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Your data matches 42 different statistics following compositions of up to 3 maps.
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Matching statistic: St000902
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
St000902: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,1] => 2
[2] => 1
[1,1,1] => 3
[1,2] => 1
[2,1] => 1
[3] => 1
[1,1,1,1] => 4
[1,1,2] => 1
[1,2,1] => 1
[1,3] => 1
[2,1,1] => 1
[2,2] => 2
[3,1] => 1
[4] => 1
[1,1,1,1,1] => 5
[1,1,1,2] => 1
[1,1,2,1] => 1
[1,1,3] => 1
[1,2,1,1] => 1
[1,2,2] => 1
[1,3,1] => 1
[1,4] => 1
[2,1,1,1] => 1
[2,1,2] => 1
[2,2,1] => 1
[2,3] => 1
[3,1,1] => 1
[3,2] => 1
[4,1] => 1
[5] => 1
[1,1,1,1,1,1] => 6
[1,1,1,1,2] => 1
[1,1,1,2,1] => 1
[1,1,1,3] => 1
[1,1,2,1,1] => 1
[1,1,2,2] => 2
[1,1,3,1] => 1
[1,1,4] => 1
[1,2,1,1,1] => 1
[1,2,1,2] => 2
[1,2,2,1] => 2
[1,2,3] => 1
[1,3,1,1] => 1
[1,3,2] => 1
[1,4,1] => 1
[1,5] => 1
[2,1,1,1,1] => 1
[2,1,1,2] => 2
[2,1,2,1] => 2
Description
The minimal number of repetitions of an integer composition.
Matching statistic: St000900
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000900: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000900: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1]]
=> [1] => 1
[1,1] => [1,1]
=> [[1],[2]]
=> [1,1] => 2
[2] => [2]
=> [[1,2]]
=> [2] => 1
[1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 3
[1,2] => [2,1]
=> [[1,3],[2]]
=> [1,2] => 1
[2,1] => [2,1]
=> [[1,3],[2]]
=> [1,2] => 1
[3] => [3]
=> [[1,2,3]]
=> [3] => 1
[1,1,1,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 4
[1,1,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
[1,2,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
[1,3] => [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
[2,1,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
[2,2] => [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
[3,1] => [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
[4] => [4]
=> [[1,2,3,4]]
=> [4] => 1
[1,1,1,1,1] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 5
[1,1,1,2] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1
[1,1,2,1] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1
[1,1,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
[1,2,1,1] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1
[1,2,2] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
[1,3,1] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
[1,4] => [4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => 1
[2,1,1,1] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1
[2,1,2] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
[2,2,1] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
[2,3] => [3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => 1
[3,1,1] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
[3,2] => [3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => 1
[4,1] => [4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => 1
[5] => [5]
=> [[1,2,3,4,5]]
=> [5] => 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => 6
[1,1,1,1,2] => [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => 1
[1,1,1,3] => [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => 1
[1,1,2,2] => [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => 2
[1,1,3,1] => [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => 1
[1,1,4] => [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => 1
[1,2,1,2] => [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => 2
[1,2,2,1] => [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => 2
[1,2,3] => [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => 1
[1,3,1,1] => [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => 1
[1,3,2] => [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => 1
[1,4,1] => [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => 1
[1,5] => [5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => 1
[2,1,1,1,1] => [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => 1
[2,1,1,2] => [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => 2
[2,1,2,1] => [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => 2
Description
The minimal number of repetitions of a part in an integer composition.
This is the smallest letter in the word obtained by applying the delta morphism.
