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Your data matches 82 different statistics following compositions of up to 3 maps.
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Matching statistic: St001801
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St001801: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 0
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 1
[2,3,4,1] => 2
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 1
[3,2,1,4] => 0
[3,2,4,1] => 1
[3,4,1,2] => 0
[3,4,2,1] => 1
[4,1,2,3] => 2
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 1
[4,3,1,2] => 1
[4,3,2,1] => 2
[1,2,3,4,5] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 1
[1,2,5,3,4] => 1
[1,2,5,4,3] => 0
[1,3,2,4,5] => 1
[1,3,2,5,4] => 2
[1,3,4,2,5] => 1
[1,3,4,5,2] => 2
[1,3,5,2,4] => 1
[1,3,5,4,2] => 1
[1,4,2,3,5] => 1
[1,4,2,5,3] => 1
[1,4,3,2,5] => 0
[1,4,3,5,2] => 1
[1,4,5,2,3] => 0
Description
Half the number of preimage-image pairs of different parity in a permutation.
Matching statistic: St001153
(load all 48 compositions to match this statistic)
(load all 48 compositions to match this statistic)
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St001153: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001153: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => {{1}}
=> 0
[1,2] => {{1},{2}}
=> 1
[2,1] => {{1,2}}
=> 0
[1,2,3] => {{1},{2},{3}}
=> 1
[1,3,2] => {{1},{2,3}}
=> 1
[2,1,3] => {{1,2},{3}}
=> 0
[2,3,1] => {{1,2,3}}
=> 0
[3,1,2] => {{1,3},{2}}
=> 1
[3,2,1] => {{1,3},{2}}
=> 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> 2
[1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,3,2,4] => {{1},{2,3},{4}}
=> 2
[1,3,4,2] => {{1},{2,3,4}}
=> 1
[1,4,2,3] => {{1},{2,4},{3}}
=> 1
[1,4,3,2] => {{1},{2,4},{3}}
=> 1
[2,1,3,4] => {{1,2},{3},{4}}
=> 1
[2,1,4,3] => {{1,2},{3,4}}
=> 0
[2,3,1,4] => {{1,2,3},{4}}
=> 1
[2,3,4,1] => {{1,2,3,4}}
=> 0
[2,4,1,3] => {{1,2,4},{3}}
=> 0
[2,4,3,1] => {{1,2,4},{3}}
=> 0
[3,1,2,4] => {{1,3},{2},{4}}
=> 2
[3,1,4,2] => {{1,3,4},{2}}
=> 1
[3,2,1,4] => {{1,3},{2},{4}}
=> 2
[3,2,4,1] => {{1,3,4},{2}}
=> 1
[3,4,1,2] => {{1,3},{2,4}}
=> 1
[3,4,2,1] => {{1,3},{2,4}}
=> 1
[4,1,2,3] => {{1,4},{2},{3}}
=> 1
[4,1,3,2] => {{1,4},{2},{3}}
=> 1
[4,2,1,3] => {{1,4},{2},{3}}
=> 1
[4,2,3,1] => {{1,4},{2},{3}}
=> 1
[4,3,1,2] => {{1,4},{2,3}}
=> 1
[4,3,2,1] => {{1,4},{2,3}}
=> 1
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 2
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 2
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 1
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> 2
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 2
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 2
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 1
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> 1
[1,3,5,2,4] => {{1},{2,3,5},{4}}
=> 2
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 2
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> 1
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> 1
Description
The number of blocks with even minimum in a set partition.
