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[1],[3]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3}
[[2],[3]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3}
[[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[2,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,1],[2]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,2}
[[1,2],[2]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,2}
[[1,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[2,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[3,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[4,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[1],[4]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[2],[4]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[3],[4]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[3,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,3],[2]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,3],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[2,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[2,3],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2}
[[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2}
[[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2}
[[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2}
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2}
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2}
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[[1,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[5,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1,1,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4}
[[1,2,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4}
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> 2
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1],[2,3]]
=> [2,2]
=> [2]
=> 1
[[1,1],[3,3]]
=> [2,2]
=> [2]
=> 1
[[1,2],[2,3]]
=> [2,2]
=> [2]
=> 1
[[1,2],[3,3]]
=> [2,2]
=> [2]
=> 1
[[2,2],[3,3]]
=> [2,2]
=> [2]
=> 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,1,1],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[1],[3],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[1],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[3],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[3],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1],[2,4]]
=> [2,2]
=> [2]
=> 1
[[1,1],[3,4]]
=> [2,2]
=> [2]
=> 1
[[1,1],[4,4]]
=> [2,2]
=> [2]
=> 1
[[1,2],[2,4]]
=> [2,2]
=> [2]
=> 1
[[1,2],[3,4]]
=> [2,2]
=> [2]
=> 1
[[1,3],[2,4]]
=> [2,2]
=> [2]
=> 1
[[1,2],[4,4]]
=> [2,2]
=> [2]
=> 1
[[1,3],[3,4]]
=> [2,2]
=> [2]
=> 1
[[1,3],[4,4]]
=> [2,2]
=> [2]
=> 1
[[2,2],[3,4]]
=> [2,2]
=> [2]
=> 1
[[2,2],[4,4]]
=> [2,2]
=> [2]
=> 1
[[2,3],[3,4]]
=> [2,2]
=> [2]
=> 1
[[2,3],[4,4]]
=> [2,2]
=> [2]
=> 1
[[3,3],[4,4]]
=> [2,2]
=> [2]
=> 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[1,1,1],[2,3]]
=> [3,2]
=> [2]
=> 1
[[1,1,1],[3,3]]
=> [3,2]
=> [2]
=> 1
Description
The number of characters of the symmetric group whose value on the partition is positive.
Matching statistic: St000993
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 67%
Values
[[1,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[2,2]]
=> [2]
=> []
=> ? ∊ {1,1,2}
[[1],[2]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2}
[[1,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[2,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[3,3]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3}
[[1],[3]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3}
[[2],[3]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3}
[[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[2,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,1],[2]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,2}
[[1,2],[2]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,2}
[[1,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[2,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[3,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[4,4]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[1],[4]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[2],[4]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[3],[4]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[3,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,3],[2]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,3],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[2,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[2,3],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2}
[[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2}
[[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2}
[[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2}
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2}
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2}
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[[1,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[5,5]]
=> [2]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1,1,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4}
[[1,2,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4}
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> 2
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1],[2,3]]
=> [2,2]
=> [2]
=> 1
[[1,1],[3,3]]
=> [2,2]
=> [2]
=> 1
[[1,2],[2,3]]
=> [2,2]
=> [2]
=> 1
[[1,2],[3,3]]
=> [2,2]
=> [2]
=> 1
[[2,2],[3,3]]
=> [2,2]
=> [2]
=> 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,1,1],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[1],[3],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[1],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[3],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[3],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1],[2,4]]
=> [2,2]
=> [2]
=> 1
[[1,1],[3,4]]
=> [2,2]
=> [2]
=> 1
[[1,1],[4,4]]
=> [2,2]
=> [2]
=> 1
[[1,2],[2,4]]
=> [2,2]
=> [2]
=> 1
[[1,2],[3,4]]
=> [2,2]
=> [2]
=> 1
[[1,3],[2,4]]
=> [2,2]
=> [2]
=> 1
[[1,2],[4,4]]
=> [2,2]
=> [2]
=> 1
[[1,3],[3,4]]
=> [2,2]
=> [2]
=> 1
[[1,3],[4,4]]
=> [2,2]
=> [2]
=> 1
[[2,2],[3,4]]
=> [2,2]
=> [2]
=> 1
[[2,2],[4,4]]
=> [2,2]
=> [2]
=> 1
[[2,3],[3,4]]
=> [2,2]
=> [2]
=> 1
[[2,3],[4,4]]
=> [2,2]
=> [2]
=> 1
[[3,3],[4,4]]
=> [2,2]
=> [2]
=> 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[1,1,1],[2,3]]
=> [3,2]
=> [2]
=> 1
[[1,1,1],[3,3]]
=> [3,2]
=> [2]
=> 1
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St000937
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000937: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000937: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 67%
Values
[[1,2]]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2}
[[2,2]]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2}
[[1],[2]]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2}
[[1,3]]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2,2,3}
[[2,3]]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2,2,3}
[[3,3]]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2,2,3}
[[1],[3]]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,2,3}
[[2],[3]]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,2,3}
[[1,1,2]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,2}
[[1,2,2]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,2}
[[2,2,2]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,2}
[[1,1],[2]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,2}
[[1,2],[2]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,2}
[[1,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[2,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[3,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[4,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[1],[4]]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[2],[4]]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[3],[4]]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4}
[[1,1,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,2,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,3,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[2,2,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[2,3,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[3,3,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,2],[3]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,3],[2]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1,3],[3]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[2,2],[3]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[2,3],[3]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3}
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[[1,1,1,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2}
[[1,1,2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2}
[[1,2,2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2}
[[2,2,2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2}
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2}
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2}
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[1,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[5,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1],[5]]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2],[5]]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3],[5]]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4],[5]]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1,1,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4}
[[1,2,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4}
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[[1,1],[2,3]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[1,1],[3,3]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[1,2],[2,3]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[1,2],[3,3]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[2,2],[3,3]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[[1,1,1],[2,2]]
