Your data matches 51 different statistics following compositions of up to 3 maps.
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Matching statistic: St000836
(load all 2 compositions to match this statistic)
Mapping
St000836: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 1
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 2
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 0
[3,2,1,4] => 1
[3,2,4,1] => 1
[3,4,1,2] => 2
[3,4,2,1] => 2
[4,1,2,3] => 1
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 2
[4,3,1,2] => 2
[4,3,2,1] => 2
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 1
[1,2,5,3,4] => 1
[1,2,5,4,3] => 1
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 2
[1,3,5,4,2] => 1
[1,4,2,3,5] => 1
[1,4,2,5,3] => 0
[1,4,3,2,5] => 1
[1,4,3,5,2] => 1
[1,4,5,2,3] => 2
[1,4,5,3,2] => 2
Description
The number of descents of distance 2 of a permutation. This is, $\operatorname{des}_2(\pi) = | \{ i : \pi(i) > \pi(i+2) \} |$.
Matching statistic: St000837
(load all 2 compositions to match this statistic)
Mapping
St000837: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0
[2,1] => 0
[1,2,3] => 1
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 2
[1,2,4,3] => 2
[1,3,2,4] => 2
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 1
[2,1,3,4] => 2
[2,1,4,3] => 2
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 0
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 2
[3,2,1,4] => 1
[3,2,4,1] => 1
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 1
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 3
[1,2,3,5,4] => 3
[1,2,4,3,5] => 3
[1,2,4,5,3] => 2
[1,2,5,3,4] => 2
[1,2,5,4,3] => 2
[1,3,2,4,5] => 3
[1,3,2,5,4] => 3
[1,3,4,2,5] => 2
[1,3,4,5,2] => 2
[1,3,5,2,4] => 1
[1,3,5,4,2] => 2
[1,4,2,3,5] => 2
[1,4,2,5,3] => 3
[1,4,3,2,5] => 2
[1,4,3,5,2] => 2
[1,4,5,2,3] => 1
[1,4,5,3,2] => 1
Description
The number of ascents of distance 2 of a permutation. This is, $\operatorname{asc}_2(\pi) = | \{ i : \pi(i) < \pi(i+2) \} |$.
Matching statistic: St000937
(load all 4 compositions to match this statistic)
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000937: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 66%●distinct values known / distinct values provided: 60%
Values
[1,2] => [1,2] => [1,1]
 => [1]
 => ? ∊ {0,0}
[2,1] => [2,1] => [2]
 => []
 => ? ∊ {0,0}
[1,2,3] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,1,1,1}
[1,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,1,1,1}
[2,1,3] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,1,1,1}
[2,3,1] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,1,1,1}
[3,1,2] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,1,1,1}
[3,2,1] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,1,1,1}
[1,2,3,4] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,2,4,3] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,3,2,4] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,3,4,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,4,2,3] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,4,3,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[2,1,3,4] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,3,1,4] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[2,3,4,1] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[2,4,1,3] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[2,4,3,1] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[3,1,2,4] => [3,1,4,2] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[3,1,4,2] => [3,1,4,2] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[3,2,1,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,2,4,1] => [3,2,4,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[3,4,1,2] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[3,4,2,1] => [3,4,2,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[4,1,2,3] => [4,1,3,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[4,1,3,2] => [4,1,3,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[4,2,1,3] => [4,2,1,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[4,2,3,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,3,1,2] => [4,3,1,2] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[4,3,2,1] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[1,2,3,4,5] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,2,3,5,4] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,2,4,3,5] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,2,4,5,3] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,2,5,3,4] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,2,5,4,3] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,3,2,4,5] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,3,2,5,4] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,3,4,2,5] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,3,4,5,2] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,3,5,2,4] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,3,5,4,2] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,4,2,3,5] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,4,2,5,3] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,4,3,2,5] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,4,3,5,2] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,4,5,2,3] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,4,5,3,2] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,5,2,3,4] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,5,2,4,3] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,5,3,2,4] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,5,3,4,2] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,5,4,2,3] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,5,4,3,2] => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[2,1,3,4,5] => [2,1,5,4,3] => [2,2,1]
 => [2,1]
 => 1
[2,1,3,5,4] => [2,1,5,4,3] => [2,2,1]
 => [2,1]
 => 1
[2,1,4,3,5] => [2,1,5,4,3] => [2,2,1]
 => [2,1]
 => 1
[2,1,4,5,3] => [2,1,5,4,3] => [2,2,1]
 => [2,1]
 => 1
[2,1,5,3,4] => [2,1,5,4,3] => [2,2,1]
 => [2,1]
 => 1
[2,1,5,4,3] => [2,1,5,4,3] => [2,2,1]
 => [2,1]
 => 1
[2,3,1,4,5] => [2,5,1,4,3] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,3,1,5,4] => [2,5,1,4,3] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,3,4,1,5] => [2,5,4,1,3] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,3,4,5,1] => [2,5,4,3,1] => [3,2]
 => [2]
 => 2