Matching statistic: St001199
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 82%●distinct values known / distinct values provided: 75%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 82%●distinct values known / distinct values provided: 75%
Values
[1] => [1]
=> [1,0]
=> [1,0]
=> ? = 0
[1,2] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[2,1] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? = 0
[1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {0,0}
[3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {0,0}
[3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[2,3,1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,3,4,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,1,2}
[2,4,1,3] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,1,2}
[2,4,3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1,4,2] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,1,2}
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[3,2,4,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,4,1,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[3,4,2,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,1,2}
[4,1,2,3] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,1,2}
[4,1,3,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[4,2,1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[4,3,1,2] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,1,2}
[4,3,2,1] => [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,3,4,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,3,4,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,2,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,4,2,5,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,4,3,2,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,4,5,3,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,5,2,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,5,4,2,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,5,4,3,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,1,3,4,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,1,4,3,5] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,3,4,5,1] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2}
[2,3,5,1,4] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2}
[2,4,1,5,3] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2}
[2,4,5,3,1] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2}
[2,5,1,3,4] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2}
[2,5,4,1,3] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2}
[3,1,4,5,2] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2}
[3,1,5,2,4] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2}
[3,4,2,5,1] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2}
[3,4,5,1,2] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2}
[3,5,2,1,4] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2}
[3,5,4,2,1] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2}
[4,1,2,5,3] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2}
[4,1,5,3,2] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2}
[4,3,1,5,2] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2}
[4,3,5,2,1] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2}
[4,5,1,2,3] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2}
[4,5,2,3,1] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2}
[5,1,2,3,4] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2}
[5,1,4,2,3] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2}
[5,3,1,2,4] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2}
[5,3,4,1,2] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2}
[5,4,1,3,2] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2}
[5,4,2,1,3] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2}
[2,3,4,5,6,1] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,4,6,1,5] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,5,1,6,4] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,5,6,4,1] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,6,1,4,5] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,6,5,1,4] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,1,5,6,3] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,1,6,3,5] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,5,3,6,1] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,5,6,1,3] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,6,3,1,5] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,6,5,3,1] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,5,1,3,6,4] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,5,1,6,4,3] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,5,4,1,6,3] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,5,4,6,3,1] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St000668
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 79%●distinct values known / distinct values provided: 50%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 79%●distinct values known / distinct values provided: 50%
Values
[1] => [] => []
=> ?
=> ? = 0
[1,2] => [1] => [1]
=> []
=> ? ∊ {0,1}
[2,1] => [1] => [1]
=> []
=> ? ∊ {0,1}
[1,2,3] => [1,2] => [2]
=> []
=> ? ∊ {0,0,1,1,1,1}
[1,3,2] => [1,2] => [2]
=> []
=> ? ∊ {0,0,1,1,1,1}
[2,1,3] => [2,1] => [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[2,3,1] => [2,1] => [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[3,1,2] => [1,2] => [2]
=> []
=> ? ∊ {0,0,1,1,1,1}
[3,2,1] => [2,1] => [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[1,2,3,4] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,2,4,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,3,2,4] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,3,4,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,4,2,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,4,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,1,3,4] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,1,4,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,3,1,4] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,3,4,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,4,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,4,3,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,1,2,4] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,1,4,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,2,1,4] => [3,2,1] => [1,1,1]
=> [1,1]
=> 1
[3,2,4,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 1
[3,4,1,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,4,2,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 1
[4,1,2,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,2,3,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,3,1,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,3,2,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 1
[1,2,3,4,5] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,3,5,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,3,5] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,5,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,3,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,4,5] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,5,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,2,5] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,3,5] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,4,3,5,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,4,5,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,5,2,3,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,4,5] => [2,1,3,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,4,3,5] => [2,1,4,3] => [2,2]
=> [2]
=> 2
[2,1,4,5,3] => [2,1,4,3] => [2,2]
=> [2]
=> 2
[2,1,5,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 2
[2,4,1,3,5] => [2,4,1,3] => [2,2]
=> [2]
=> 2
[2,4,1,5,3] => [2,4,1,3] => [2,2]
=> [2]
=> 2
[2,4,3,1,5] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 1
[2,4,3,5,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 1
[2,4,5,1,3] => [2,4,1,3] => [2,2]
=> [2]
=> 2
[2,4,5,3,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 1
[2,5,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 2
[2,5,4,1,3] => [2,4,1,3] => [2,2]
=> [2]
=> 2
[2,5,4,3,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 1
[3,1,4,2,5] => [3,1,4,2] => [2,2]
=> [2]
=> 2
[3,1,4,5,2] => [3,1,4,2] => [2,2]
=> [2]
=> 2
[3,1,5,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> 2
[3,2,1,4,5] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,2,1,5,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,2,4,1,5] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 1
[3,2,4,5,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 1
[3,2,5,1,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,2,5,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 1
[3,4,1,2,5] => [3,4,1,2] => [2,2]
=> [2]
=> 2
[3,4,1,5,2] => [3,4,1,2] => [2,2]
=> [2]
=> 2
[3,4,2,1,5] => [3,4,2,1] => [2,1,1]
=> [1,1]
=> 1
[3,4,2,5,1] => [3,4,2,1] => [2,1,1]
=> [1,1]
=> 1
[3,4,5,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 2
[3,4,5,2,1] => [3,4,2,1] => [2,1,1]
=> [1,1]
=> 1
[3,5,1,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> 2
[3,5,2,1,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,5,2,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 1
[3,5,4,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 2
[3,5,4,2,1] => [3,4,2,1] => [2,1,1]
=> [1,1]
=> 1
[4,1,3,2,5] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 1
[4,1,3,5,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 1
[4,1,5,3,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 1
[4,2,1,3,5] => [4,2,1,3] => [2,1,1]
=> [1,1]
=> 1
[4,2,1,5,3] => [4,2,1,3] => [2,1,1]
=> [1,1]
=> 1
[4,2,3,1,5] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[4,2,3,5,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[4,2,5,1,3] => [4,2,1,3] => [2,1,1]
=> [1,1]
=> 1
[4,2,5,3,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[4,3,1,2,5] => [4,3,1,2] => [2,1,1]
=> [1,1]
=> 1
Description
The least common multiple of the parts of the partition.
