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Your data matches 95 different statistics following compositions of up to 3 maps.
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Matching statistic: St001403
St001403: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 2
[1,3,4,2] => 1
[1,4,2,3] => 2
[1,4,3,2] => 0
[2,1,3,4] => 1
[2,1,4,3] => 0
[2,3,1,4] => 2
[2,3,4,1] => 0
[2,4,1,3] => 2
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 2
[3,2,1,4] => 0
[3,2,4,1] => 2
[3,4,1,2] => 0
[3,4,2,1] => 1
[4,1,2,3] => 0
[4,1,3,2] => 2
[4,2,1,3] => 1
[4,2,3,1] => 2
[4,3,1,2] => 1
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 2
[1,2,4,5,3] => 1
[1,2,5,3,4] => 2
[1,2,5,4,3] => 0
[1,3,2,4,5] => 2
[1,3,2,5,4] => 1
[1,3,4,2,5] => 2
[1,3,4,5,2] => 0
[1,3,5,2,4] => 2
[1,3,5,4,2] => 1
[1,4,2,3,5] => 2
[1,4,2,5,3] => 3
[1,4,3,2,5] => 0
[1,4,3,5,2] => 2
[1,4,5,2,3] => 0
Description
The number of vertical separators in a permutation.
Let $\sigma$ be a permutation. Then $\sigma_i$ is a vertical separator if $|\sigma_{i-1} - \sigma_{i+1}| = 1$.
Matching statistic: St000698
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 80%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => [[1],[]]
=> []
=> ? = 0
[1,2] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0}
[2,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0}
[1,2,3] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1,1,1}
[1,3,2] => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[2,1,3] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1}
[2,3,1] => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[3,1,2] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1}
[3,2,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1}
[1,2,3,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2}
[1,2,4,3] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,3,2,4] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2}
[1,3,4,2] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,4,2,3] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2}
[1,4,3,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[2,1,3,4] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2}
[2,1,4,3] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2}
[2,3,1,4] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2}
[2,3,4,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[2,4,1,3] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2}
[2,4,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[3,1,2,4] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2}
[3,1,4,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2}
[3,2,1,4] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2}
[3,2,4,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2}
[3,4,1,2] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2}
[3,4,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[4,1,2,3] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2}
[4,1,3,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2}
[4,2,1,3] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2}
[4,2,3,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2}
[4,3,1,2] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2}
[4,3,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2}
[1,2,3,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,2,3,5,4] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,2,4,3,5] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[1,2,4,5,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,2,5,3,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[1,2,5,4,3] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,3,2,4,5] => [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,3,2,5,4] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[1,3,4,2,5] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[1,3,4,5,2] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,3,5,2,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[1,3,5,4,2] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,4,2,3,5] => [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,4,2,5,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[1,4,3,2,5] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,4,3,5,2] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[1,4,5,2,3] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[1,4,5,3,2] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,5,2,3,4] => [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,5,2,4,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[1,5,3,2,4] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,5,3,4,2] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[1,5,4,2,3] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,5,4,3,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[2,1,3,4,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,1,3,5,4] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[2,1,4,3,5] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,1,4,5,3] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[2,1,5,3,4] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,1,5,4,3] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[2,3,1,4,5] => [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,3,1,5,4] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[2,3,4,1,5] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[2,3,4,5,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[2,3,5,1,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[2,3,5,4,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[2,4,1,3,5] => [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,4,1,5,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[2,4,3,1,5] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
[2,4,3,5,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[2,4,5,1,3] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[2,4,5,3,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[2,5,1,3,4] => [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,5,1,4,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[2,5,3,1,4] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
[2,5,3,4,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[2,5,4,1,3] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
[2,5,4,3,1] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[3,1,2,4,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,1,2,5,4] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[3,1,4,2,5] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,1,4,5,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[3,1,5,2,4] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,1,5,4,2] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[3,2,1,4,5] => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,2,1,5,4] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,2,4,1,5] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,2,4,5,1] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[3,2,5,1,4] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,2,5,4,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[3,4,1,2,5] => [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,4,1,5,2] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[3,5,1,2,4] => [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[4,1,2,3,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[4,1,3,2,5] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[4,1,5,2,3] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[4,2,1,3,5] => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
Description
The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core.
For any positive integer $k$, one associates a $k$-core to a partition by repeatedly removing all rim hooks of size $k$.