[2,3,5,1,4] => [2,5,4,1,3] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,3,5,4,1] => [2,5,4,3,1] => [3,2]
 => [2]
 => 2
[2,4,1,3,5] => [2,5,1,4,3] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,4,1,5,3] => [2,5,1,4,3] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,4,3,1,5] => [2,5,4,1,3] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,4,3,5,1] => [2,5,4,3,1] => [3,2]
 => [2]
 => 2
[2,4,5,1,3] => [2,5,4,1,3] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,4,5,3,1] => [2,5,4,3,1] => [3,2]
 => [2]
 => 2
[2,5,1,3,4] => [2,5,1,4,3] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,5,1,4,3] => [2,5,1,4,3] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,5,3,1,4] => [2,5,4,1,3] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,5,3,4,1] => [2,5,4,3,1] => [3,2]
 => [2]
 => 2
[2,5,4,1,3] => [2,5,4,1,3] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,5,4,3,1] => [2,5,4,3,1] => [3,2]
 => [2]
 => 2
[3,1,2,4,5] => [3,1,5,4,2] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,1,2,5,4] => [3,1,5,4,2] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,1,4,2,5] => [3,1,5,4,2] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,1,4,5,2] => [3,1,5,4,2] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,1,5,2,4] => [3,1,5,4,2] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,1,5,4,2] => [3,1,5,4,2] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,2,1,4,5] => [3,2,1,5,4] => [2,2,1]
 => [2,1]
 => 1
[3,2,1,5,4] => [3,2,1,5,4] => [2,2,1]
 => [2,1]
 => 1
[3,2,4,1,5] => [3,2,5,1,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,2,5,1,4] => [3,2,5,1,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,4,2,1,5] => [3,5,2,1,4] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,4,2,5,1] => [3,5,2,4,1] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,4,5,2,1] => [3,5,4,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,5,2,1,4] => [3,5,2,1,4] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,5,2,4,1] => [3,5,2,4,1] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,5,4,2,1] => [3,5,4,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[4,1,2,3,5] => [4,1,5,3,2] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[4,1,2,5,3] => [4,1,5,3,2] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[4,1,3,2,5] => [4,1,5,3,2] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[4,1,3,5,2] => [4,1,5,3,2] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
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: St000260
(load all 13 compositions to match this statistic)
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 65%●distinct values known / distinct values provided: 40%
Values
[1,2] => [1,2] => [1,2] => ([],2)
 => ? ∊ {0,0}
[2,1] => [1,2] => [1,2] => ([],2)
 => ? ∊ {0,0}
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,1}
[1,3,2] => [1,2,3] => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,1}
[2,1,3] => [1,2,3] => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,1}
[2,3,1] => [1,2,3] => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,1}
[3,1,2] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[3,2,1] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1}
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1}
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1}
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1}
[1,4,2,3] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,4,3,2] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1}
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1}
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1}
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1}
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 1
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1}
[3,1,4,2] => [1,3,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 2
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1}
[3,2,4,1] => [1,3,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 2
[3,4,1,2] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1}
[3,4,2,1] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1}
[4,1,2,3] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,1,3,2] => [1,4,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[4,2,1,3] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,2,3,1] => [1,4,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[4,3,1,2] => [1,4,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[4,3,2,1] => [1,4,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,2,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,2,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,3,5,2,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,3,5,4,2] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,4,2,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,4,2,5,3] => [1,2,4,5,3] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
[1,4,3,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,4,3,5,2] => [1,2,4,5,3] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
[1,4,5,2,3] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,4,5,3,2] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,5,2,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,5,2,4,3] => [1,2,5,3,4] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[1,5,3,2,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,5,3,4,2] => [1,2,5,3,4] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[1,5,4,2,3] => [1,2,5,3,4] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[1,5,4,3,2] => [1,2,5,3,4] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,1,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[2,1,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,3,5,1,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[2,4,1,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,4,1,5,3] => [1,2,4,5,3] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
[2,4,3,1,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,4,3,5,1] => [1,2,4,5,3] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