Matching statistic: St000707
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000707: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 79%●distinct values known / distinct values provided: 50%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000707: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 79%●distinct values known / distinct values provided: 50%
Values
[1] => [] => []
=> ?
=> ? = 0
[1,2] => [1] => [1]
=> []
=> ? ∊ {0,1}
[2,1] => [1] => [1]
=> []
=> ? ∊ {0,1}
[1,2,3] => [1,2] => [2]
=> []
=> ? ∊ {0,0,1,1,1,1}
[1,3,2] => [1,2] => [2]
=> []
=> ? ∊ {0,0,1,1,1,1}
[2,1,3] => [2,1] => [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[2,3,1] => [2,1] => [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[3,1,2] => [1,2] => [2]
=> []
=> ? ∊ {0,0,1,1,1,1}
[3,2,1] => [2,1] => [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[1,2,3,4] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,2,4,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,3,2,4] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,3,4,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,4,2,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,4,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,1,3,4] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,1,4,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,3,1,4] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,3,4,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,4,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,4,3,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,1,2,4] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,1,4,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,2,1,4] => [3,2,1] => [1,1,1]
=> [1,1]
=> 1
[3,2,4,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 1
[3,4,1,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,4,2,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 1
[4,1,2,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,2,3,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,3,1,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,3,2,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 1
[1,2,3,4,5] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,3,5,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,3,5] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,5,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,3,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,4,5] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,5,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,2,5] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,3,5] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,4,3,5,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,4,5,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,5,2,3,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,4,5] => [2,1,3,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,4,3,5] => [2,1,4,3] => [2,2]
=> [2]
=> 2
[2,1,4,5,3] => [2,1,4,3] => [2,2]
=> [2]
=> 2
[2,1,5,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 2
[2,4,1,3,5] => [2,4,1,3] => [2,2]
=> [2]
=> 2
[2,4,1,5,3] => [2,4,1,3] => [2,2]
=> [2]
=> 2
[2,4,3,1,5] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 1
[2,4,3,5,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 1
[2,4,5,1,3] => [2,4,1,3] => [2,2]
=> [2]
=> 2
[2,4,5,3,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 1
[2,5,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 2
[2,5,4,1,3] => [2,4,1,3] => [2,2]
=> [2]
=> 2
[2,5,4,3,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 1
[3,1,4,2,5] => [3,1,4,2] => [2,2]
=> [2]
=> 2
[3,1,4,5,2] => [3,1,4,2] => [2,2]
=> [2]
=> 2
[3,1,5,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> 2
[3,2,1,4,5] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,2,1,5,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,2,4,1,5] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 1
[3,2,4,5,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 1
[3,2,5,1,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,2,5,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 1
[3,4,1,2,5] => [3,4,1,2] => [2,2]
=> [2]
=> 2
[3,4,1,5,2] => [3,4,1,2] => [2,2]
=> [2]
=> 2
[3,4,2,1,5] => [3,4,2,1] => [2,1,1]
=> [1,1]
=> 1
[3,4,2,5,1] => [3,4,2,1] => [2,1,1]
=> [1,1]
=> 1
[3,4,5,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 2
[3,4,5,2,1] => [3,4,2,1] => [2,1,1]
=> [1,1]
=> 1
[3,5,1,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> 2
[3,5,2,1,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,5,2,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 1
[3,5,4,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 2
[3,5,4,2,1] => [3,4,2,1] => [2,1,1]
=> [1,1]
=> 1
[4,1,3,2,5] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 1
[4,1,3,5,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 1
[4,1,5,3,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 1
[4,2,1,3,5] => [4,2,1,3] => [2,1,1]
=> [1,1]
=> 1
[4,2,1,5,3] => [4,2,1,3] => [2,1,1]
=> [1,1]
=> 1
[4,2,3,1,5] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[4,2,3,5,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[4,2,5,1,3] => [4,2,1,3] => [2,1,1]
=> [1,1]
=> 1
[4,2,5,3,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[4,3,1,2,5] => [4,3,1,2] => [2,1,1]
=> [1,1]
=> 1
Description
The product of the factorials of the parts.