This statistic counts the $2$-rim hooks that are removed in this process to obtain a $2$-core.
Matching statistic: St000668
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 67%●distinct values known / distinct values provided: 60%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 67%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => [1]
=> []
=> ? = 0
[1,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0}
[2,1] => [2,1] => [2]
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[2,3,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[3,1,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[3,2,1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,1,1,1,1}
[1,2,3,4] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[1,2,4,3] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[1,3,2,4] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[1,3,4,2] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[1,4,2,3] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[1,4,3,2] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[2,1,3,4] => [2,1,4,3] => [2,2]
=> [2]
=> 2
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 2
[2,3,1,4] => [2,4,1,3] => [2,1,1]
=> [1,1]
=> 1
[2,3,4,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[2,4,1,3] => [2,4,1,3] => [2,1,1]
=> [1,1]
=> 1
[2,4,3,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[3,1,2,4] => [3,1,4,2] => [2,2]
=> [2]
=> 2
[3,1,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> 2
[3,2,1,4] => [3,2,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[3,2,4,1] => [3,2,4,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[3,4,1,2] => [3,4,1,2] => [2,1,1]
=> [1,1]
=> 1
[3,4,2,1] => [3,4,2,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[4,1,2,3] => [4,1,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[4,1,3,2] => [4,1,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[4,2,1,3] => [4,2,1,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[4,2,3,1] => [4,2,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[4,3,1,2] => [4,3,1,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[4,3,2,1] => [4,3,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[1,2,3,4,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,2,3,5,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,2,4,3,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,2,4,5,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,2,5,3,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,2,5,4,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,3,2,4,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,3,2,5,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,3,4,2,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,3,4,5,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,3,5,2,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,3,5,4,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,4,2,3,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,4,2,5,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,4,3,2,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,4,3,5,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,4,5,2,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,4,5,3,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,5,2,3,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,5,2,4,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,5,3,2,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,5,3,4,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,5,4,2,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,5,4,3,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[2,1,3,4,5] => [2,1,5,4,3] => [3,2]
=> [2]
=> 2
[2,1,3,5,4] => [2,1,5,4,3] => [3,2]
=> [2]
=> 2
[2,1,4,3,5] => [2,1,5,4,3] => [3,2]
=> [2]
=> 2
[2,1,4,5,3] => [2,1,5,4,3] => [3,2]
=> [2]
=> 2
[2,1,5,3,4] => [2,1,5,4,3] => [3,2]
=> [2]
=> 2
[2,1,5,4,3] => [2,1,5,4,3] => [3,2]
=> [2]
=> 2
[2,3,1,4,5] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,3,1,5,4] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,3,4,1,5] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,3,5,1,4] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,1,3,5] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,1,5,3] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,3,1,5] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,5,1,3] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,1,3,4] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,1,4,3] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,3,1,4] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,4,1,3] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[3,1,2,4,5] => [3,1,5,4,2] => [3,2]
=> [2]
=> 2
[3,1,2,5,4] => [3,1,5,4,2] => [3,2]
=> [2]
=> 2
[3,1,4,2,5] => [3,1,5,4,2] => [3,2]
=> [2]
=> 2
[3,1,4,5,2] => [3,1,5,4,2] => [3,2]
=> [2]
=> 2
[3,1,5,2,4] => [3,1,5,4,2] => [3,2]
=> [2]
=> 2
[3,1,5,4,2] => [3,1,5,4,2] => [3,2]
=> [2]
=> 2
[3,2,1,4,5] => [3,2,1,5,4] => [3,2]
=> [2]
=> 2
[3,2,1,5,4] => [3,2,1,5,4] => [3,2]
=> [2]
=> 2
[3,2,4,1,5] => [3,2,5,1,4] => [3,1,1]
=> [1,1]
=> 1
[3,2,4,5,1] => [3,2,5,4,1] => [3,2]
=> [2]
=> 2
[3,2,5,1,4] => [3,2,5,1,4] => [3,1,1]
=> [1,1]
=> 1
[3,2,5,4,1] => [3,2,5,4,1] => [3,2]
=> [2]
=> 2
[3,4,1,2,5] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,4,1,5,2] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,4,2,1,5] => [3,5,2,1,4] => [3,1,1]
=> [1,1]
=> 1
[3,4,2,5,1] => [3,5,2,4,1] => [3,1,1]
=> [1,1]
=> 1
[3,4,5,1,2] => [3,5,4,1,2] => [3,1,1]
=> [1,1]
=> 1
[3,5,1,2,4] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,5,1,4,2] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,5,2,1,4] => [3,5,2,1,4] => [3,1,1]
=> [1,1]
=> 1
[3,5,2,4,1] => [3,5,2,4,1] => [3,1,1]
=> [1,1]
=> 1
[3,5,4,1,2] => [3,5,4,1,2] => [3,1,1]
=> [1,1]
=> 1
[4,1,2,3,5] => [4,1,5,3,2] => [3,2]
=> [2]
=> 2
[4,1,2,5,3] => [4,1,5,3,2] => [3,2]
=> [2]
=> 2
[4,1,3,2,5] => [4,1,5,3,2] => [3,2]
=> [2]
=> 2
Description
The least common multiple of the parts of the partition.