[2,4,5,1,3] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,4,5,3,1] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,5,1,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[2,5,1,4,3] => [1,2,5,3,4] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[2,5,3,1,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[2,5,3,4,1] => [1,2,5,3,4] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[2,5,4,1,3] => [1,2,5,3,4] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[2,5,4,3,1] => [1,2,5,3,4] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[3,1,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,1,2,5,4] => [1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,1,4,2,5] => [1,3,4,2,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,1,4,5,2] => [1,3,4,5,2] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
[3,1,5,2,4] => [1,3,5,4,2] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[3,1,5,4,2] => [1,3,5,2,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
[3,2,1,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,2,1,5,4] => [1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,2,4,1,5] => [1,3,4,2,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,2,4,5,1] => [1,3,4,5,2] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
[3,2,5,1,4] => [1,3,5,4,2] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[3,2,5,4,1] => [1,3,5,2,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
[3,4,1,2,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,4,1,5,2] => [1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,4,5,1,2] => [1,3,5,2,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
[3,4,5,2,1] => [1,3,5,2,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
[3,5,1,2,4] => [1,3,2,5,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
[3,5,1,4,2] => [1,3,2,5,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
[3,5,2,1,4] => [1,3,2,5,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
[3,5,2,4,1] => [1,3,2,5,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
Description
The radius of a connected graph. This is the minimum eccentricity of any vertex.
Matching statistic: St000668
(load all 4 compositions to match this statistic)
Mapping
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 61%●distinct values known / distinct values provided: 60%
Values
[1,2] => [1,1]
 => [1]
 => ? ∊ {0,0}
[2,1] => [2]
 => []
 => ? ∊ {0,0}
[1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,1,1}
[2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,1,1}
[2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,1,1}
[3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,1,1}
[3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,1,1}
[1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,1,4,3] => [2,2]
 => [2]
 => 2
[2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[3,1,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[3,1,4,2] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,2,4,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[3,4,1,2] => [2,2]
 => [2]
 => 2
[3,4,2,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[4,1,2,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[4,1,3,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[4,2,1,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,3,1,2] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[4,3,2,1] => [2,2]
 => [2]
 => 2
[1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
[1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,2,4,5,3] => [3,1,1]
 => [1,1]
 => 1
[1,2,5,3,4] => [3,1,1]
 => [1,1]
 => 1
[1,2,5,4,3] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,3,2,5,4] => [2,2,1]
 => [2,1]
 => 2
[1,3,4,2,5] => [3,1,1]
 => [1,1]
 => 1
[1,3,4,5,2] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,3,5,2,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 1
[1,4,2,3,5] => [3,1,1]
 => [1,1]
 => 1
[1,4,2,5,3] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,4,3,2,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,4,3,5,2] => [3,1,1]
 => [1,1]
 => 1
[1,4,5,2,3] => [2,2,1]
 => [2,1]
 => 2
[1,4,5,3,2] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,5,2,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,5,2,4,3] => [3,1,1]
 => [1,1]
 => 1
[1,5,3,2,4] => [3,1,1]
 => [1,1]
 => 1
[1,5,3,4,2] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,5,4,2,3] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 2
[2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[2,1,3,5,4] => [2,2,1]
 => [2,1]
 => 2
[2,1,4,3,5] => [2,2,1]
 => [2,1]
 => 2
[2,1,4,5,3] => [3,2]
 => [2]
 => 2
[2,1,5,3,4] => [3,2]
 => [2]
 => 2
[2,1,5,4,3] => [2,2,1]
 => [2,1]
 => 2
[2,3,1,4,5] => [3,1,1]
 => [1,1]
 => 1
[2,3,1,5,4] => [3,2]
 => [2]
 => 2
[2,3,4,1,5] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,3,4,5,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,3,5,1,4] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,3,5,4,1] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,4,1,3,5] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,4,1,5,3] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,4,3,1,5] => [3,1,1]
 => [1,1]
 => 1
[2,4,3,5,1] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,4,5,1,3] => [3,2]
 => [2]
 => 2
[2,4,5,3,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,5,1,3,4] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,5,1,4,3] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,5,3,1,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,5,3,4,1] => [3,1,1]
 => [1,1]
 => 1
[2,5,4,1,3] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,5,4,3,1] => [3,2]
 => [2]
 => 2
[3,1,2,4,5] => [3,1,1]
 => [1,1]
 => 1
[3,1,2,5,4] => [3,2]
 => [2]
 => 2
[3,1,4,2,5] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,1,4,5,2] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,1,5,2,4] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,1,5,4,2] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,2,1,4,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[3,2,1,5,4] => [2,2,1]
 => [2,1]
 => 2
[3,2,4,1,5] => [3,1,1]
 => [1,1]
 => 1
[3,2,4,5,1] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,2,5,1,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,2,5,4,1] => [3,1,1]
 => [1,1]
 => 1
[3,4,1,2,5] => [2,2,1]
 => [2,1]
 => 2
[3,4,1,5,2] => [3,2]
 => [2]
 => 2
[3,4,2,1,5] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,4,2,5,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,4,5,1,2] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,4,5,2,1] => [3,2]
 => [2]
 => 2
[3,5,2,1,4] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,5,2,4,1] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
Description
The least common multiple of the parts of the partition.