=> [3,2]
=> [2]
=> [1,1]
=> 1
[[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> [1,1]
=> 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[[1],[3],[5]]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[[1],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[[2],[3],[5]]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[[2],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[[3],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[[1,1],[2,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[1,1],[3,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[1,1],[4,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[1,2],[2,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[1,2],[3,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[1,3],[2,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[1,2],[4,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[1,3],[3,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[1,3],[4,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[2,2],[3,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[2,2],[4,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[2,3],[3,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[2,3],[4,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[3,3],[4,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[[1,1,1],[2,3]]
=> [3,2]
=> [2]
=> [1,1]
=> 1
[[1,1,1],[3,3]]
=> [3,2]
=> [2]
=> [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: St001199
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 33%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 33%
Values
[[1,2]]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2}
[[2,2]]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2}
[[1],[2]]
=> [1,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,2}
[[1,3]]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2,2,3}
[[2,3]]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2,2,3}
[[3,3]]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2,2,3}
[[1],[3]]
=> [1,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,2,2,3}
[[2],[3]]
=> [1,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,2,2,3}
[[1,1,2]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,2}
[[1,2,2]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,2}
[[2,2,2]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,2}
[[1,1],[2]]
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,2}
[[1,2],[2]]
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,2}
[[1,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[2,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[3,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[4,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[1],[4]]
=> [1,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,2,2,3,3,4}
[[2],[4]]
=> [1,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,2,2,3,3,4}
[[3],[4]]
=> [1,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,2,2,3,3,4}
[[1,1,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,2,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,3,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3}
[[3,3,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2],[3]]
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3],[2]]
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3],[3]]
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,2],[3]]
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3}
[[2,3],[3]]
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1,1,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2}
[[1,1,2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2}
[[1,2,2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2}
[[2,2,2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,2,2}
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,2,2}
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,2,2}
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[5,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1],[5]]
=> [1,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2],[5]]
=> [1,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3],[5]]
=> [1,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4],[5]]
=> [1,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1,1,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1,1],[2,3]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1],[3,3]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,2],[2,3]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,2],[3,3]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[2,2],[3,3]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1,1],[2,2]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1],[2,4]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1],[3,4]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1],[4,4]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,2],[2,4]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,2],[3,4]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,3],[2,4]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,2],[4,4]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,3],[3,4]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,3],[4,4]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[2,2],[3,4]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[2,2],[4,4]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[2,3],[3,4]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[2,3],[4,4]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[3,3],[4,4]]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[[1,1,1],[2,3]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1,1],[3,3]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1,2],[2,3]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1,3],[2,2]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1,2],[3,3]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1,3],[2,3]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1,3],[3,3]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,2,2],[2,3]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,2,2],[3,3]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,2,3],[2,3]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,2,3],[3,3]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[[2,2,2],[3,3]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[[2,2,3],[3,3]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1],[2,2],[3]]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[[1,1],[2,3],[3]]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[[1,2],[2,3],[3]]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[[1,1,1,1],[2,2]]
=> [4,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1,1,2],[2,2]]
=> [4,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1,2,2],[2,2]]
=> [4,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1,1],[2,2,2]]
=> [3,3]
=> [3]
=> [1,0,1,0,1,0]
=> 2
[[1,1,1],[2,4]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1,1],[3,4]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1,1],[4,4]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1,2],[2,4]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1,4],[2,2]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1,2],[3,4]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1,3],[2,4]]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St000207
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000207: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 25%●distinct values known / distinct values provided: 17%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000207: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 25%●distinct values known / distinct values provided: 17%
Values
[[1,2]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2}
[[2,2]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2}
[[1],[2]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2}
[[1,3]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3}
[[2,3]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3}
[[3,3]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3}
[[1],[3]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3}
[[2],[3]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3}
[[1,1,2]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,2}
[[1,2,2]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,2}
[[2,2,2]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,2}
[[1,1],[2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,2],[2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,4]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4}
[[2,4]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4}
[[3,4]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4}
[[4,4]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4}
[[1],[4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[2],[4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[3],[4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[1,1,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[2,2,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[2,3,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[3,3,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3],[2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[2,2],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[2,3],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,1,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,1,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,2,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2}