Matching statistic: St000708
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 67%●distinct values known / distinct values provided: 60%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 67%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => [1]
=> []
=> ? = 0
[1,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0}
[2,1] => [2,1] => [2]
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[2,3,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[3,1,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[3,2,1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,1,1,1,1}
[1,2,3,4] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[1,2,4,3] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[1,3,2,4] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[1,3,4,2] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[1,4,2,3] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[1,4,3,2] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[2,1,3,4] => [2,1,4,3] => [2,2]
=> [2]
=> 2
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 2
[2,3,1,4] => [2,4,1,3] => [2,1,1]
=> [1,1]
=> 1
[2,3,4,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[2,4,1,3] => [2,4,1,3] => [2,1,1]
=> [1,1]
=> 1
[2,4,3,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[3,1,2,4] => [3,1,4,2] => [2,2]
=> [2]
=> 2
[3,1,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> 2
[3,2,1,4] => [3,2,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[3,2,4,1] => [3,2,4,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[3,4,1,2] => [3,4,1,2] => [2,1,1]
=> [1,1]
=> 1
[3,4,2,1] => [3,4,2,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[4,1,2,3] => [4,1,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[4,1,3,2] => [4,1,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[4,2,1,3] => [4,2,1,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[4,2,3,1] => [4,2,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[4,3,1,2] => [4,3,1,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[4,3,2,1] => [4,3,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[1,2,3,4,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,2,3,5,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,2,4,3,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,2,4,5,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,2,5,3,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,2,5,4,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,3,2,4,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,3,2,5,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,3,4,2,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,3,4,5,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,3,5,2,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,3,5,4,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,4,2,3,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,4,2,5,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,4,3,2,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,4,3,5,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,4,5,2,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,4,5,3,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,5,2,3,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,5,2,4,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,5,3,2,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,5,3,4,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,5,4,2,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,5,4,3,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[2,1,3,4,5] => [2,1,5,4,3] => [3,2]
=> [2]
=> 2
[2,1,3,5,4] => [2,1,5,4,3] => [3,2]
=> [2]
=> 2
[2,1,4,3,5] => [2,1,5,4,3] => [3,2]
=> [2]
=> 2
[2,1,4,5,3] => [2,1,5,4,3] => [3,2]
=> [2]
=> 2
[2,1,5,3,4] => [2,1,5,4,3] => [3,2]
=> [2]
=> 2
[2,1,5,4,3] => [2,1,5,4,3] => [3,2]
=> [2]
=> 2
[2,3,1,4,5] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,3,1,5,4] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,3,4,1,5] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,3,5,1,4] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,1,3,5] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,1,5,3] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,3,1,5] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,5,1,3] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,1,3,4] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,1,4,3] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,3,1,4] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,4,1,3] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[3,1,2,4,5] => [3,1,5,4,2] => [3,2]
=> [2]
=> 2
[3,1,2,5,4] => [3,1,5,4,2] => [3,2]
=> [2]
=> 2
[3,1,4,2,5] => [3,1,5,4,2] => [3,2]
=> [2]
=> 2
[3,1,4,5,2] => [3,1,5,4,2] => [3,2]
=> [2]
=> 2
[3,1,5,2,4] => [3,1,5,4,2] => [3,2]
=> [2]
=> 2
[3,1,5,4,2] => [3,1,5,4,2] => [3,2]
=> [2]
=> 2
[3,2,1,4,5] => [3,2,1,5,4] => [3,2]
=> [2]
=> 2
[3,2,1,5,4] => [3,2,1,5,4] => [3,2]
=> [2]
=> 2
[3,2,4,1,5] => [3,2,5,1,4] => [3,1,1]
=> [1,1]
=> 1
[3,2,4,5,1] => [3,2,5,4,1] => [3,2]
=> [2]
=> 2
[3,2,5,1,4] => [3,2,5,1,4] => [3,1,1]
=> [1,1]
=> 1
[3,2,5,4,1] => [3,2,5,4,1] => [3,2]
=> [2]
=> 2
[3,4,1,2,5] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,4,1,5,2] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,4,2,1,5] => [3,5,2,1,4] => [3,1,1]
=> [1,1]
=> 1
[3,4,2,5,1] => [3,5,2,4,1] => [3,1,1]
=> [1,1]
=> 1
[3,4,5,1,2] => [3,5,4,1,2] => [3,1,1]
=> [1,1]
=> 1
[3,5,1,2,4] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,5,1,4,2] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,5,2,1,4] => [3,5,2,1,4] => [3,1,1]
=> [1,1]
=> 1
[3,5,2,4,1] => [3,5,2,4,1] => [3,1,1]
=> [1,1]
=> 1
[3,5,4,1,2] => [3,5,4,1,2] => [3,1,1]
=> [1,1]
=> 1
[4,1,2,3,5] => [4,1,5,3,2] => [3,2]
=> [2]
=> 2
[4,1,2,5,3] => [4,1,5,3,2] => [3,2]
=> [2]
=> 2
[4,1,3,2,5] => [4,1,5,3,2] => [3,2]
=> [2]
=> 2
Description
The product of the parts of an integer partition.
Matching statistic: St000933