Matching statistic: St000708
(load all 4 compositions to match this statistic)
Mapping
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 80%
Values
[1,2] => [1,1]
 => [1]
 => ? ∊ {0,0}
[2,1] => [2]
 => []
 => ? ∊ {0,0}
[1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,1,1}
[2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,1,1}
[2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,1,1}
[3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,1,1}
[3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,1,1}
[1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,1,4,3] => [2,2]
 => [2]
 => 2
[2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[3,1,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[3,1,4,2] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,2,4,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[3,4,1,2] => [2,2]
 => [2]
 => 2
[3,4,2,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[4,1,2,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[4,1,3,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[4,2,1,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,3,1,2] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[4,3,2,1] => [2,2]
 => [2]
 => 2
[1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
[1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,2,4,5,3] => [3,1,1]
 => [1,1]
 => 1
[1,2,5,3,4] => [3,1,1]
 => [1,1]
 => 1
[1,2,5,4,3] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,3,2,5,4] => [2,2,1]
 => [2,1]
 => 2
[1,3,4,2,5] => [3,1,1]
 => [1,1]
 => 1
[1,3,4,5,2] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,3,5,2,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 1
[1,4,2,3,5] => [3,1,1]
 => [1,1]
 => 1
[1,4,2,5,3] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,4,3,2,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,4,3,5,2] => [3,1,1]
 => [1,1]
 => 1
[1,4,5,2,3] => [2,2,1]
 => [2,1]
 => 2
[1,4,5,3,2] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,5,2,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,5,2,4,3] => [3,1,1]
 => [1,1]
 => 1
[1,5,3,2,4] => [3,1,1]
 => [1,1]
 => 1
[1,5,3,4,2] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,5,4,2,3] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 2
[2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[2,1,3,5,4] => [2,2,1]
 => [2,1]
 => 2
[2,1,4,3,5] => [2,2,1]
 => [2,1]
 => 2
[2,1,4,5,3] => [3,2]
 => [2]
 => 2
[2,1,5,3,4] => [3,2]
 => [2]
 => 2
[2,1,5,4,3] => [2,2,1]
 => [2,1]
 => 2
[2,3,1,4,5] => [3,1,1]
 => [1,1]
 => 1
[2,3,1,5,4] => [3,2]
 => [2]
 => 2
[2,3,4,1,5] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,3,4,5,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,3,5,1,4] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,3,5,4,1] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,4,1,3,5] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,4,1,5,3] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,4,3,1,5] => [3,1,1]
 => [1,1]
 => 1
[2,4,3,5,1] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,4,5,1,3] => [3,2]
 => [2]
 => 2
[2,4,5,3,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,5,1,3,4] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,5,1,4,3] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,5,3,1,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,5,3,4,1] => [3,1,1]
 => [1,1]
 => 1
[2,5,4,1,3] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,5,4,3,1] => [3,2]
 => [2]
 => 2
[3,1,2,4,5] => [3,1,1]
 => [1,1]
 => 1
[3,1,2,5,4] => [3,2]
 => [2]
 => 2
[3,1,4,2,5] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,1,4,5,2] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,1,5,2,4] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,1,5,4,2] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,2,1,4,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[3,2,1,5,4] => [2,2,1]
 => [2,1]
 => 2
[3,2,4,1,5] => [3,1,1]
 => [1,1]
 => 1
[3,2,4,5,1] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,2,5,1,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,2,5,4,1] => [3,1,1]
 => [1,1]
 => 1
[3,4,1,2,5] => [2,2,1]
 => [2,1]
 => 2
[3,4,1,5,2] => [3,2]
 => [2]
 => 2
[3,4,2,1,5] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,4,2,5,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,4,5,1,2] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,4,5,2,1] => [3,2]
 => [2]
 => 2
[3,5,2,1,4] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,5,2,4,1] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
Description
The product of the parts of an integer partition.
