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Your data matches 15 different statistics following compositions of up to 3 maps.
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Matching statistic: St001450
St001450: Plane partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> 1
[[1],[1]]
=> 1
[[2]]
=> 2
[[1,1]]
=> 1
[[1],[1],[1]]
=> 1
[[2],[1]]
=> 1
[[1,1],[1]]
=> 1
[[3]]
=> 3
[[2,1]]
=> 1
[[1,1,1]]
=> 1
[[1],[1],[1],[1]]
=> 1
[[2],[1],[1]]
=> 1
[[2],[2]]
=> 2
[[1,1],[1],[1]]
=> 1
[[1,1],[1,1]]
=> 1
[[3],[1]]
=> 1
[[2,1],[1]]
=> 1
[[1,1,1],[1]]
=> 1
[[4]]
=> 4
[[3,1]]
=> 1
[[2,2]]
=> 2
[[2,1,1]]
=> 1
[[1,1,1,1]]
=> 1
[[1],[1],[1],[1],[1]]
=> 1
[[2],[1],[1],[1]]
=> 1
[[2],[2],[1]]
=> 1
[[1,1],[1],[1],[1]]
=> 1
[[1,1],[1,1],[1]]
=> 1
[[3],[1],[1]]
=> 1
[[3],[2]]
=> 2
[[2,1],[1],[1]]
=> 1
[[2,1],[2]]
=> 2
[[2,1],[1,1]]
=> 1
[[1,1,1],[1],[1]]
=> 1
[[1,1,1],[1,1]]
=> 1
[[4],[1]]
=> 1
[[3,1],[1]]
=> 1
[[2,2],[1]]
=> 1
[[2,1,1],[1]]
=> 1
[[1,1,1,1],[1]]
=> 1
[[5]]
=> 5
[[4,1]]
=> 1
[[3,2]]
=> 2
[[3,1,1]]
=> 1
[[2,2,1]]
=> 1
[[2,1,1,1]]
=> 1
[[1,1,1,1,1]]
=> 1
[[1],[1],[1],[1],[1],[1]]
=> 1
[[2],[1],[1],[1],[1]]
=> 1
[[2],[2],[1],[1]]
=> 1
Description
The minimum height of a plane partition.
Matching statistic: St000627
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000627: Binary words ⟶ ℤResult quality: 30% ●values known / values provided: 88%●distinct values known / distinct values provided: 30%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000627: Binary words ⟶ ℤResult quality: 30% ●values known / values provided: 88%●distinct values known / distinct values provided: 30%
Values
[[1]]
=> [1]
=> []
=> => ? = 1
[[1],[1]]
=> [1,1]
=> [1]
=> 10 => 1
[[2]]
=> [2]
=> []
=> => ? ∊ {1,2}
[[1,1]]
=> [2]
=> []
=> => ? ∊ {1,2}
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> 110 => 1
[[2],[1]]
=> [2,1]
=> [1]
=> 10 => 1
[[1,1],[1]]
=> [2,1]
=> [1]
=> 10 => 1
[[3]]
=> [3]
=> []
=> => ? ∊ {1,1,3}
[[2,1]]
=> [3]
=> []
=> => ? ∊ {1,1,3}
[[1,1,1]]
=> [3]
=> []
=> => ? ∊ {1,1,3}
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> 110 => 1
[[2],[2]]
=> [2,2]
=> [2]
=> 100 => 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> 110 => 1
[[1,1],[1,1]]
=> [2,2]
=> [2]
=> 100 => 1
[[3],[1]]
=> [3,1]
=> [1]
=> 10 => 1
[[2,1],[1]]
=> [3,1]
=> [1]
=> 10 => 1
[[1,1,1],[1]]
=> [3,1]
=> [1]
=> 10 => 1
[[4]]
=> [4]
=> []
=> => ? ∊ {1,1,2,2,4}
[[3,1]]
=> [4]
=> []
=> => ? ∊ {1,1,2,2,4}
[[2,2]]
=> [4]
=> []
=> => ? ∊ {1,1,2,2,4}
[[2,1,1]]
=> [4]
=> []
=> => ? ∊ {1,1,2,2,4}
[[1,1,1,1]]
=> [4]
=> []
=> => ? ∊ {1,1,2,2,4}
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 1
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> 1010 => 2
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> 1010 => 2
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> 110 => 1
[[3],[2]]
=> [3,2]
=> [2]
=> 100 => 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> 110 => 1
[[2,1],[2]]
=> [3,2]
=> [2]
=> 100 => 1
[[2,1],[1,1]]
=> [3,2]
=> [2]
=> 100 => 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> 110 => 1
[[1,1,1],[1,1]]
=> [3,2]
=> [2]
=> 100 => 1
[[4],[1]]
=> [4,1]
=> [1]
=> 10 => 1
[[3,1],[1]]
=> [4,1]
=> [1]
=> 10 => 1
[[2,2],[1]]
=> [4,1]
=> [1]
=> 10 => 1
[[2,1,1],[1]]
=> [4,1]
=> [1]
=> 10 => 1
[[1,1,1,1],[1]]
=> [4,1]
=> [1]
=> 10 => 1
[[5]]
=> [5]
=> []
=> => ? ∊ {1,1,1,1,1,2,5}
[[4,1]]
=> [5]
=> []
=> => ? ∊ {1,1,1,1,1,2,5}
[[3,2]]
=> [5]
=> []
=> => ? ∊ {1,1,1,1,1,2,5}
[[3,1,1]]
=> [5]
=> []
=> => ? ∊ {1,1,1,1,1,2,5}
[[2,2,1]]
=> [5]
=> []
=> => ? ∊ {1,1,1,1,1,2,5}
[[2,1,1,1]]
=> [5]
=> []
=> => ? ∊ {1,1,1,1,1,2,5}
[[1,1,1,1,1]]
=> [5]
=> []
=> => ? ∊ {1,1,1,1,1,2,5}
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> 10110 => 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> 1100 => 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> 10110 => 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> 1100 => 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> 1110 => 1
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> 1010 => 2
[[3],[3]]
=> [3,3]
=> [3]
=> 1000 => 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> 1110 => 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> 1010 => 2
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> 1010 => 2