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000933: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 67%●distinct values known / distinct values provided: 60%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000933: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 67%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => [1]
=> []
=> ? = 0
[1,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0}
[2,1] => [2,1] => [2]
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[2,3,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[3,1,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[3,2,1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,1,1,1,1}
[1,2,3,4] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[1,2,4,3] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[1,3,2,4] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[1,3,4,2] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[1,4,2,3] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[1,4,3,2] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[2,1,3,4] => [2,1,4,3] => [2,2]
=> [2]
=> 2
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 2
[2,3,1,4] => [2,4,1,3] => [2,1,1]
=> [1,1]
=> 1
[2,3,4,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[2,4,1,3] => [2,4,1,3] => [2,1,1]
=> [1,1]
=> 1
[2,4,3,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[3,1,2,4] => [3,1,4,2] => [2,2]
=> [2]
=> 2
[3,1,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> 2
[3,2,1,4] => [3,2,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[3,2,4,1] => [3,2,4,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[3,4,1,2] => [3,4,1,2] => [2,1,1]
=> [1,1]
=> 1
[3,4,2,1] => [3,4,2,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[4,1,2,3] => [4,1,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[4,1,3,2] => [4,1,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[4,2,1,3] => [4,2,1,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[4,2,3,1] => [4,2,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[4,3,1,2] => [4,3,1,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[4,3,2,1] => [4,3,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[1,2,3,4,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,2,3,5,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,2,4,3,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,2,4,5,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,2,5,3,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,2,5,4,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,3,2,4,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,3,2,5,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,3,4,2,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,3,4,5,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,3,5,2,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,3,5,4,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,4,2,3,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,4,2,5,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,4,3,2,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,4,3,5,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,4,5,2,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,4,5,3,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,5,2,3,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,5,2,4,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,5,3,2,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,5,3,4,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,5,4,2,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,5,4,3,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[2,1,3,4,5] => [2,1,5,4,3] => [3,2]
=> [2]
=> 2
[2,1,3,5,4] => [2,1,5,4,3] => [3,2]
=> [2]
=> 2
[2,1,4,3,5] => [2,1,5,4,3] => [3,2]
=> [2]
=> 2
[2,1,4,5,3] => [2,1,5,4,3] => [3,2]
=> [2]
=> 2
[2,1,5,3,4] => [2,1,5,4,3] => [3,2]
=> [2]
=> 2
[2,1,5,4,3] => [2,1,5,4,3] => [3,2]
=> [2]
=> 2
[2,3,1,4,5] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,3,1,5,4] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,3,4,1,5] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,3,5,1,4] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,1,3,5] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,1,5,3] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,3,1,5] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,5,1,3] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,1,3,4] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,1,4,3] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,3,1,4] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,4,1,3] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[3,1,2,4,5] => [3,1,5,4,2] => [3,2]
=> [2]
=> 2
[3,1,2,5,4] => [3,1,5,4,2] => [3,2]
=> [2]
=> 2
[3,1,4,2,5] => [3,1,5,4,2] => [3,2]
=> [2]
=> 2
[3,1,4,5,2] => [3,1,5,4,2] => [3,2]
=> [2]
=> 2
[3,1,5,2,4] => [3,1,5,4,2] => [3,2]
=> [2]
=> 2
[3,1,5,4,2] => [3,1,5,4,2] => [3,2]
=> [2]
=> 2
[3,2,1,4,5] => [3,2,1,5,4] => [3,2]
=> [2]
=> 2
[3,2,1,5,4] => [3,2,1,5,4] => [3,2]
=> [2]
=> 2
[3,2,4,1,5] => [3,2,5,1,4] => [3,1,1]
=> [1,1]
=> 1
[3,2,4,5,1] => [3,2,5,4,1] => [3,2]
=> [2]
=> 2
[3,2,5,1,4] => [3,2,5,1,4] => [3,1,1]
=> [1,1]
=> 1
[3,2,5,4,1] => [3,2,5,4,1] => [3,2]
=> [2]
=> 2
[3,4,1,2,5] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,4,1,5,2] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,4,2,1,5] => [3,5,2,1,4] => [3,1,1]
=> [1,1]
=> 1
[3,4,2,5,1] => [3,5,2,4,1] => [3,1,1]
=> [1,1]
=> 1
[3,4,5,1,2] => [3,5,4,1,2] => [3,1,1]
=> [1,1]
=> 1
[3,5,1,2,4] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,5,1,4,2] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,5,2,1,4] => [3,5,2,1,4] => [3,1,1]
=> [1,1]
=> 1
[3,5,2,4,1] => [3,5,2,4,1] => [3,1,1]
=> [1,1]
=> 1
[3,5,4,1,2] => [3,5,4,1,2] => [3,1,1]
=> [1,1]
=> 1
[4,1,2,3,5] => [4,1,5,3,2] => [3,2]
=> [2]
=> 2
[4,1,2,5,3] => [4,1,5,3,2] => [3,2]
=> [2]
=> 2
[4,1,3,2,5] => [4,1,5,3,2] => [3,2]
=> [2]
=> 2
Description
The number of multipartitions of sizes given by an integer partition.
This is, for $\lambda = (\lambda_1,\ldots,\lambda_n)$, this is the number of $n$-tuples $(\lambda^{(1)},\ldots,\lambda^{(n)})$ of partitions $\lambda^{(i)}$ such that $\lambda^{(i)} \vdash \lambda_i$.
Matching statistic: St000620
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000620: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 66%●distinct values known / distinct values provided: 60%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000620: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 66%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => [1]
=> []
=> ? = 0
[1,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0}
[2,1] => [2,1] => [2]
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[2,3,1] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0,1,1,1,1}
[3,1,2] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0,1,1,1,1}
[3,2,1] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[1,2,3,4] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,2,4,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,4] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,4,2,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,4] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,3,1,4] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2}
[2,3,4,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2}
[2,4,1,3] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2}
[2,4,3,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2}
[3,1,2,4] => [3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2}
[3,1,4,2] => [3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2}
[3,2,1,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,2,4,1] => [3,2,4,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2}
[3,4,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 0
[3,4,2,1] => [3,4,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2}
[4,1,2,3] => [4,1,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2}
[4,1,3,2] => [4,1,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2}
[4,2,1,3] => [4,2,1,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2}
[4,2,3,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[4,3,1,2] => [4,3,1,2] => [4]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2}
[4,3,2,1] => [4,3,2,1] => [2,2]
=> [2]
=> 0
[1,2,3,4,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,2,3,5,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,2,4,3,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,2,4,5,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,2,5,3,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,2,5,4,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,3,2,4,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,3,2,5,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,3,4,2,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,3,4,5,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,3,5,2,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,3,5,4,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,4,2,3,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,4,2,5,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,4,3,2,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,4,3,5,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,4,5,2,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,4,5,3,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,5,2,3,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,5,2,4,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,5,3,2,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,5,3,4,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,5,4,2,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,5,4,3,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[2,1,3,4,5] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1
[2,1,3,5,4] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1
[2,1,4,3,5] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1
[2,1,4,5,3] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1
[2,1,5,3,4] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1
[2,1,5,4,3] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1
[2,3,1,4,5] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,3,1,5,4] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,3,4,1,5] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,3,4,5,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 0
[2,3,5,1,4] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,3,5,4,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 0
[2,4,1,3,5] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,4,1,5,3] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,4,3,1,5] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,4,3,5,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 0
[2,4,5,1,3] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,4,5,3,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 0
[2,5,1,3,4] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,5,1,4,3] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,5,3,1,4] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,5,3,4,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 0
[2,5,4,1,3] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,5,4,3,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 0
[3,1,2,4,5] => [3,1,5,4,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,1,2,5,4] => [3,1,5,4,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,1,4,2,5] => [3,1,5,4,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,1,4,5,2] => [3,1,5,4,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,1,5,2,4] => [3,1,5,4,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,1,5,4,2] => [3,1,5,4,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,2,1,4,5] => [3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 1
[3,2,1,5,4] => [3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 1