Matching statistic: St000933
(load all 4 compositions to match this statistic)
Mapping
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000933: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 80%
Values
[1,2] => [1,1]
 => [1]
 => ? ∊ {0,0}
[2,1] => [2]
 => []
 => ? ∊ {0,0}
[1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,1,1}
[2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,1,1}
[2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,1,1}
[3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,1,1}
[3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,1,1}
[1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,1,4,3] => [2,2]
 => [2]
 => 2
[2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[3,1,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[3,1,4,2] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,2,4,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[3,4,1,2] => [2,2]
 => [2]
 => 2
[3,4,2,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[4,1,2,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[4,1,3,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[4,2,1,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,3,1,2] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[4,3,2,1] => [2,2]
 => [2]
 => 2
[1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
[1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,2,4,5,3] => [3,1,1]
 => [1,1]
 => 1
[1,2,5,3,4] => [3,1,1]
 => [1,1]
 => 1
[1,2,5,4,3] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,3,2,5,4] => [2,2,1]
 => [2,1]
 => 2
[1,3,4,2,5] => [3,1,1]
 => [1,1]
 => 1
[1,3,4,5,2] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,3,5,2,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 1
[1,4,2,3,5] => [3,1,1]
 => [1,1]
 => 1
[1,4,2,5,3] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,4,3,2,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,4,3,5,2] => [3,1,1]
 => [1,1]
 => 1
[1,4,5,2,3] => [2,2,1]
 => [2,1]
 => 2
[1,4,5,3,2] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,5,2,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,5,2,4,3] => [3,1,1]
 => [1,1]
 => 1
[1,5,3,2,4] => [3,1,1]
 => [1,1]
 => 1
[1,5,3,4,2] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,5,4,2,3] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 2
[2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[2,1,3,5,4] => [2,2,1]
 => [2,1]
 => 2
[2,1,4,3,5] => [2,2,1]
 => [2,1]
 => 2
[2,1,4,5,3] => [3,2]
 => [2]
 => 2
[2,1,5,3,4] => [3,2]
 => [2]
 => 2
[2,1,5,4,3] => [2,2,1]
 => [2,1]
 => 2
[2,3,1,4,5] => [3,1,1]
 => [1,1]
 => 1
[2,3,1,5,4] => [3,2]
 => [2]
 => 2
[2,3,4,1,5] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,3,4,5,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,3,5,1,4] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,3,5,4,1] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,4,1,3,5] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,4,1,5,3] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,4,3,1,5] => [3,1,1]
 => [1,1]
 => 1
[2,4,3,5,1] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,4,5,1,3] => [3,2]
 => [2]
 => 2
[2,4,5,3,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,5,1,3,4] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,5,1,4,3] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,5,3,1,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,5,3,4,1] => [3,1,1]
 => [1,1]
 => 1
[2,5,4,1,3] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,5,4,3,1] => [3,2]
 => [2]
 => 2
[3,1,2,4,5] => [3,1,1]
 => [1,1]
 => 1
[3,1,2,5,4] => [3,2]
 => [2]
 => 2
[3,1,4,2,5] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,1,4,5,2] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,1,5,2,4] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,1,5,4,2] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,2,1,4,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[3,2,1,5,4] => [2,2,1]
 => [2,1]
 => 2
[3,2,4,1,5] => [3,1,1]
 => [1,1]
 => 1
[3,2,4,5,1] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,2,5,1,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,2,5,4,1] => [3,1,1]
 => [1,1]
 => 1
[3,4,1,2,5] => [2,2,1]
 => [2,1]
 => 2
[3,4,1,5,2] => [3,2]
 => [2]
 => 2
[3,4,2,1,5] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,4,2,5,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,4,5,1,2] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,4,5,2,1] => [3,2]
 => [2]
 => 2
[3,5,2,1,4] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,5,2,4,1] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
Description
The number of multipartitions of sizes given by an integer partition. This is, for $\lambda = (\lambda_1,\ldots,\lambda_n)$, this is the number of $n$-tuples $(\lambda^{(1)},\ldots,\lambda^{(n)})$ of partitions $\lambda^{(i)}$ such that $\lambda^{(i)} \vdash \lambda_i$.