[[2,1],[2,1]]
=> [3,3]
=> [3]
=> 1000 => 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> 1110 => 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> 1010 => 2
[[1,1,1],[1,1,1]]
=> [3,3]
=> [3]
=> 1000 => 1
[[4],[1],[1]]
=> [4,1,1]
=> [1,1]
=> 110 => 1
[[4],[2]]
=> [4,2]
=> [2]
=> 100 => 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> 110 => 1
[[3,1],[2]]
=> [4,2]
=> [2]
=> 100 => 1
[[6]]
=> [6]
=> []
=> => ? ∊ {1,1,1,1,1,2,2,2,3,3,6}
[[5,1]]
=> [6]
=> []
=> => ? ∊ {1,1,1,1,1,2,2,2,3,3,6}
[[4,2]]
=> [6]
=> []
=> => ? ∊ {1,1,1,1,1,2,2,2,3,3,6}
[[4,1,1]]
=> [6]
=> []
=> => ? ∊ {1,1,1,1,1,2,2,2,3,3,6}
[[3,3]]
=> [6]
=> []
=> => ? ∊ {1,1,1,1,1,2,2,2,3,3,6}
[[3,2,1]]
=> [6]
=> []
=> => ? ∊ {1,1,1,1,1,2,2,2,3,3,6}
[[3,1,1,1]]
=> [6]
=> []
=> => ? ∊ {1,1,1,1,1,2,2,2,3,3,6}
[[2,2,2]]
=> [6]
=> []
=> => ? ∊ {1,1,1,1,1,2,2,2,3,3,6}
[[2,2,1,1]]
=> [6]
=> []
=> => ? ∊ {1,1,1,1,1,2,2,2,3,3,6}
[[2,1,1,1,1]]
=> [6]
=> []
=> => ? ∊ {1,1,1,1,1,2,2,2,3,3,6}
[[1,1,1,1,1,1]]
=> [6]
=> []
=> => ? ∊ {1,1,1,1,1,2,2,2,3,3,6}
[[7]]
=> [7]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,3,3,3,7}
[[6,1]]
=> [7]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,3,3,3,7}
[[5,2]]
=> [7]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,3,3,3,7}
[[5,1,1]]
=> [7]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,3,3,3,7}
[[4,3]]
=> [7]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,3,3,3,7}
[[4,2,1]]
=> [7]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,3,3,3,7}
[[4,1,1,1]]
=> [7]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,3,3,3,7}
[[3,3,1]]
=> [7]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,3,3,3,7}
[[3,2,2]]
=> [7]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,3,3,3,7}
[[3,2,1,1]]
=> [7]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,3,3,3,7}
[[3,1,1,1,1]]
=> [7]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,3,3,3,7}
[[2,2,2,1]]
=> [7]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,3,3,3,7}
[[2,2,1,1,1]]
=> [7]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,3,3,3,7}
[[2,1,1,1,1,1]]
=> [7]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,3,3,3,7}
[[1,1,1,1,1,1,1]]
=> [7]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,3,3,3,7}
[[8]]
=> [8]
=> []
=> => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,8}
[[7,1]]
=> [8]
=> []
=> => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,8}
[[6,2]]
=> [8]
=> []
=> => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,8}
[[6,1,1]]
=> [8]
=> []
=> => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,8}
[[5,3]]
=> [8]
=> []
=> => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,8}
[[5,2,1]]
=> [8]
=> []
=> => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,8}
Description
The exponent of a binary word.
This is the largest number $e$ such that $w$ is the concatenation of $e$ identical factors. This statistic is also called '''frequency'''.
Matching statistic: St001481
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001481: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 85%●distinct values known / distinct values provided: 20%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001481: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 85%●distinct values known / distinct values provided: 20%
Values
[[1]]
=> [1]
=> []
=> []
=> ? = 1
[[1],[1]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[[2]]
=> [2]
=> []
=> []
=> ? ∊ {1,2}
[[1,1]]
=> [2]
=> []
=> []
=> ? ∊ {1,2}
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[2],[1]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,1],[1]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[[3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,3}
[[2,1]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,3}
[[1,1,1]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,3}
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[2],[2]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1,1],[1,1]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[3],[1]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[[2,1],[1]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,1,1],[1]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[[4]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,2,2,4}