Matching statistic: St000708
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 79%●distinct values known / distinct values provided: 50%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 79%●distinct values known / distinct values provided: 50%
Values
[1] => [] => []
=> ?
=> ? = 0
[1,2] => [1] => [1]
=> []
=> ? ∊ {0,1}
[2,1] => [1] => [1]
=> []
=> ? ∊ {0,1}
[1,2,3] => [1,2] => [2]
=> []
=> ? ∊ {0,0,1,1,1,1}
[1,3,2] => [1,2] => [2]
=> []
=> ? ∊ {0,0,1,1,1,1}
[2,1,3] => [2,1] => [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[2,3,1] => [2,1] => [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[3,1,2] => [1,2] => [2]
=> []
=> ? ∊ {0,0,1,1,1,1}
[3,2,1] => [2,1] => [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[1,2,3,4] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,2,4,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,3,2,4] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,3,4,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,4,2,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,4,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,1,3,4] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,1,4,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,3,1,4] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,3,4,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,4,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,4,3,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,1,2,4] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,1,4,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,2,1,4] => [3,2,1] => [1,1,1]
=> [1,1]
=> 1
[3,2,4,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 1
[3,4,1,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,4,2,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 1
[4,1,2,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,2,3,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,3,1,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,3,2,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 1
[1,2,3,4,5] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,3,5,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,3,5] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,5,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,3,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,4,5] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,5,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,2,5] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,3,5] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,4,3,5,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,4,5,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,5,2,3,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,4,5] => [2,1,3,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,4,3,5] => [2,1,4,3] => [2,2]
=> [2]
=> 2
[2,1,4,5,3] => [2,1,4,3] => [2,2]
=> [2]
=> 2
[2,1,5,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 2
[2,4,1,3,5] => [2,4,1,3] => [2,2]
=> [2]
=> 2
[2,4,1,5,3] => [2,4,1,3] => [2,2]
=> [2]
=> 2
[2,4,3,1,5] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 1
[2,4,3,5,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 1
[2,4,5,1,3] => [2,4,1,3] => [2,2]
=> [2]
=> 2
[2,4,5,3,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 1
[2,5,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 2
[2,5,4,1,3] => [2,4,1,3] => [2,2]
=> [2]
=> 2
[2,5,4,3,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 1
[3,1,4,2,5] => [3,1,4,2] => [2,2]
=> [2]
=> 2
[3,1,4,5,2] => [3,1,4,2] => [2,2]
=> [2]
=> 2
[3,1,5,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> 2
[3,2,1,4,5] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,2,1,5,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,2,4,1,5] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 1
[3,2,4,5,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 1
[3,2,5,1,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,2,5,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 1
[3,4,1,2,5] => [3,4,1,2] => [2,2]
=> [2]
=> 2
[3,4,1,5,2] => [3,4,1,2] => [2,2]
=> [2]
=> 2
[3,4,2,1,5] => [3,4,2,1] => [2,1,1]
=> [1,1]
=> 1
[3,4,2,5,1] => [3,4,2,1] => [2,1,1]
=> [1,1]
=> 1
[3,4,5,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 2
[3,4,5,2,1] => [3,4,2,1] => [2,1,1]
=> [1,1]
=> 1
[3,5,1,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> 2
[3,5,2,1,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,5,2,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 1
[3,5,4,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 2
[3,5,4,2,1] => [3,4,2,1] => [2,1,1]
=> [1,1]
=> 1
[4,1,3,2,5] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 1
[4,1,3,5,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 1
[4,1,5,3,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 1
[4,2,1,3,5] => [4,2,1,3] => [2,1,1]
=> [1,1]
=> 1
[4,2,1,5,3] => [4,2,1,3] => [2,1,1]
=> [1,1]
=> 1
[4,2,3,1,5] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[4,2,3,5,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[4,2,5,1,3] => [4,2,1,3] => [2,1,1]
=> [1,1]
=> 1
[4,2,5,3,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[4,3,1,2,5] => [4,3,1,2] => [2,1,1]
=> [1,1]
=> 1
Description
The product of the parts of an integer partition.