[[2,2,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[5,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1],[5]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2],[5]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3],[5]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4],[5]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1,1,4]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[1],[3],[5]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[1],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[2],[3],[5]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[2],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[3],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,1],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,2],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,3],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,2],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,3],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,3,3],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1],[2,2],[3]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[1,1],[2,3],[3]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[1,2],[2,3],[3]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[1,1,1],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,1],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,2],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,3],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,4],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,4],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,3],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,4],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,2],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,3],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,4],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,4],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,3],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has all vertices in integer lattice points.
Matching statistic: St000208
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 25%●distinct values known / distinct values provided: 17%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 25%●distinct values known / distinct values provided: 17%
Values
[[1,2]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2}
[[2,2]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2}
[[1],[2]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2}
[[1,3]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3}
[[2,3]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3}
[[3,3]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3}
[[1],[3]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3}
[[2],[3]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3}
[[1,1,2]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,2}
[[1,2,2]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,2}
[[2,2,2]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,2}
[[1,1],[2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,2],[2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,4]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4}
[[2,4]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4}
[[3,4]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4}
[[4,4]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4}
[[1],[4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[2],[4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[3],[4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[1,1,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[2,2,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[2,3,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[3,3,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3],[2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[2,2],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[2,3],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,1,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,1,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,2,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2}
[[2,2,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[5,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1],[5]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2],[5]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3],[5]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4],[5]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1,1,4]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[1],[3],[5]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[1],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[2],[3],[5]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[2],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[3],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,1],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,2],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,3],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,2],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,3],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,3,3],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1],[2,2],[3]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[1,1],[2,3],[3]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[1,2],[2,3],[3]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[1,1,1],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,1],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,2],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,3],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,4],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,4],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,3],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,4],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,2],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,3],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,4],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,4],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,3],[3],[4]]
=> [3,1,1]
=> [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: St000460
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 50%
Values
[[1,2]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2}
[[2,2]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2}
[[1],[2]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2}
[[1,3]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3}
[[2,3]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3}
[[3,3]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3}
[[1],[3]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3}
[[2],[3]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3}
[[1,1,2]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,2}
[[1,2,2]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,2}
[[2,2,2]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,2}
[[1,1],[2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,2],[2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,4]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4}
[[2,4]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4}
[[3,4]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4}
[[4,4]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4}
[[1],[4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[2],[4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[3],[4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[1,1,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[2,2,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[2,3,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[3,3,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3],[2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[2,2],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[2,3],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,1,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,1,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,2,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2}
[[2,2,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[5,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1],[5]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2],[5]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3],[5]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4],[5]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1,1,4]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[1],[3],[5]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[1],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[2],[3],[5]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[2],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[3],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1,1],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,2],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,3],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,2],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,3],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,3,3],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1],[2,2],[3]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[1,1],[2,3],[3]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[1,2],[2,3],[3]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[1,1,1],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,1],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,2],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,3],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,4],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,4],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,3],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,4],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,2],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,3],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,4],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,4],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,3],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
Description
The hook length of the last cell along the main diagonal of an integer partition.