[[3,1]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,2,2,4}
[[2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,2,2,4}
[[2,1,1]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,2,2,4}
[[1,1,1,1]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,2,2,4}
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[3],[2]]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[2,1],[2]]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[2,1],[1,1]]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1,1,1],[1,1]]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[4],[1]]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 1
[[3,1],[1]]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 1
[[2,2],[1]]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 1
[[2,1,1],[1]]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 1
[[1,1,1,1],[1]]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 1
[[5]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,5}
[[4,1]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,5}
[[3,2]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,5}
[[3,1,1]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,5}
[[2,2,1]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,5}
[[2,1,1,1]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,5}
[[1,1,1,1,1]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,5}
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[[3],[3]]
=> [3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[[2,1],[2,1]]
=> [3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[[1,1,1],[1,1,1]]
=> [3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[4],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[4],[2]]
=> [4,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[3,1],[2]]
=> [4,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[6]]
=> [6]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,6}
[[5,1]]
=> [6]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,6}
[[4,2]]
=> [6]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,6}
[[4,1,1]]
=> [6]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,6}
[[3,3]]
=> [6]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,6}
[[3,2,1]]
=> [6]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,6}
[[3,1,1,1]]
=> [6]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,6}
[[2,2,2]]
=> [6]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,6}
[[2,2,1,1]]
=> [6]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,6}
[[2,1,1,1,1]]
=> [6]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,6}
[[1,1,1,1,1,1]]
=> [6]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,6}
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,7}
[[7]]
=> [7]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,7}
[[6,1]]
=> [7]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,7}
[[5,2]]
=> [7]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,7}
[[5,1,1]]
=> [7]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,7}
[[4,3]]
=> [7]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,7}
[[4,2,1]]
=> [7]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,7}
[[4,1,1,1]]
=> [7]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,7}
[[3,3,1]]
=> [7]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,7}
[[3,2,2]]
=> [7]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,7}
[[3,2,1,1]]
=> [7]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,7}
[[3,1,1,1,1]]
=> [7]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,7}
[[2,2,2,1]]
=> [7]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,7}
[[2,2,1,1,1]]
=> [7]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,7}
[[2,1,1,1,1,1]]
=> [7]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,7}
[[1,1,1,1,1,1,1]]
=> [7]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,7}
[[1],[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,8}
[[2],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,8}
[[1,1],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,8}
[[8]]
=> [8]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,8}
[[7,1]]
=> [8]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,8}
Description
The minimal height of a peak of a Dyck path.