[3,2,4,1,5] => [3,2,5,1,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,2,5,1,4] => [3,2,5,1,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,4,2,1,5] => [3,5,2,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,4,2,5,1] => [3,5,2,4,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,4,5,2,1] => [3,5,4,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,5,2,1,4] => [3,5,2,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,5,2,4,1] => [3,5,2,4,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,5,4,2,1] => [3,5,4,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[4,1,2,3,5] => [4,1,5,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[4,1,2,5,3] => [4,1,5,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[4,1,3,2,5] => [4,1,5,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
Description
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd.
To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is odd.
The case of an even minimum is [[St000621]].
Matching statistic: St000771
(load all 42 compositions to match this statistic)
(load all 42 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,2] => ([],2)
=> ([],2)
=> ? = 0 + 1
[2,1] => ([(0,1)],2)
=> ([],1)
=> 1 = 0 + 1
[1,2,3] => ([],3)
=> ([],3)
=> ? ∊ {1,1,1} + 1
[1,3,2] => ([(1,2)],3)
=> ([],2)
=> ? ∊ {1,1,1} + 1
[2,1,3] => ([(1,2)],3)
=> ([],2)
=> ? ∊ {1,1,1} + 1
[2,3,1] => ([(0,2),(1,2)],3)
=> ([],1)
=> 1 = 0 + 1
[3,1,2] => ([(0,2),(1,2)],3)
=> ([],1)
=> 1 = 0 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2,3,4] => ([],4)
=> ([],4)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2} + 1
[1,2,4,3] => ([(2,3)],4)
=> ([],3)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2} + 1
[1,3,2,4] => ([(2,3)],4)
=> ([],3)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2} + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([],2)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2} + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([],2)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2} + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2} + 1
[2,1,3,4] => ([(2,3)],4)
=> ([],3)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2} + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([],2)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2} + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([],2)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2} + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> 1 = 0 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([],1)
=> 1 = 0 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([],2)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2} + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([],1)
=> 1 = 0 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2} + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> 1 = 0 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,2,3,4,5] => ([],5)
=> ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[1,2,3,5,4] => ([(3,4)],5)
=> ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[1,2,4,3,5] => ([(3,4)],5)
=> ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[1,3,2,4,5] => ([(3,4)],5)
=> ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[2,1,3,4,5] => ([(3,4)],5)
=> ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> 1 = 0 + 1
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([],1)
=> 1 = 0 + 1
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> 1 = 0 + 1
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3} + 1
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([],1)
=> 1 = 0 + 1
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([],1)
=> 1 = 0 + 1
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> 1 = 0 + 1
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 1 + 1
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1 = 0 + 1
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([],1)
=> 1 = 0 + 1
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $2$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St000937
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000937: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 80%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000937: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 80%
Values
[1] => [1]
=> []
=> ? = 0
[1,2] => [1,1]
=> [1]
=> ? ∊ {0,0}
[2,1] => [2]
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1}
[2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1}
[2,3,1] => [3]
=> []
=> ? ∊ {0,0,1,1,1}
[3,1,2] => [3]
=> []
=> ? ∊ {0,0,1,1,1}
[3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 2
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2}
[1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> 2
[2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2}
[2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2}
[2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2}
[2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2}
[3,1,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2}
[3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2}
[3,4,1,2] => [2,2]
=> [2]
=> 2
[3,4,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2}
[4,1,2,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2}
[4,1,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2}
[4,2,1,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[4,3,1,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2}
[4,3,2,1] => [2,2]
=> [2]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 3
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[1,3,5,2,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> 1
[1,4,2,5,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> 1
[1,4,5,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,5,4,2,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 1
[2,1,4,5,3] => [3,2]
=> [2]
=> 2
[2,1,5,3,4] => [3,2]
=> [2]
=> 2
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> 1
[2,3,1,5,4] => [3,2]
=> [2]
=> 2
[2,3,4,1,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[2,3,4,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[2,3,5,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[2,3,5,4,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[2,4,1,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[2,4,1,5,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> 1
[2,4,3,5,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[2,4,5,1,3] => [3,2]
=> [2]
=> 2
[2,4,5,3,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[2,5,1,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[2,5,1,4,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[2,5,3,1,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> 1
[2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[2,5,4,3,1] => [3,2]
=> [2]
=> 2
[3,1,2,4,5] => [3,1,1]
=> [1,1]
=> 1
[3,1,2,5,4] => [3,2]
=> [2]
=> 2
[3,1,4,2,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[3,1,4,5,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[3,1,5,2,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[3,1,5,4,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 1
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> 1
[3,2,4,5,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[3,2,5,1,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[3,2,5,4,1] => [3,1,1]
=> [1,1]
=> 1
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> 1
[3,4,1,5,2] => [3,2]
=> [2]
=> 2
[3,4,2,1,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[3,4,2,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[3,4,5,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[3,4,5,2,1] => [3,2]
=> [2]
=> 2
[3,5,2,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
Description
The number of positive values of the symmetric group character corresponding to the partition.
For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugacy class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $2$.
Matching statistic: St000476
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 80%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 80%
Values
[1] => [1]
=> []
=> []
=> ? = 0
[1,2] => [1,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0}
[2,1] => [2]
=> []
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,3,2] => [2,1]
=> [1]
=> [1,0]
=> ? ∊ {0,1,1,1,1}
[2,1,3] => [2,1]
=> [1]
=> [1,0]
=> ? ∊ {0,1,1,1,1}
[2,3,1] => [3]
=> []
=> []
=> ? ∊ {0,1,1,1,1}
[3,1,2] => [3]
=> []
=> []
=> ? ∊ {0,1,1,1,1}
[3,2,1] => [2,1]
=> [1]
=> [1,0]
=> ? ∊ {0,1,1,1,1}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,3,4,2] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2}
[1,4,2,3] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,1,4,3] => [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[2,3,1,4] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2}
[2,3,4,1] => [4]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2}
[2,4,1,3] => [4]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2}
[2,4,3,1] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2}
[3,1,2,4] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2}
[3,1,4,2] => [4]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[3,2,4,1] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2}
[3,4,1,2] => [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[3,4,2,1] => [4]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2}
[4,1,2,3] => [4]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2}
[4,1,3,2] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2}
[4,2,1,3] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[4,3,1,2] => [4]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2}
[4,3,2,1] => [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,3,4,5,2] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,3,5,2,4] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,4,2,5,3] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,4,5,3,2] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,5,2,3,4] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,5,4,2,3] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,4,5,3] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[2,1,5,3,4] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,3,1,5,4] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[2,3,4,1,5] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,3,4,5,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,3,5,1,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,3,5,4,1] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,4,1,3,5] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,4,1,5,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,4,3,5,1] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,4,5,1,3] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[2,4,5,3,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,5,1,3,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,5,1,4,3] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,5,3,1,4] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,5,4,1,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,5,4,3,1] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[3,1,2,4,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[3,1,2,5,4] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[3,1,4,2,5] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,1,4,5,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,1,5,2,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,1,5,4,2] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[3,2,4,5,1] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,2,5,1,4] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,2,5,4,1] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[3,4,1,5,2] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[3,4,2,1,5] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,4,2,5,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,4,5,1,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,4,5,2,1] => [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[3,5,2,1,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
Description
The sum of the semi-lengths of tunnels before a valley of a Dyck path.