Matching statistic: St000781
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 47%●distinct values known / distinct values provided: 11%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 47%●distinct values known / distinct values provided: 11%
Values
[1] => [1] => [[1],[]]
=> []
=> ? = 1
[1,1] => [2] => [[2],[]]
=> []
=> ? ∊ {1,2}
[2] => [1] => [[1],[]]
=> []
=> ? ∊ {1,2}
[1,1,1] => [3] => [[3],[]]
=> []
=> ? ∊ {1,1,1,3}
[1,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {1,1,1,3}
[2,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {1,1,1,3}
[3] => [1] => [[1],[]]
=> []
=> ? ∊ {1,1,1,3}
[1,1,1,1] => [4] => [[4],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,4}
[1,1,2] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,4}
[1,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,4}
[2,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,4}
[2,2] => [2] => [[2],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,4}
[3,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,4}
[4] => [1] => [[1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,4}
[1,1,1,1,1] => [5] => [[5],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,5}
[1,1,1,2] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,3] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,5}
[1,2,2] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,5}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,5}
[1,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,5}
[2,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,5}
[2,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,5}
[2,2,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[2,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,5}
[3,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,5}
[3,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,5}
[4,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,5}
[5] => [1] => [[1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,5}
[1,1,1,1,1,1] => [6] => [[6],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,1,1,1,2] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
[1,1,1,3] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,2,2] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,4] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,1,1,1] => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,2,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,2,2,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,2,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,3,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,3,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,4,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,5] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[2,1,1,1,1] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[2,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,1,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[2,1,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[2,2,1,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[2,2,2] => [3] => [[3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[2,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[2,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[3,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[3,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[3,2,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[3,3] => [2] => [[2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[4,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[4,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[5,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[6] => [1] => [[1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,1,1,1,1,1,1] => [7] => [[7],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
=> [4]
=> 1
[1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 1
[1,1,1,1,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1,1] => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 1
[1,1,1,2,2] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[1,1,1,3,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
[1,1,1,4] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1,1,1] => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,2,1,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,2,3] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,3,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,3,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,5] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,1,1,1,1] => [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,2,1,1,2] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 1
[1,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 1
[2,1,1,1,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[2,1,1,2,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[2,1,1,3] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,2,1,1,1] => [2,3] => [[4,2],[1]]
=> [1]
=> 1
[2,2,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[2,2,2,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[2,2,3] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[3,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[3,3,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,1,1,1,1,1,2] => [6,1] => [[6,6],[5]]
=> [5]
=> 1
[1,1,1,1,1,2,1] => [5,1,1] => [[5,5,5],[4,4]]
=> [4,4]
=> 1
[1,1,1,1,1,3] => [5,1] => [[5,5],[4]]
=> [4]
=> 1
[1,1,1,1,2,1,1] => [4,1,2] => [[5,4,4],[3,3]]
=> [3,3]
=> 1
[1,1,1,1,2,2] => [4,2] => [[5,4],[3]]
=> [3]
=> 1
[1,1,1,1,3,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 1
[1,1,1,1,4] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> [2,2]
=> 1
[1,1,1,2,1,2] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 1
Description
The number of proper colouring schemes of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1].
This statistic is the number of distinct such integer partitions that occur.
Matching statistic: St001568
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 28%●distinct values known / distinct values provided: 11%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 28%●distinct values known / distinct values provided: 11%
Values
[1] => [1,0]
=> [1,0]
=> []
=> ? = 1
[1,1] => [1,0,1,0]
=> [1,0,1,0]
=> [1]
=> ? ∊ {1,2}
[2] => [1,1,0,0]
=> [1,1,0,0]
=> []
=> ? ∊ {1,2}
[1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {1,3}
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2]
=> 1
[3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {1,3}
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {2,4}
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {2,4}
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> ? ∊ {1,5}
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,5}
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> ? ∊ {2,2,2,2,2,2,2,3,6}
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> ? ∊ {2,2,2,2,2,2,2,3,6}
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> ? ∊ {2,2,2,2,2,2,2,3,6}
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,2,1]
=> 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,2,1]
=> 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> ? ∊ {2,2,2,2,2,2,2,3,6}
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,3,1]
=> 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [5,3,1]
=> 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [5,4,1]
=> 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> ? ∊ {2,2,2,2,2,2,2,3,6}
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> ? ∊ {2,2,2,2,2,2,2,3,6}
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,3,2]
=> 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> ? ∊ {2,2,2,2,2,2,2,3,6}
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,2]
=> 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> ? ∊ {2,2,2,2,2,2,2,3,6}
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,3]
=> 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {2,2,2,2,2,2,2,3,6}
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1]
=> 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [3,2,1]
=> 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [5,4,2,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [6,4,2,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [4,2,1]
=> 1
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [5,2,1]
=> 1
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [6,2,1]
=> 1
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [5,4,3,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [6,4,3,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [6,5,3,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [6,4,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [6,5,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [6,3,2]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [5,4,2]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [6,4,2]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [6,5,2]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [5,4,3]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [6,4,3]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [6,5,3]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
Description
The smallest positive integer that does not appear twice in the partition.
Matching statistic: St000914
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000914: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 24%●distinct values known / distinct values provided: 11%
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000914: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 24%●distinct values known / distinct values provided: 11%
Values
[1] => [[1],[]]
=> ([],1)
=> ([],1)
=> ? = 1
[1,1] => [[1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2}
[2] => [[2],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2}
[1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,3}
[1,2] => [[2,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,3}
[2,1] => [[2,2],[1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,3}
[3] => [[3],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,3}
[1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,4}
[1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,4}
[1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,3] => [[3,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,4}
[2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,4}
[2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,1] => [[3,3],[2]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,4}
[4] => [[4],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,4}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,5}
[1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,5}
[1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,3] => [[3,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,5}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,4] => [[4,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,5}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,5}
[2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,5}
[3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[4,1] => [[4,4],[3]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,5}
[5] => [[5],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,5}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,3,6}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,3,6}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,1,3] => [[3,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,3,6}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,3,1] => [[3,3,1,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,4] => [[4,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,3,6}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,4,1] => [[4,4,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,5] => [[5,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,3,6}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,3,6}
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,1,3] => [[4,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,4] => [[5,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,3,6}
[3,1,2] => [[4,3,3],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[4,1,1] => [[4,4,4],[3,3]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,3,6}
[4,2] => [[5,4],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[5,1] => [[5,5],[4]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,3,6}
[6] => [[6],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,3,6}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,1,1,3] => [[3,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,1,4] => [[4,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1
[1,1,2,3] => [[4,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,3,2] => [[4,3,1,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,1,4,1] => [[4,4,1,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,5] => [[5,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1
[1,2,1,3] => [[4,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1
[1,2,2,2] => [[4,3,2,1],[2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,2,3,1] => [[4,4,2,1],[3,1]]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 1
[1,2,4] => [[5,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,6] => [[6,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[4,1,1,1] => [[4,4,4,4],[3,3,3]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[5,1,1] => [[5,5,5],[4,4]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[6,1] => [[6,6],[5]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[7] => [[7],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,8}
[1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,8}
[1,1,1,1,1,3] => [[3,1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,8}
[1,1,1,1,3,1] => [[3,3,1,1,1,1],[2]]
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,8}
[1,1,1,1,4] => [[4,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,8}
Description
The sum of the values of the Möbius function of a poset.
The Möbius function $\mu$ of a finite poset is defined as
$$\mu (x,y)=\begin{cases} 1& \text{if }x = y\\
-\sum _{z: x\leq z < y}\mu (x,z)& \text{for }x < y\\
0&\text{otherwise}.
\end{cases}
$$
Since $\mu(x,y)=0$ whenever $x\not\leq y$, this statistic is
$$
\sum_{x\leq y} \mu(x,y).
$$
If the poset has a minimal or a maximal element, then the definition implies immediately that the statistic equals $1$. Moreover, the statistic equals the sum of the statistics of the connected components.
This statistic is also called the magnitude of a poset.
Matching statistic: St000207
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000207: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 20%●distinct values known / distinct values provided: 11%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000207: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 20%●distinct values known / distinct values provided: 11%
Values
[1] => ([],1)
=> [1]
=> []
=> ? = 1
[1,1] => ([(0,1)],2)
=> [2]
=> []
=> ? = 2
[2] => ([],2)
=> [1,1]
=> [1]
=> 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {1,3}
[1,2] => ([(1,2)],3)
=> [2,1]
=> [1]
=> 1
[2,1] => ([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {1,3}
[3] => ([],3)
=> [1,1,1]
=> [1,1]
=> 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,2,4}
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,2,4}
[1,3] => ([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,2,4}
[2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,2,4}
[4] => ([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[2,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[5] => ([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,5] => ([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[2,4] => ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[6] => ([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 1
[1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> 1
[1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 1
[1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 1
[1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> 1
[1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 1
[1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,6] => ([(5,6)],7)
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 1
[2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has all vertices in integer lattice points.
Matching statistic: St000208
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 20%●distinct values known / distinct values provided: 11%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 20%●distinct values known / distinct values provided: 11%
Values
[1] => ([],1)
=> [1]
=> []
=> ? = 1
[1,1] => ([(0,1)],2)
=> [2]
=> []
=> ? = 2
[2] => ([],2)
=> [1,1]
=> [1]
=> 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {1,3}
[1,2] => ([(1,2)],3)
=> [2,1]
=> [1]
=> 1
[2,1] => ([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {1,3}
[3] => ([],3)
=> [1,1,1]
=> [1,1]
=> 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,2,4}
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,2,4}
[1,3] => ([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,2,4}
[2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,2,4}
[4] => ([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[2,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[5] => ([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,5] => ([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[2,4] => ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[6] => ([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 1
[1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> 1
[1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 1
[1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 1
[1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> 1
[1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 1
[1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,6] => ([(5,6)],7)
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 1
[2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has only integer lattice points as vertices.
See also [[St000205]], [[St000206]] and [[St000207]].
Matching statistic: St000618
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000618: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 20%●distinct values known / distinct values provided: 11%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000618: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 20%●distinct values known / distinct values provided: 11%
Values
[1] => ([],1)
=> [1]
=> []
=> ? = 1
[1,1] => ([(0,1)],2)
=> [2]
=> []
=> ? = 2
[2] => ([],2)
=> [1,1]
=> [1]
=> 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {1,3}
[1,2] => ([(1,2)],3)
=> [2,1]
=> [1]
=> 1
[2,1] => ([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {1,3}
[3] => ([],3)
=> [1,1,1]
=> [1,1]
=> 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,2,4}
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,2,4}
[1,3] => ([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,2,4}
[2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,2,4}
[4] => ([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[2,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[5] => ([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,5] => ([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[2,4] => ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[6] => ([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 1
[1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> 1
[1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 1
[1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 1
[1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> 1
[1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 1
[1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,6] => ([(5,6)],7)
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 1
[2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
Description
The number of self-evacuating tableaux of given shape.
This is the same as the number of standard domino tableaux of the given shape.
Matching statistic: St000667
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 20%●distinct values known / distinct values provided: 11%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 20%●distinct values known / distinct values provided: 11%
Values
[1] => ([],1)
=> [1]
=> []
=> ? = 1
[1,1] => ([(0,1)],2)
=> [2]
=> []
=> ? = 2
[2] => ([],2)
=> [1,1]
=> [1]
=> 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {1,3}
[1,2] => ([(1,2)],3)
=> [2,1]
=> [1]
=> 1
[2,1] => ([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {1,3}
[3] => ([],3)
=> [1,1,1]
=> [1,1]
=> 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,2,4}
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,2,4}
[1,3] => ([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,2,4}
[2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,2,4}
[4] => ([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[2,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[5] => ([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,5] => ([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[2,4] => ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[6] => ([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 1
[1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> 1
[1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 1
[1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 1
[1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> 1
[1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 1
[1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,6] => ([(5,6)],7)
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 1
[2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
Description
The greatest common divisor of the parts of the partition.
Matching statistic: St000755
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000755: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 20%●distinct values known / distinct values provided: 11%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000755: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 20%●distinct values known / distinct values provided: 11%
Values
[1] => ([],1)
=> [1]
=> []
=> ? = 1
[1,1] => ([(0,1)],2)
=> [2]
=> []
=> ? = 2
[2] => ([],2)
=> [1,1]
=> [1]
=> 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {1,3}
[1,2] => ([(1,2)],3)
=> [2,1]
=> [1]
=> 1
[2,1] => ([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {1,3}
[3] => ([],3)
=> [1,1,1]
=> [1,1]
=> 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,2,4}
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,2,4}
[1,3] => ([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,2,4}
[2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,2,4}
[4] => ([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[2,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,5}
[5] => ([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,5] => ([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[2,4] => ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[6] => ([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 1
[1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> 1
[1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 1
[1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 1
[1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> 1
[1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 1
[1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[1,6] => ([(5,6)],7)
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
[2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,7}
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 1
[2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 1
Description
The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition.
Consider the recurrence $$f(n)=\sum_{p\in\lambda} f(n-p).$$ This statistic returns the number of distinct real roots of the associated characteristic polynomial.
For example, the partition $(2,1)$ corresponds to the recurrence $f(n)=f(n-1)+f(n-2)$ with associated characteristic polynomial $x^2-x-1$, which has two real roots.
The following 32 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001571The Cartan determinant of the integer partition. 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. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001890The maximum magnitude of the Möbius function of a poset. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000782The indicator function of whether a given perfect matching is an L & P matching. St001877Number of indecomposable injective modules with projective dimension 2. 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. 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. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St000617The number of global maxima of a Dyck path. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph.
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