Matching statistic: St000933
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000933: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 79%●distinct values known / distinct values provided: 50%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000933: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 79%●distinct values known / distinct values provided: 50%
Values
[1] => [] => []
=> ?
=> ? = 0
[1,2] => [1] => [1]
=> []
=> ? ∊ {0,1}
[2,1] => [1] => [1]
=> []
=> ? ∊ {0,1}
[1,2,3] => [1,2] => [2]
=> []
=> ? ∊ {0,0,1,1,1,1}
[1,3,2] => [1,2] => [2]
=> []
=> ? ∊ {0,0,1,1,1,1}
[2,1,3] => [2,1] => [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[2,3,1] => [2,1] => [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[3,1,2] => [1,2] => [2]
=> []
=> ? ∊ {0,0,1,1,1,1}
[3,2,1] => [2,1] => [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[1,2,3,4] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,2,4,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,3,2,4] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,3,4,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,4,2,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,4,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,1,3,4] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,1,4,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,3,1,4] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,3,4,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,4,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,4,3,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,1,2,4] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,1,4,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,2,1,4] => [3,2,1] => [1,1,1]
=> [1,1]
=> 1
[3,2,4,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 1
[3,4,1,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,4,2,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 1
[4,1,2,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,2,3,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,3,1,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,3,2,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 1
[1,2,3,4,5] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,3,5,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,3,5] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,5,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,3,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,4,5] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,5,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,2,5] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,3,5] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,4,3,5,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,4,5,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,5,2,3,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,4,5] => [2,1,3,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,4,3,5] => [2,1,4,3] => [2,2]
=> [2]
=> 2
[2,1,4,5,3] => [2,1,4,3] => [2,2]
=> [2]
=> 2
[2,1,5,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 2
[2,4,1,3,5] => [2,4,1,3] => [2,2]
=> [2]
=> 2
[2,4,1,5,3] => [2,4,1,3] => [2,2]
=> [2]
=> 2
[2,4,3,1,5] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 1
[2,4,3,5,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 1
[2,4,5,1,3] => [2,4,1,3] => [2,2]
=> [2]
=> 2
[2,4,5,3,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 1
[2,5,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 2
[2,5,4,1,3] => [2,4,1,3] => [2,2]
=> [2]
=> 2
[2,5,4,3,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 1
[3,1,4,2,5] => [3,1,4,2] => [2,2]
=> [2]
=> 2
[3,1,4,5,2] => [3,1,4,2] => [2,2]
=> [2]
=> 2
[3,1,5,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> 2
[3,2,1,4,5] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,2,1,5,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,2,4,1,5] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 1
[3,2,4,5,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 1
[3,2,5,1,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,2,5,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 1
[3,4,1,2,5] => [3,4,1,2] => [2,2]
=> [2]
=> 2
[3,4,1,5,2] => [3,4,1,2] => [2,2]
=> [2]
=> 2
[3,4,2,1,5] => [3,4,2,1] => [2,1,1]
=> [1,1]
=> 1
[3,4,2,5,1] => [3,4,2,1] => [2,1,1]
=> [1,1]
=> 1
[3,4,5,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 2
[3,4,5,2,1] => [3,4,2,1] => [2,1,1]
=> [1,1]
=> 1
[3,5,1,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> 2
[3,5,2,1,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,5,2,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 1
[3,5,4,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 2
[3,5,4,2,1] => [3,4,2,1] => [2,1,1]
=> [1,1]
=> 1
[4,1,3,2,5] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 1
[4,1,3,5,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 1
[4,1,5,3,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 1
[4,2,1,3,5] => [4,2,1,3] => [2,1,1]
=> [1,1]
=> 1
[4,2,1,5,3] => [4,2,1,3] => [2,1,1]
=> [1,1]
=> 1
[4,2,3,1,5] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[4,2,3,5,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[4,2,5,1,3] => [4,2,1,3] => [2,1,1]
=> [1,1]
=> 1
[4,2,5,3,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[4,3,1,2,5] => [4,3,1,2] => [2,1,1]
=> [1,1]
=> 1
Description
The number of multipartitions of sizes given by an integer partition.
This is, for $\lambda = (\lambda_1,\ldots,\lambda_n)$, this is the number of $n$-tuples $(\lambda^{(1)},\ldots,\lambda^{(n)})$ of partitions $\lambda^{(i)}$ such that $\lambda^{(i)} \vdash \lambda_i$.
Matching statistic: St000260
(load all 55 compositions to match this statistic)
(load all 55 compositions to match this statistic)
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,2] => [1,2] => [2,1] => ([(0,1)],2)
=> 1
[2,1] => [2,1] => [1,2] => ([],2)
=> ? = 0
[1,2,3] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[1,3,2] => [1,3,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[2,1,3] => [2,1,3] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0}
[2,3,1] => [1,3,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[3,1,2] => [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[3,2,1] => [3,2,1] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0}
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,2,4,3] => [1,2,4,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,3,2,4] => [1,3,2,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,3,4,2] => [1,2,4,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,4,2,3] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,4,3,2] => [1,4,3,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,3,4] => [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,1,1,1,2}
[2,1,4,3] => [2,1,4,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,1,1,1,2}
[2,3,1,4] => [1,3,2,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,3,4,1] => [1,2,4,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,4,1,3] => [2,4,1,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,1,1,1,2}
[2,4,3,1] => [1,4,3,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,1,2,4] => [3,1,2,4] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[3,1,4,2] => [2,1,4,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,1,1,1,2}
[3,2,1,4] => [3,2,1,4] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,0,0,1,1,1,2}
[3,2,4,1] => [2,1,4,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,1,1,1,2}
[3,4,1,2] => [2,4,1,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,1,1,1,2}
[3,4,2,1] => [1,4,3,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,1,2,3] => [4,1,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[4,1,3,2] => [4,1,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,2,1,3] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[4,2,3,1] => [4,1,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => [4,3,1,2] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,3,2,1] => [4,3,2,1] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,1,1,1,2}
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,4,5,3] => [1,2,3,5,4] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,5,4,3] => [1,2,5,4,3] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[1,3,4,2,5] => [1,2,4,3,5] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,4,5,2] => [1,2,3,5,4] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,5,4,2] => [1,2,5,4,3] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,2,5,3] => [1,3,2,5,4] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,5,2] => [1,3,2,5,4] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[1,4,5,2,3] => [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,5,3,2] => [1,2,5,4,3] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,5,2,3,4] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,5,2,4,3] => [1,5,2,4,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,5,3,2,4] => [1,5,3,2,4] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,5,3,4,2] => [1,5,2,4,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,5,4,2,3] => [1,5,4,2,3] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,1,3,5,4] => [2,1,3,5,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,1,4,3,5] => [2,1,4,3,5] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,1,4,5,3] => [2,1,3,5,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,1,5,3,4] => [2,1,5,3,4] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,1,5,4,3] => [2,1,5,4,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,3,1,4,5] => [1,3,2,4,5] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,3,1,5,4] => [1,3,2,5,4] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[2,3,4,1,5] => [1,2,4,3,5] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,3,4,5,1] => [1,2,3,5,4] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,4,1,3,5] => [2,4,1,3,5] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,5,1,3,4] => [2,5,1,3,4] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,5,1,4,3] => [2,5,1,4,3] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,5,4,1,3] => [2,5,4,1,3] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,1,4,2,5] => [2,1,4,3,5] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,1,4,5,2] => [2,1,3,5,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,1,5,4,2] => [2,1,5,4,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,2,1,4,5] => [3,2,1,4,5] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,2,1,5,4] => [3,2,1,5,4] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,2,4,1,5] => [2,1,4,3,5] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,2,4,5,1] => [2,1,3,5,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,2,5,1,4] => [3,2,5,1,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,2,5,4,1] => [2,1,5,4,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,4,1,2,5] => [2,4,1,3,5] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,5,1,4,2] => [2,5,1,4,3] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,5,2,4,1] => [2,5,1,4,3] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,5,4,1,2] => [2,5,4,1,3] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,1,5,3,2] => [2,1,5,4,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,2,1,5,3] => [3,2,1,5,4] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,2,5,1,3] => [3,2,5,1,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,2,5,3,1] => [2,1,5,4,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,3,1,5,2] => [3,2,1,5,4] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,3,2,1,5] => [4,3,2,1,5] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,3,2,5,1] => [3,2,1,5,4] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,3,5,1,2] => [3,2,5,1,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,3,5,2,1] => [2,1,5,4,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,5,1,3,2] => [2,5,1,4,3] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,5,2,3,1] => [2,5,1,4,3] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,5,3,1,2] => [2,5,4,1,3] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[5,4,3,2,1] => [5,4,3,2,1] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,1,3,4,5,6] => [2,1,3,4,5,6] => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,3,4,6,5] => [2,1,3,4,6,5] => [1,3,4,6,5,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,3,5,4,6] => [2,1,3,5,4,6] => [1,3,5,4,6,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St001568
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 75%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 75%
Values
[1] => [] => []
=> []
=> ? = 0
[1,2] => [1] => [1,0]
=> []
=> ? ∊ {0,1}
[2,1] => [1] => [1,0]
=> []
=> ? ∊ {0,1}
[1,2,3] => [1,2] => [1,0,1,0]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[1,3,2] => [1,2] => [1,0,1,0]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[2,1,3] => [2,1] => [1,1,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1}
[2,3,1] => [2,1] => [1,1,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1}
[3,1,2] => [1,2] => [1,0,1,0]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[3,2,1] => [2,1] => [1,1,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1}
[1,2,3,4] => [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 1
[1,2,4,3] => [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 1
[1,3,2,4] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 2
[1,3,4,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 2
[1,4,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 1
[1,4,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 2
[2,1,3,4] => [2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 1
[2,1,4,3] => [2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 1
[2,3,1,4] => [2,3,1] => [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1}
[2,3,4,1] => [2,3,1] => [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1}
[2,4,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 1
[2,4,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1}
[3,1,2,4] => [3,1,2] => [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1}
[3,1,4,2] => [3,1,2] => [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1}
[3,2,1,4] => [3,2,1] => [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1}
[3,2,4,1] => [3,2,1] => [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1}
[3,4,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1}
[3,4,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1}
[4,1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 1
[4,1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 2
[4,2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 1
[4,2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1}
[4,3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1}
[4,3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1}
[1,2,3,4,5] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
[1,2,3,5,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
[1,2,4,3,5] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1
[1,2,4,5,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1
[1,2,5,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
[1,2,5,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1
[1,3,2,4,5] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 2
[1,3,2,5,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 2
[1,3,4,2,5] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[1,3,4,5,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[1,3,5,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 2
[1,3,5,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[1,4,2,3,5] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 2
[1,4,2,5,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 2
[1,4,3,2,5] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 2
[1,4,3,5,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 2
[1,4,5,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 2
[1,4,5,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 2
[1,5,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
[1,5,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1
[1,5,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 2
[1,5,3,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[1,5,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 2
[1,5,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 2
[2,1,3,4,5] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[2,1,3,5,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[2,1,4,3,5] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
[2,1,4,5,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
[2,1,5,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[2,1,5,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
[2,3,1,4,5] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[2,3,1,5,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[2,3,4,1,5] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[2,3,4,5,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[2,3,5,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[2,3,5,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[2,4,1,3,5] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[2,4,1,5,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[3,4,1,2,5] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,4,1,5,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,4,2,1,5] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,4,2,5,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,4,5,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,4,5,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,5,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,5,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,1,2,3,5] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,1,2,5,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,1,3,2,5] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,1,3,5,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,1,5,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,1,5,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,2,1,3,5] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,2,1,5,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,2,3,1,5] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,2,3,5,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,2,5,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,2,5,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,3,1,2,5] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,3,1,5,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,3,2,1,5] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,3,2,5,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,3,5,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,3,5,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,5,1,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,5,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,5,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
Description
The smallest positive integer that does not appear twice in the partition.
Matching statistic: St000937
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000937: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 75%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000937: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 75%
Values
[1] => [1] => [1]
=> []
=> ? = 0
[1,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,1}
[2,1] => [2,1] => [2]
=> []
=> ? ∊ {0,1}
[1,2,3] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[2,3,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[3,1,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[3,2,1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,1,1,1,1}