For each valley $v$ in a Dyck path $D$ there is a corresponding tunnel, which
is the factor $T_v = s_i\dots s_j$ of $D$ where $s_i$ is the step after the first intersection of $D$ with the line $y = ht(v)$ to the left of $s_j$. This statistic is
$$
\sum_v (j_v-i_v)/2.
$$
Matching statistic: St000678
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 80%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 80%
Values
[1] => [1]
=> []
=> []
=> ? = 0
[1,2] => [1,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0}
[2,1] => [2]
=> []
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,3,2] => [2,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,1,1,1}
[2,1,3] => [2,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,1,1,1}
[2,3,1] => [3]
=> []
=> []
=> ? ∊ {0,0,1,1,1}
[3,1,2] => [3]
=> []
=> []
=> ? ∊ {0,0,1,1,1}
[3,2,1] => [2,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,1,1,1}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2}
[1,4,2,3] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,3,1,4] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2}
[2,3,4,1] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2}
[2,4,1,3] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2}
[2,4,3,1] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2}
[3,1,2,4] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2}
[3,1,4,2] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2}
[3,4,1,2] => [2,2]
=> [2]
=> [1,0,1,0]
=> 2
[3,4,2,1] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2}
[4,1,2,3] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2}
[4,1,3,2] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2}
[4,2,1,3] => [3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[4,3,1,2] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,2,2,2,2}
[4,3,2,1] => [2,2]
=> [2]
=> [1,0,1,0]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[1,3,5,2,4] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,4,2,5,3] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,4,5,3,2] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[1,5,2,3,4] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,5,4,2,3] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,4,5,3] => [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,1,5,3,4] => [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,3,1,5,4] => [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,3,4,1,5] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[2,3,4,5,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[2,3,5,1,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[2,3,5,4,1] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[2,4,1,3,5] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[2,4,1,5,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,4,3,5,1] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[2,4,5,1,3] => [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,4,5,3,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[2,5,1,3,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[2,5,1,4,3] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[2,5,3,1,4] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,5,4,1,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[2,5,4,3,1] => [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[3,1,2,4,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,1,2,5,4] => [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[3,1,4,2,5] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[3,1,4,5,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[3,1,5,2,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[3,1,5,4,2] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,2,4,5,1] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[3,2,5,1,4] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[3,2,5,4,1] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[3,4,1,5,2] => [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[3,4,2,1,5] => [4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[3,4,2,5,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[3,4,5,1,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
[3,4,5,2,1] => [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[3,5,2,1,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3}
Description
The number of up steps after the last double rise of a Dyck path.
The following 85 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000932The number of occurrences of the pattern UDU in a Dyck path. St000939The number of characters of the symmetric group whose value on the partition is positive. St000946The sum of the skew hook positions in a Dyck path. St000984The number of boxes below precisely one peak. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St001568The smallest positive integer that does not appear twice in the partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000707The product of the factorials of the parts. St000993The multiplicity of the largest part of an integer partition. St000260The radius of a connected graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000929The constant term of the character polynomial of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000137The Grundy value of an integer partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001249Sum of the odd parts of a partition. St001360The number of covering relations in Young's lattice below a partition. St001383The BG-rank of an integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001933The largest multiplicity of a part in an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000456The monochromatic index of a connected graph. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000284The Plancherel distribution on integer partitions. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. 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. St001128The exponens consonantiae of a partition. St001060The distinguishing index of a graph. St001877Number of indecomposable injective modules with projective dimension 2. St000259The diameter of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000478Another weight of a partition according to Alladi. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001570The minimal number of edges to add to make a graph Hamiltonian. St001820The size of the image of the pop stack sorting operator. St001720The minimal length of a chain of small intervals in a lattice. St000454The largest eigenvalue of a graph if it is integral. St001875The number of simple modules with projective dimension at most 1. St001904The length of the initial strictly increasing segment of a parking function. St000850The number of 1/2-balanced pairs in a poset. St000307The number of rowmotion orbits of a poset. St000632The jump number of the poset. St001117The game chromatic index of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001742The difference of the maximal and the minimal degree in a graph. St001734The lettericity of a graph. St000077The number of boxed and circled entries. St001575The minimal number of edges to add or remove to make a graph edge transitive.
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