Matching statistic: St001487
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 10% ●values known / values provided: 74%●distinct values known / distinct values provided: 10%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 10% ●values known / values provided: 74%●distinct values known / distinct values provided: 10%
Values
[[1]]
=> [1]
=> []
=> [[],[]]
=> ? = 1
[[1],[1]]
=> [1,1]
=> [1]
=> [[1],[]]
=> 1
[[2]]
=> [2]
=> []
=> [[],[]]
=> ? ∊ {1,2}
[[1,1]]
=> [2]
=> []
=> [[],[]]
=> ? ∊ {1,2}
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[[2],[1]]
=> [2,1]
=> [1]
=> [[1],[]]
=> 1
[[1,1],[1]]
=> [2,1]
=> [1]
=> [[1],[]]
=> 1
[[3]]
=> [3]
=> []
=> [[],[]]
=> ? ∊ {1,1,3}
[[2,1]]
=> [3]
=> []
=> [[],[]]
=> ? ∊ {1,1,3}
[[1,1,1]]
=> [3]
=> []
=> [[],[]]
=> ? ∊ {1,1,3}
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[[2],[2]]
=> [2,2]
=> [2]
=> [[2],[]]
=> 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[[1,1],[1,1]]
=> [2,2]
=> [2]
=> [[2],[]]
=> 1
[[3],[1]]
=> [3,1]
=> [1]
=> [[1],[]]
=> 1
[[2,1],[1]]
=> [3,1]
=> [1]
=> [[1],[]]
=> 1
[[1,1,1],[1]]
=> [3,1]
=> [1]
=> [[1],[]]
=> 1
[[4]]
=> [4]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,4}
[[3,1]]
=> [4]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,4}
[[2,2]]
=> [4]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,4}
[[2,1,1]]
=> [4]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,4}
[[1,1,1,1]]
=> [4]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,4}
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [[2,1],[]]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> [[2,1],[]]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[[3],[2]]
=> [3,2]
=> [2]
=> [[2],[]]
=> 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[[2,1],[2]]
=> [3,2]
=> [2]
=> [[2],[]]
=> 1
[[2,1],[1,1]]
=> [3,2]
=> [2]
=> [[2],[]]
=> 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[[1,1,1],[1,1]]
=> [3,2]
=> [2]
=> [[2],[]]
=> 1
[[4],[1]]
=> [4,1]
=> [1]
=> [[1],[]]
=> 1
[[3,1],[1]]
=> [4,1]
=> [1]
=> [[1],[]]
=> 1
[[2,2],[1]]
=> [4,1]
=> [1]
=> [[1],[]]
=> 1
[[2,1,1],[1]]
=> [4,1]
=> [1]
=> [[1],[]]
=> 1
[[1,1,1,1],[1]]
=> [4,1]
=> [1]
=> [[1],[]]
=> 1
[[5]]
=> [5]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,2,2,2,5}
[[4,1]]
=> [5]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,2,2,2,5}
[[3,2]]
=> [5]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,2,2,2,5}
[[3,1,1]]
=> [5]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,2,2,2,5}
[[2,2,1]]
=> [5]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,2,2,2,5}
[[2,1,1,1]]
=> [5]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,2,2,2,5}
[[1,1,1,1,1]]
=> [5]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,2,2,2,5}
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [[2,1,1],[]]
=> 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [[2,2],[]]
=> 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [[2,1,1],[]]
=> 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [[2,2],[]]
=> 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [[2,1],[]]
=> 1
[[3],[3]]
=> [3,3]
=> [3]
=> [[3],[]]
=> 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [[2,1],[]]
=> 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [[2,1],[]]
=> 1
[[2,1],[2,1]]
=> [3,3]
=> [3]
=> [[3],[]]
=> 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [[2,1],[]]
=> 1
[[1,1,1],[1,1,1]]
=> [3,3]
=> [3]
=> [[3],[]]
=> 1
[[4],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[[4],[2]]
=> [4,2]
=> [2]
=> [[2],[]]
=> 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[[3,1],[2]]
=> [4,2]
=> [2]
=> [[2],[]]
=> 1
[[6]]
=> [6]
=> []
=> [[],[]]
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,6}
[[5,1]]
=> [6]
=> []
=> [[],[]]
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,6}
[[4,2]]
=> [6]
=> []
=> [[],[]]
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,6}
[[4,1,1]]
=> [6]
=> []
=> [[],[]]
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,6}
[[3,3]]
=> [6]
=> []
=> [[],[]]
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,6}
[[3,2,1]]
=> [6]
=> []
=> [[],[]]
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,6}
[[3,1,1,1]]
=> [6]
=> []
=> [[],[]]
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,6}
[[2,2,2]]
=> [6]
=> []
=> [[],[]]
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,6}
[[2,2,1,1]]
=> [6]
=> []
=> [[],[]]
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,6}
[[2,1,1,1,1]]
=> [6]
=> []
=> [[],[]]
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,6}
[[1,1,1,1,1,1]]
=> [6]
=> []
=> [[],[]]
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,6}
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[[7]]
=> [7]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[[6,1]]
=> [7]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[[5,2]]
=> [7]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[[5,1,1]]
=> [7]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[[4,3]]
=> [7]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[[4,2,1]]
=> [7]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[[4,1,1,1]]
=> [7]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[[3,3,1]]
=> [7]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[[3,2,2]]
=> [7]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[[3,2,1,1]]
=> [7]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[[3,1,1,1,1]]
=> [7]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[[2,2,2,1]]
=> [7]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[[2,2,1,1,1]]
=> [7]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[[2,1,1,1,1,1]]
=> [7]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[[1,1,1,1,1,1,1]]
=> [7]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[[1],[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,8}
[[2],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,8}
[[2],[2],[1],[1],[1],[1]]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,8}
[[2],[2],[2],[1],[1]]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,8}
[[2],[2],[2],[2]]
=> [2,2,2,2]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,8}
Description
The number of inner corners of a skew partition.
Matching statistic: St001490
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 10% ●values known / values provided: 74%●distinct values known / distinct values provided: 10%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 10% ●values known / values provided: 74%●distinct values known / distinct values provided: 10%
Values
[[1]]
=> [1]
=> []
=> [[],[]]
=> ? = 1
[[1],[1]]
=> [1,1]
=> [1]
=> [[1],[]]
=> 1
[[2]]
=> [2]
=> []
=> [[],[]]
=> ? ∊ {1,2}
[[1,1]]
=> [2]
=> []
=> [[],[]]
=> ? ∊ {1,2}
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[[2],[1]]
=> [2,1]
=> [1]
=> [[1],[]]
=> 1
[[1,1],[1]]
=> [2,1]
=> [1]
=> [[1],[]]
=> 1
[[3]]
=> [3]
=> []
=> [[],[]]
=> ? ∊ {1,1,3}
[[2,1]]
=> [3]
=> []
=> [[],[]]
=> ? ∊ {1,1,3}
[[1,1,1]]
=> [3]
=> []
=> [[],[]]
=> ? ∊ {1,1,3}
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[[2],[2]]
=> [2,2]
=> [2]
=> [[2],[]]
=> 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[[1,1],[1,1]]
=> [2,2]
=> [2]
=> [[2],[]]
=> 1
[[3],[1]]
=> [3,1]
=> [1]
=> [[1],[]]
=> 1
[[2,1],[1]]
=> [3,1]
=> [1]
=> [[1],[]]
=> 1
[[1,1,1],[1]]
=> [3,1]
=> [1]
=> [[1],[]]
=> 1
[[4]]
=> [4]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,4}
[[3,1]]
=> [4]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,4}
[[2,2]]
=> [4]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,4}
[[2,1,1]]
=> [4]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,4}
[[1,1,1,1]]
=> [4]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,4}
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [[2,1],[]]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> [[2,1],[]]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[[3],[2]]
=> [3,2]
=> [2]
=> [[2],[]]
=> 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[[2,1],[2]]
=> [3,2]
=> [2]
=> [[2],[]]
=> 1
[[2,1],[1,1]]
=> [3,2]
=> [2]
=> [[2],[]]
=> 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[[1,1,1],[1,1]]
=> [3,2]
=> [2]
=> [[2],[]]
=> 1
[[4],[1]]
=> [4,1]
=> [1]
=> [[1],[]]
=> 1
[[3,1],[1]]
=> [4,1]
=> [1]
=> [[1],[]]
=> 1
[[2,2],[1]]
=> [4,1]
=> [1]
=> [[1],[]]
=> 1
[[2,1,1],[1]]
=> [4,1]
=> [1]
=> [[1],[]]
=> 1
[[1,1,1,1],[1]]
=> [4,1]
=> [1]
=> [[1],[]]
=> 1
[[5]]
=> [5]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,2,2,2,5}
[[4,1]]
=> [5]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,2,2,2,5}
[[3,2]]
=> [5]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,2,2,2,5}
[[3,1,1]]
=> [5]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,2,2,2,5}
[[2,2,1]]
=> [5]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,2,2,2,5}
[[2,1,1,1]]
=> [5]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,2,2,2,5}
[[1,1,1,1,1]]
=> [5]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,2,2,2,5}
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [[2,1,1],[]]
=> 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [[2,2],[]]
=> 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [[2,1,1],[]]
=> 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [[2,2],[]]
=> 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [[2,1],[]]
=> 1
[[3],[3]]
=> [3,3]
=> [3]
=> [[3],[]]
=> 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [[2,1],[]]
=> 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [[2,1],[]]
=> 1
[[2,1],[2,1]]
=> [3,3]
=> [3]
=> [[3],[]]
=> 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [[2,1],[]]
=> 1
[[1,1,1],[1,1,1]]
=> [3,3]
=> [3]
=> [[3],[]]
=> 1
[[4],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[[4],[2]]
=> [4,2]
=> [2]
=> [[2],[]]
=> 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[[3,1],[2]]
=> [4,2]
=> [2]
=> [[2],[]]
=> 1
[[6]]
=> [6]
=> []
=> [[],[]]
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,6}
[[5,1]]
=> [6]
=> []
=> [[],[]]
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,6}
[[4,2]]
=> [6]
=> []
=> [[],[]]
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,6}
[[4,1,1]]
=> [6]
=> []
=> [[],[]]
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,6}
[[3,3]]
=> [6]
=> []
=> [[],[]]
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,6}
[[3,2,1]]
=> [6]
=> []
=> [[],[]]
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,6}
[[3,1,1,1]]
=> [6]
=> []
=> [[],[]]
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,6}
[[2,2,2]]
=> [6]
=> []
=> [[],[]]
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,6}
[[2,2,1,1]]
=> [6]
=> []
=> [[],[]]
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,6}
[[2,1,1,1,1]]
=> [6]
=> []
=> [[],[]]
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,6}
[[1,1,1,1,1,1]]
=> [6]
=> []
=> [[],[]]
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,6}
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[[7]]
=> [7]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[[6,1]]
=> [7]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[[5,2]]
=> [7]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[[5,1,1]]
=> [7]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[[4,3]]
=> [7]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[[4,2,1]]
=> [7]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[[4,1,1,1]]
=> [7]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[[3,3,1]]
=> [7]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[[3,2,2]]
=> [7]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[[3,2,1,1]]
=> [7]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[[3,1,1,1,1]]
=> [7]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[[2,2,2,1]]
=> [7]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[[2,2,1,1,1]]
=> [7]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[[2,1,1,1,1,1]]
=> [7]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[[1,1,1,1,1,1,1]]
=> [7]
=> []
=> [[],[]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,7}
[[1],[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,8}
[[2],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,8}
[[2],[2],[1],[1],[1],[1]]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,8}
[[2],[2],[2],[1],[1]]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,8}
[[2],[2],[2],[2]]
=> [2,2,2,2]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,8}
Description
The number of connected components of a skew partition.