Matching statistic: St000939
Mapping
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000939: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 61%●distinct values known / distinct values provided: 60%
Values
[1,2] => [1,1]
 => [1]
 => [1]
 => ? ∊ {0,0}
[2,1] => [2]
 => []
 => []
 => ? ∊ {0,0}
[1,2,3] => [1,1,1]
 => [1,1]
 => [2]
 => 1
[1,3,2] => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,1,1}
[2,1,3] => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,1,1}
[2,3,1] => [3]
 => []
 => []
 => ? ∊ {0,0,0,1,1}
[3,1,2] => [3]
 => []
 => []
 => ? ∊ {0,0,0,1,1}
[3,2,1] => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,1,1}
[1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[1,2,4,3] => [2,1,1]
 => [1,1]
 => [2]
 => 1
[1,3,2,4] => [2,1,1]
 => [1,1]
 => [2]
 => 1
[1,3,4,2] => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[1,4,2,3] => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[1,4,3,2] => [2,1,1]
 => [1,1]
 => [2]
 => 1
[2,1,3,4] => [2,1,1]
 => [1,1]
 => [2]
 => 1
[2,1,4,3] => [2,2]
 => [2]
 => [1,1]
 => 2
[2,3,1,4] => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[2,3,4,1] => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[2,4,1,3] => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[2,4,3,1] => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[3,1,2,4] => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[3,1,4,2] => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[3,2,1,4] => [2,1,1]
 => [1,1]
 => [2]
 => 1
[3,2,4,1] => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[3,4,1,2] => [2,2]
 => [2]
 => [1,1]
 => 2
[3,4,2,1] => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[4,1,2,3] => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[4,1,3,2] => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[4,2,1,3] => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[4,2,3,1] => [2,1,1]
 => [1,1]
 => [2]
 => 1
[4,3,1,2] => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[4,3,2,1] => [2,2]
 => [2]
 => [1,1]
 => 2
[1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => [3,1]
 => 2
[1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[1,2,4,5,3] => [3,1,1]
 => [1,1]
 => [2]
 => 1
[1,2,5,3,4] => [3,1,1]
 => [1,1]
 => [2]
 => 1
[1,2,5,4,3] => [2,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[1,3,2,5,4] => [2,2,1]
 => [2,1]
 => [3]
 => 2
[1,3,4,2,5] => [3,1,1]
 => [1,1]
 => [2]
 => 1
[1,3,4,5,2] => [4,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,3,5,2,4] => [4,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,3,5,4,2] => [3,1,1]
 => [1,1]
 => [2]
 => 1
[1,4,2,3,5] => [3,1,1]
 => [1,1]
 => [2]
 => 1
[1,4,2,5,3] => [4,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,4,3,2,5] => [2,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[1,4,3,5,2] => [3,1,1]
 => [1,1]
 => [2]
 => 1
[1,4,5,2,3] => [2,2,1]
 => [2,1]
 => [3]
 => 2
[1,4,5,3,2] => [4,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,5,2,3,4] => [4,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,5,2,4,3] => [3,1,1]
 => [1,1]
 => [2]
 => 1
[1,5,3,2,4] => [3,1,1]
 => [1,1]
 => [2]
 => 1
[1,5,3,4,2] => [2,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[1,5,4,2,3] => [4,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,5,4,3,2] => [2,2,1]
 => [2,1]
 => [3]
 => 2
[2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[2,1,3,5,4] => [2,2,1]
 => [2,1]
 => [3]
 => 2
[2,1,4,3,5] => [2,2,1]
 => [2,1]
 => [3]
 => 2
[2,1,4,5,3] => [3,2]
 => [2]
 => [1,1]
 => 2
[2,1,5,3,4] => [3,2]
 => [2]
 => [1,1]
 => 2
[2,1,5,4,3] => [2,2,1]
 => [2,1]
 => [3]
 => 2
[2,3,1,4,5] => [3,1,1]
 => [1,1]
 => [2]
 => 1
[2,3,1,5,4] => [3,2]
 => [2]
 => [1,1]
 => 2
[2,3,4,1,5] => [4,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,3,4,5,1] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,3,5,1,4] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,3,5,4,1] => [4,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,4,1,3,5] => [4,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,4,1,5,3] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,4,3,1,5] => [3,1,1]
 => [1,1]
 => [2]
 => 1
[2,4,3,5,1] => [4,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,4,5,1,3] => [3,2]
 => [2]
 => [1,1]
 => 2
[2,4,5,3,1] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,5,1,3,4] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,5,1,4,3] => [4,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,5,3,1,4] => [4,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,5,3,4,1] => [3,1,1]
 => [1,1]
 => [2]
 => 1
[2,5,4,1,3] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,5,4,3,1] => [3,2]
 => [2]
 => [1,1]
 => 2
[3,1,2,4,5] => [3,1,1]
 => [1,1]
 => [2]
 => 1
[3,1,2,5,4] => [3,2]
 => [2]
 => [1,1]
 => 2
[3,1,4,2,5] => [4,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,1,4,5,2] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,1,5,2,4] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,1,5,4,2] => [4,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,2,1,4,5] => [2,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[3,2,1,5,4] => [2,2,1]
 => [2,1]
 => [3]
 => 2
[3,2,4,1,5] => [3,1,1]
 => [1,1]
 => [2]
 => 1
[3,2,4,5,1] => [4,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,2,5,1,4] => [4,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,2,5,4,1] => [3,1,1]
 => [1,1]
 => [2]
 => 1
[3,4,1,2,5] => [2,2,1]
 => [2,1]
 => [3]
 => 2
[3,4,1,5,2] => [3,2]
 => [2]
 => [1,1]
 => 2
[3,4,2,1,5] => [4,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,4,2,5,1] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,4,5,1,2] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,4,5,2,1] => [3,2]
 => [2]
 => [1,1]
 => 2
[3,5,2,1,4] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,5,2,4,1] => [4,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
Description
The number of characters of the symmetric group whose value on the partition is positive.
Matching statistic: St001039
Mapping
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 61%●distinct values known / distinct values provided: 60%
Values
[1,2] => [1,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0}
[2,1] => [2]
 => []
 => []
 => ? ∊ {0,0}
[1,2,3] => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[1,3,2] => [2,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,1,1}
[2,1,3] => [2,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,1,1}
[2,3,1] => [3]
 => []
 => []
 => ? ∊ {0,0,0,1,1}
[3,1,2] => [3]
 => []
 => []
 => ? ∊ {0,0,0,1,1}