Matching statistic: St000618
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000618: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 25%●distinct values known / distinct values provided: 17%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000618: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 25%●distinct values known / distinct values provided: 17%
Values
[[1,2]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2}
[[2,2]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2}
[[1],[2]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2}
[[1,3]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3}
[[2,3]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3}
[[3,3]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3}
[[1],[3]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3}
[[2],[3]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3}
[[1,1,2]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,2}
[[1,2,2]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,2}
[[2,2,2]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,2}
[[1,1],[2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,2],[2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,4]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4}
[[2,4]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4}
[[3,4]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4}
[[4,4]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4}
[[1],[4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[2],[4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[3],[4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[1,1,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[2,2,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[2,3,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[3,3,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3],[2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[2,2],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[2,3],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,1,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,1,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,2,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2}
[[2,2,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[5,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1],[5]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2],[5]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3],[5]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4],[5]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1,1,4]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[1],[3],[5]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[1],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[2],[3],[5]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[2],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[3],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,1],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,2],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,3],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,2],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,3],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,3,3],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1],[2,2],[3]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[1,1],[2,3],[3]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[1,2],[2,3],[3]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[1,1,1],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,1],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,2],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,3],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,4],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,4],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,3],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,4],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,2],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,3],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,4],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,4],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,3],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
Description
The number of self-evacuating tableaux of given shape.
This is the same as the number of standard domino tableaux of the given shape.
Matching statistic: St000667
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 25%●distinct values known / distinct values provided: 17%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 25%●distinct values known / distinct values provided: 17%
Values
[[1,2]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2}
[[2,2]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2}
[[1],[2]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2}
[[1,3]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3}
[[2,3]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3}
[[3,3]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3}
[[1],[3]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3}
[[2],[3]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3}
[[1,1,2]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,2}
[[1,2,2]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,2}
[[2,2,2]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,2}
[[1,1],[2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,2],[2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[[1,4]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4}
[[2,4]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4}
[[3,4]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4}
[[4,4]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4}
[[1],[4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[2],[4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[3],[4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4}
[[1,1,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[2,2,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[2,3,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[3,3,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,2],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3],[2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1,3],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[2,2],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[2,3],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3}
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,1,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,1,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,2,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2}
[[2,2,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2}
[[1,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[5,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1],[5]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[2],[5]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[3],[5]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[4],[5]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,3,3,4,4,5}
[[1,1,4]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[1],[3],[5]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[1],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[2],[3],[5]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[2],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[3],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,1],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,2],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,3],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,2],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,3],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,3,3],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1],[2,2],[3]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[1,1],[2,3],[3]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[1,2],[2,3],[3]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[1,1,1],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,1],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,2],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,3],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,4],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,4],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,3],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,4],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,2],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,3],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,4],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,4],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,2,3],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
Description
The greatest common divisor of the parts of the partition.
The following 71 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001118The acyclic chromatic index of a graph. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000260The radius of a connected graph. St001877Number of indecomposable injective modules with projective dimension 2. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001624The breadth of a lattice. St000454The largest eigenvalue of a graph if it is integral. St000456The monochromatic index of a connected graph. St000464The Schultz index of a connected graph. St001281The normalized isoperimetric number of a graph. St001545The second Elser number of a connected graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000264The girth of a graph, which is not a tree. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001060The distinguishing index of a graph. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000929The constant term of the character polynomial of an integer partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees.
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