Matching statistic: St000667
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 57%●distinct values known / distinct values provided: 30%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 57%●distinct values known / distinct values provided: 30%
Values
[[1]]
=> [1]
=> []
=> ?
=> ? = 1
[[1],[1]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2}
[[2]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2}
[[1,1]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2}
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[2],[1]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,3}
[[1,1],[1]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,3}
[[3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,3}
[[2,1]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,3}
[[1,1,1]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,3}
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[2],[2]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1],[1,1]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[3],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[2,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[1,1,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[4]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[3,1]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[2,1,1]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[1,1,1,1]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[3],[2]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[2,1],[2]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[2,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[4],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[3,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[2,2],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[2,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[1,1,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[5]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[4,1]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[3,2]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[3,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[2,2,1]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[2,1,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[1,1,1,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[3],[3]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,6}
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[2,1],[2,1]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,6}
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[1,1,1],[1,1,1]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,6}
[[4],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[4],[2]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,6}
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[3,1],[2]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,6}
[[3,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,6}
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[2,2],[2]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,6}
[[2,2],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,6}
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[2,1,1],[2]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,6}
[[2,1,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,6}
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,1,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,6}
[[5],[1]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,6}
[[4,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,6}
[[3,2],[1]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,6}
[[3,1,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,6}
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[3],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[3],[3],[1]]
=> [3,3,1]
=> [3,1]
=> [1]
=> 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [3,1]
=> [1]
=> 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
Description
The greatest common divisor of the parts of the partition.
Matching statistic: St000781
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 10% ●values known / values provided: 57%●distinct values known / distinct values provided: 10%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 10% ●values known / values provided: 57%●distinct values known / distinct values provided: 10%
Values
[[1]]
=> [1]
=> []
=> ?
=> ? = 1
[[1],[1]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2}
[[2]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2}
[[1,1]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2}
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[2],[1]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,3}
[[1,1],[1]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,3}
[[3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,3}
[[2,1]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,3}
[[1,1,1]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,3}
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[2],[2]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1],[1,1]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[3],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[2,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[1,1,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[4]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[3,1]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[2,1,1]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[1,1,1,1]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[3],[2]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[2,1],[2]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[2,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[4],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[3,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[2,2],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[2,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[1,1,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[5]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[4,1]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[3,2]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[3,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[2,2,1]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[2,1,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[1,1,1,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[3],[3]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[2,1],[2,1]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[1,1,1],[1,1,1]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[4],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[4],[2]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[3,1],[2]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[3,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[2,2],[2]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[2,2],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[2,1,1],[2]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[2,1,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,1,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[5],[1]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[4,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[3,2],[1]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[3,1,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[3],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[3],[3],[1]]
=> [3,3,1]
=> [3,1]
=> [1]
=> 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [3,1]
=> [1]
=> 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
Description
The number of proper colouring schemes of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1].
This statistic is the number of distinct such integer partitions that occur.
Matching statistic: St001432
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 57%●distinct values known / distinct values provided: 20%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 57%●distinct values known / distinct values provided: 20%
Values
[[1]]
=> [1]
=> []
=> ?
=> ? = 1
[[1],[1]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2}
[[2]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2}
[[1,1]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2}
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[2],[1]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,3}
[[1,1],[1]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,3}
[[3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,3}
[[2,1]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,3}
[[1,1,1]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,3}
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[2],[2]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1],[1,1]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[3],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[2,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[1,1,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[4]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[3,1]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[2,1,1]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[1,1,1,1]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[3],[2]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[2,1],[2]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[2,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[4],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[3,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[2,2],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[2,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[1,1,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[5]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[4,1]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[3,2]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[3,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[2,2,1]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[2,1,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[1,1,1,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[3],[3]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[2,1],[2,1]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[1,1,1],[1,1,1]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[4],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[4],[2]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[3,1],[2]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[3,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[2,2],[2]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[2,2],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[2,1,1],[2]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[2,1,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,1,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[5],[1]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[4,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[3,2],[1]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[3,1,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 2
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 2
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[3],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[3],[3],[1]]
=> [3,3,1]
=> [3,1]
=> [1]
=> 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [3,1]
=> [1]
=> 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
Description
The order dimension of the partition.