[3,2,1] => [2,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,1,1}
[1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,2,4,3] => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[1,3,2,4] => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[1,3,4,2] => [3,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[1,4,2,3] => [3,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[1,4,3,2] => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[2,1,3,4] => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[2,1,4,3] => [2,2]
 => [2]
 => [1,0,1,0]
 => 2
[2,3,1,4] => [3,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[2,3,4,1] => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[2,4,1,3] => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[2,4,3,1] => [3,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[3,1,2,4] => [3,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[3,1,4,2] => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[3,2,1,4] => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[3,2,4,1] => [3,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[3,4,1,2] => [2,2]
 => [2]
 => [1,0,1,0]
 => 2
[3,4,2,1] => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[4,1,2,3] => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[4,1,3,2] => [3,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[4,2,1,3] => [3,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[4,2,3,1] => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[4,3,1,2] => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2,2,2}
[4,3,2,1] => [2,2]
 => [2]
 => [1,0,1,0]
 => 2
[1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,2,4,5,3] => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[1,2,5,3,4] => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[1,2,5,4,3] => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,3,2,5,4] => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[1,3,4,2,5] => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[1,3,4,5,2] => [4,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,3,5,2,4] => [4,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,3,5,4,2] => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[1,4,2,3,5] => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[1,4,2,5,3] => [4,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,4,3,2,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,4,3,5,2] => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[1,4,5,2,3] => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[1,4,5,3,2] => [4,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,5,2,3,4] => [4,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,5,2,4,3] => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[1,5,3,2,4] => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[1,5,3,4,2] => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,5,4,2,3] => [4,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,5,4,3,2] => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[2,1,3,5,4] => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[2,1,4,3,5] => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[2,1,4,5,3] => [3,2]
 => [2]
 => [1,0,1,0]
 => 2
[2,1,5,3,4] => [3,2]
 => [2]
 => [1,0,1,0]
 => 2
[2,1,5,4,3] => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[2,3,1,4,5] => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[2,3,1,5,4] => [3,2]
 => [2]
 => [1,0,1,0]
 => 2
[2,3,4,1,5] => [4,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,3,4,5,1] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,3,5,1,4] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,3,5,4,1] => [4,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,4,1,3,5] => [4,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,4,1,5,3] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,4,3,1,5] => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[2,4,3,5,1] => [4,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,4,5,1,3] => [3,2]
 => [2]
 => [1,0,1,0]
 => 2
[2,4,5,3,1] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,5,1,3,4] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,5,1,4,3] => [4,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,5,3,1,4] => [4,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,5,3,4,1] => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[2,5,4,1,3] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,5,4,3,1] => [3,2]
 => [2]
 => [1,0,1,0]
 => 2
[3,1,2,4,5] => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[3,1,2,5,4] => [3,2]
 => [2]
 => [1,0,1,0]
 => 2
[3,1,4,2,5] => [4,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,1,4,5,2] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,1,5,2,4] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,1,5,4,2] => [4,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,2,1,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[3,2,1,5,4] => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[3,2,4,1,5] => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[3,2,4,5,1] => [4,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,2,5,1,4] => [4,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,2,5,4,1] => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[3,4,1,2,5] => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[3,4,1,5,2] => [3,2]
 => [2]
 => [1,0,1,0]
 => 2
[3,4,2,1,5] => [4,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,4,2,5,1] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,4,5,1,2] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,4,5,2,1] => [3,2]
 => [2]
 => [1,0,1,0]
 => 2
[3,5,2,1,4] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,5,2,4,1] => [4,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Matching statistic: St000259
(load all 3 compositions to match this statistic)
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00264: Graphs —delete endpoints⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 60%
Values
[1,2] => [2] => ([],2)
 => ([],2)
 => ? = 0
[2,1] => [1,1] => ([(0,1)],2)
 => ([],1)
 => 0
[1,2,3] => [3] => ([],3)
 => ([],3)
 => ? ∊ {0,1,1}
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => ([],1)
 => 0
[2,1,3] => [1,2] => ([(1,2)],3)
 => ([],2)
 => ? ∊ {0,1,1}
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ([],1)
 => 0
[3,1,2] => [1,2] => ([(1,2)],3)
 => ([],2)
 => ? ∊ {0,1,1}
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,2,3,4] => [4] => ([],4)
 => ([],4)
 => ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2}
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => 0