[1,2,3,4] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,4,3] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,2,4] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,4,2] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,2,3] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,3,2] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[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] => [2,1,1]
=> [1,1]
=> 1
[2,3,4,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,1,3] => [2,4,1,3] => [2,1,1]
=> [1,1]
=> 1
[2,4,3,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,2,4] => [3,1,4,2] => [2,2]
=> [2]
=> 2
[3,1,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> 2
[3,2,1,4] => [3,2,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,2,4,1] => [3,2,4,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,4,1,2] => [3,4,1,2] => [2,1,1]
=> [1,1]
=> 1
[3,4,2,1] => [3,4,2,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,2,3] => [4,1,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,3,2] => [4,1,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,2,1,3] => [4,2,1,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,2,3,1] => [4,2,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,1,2] => [4,3,1,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,2,1] => [4,3,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,3,4,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,2,3,5,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,2,4,3,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,2,4,5,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,2,5,3,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,2,5,4,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,3,2,4,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,3,2,5,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,3,4,2,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,3,4,5,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,3,5,2,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,3,5,4,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,4,2,3,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,4,2,5,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,4,3,2,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,4,3,5,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,4,5,2,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,4,5,3,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,5,2,3,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,5,2,4,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,5,3,2,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,5,3,4,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,5,4,2,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,5,4,3,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,1,3,4,5] => [2,1,5,4,3] => [3,2]
=> [2]
=> 2
[2,1,3,5,4] => [2,1,5,4,3] => [3,2]
=> [2]
=> 2
[2,1,4,3,5] => [2,1,5,4,3] => [3,2]
=> [2]
=> 2
[2,1,4,5,3] => [2,1,5,4,3] => [3,2]
=> [2]
=> 2
[2,1,5,3,4] => [2,1,5,4,3] => [3,2]
=> [2]
=> 2
[2,1,5,4,3] => [2,1,5,4,3] => [3,2]
=> [2]
=> 2
[2,3,1,4,5] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,3,1,5,4] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,3,4,1,5] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,3,5,1,4] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,1,3,5] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,1,5,3] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,3,1,5] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,5,1,3] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,1,3,4] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,1,4,3] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,3,1,4] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,4,1,3] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[3,1,2,4,5] => [3,1,5,4,2] => [3,2]
=> [2]
=> 2
[3,1,2,5,4] => [3,1,5,4,2] => [3,2]
=> [2]
=> 2
[3,1,4,2,5] => [3,1,5,4,2] => [3,2]
=> [2]
=> 2
[3,1,4,5,2] => [3,1,5,4,2] => [3,2]
=> [2]
=> 2
[3,1,5,2,4] => [3,1,5,4,2] => [3,2]
=> [2]
=> 2
[3,1,5,4,2] => [3,1,5,4,2] => [3,2]
=> [2]
=> 2
[3,2,1,4,5] => [3,2,1,5,4] => [3,2]
=> [2]
=> 2
[3,2,1,5,4] => [3,2,1,5,4] => [3,2]
=> [2]
=> 2
[3,2,4,1,5] => [3,2,5,1,4] => [3,1,1]
=> [1,1]
=> 1
[3,2,4,5,1] => [3,2,5,4,1] => [3,2]
=> [2]
=> 2
[3,2,5,1,4] => [3,2,5,1,4] => [3,1,1]
=> [1,1]
=> 1
[3,2,5,4,1] => [3,2,5,4,1] => [3,2]
=> [2]
=> 2
[3,4,1,2,5] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,4,1,5,2] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,4,2,1,5] => [3,5,2,1,4] => [3,1,1]
=> [1,1]
=> 1
[3,4,2,5,1] => [3,5,2,4,1] => [3,1,1]
=> [1,1]
=> 1
[3,4,5,1,2] => [3,5,4,1,2] => [3,1,1]
=> [1,1]
=> 1
[3,5,1,2,4] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,5,1,4,2] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,5,2,1,4] => [3,5,2,1,4] => [3,1,1]
=> [1,1]
=> 1
[3,5,2,4,1] => [3,5,2,4,1] => [3,1,1]
=> [1,1]
=> 1
[3,5,4,1,2] => [3,5,4,1,2] => [3,1,1]
=> [1,1]
=> 1
[4,1,2,3,5] => [4,1,5,3,2] => [3,2]
=> [2]
=> 2
[4,1,2,5,3] => [4,1,5,3,2] => [3,2]
=> [2]
=> 2
[4,1,3,2,5] => [4,1,5,3,2] => [3,2]
=> [2]
=> 2
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 $2$.
The following 72 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000259The diameter of a connected graph. St001128The exponens consonantiae of a partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000456The monochromatic index of a connected graph. St001597The Frobenius rank of a skew partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001432The order dimension of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000706The product of the factorials of the multiplicities of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000681The Grundy value of Chomp on Ferrers diagrams. St000618The number of self-evacuating tableaux of given shape. St001280The number of parts of an integer partition that are at least two. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000667The greatest common divisor of the parts of the partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001389The number of partitions of the same length below the given integer partition. St001571The Cartan determinant of the integer partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001271The competition number of a graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000781The number of proper colouring schemes of a Ferrers diagram. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001877Number of indecomposable injective modules with projective dimension 2. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000284The Plancherel distribution on integer partitions. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. 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. St000455The second largest eigenvalue of a graph if it is integral. St000741The Colin de Verdière graph invariant. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000454The largest eigenvalue of a graph if it is integral. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001890The maximum magnitude of the Möbius function of a poset. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001624The breadth of a lattice.
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