Given a partition $\lambda$, let $I(\lambda)$ be the principal order ideal in the Young lattice generated by $\lambda$. The order dimension of a partition is defined as the order dimension of the poset $I(\lambda)$.
Matching statistic: St001571
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001571: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 57%●distinct values known / distinct values provided: 30%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001571: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 57%●distinct values known / distinct values provided: 30%
Values
[[1]]
=> [1]
=> []
=> ?
=> ? = 1
[[1],[1]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2}
[[2]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2}
[[1,1]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2}
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[2],[1]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,3}
[[1,1],[1]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,3}
[[3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,3}
[[2,1]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,3}
[[1,1,1]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,3}
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[2],[2]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1],[1,1]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[3],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[2,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[1,1,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[4]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[3,1]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[2,1,1]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[1,1,1,1]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[3],[2]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[2,1],[2]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[2,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[4],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[3,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[2,2],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[2,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[1,1,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[5]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[4,1]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[3,2]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[3,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[2,2,1]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[2,1,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[1,1,1,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[3],[3]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,6}
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[2,1],[2,1]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,6}
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[1,1,1],[1,1,1]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,6}
[[4],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[4],[2]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,6}
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[3,1],[2]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,6}
[[3,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,6}
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[2,2],[2]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,6}
[[2,2],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,6}
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[2,1,1],[2]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,6}
[[2,1,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,6}
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,1,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,6}
[[5],[1]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,6}
[[4,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,6}
[[3,2],[1]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,6}
[[3,1,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,6}
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[3],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[3],[3],[1]]
=> [3,3,1]
=> [3,1]
=> [1]
=> 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [3,1]
=> [1]
=> 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
Description
The Cartan determinant of the integer partition.
Let $p=[p_1,...,p_r]$ be a given integer partition with highest part t. Let $A=K[x]/(x^t)$ be the finite dimensional algebra over the field $K$ and $M$ the direct sum of the indecomposable $A$-modules of vector space dimension $p_i$ for each $i$. Then the Cartan determinant of $p$ is the Cartan determinant of the endomorphism algebra of $M$ over $A$.
Explicitly, this is the determinant of the matrix $\left(\min(\bar p_i, \bar p_j)\right)_{i,j}$, where $\bar p$ is the set of distinct parts of the partition.
Matching statistic: St001899
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001899: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 57%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001899: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 57%●distinct values known / distinct values provided: 50%
Values
[[1]]
=> [1]
=> []
=> ?
=> ? = 1
[[1],[1]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2}
[[2]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2}
[[1,1]]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2}
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[2],[1]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,3}
[[1,1],[1]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,3}
[[3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,3}
[[2,1]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,3}
[[1,1,1]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,3}
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[2],[2]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1],[1,1]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[3],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[2,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[1,1,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[4]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[3,1]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[2,1,1]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[1,1,1,1]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[3],[2]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[2,1],[2]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[2,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[4],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[3,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[2,2],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[2,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[1,1,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[5]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[4,1]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[3,2]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[3,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[2,2,1]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[2,1,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[1,1,1,1,1]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,5}
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[3],[3]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[2,1],[2,1]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[1,1,1],[1,1,1]]
=> [3,3]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[4],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[4],[2]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[3,1],[2]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[3,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[2,2],[2]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[2,2],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[2,1,1],[2]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[2,1,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,1,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[5],[1]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[4,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[3,2],[1]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[3,1,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,6}
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 2
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 2
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[3],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[3],[3],[1]]
=> [3,3,1]
=> [3,1]
=> [1]
=> 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [3,1]
=> [1]
=> 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
Description
The total number of irreducible representations contained in the higher Lie character for an integer partition.
The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001128The exponens consonantiae of a partition. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra.
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