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => ([],2)
 => ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2}
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => 0
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ([],2)
 => ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2}
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,1,3,4] => [1,3] => ([(2,3)],4)
 => ([],3)
 => ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2}
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ([],2)
 => ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2}
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => 0
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ([],2)
 => ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2}
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,1,2,4] => [1,3] => ([(2,3)],4)
 => ([],3)
 => ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2}
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2}
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ([],2)
 => ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2}
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,1,2,3] => [1,3] => ([(2,3)],4)
 => ([],3)
 => ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2}
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2}
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2}
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,2,3,4,5] => [5] => ([],5)
 => ([],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 0
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([],2)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 0
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([],2)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => ([],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([],2)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 0
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([],2)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => ([],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([],2)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => ([],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,5,3,2,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,5,3,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,5,4,2,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,1,3,4,5] => [1,4] => ([(3,4)],5)
 => ([],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,1,3,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[2,1,4,3,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,1,4,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[2,1,5,3,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,1,5,4,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,3,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => ([],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,3,4,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([],2)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 0
[2,3,5,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([],2)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => ([],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,4,1,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,4,3,1,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,4,3,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,4,5,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([],2)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => ([],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,5,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,5,3,1,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,5,3,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,5,4,1,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,5,4,3,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,1,2,4,5] => [1,4] => ([(3,4)],5)
 => ([],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,1,2,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[3,1,4,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,1,4,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[3,1,5,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,1,5,4,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,2,1,4,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,2,1,5,4] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[3,2,4,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,2,4,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[3,2,5,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,2,5,4,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
 => ([],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,4,1,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,4,2,1,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,4,2,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,4,5,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([],2)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,5,1,2,4] => [2,3] => ([(2,4),(3,4)],5)
 => ([],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,5,1,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
Description
The diameter of a connected graph. This is the greatest distance between any pair of vertices.
The following 41 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000707The product of the factorials of the parts. 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. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000454The largest eigenvalue of a graph if it is integral. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. 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. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001060The distinguishing index of a graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000741The Colin de Verdière graph invariant. St000455The second largest eigenvalue of a graph if it is integral. St001877Number of indecomposable injective modules with projective dimension 2. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. 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. St001128The exponens consonantiae of a partition. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000264The girth of a graph, which is not a tree. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001624The breadth of a lattice. St001875The number of simple modules with projective dimension at most 1. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000422The energy of a graph, if it is integral. St000077The number of boxed and circled entries. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree.