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Your data matches 139 different statistics following compositions of up to 3 maps.
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Matching statistic: St000665
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
St000665: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 1
[2,1] => 0
[1,2,3] => 1
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 0
[1,2,3,4] => 1
[1,2,4,3] => 1
[1,3,2,4] => 0
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 0
[2,1,3,4] => 1
[2,1,4,3] => 0
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 0
[2,4,3,1] => 0
[3,1,2,4] => 1
[3,1,4,2] => 0
[3,2,1,4] => 0
[3,2,4,1] => 0
[3,4,1,2] => 2
[3,4,2,1] => 1
[4,1,2,3] => 1
[4,1,3,2] => 0
[4,2,1,3] => 0
[4,2,3,1] => 1
[4,3,1,2] => 1
[4,3,2,1] => 0
[1,2,3,4,5] => 1
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 2
[1,2,5,3,4] => 2
[1,2,5,4,3] => 1
[1,3,2,4,5] => 1
[1,3,2,5,4] => 0
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 0
[1,3,5,4,2] => 0
[1,4,2,3,5] => 1
[1,4,2,5,3] => 0
[1,4,3,2,5] => 0
[1,4,3,5,2] => 0
[1,4,5,2,3] => 2
Description
The number of rafts of a permutation.
Let $\pi$ be a permutation of length $n$. A small ascent of $\pi$ is an index $i$ such that $\pi(i+1)= \pi(i)+1$, see [[St000441]], and a raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents.
Matching statistic: St001587
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001587: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 86%●distinct values known / distinct values provided: 75%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001587: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 86%●distinct values known / distinct values provided: 75%
Values
[1] => [1] => [[1],[]]
=> []
=> ? = 0
[1,2] => [2] => [[2],[]]
=> []
=> ? ∊ {0,1}
[2,1] => [2] => [[2],[]]
=> []
=> ? ∊ {0,1}
[1,2,3] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1,1}
[1,3,2] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,1,1,1}
[2,1,3] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1,1}
[2,3,1] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1,1}
[3,1,2] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1,1}
[3,2,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[1,2,3,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2}
[1,2,4,3] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[1,3,2,4] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2}
[1,3,4,2] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2}
[1,4,2,3] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2}
[1,4,3,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[2,1,3,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2}
[2,1,4,3] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[2,3,1,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2}
[2,3,4,1] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2}
[2,4,1,3] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2}
[2,4,3,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[3,1,2,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2}
[3,1,4,2] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[3,2,1,4] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[3,2,4,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[3,4,1,2] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2}
[3,4,2,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[4,1,2,3] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2}
[4,1,3,2] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[4,2,1,3] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[4,2,3,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[4,3,1,2] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,2}
[4,3,2,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[1,2,3,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,2,3,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[1,2,4,3,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[1,2,4,5,3] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[1,2,5,3,4] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[1,2,5,4,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,3,2,4,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,5,4] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,3,4,2,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[1,4,2,3,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,4,3,2,5] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,4,3,5,2] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,4,5,2,3] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,3,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[1,5,2,3,4] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,4,3] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[1,5,3,2,4] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,5,3,4,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[1,5,4,2,3] => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[2,1,3,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,1,3,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[2,1,4,3,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,1,4,5,3] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,1,5,3,4] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,1,5,4,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[2,3,1,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,3,1,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[2,3,4,1,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,5,1] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,4,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[2,4,1,3,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[2,4,3,1,5] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[2,4,3,5,1] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[2,4,5,1,3] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,3,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[2,5,1,3,4] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,4,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[2,5,3,1,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[2,5,3,4,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[2,5,4,1,3] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,5,4,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[3,1,2,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,1,2,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[3,1,4,2,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,1,4,5,2] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,1,5,2,4] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,1,5,4,2] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[3,2,1,4,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,4,1,2,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,4,5,1,2] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,5,1,2,4] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[4,1,2,3,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[4,3,1,2,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[4,3,5,1,2] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[4,5,1,2,3] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[5,1,2,3,4] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[5,3,1,2,4] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[5,3,4,1,2] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[5,4,1,2,3] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,2,3,4,5,6] => [6] => [[6],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
Description
Half of the largest even part of an integer partition.
The largest even part is recorded by [[St000995]].
Matching statistic: St001767
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001767: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 82%●distinct values known / distinct values provided: 75%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001767: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 82%●distinct values known / distinct values provided: 75%
Values
[1] => [1]
=> []
=> ? = 0
[1,2] => [1,1]
=> [1]
=> 0
[2,1] => [2]
=> []
=> ? = 1
[1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,3,2] => [2,1]
=> [1]
=> 0
[2,1,3] => [2,1]
=> [1]
=> 0
[2,3,1] => [3]
=> []
=> ? ∊ {1,1}
[3,1,2] => [3]
=> []
=> ? ∊ {1,1}
[3,2,1] => [2,1]
=> [1]
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> 0
[1,4,2,3] => [3,1]
=> [1]
=> 0
[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]
=> 1
[2,3,1,4] => [3,1]
=> [1]
=> 0
[2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,1,1,2}
[2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,1,1,2}
[2,4,3,1] => [3,1]
=> [1]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> 0
[3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,0,1,1,2}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> 1
[3,4,2,1] => [4]
=> []
=> ? ∊ {0,0,0,1,1,2}
[4,1,2,3] => [4]
=> []
=> ? ∊ {0,0,0,1,1,2}
[4,1,3,2] => [3,1]
=> [1]
=> 0
[4,2,1,3] => [3,1]
=> [1]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[4,3,1,2] => [4]
=> []
=> ? ∊ {0,0,0,1,1,2}
[4,3,2,1] => [2,2]
=> [2]
=> 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 2
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 0
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> 0
[1,3,5,2,4] => [4,1]
=> [1]
=> 0
[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
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> 0
[1,4,5,3,2] => [4,1]
=> [1]
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,5,4,2,3] => [4,1]
=> [1]
=> 0
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 0
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 0
[2,3,4,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,4,5,3,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,5,1,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[3,1,4,5,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[3,1,5,2,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[3,4,2,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[3,4,5,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[3,5,2,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[3,5,4,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[4,1,2,5,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[4,1,5,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[4,3,1,5,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[4,3,5,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[4,5,1,2,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[4,5,2,3,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[5,1,2,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[5,1,4,2,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[5,3,1,2,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[5,3,4,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[5,4,1,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[5,4,2,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,3,4,5,6,1] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,4,6,1,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,5,1,6,4] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,5,6,4,1] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,6,1,4,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,6,5,1,4] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,1,5,6,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,1,6,3,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,5,3,6,1] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,5,6,1,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,6,3,1,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,6,5,3,1] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,5,1,3,6,4] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,5,1,6,4,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,5,4,1,6,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,5,4,6,3,1] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
Description
The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment.
Assign to each cell of the Ferrers diagram an arrow pointing north, east, south or west. Then compute for each cell the number of arrows pointing towards it, and take the minimum of those. This statistic is the maximal minimum that can be obtained by assigning arrows in any way.
Matching statistic: St000147
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 82%●distinct values known / distinct values provided: 75%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 82%●distinct values known / distinct values provided: 75%
Values
[1] => [1]
=> []
=> ?
=> ? = 0
[1,2] => [1,1]
=> [1]
=> []
=> 0
[2,1] => [2]
=> []
=> ?
=> ? = 1
[1,2,3] => [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,2] => [2,1]
=> [1]
=> []
=> 0
[2,1,3] => [2,1]
=> [1]
=> []
=> 0
[2,3,1] => [3]
=> []
=> ?
=> ? ∊ {1,1}
[3,1,2] => [3]
=> []
=> ?
=> ? ∊ {1,1}
[3,2,1] => [2,1]
=> [1]
=> []
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> []
=> 0
[1,4,2,3] => [3,1]
=> [1]
=> []
=> 0
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> []
=> 0
[2,3,1,4] => [3,1]
=> [1]
=> []
=> 0
[2,3,4,1] => [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,2}
[2,4,1,3] => [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,2}
[2,4,3,1] => [3,1]
=> [1]
=> []
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> []
=> 0
[3,1,4,2] => [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,2}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> []
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> []
=> 0
[3,4,2,1] => [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,2}
[4,1,2,3] => [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,2}
[4,1,3,2] => [3,1]
=> [1]
=> []
=> 0
[4,2,1,3] => [3,1]
=> [1]
=> []
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,3,1,2] => [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,2}
[4,3,2,1] => [2,2]
=> [2]
=> []
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> 0
[1,3,5,2,4] => [4,1]
=> [1]
=> []
=> 0
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,2,5,3] => [4,1]
=> [1]
=> []
=> 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[1,4,5,3,2] => [4,1]
=> [1]
=> []
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> 0
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,5,4,2,3] => [4,1]
=> [1]
=> []
=> 0
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,3,4,5,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,3,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,3,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,5,4,1,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,1,4,5,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,1,5,2,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,4,2,5,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,4,5,1,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,5,2,1,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,5,4,2,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[4,1,2,5,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[4,1,5,3,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[4,3,1,5,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[4,3,5,2,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[4,5,1,2,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[4,5,2,3,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[5,1,2,3,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[5,1,4,2,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[5,3,1,2,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[5,3,4,1,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[5,4,1,3,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[5,4,2,1,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,5,6,1] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,4,6,1,5] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,5,1,6,4] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,5,6,4,1] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,6,1,4,5] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,6,5,1,4] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,1,5,6,3] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,1,6,3,5] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,5,3,6,1] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,5,6,1,3] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,6,3,1,5] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,6,5,3,1] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,5,1,3,6,4] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,5,1,6,4,3] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,5,4,1,6,3] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,5,4,6,3,1] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
Description
The largest part of an integer partition.
Matching statistic: St000257
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 82%●distinct values known / distinct values provided: 75%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 82%●distinct values known / distinct values provided: 75%
Values
[1] => [1,0]
=> []
=> ?
=> ? = 0
[1,2] => [1,0,1,0]
=> [1]
=> [1]
=> 0
[2,1] => [1,1,0,0]
=> []
=> ?
=> ? = 1
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> [1,1,1]
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> [2]
=> 0
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> [1,1]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> [1]
=> 0
[3,1,2] => [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,1}
[3,2,1] => [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,1}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [3,1,1,1]
=> 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,1,1]
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [3,2]
=> 0
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [2,1,1]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [2,1]
=> 0
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [2,1]
=> 0
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> [3,1,1]
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,1]
=> 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> [3,1]
=> 0
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,1,1]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [2]
=> 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [2]
=> 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [3]
=> 0
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [3]
=> 0
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 0
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 0
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,2}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,2}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,2}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,2}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,2}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,2}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,2,1,1,1]
=> 2
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [6,1,1,1]
=> 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1,1,1,1,1]
=> 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [3,1,1,1,1,1]
=> 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,1,1,1]
=> 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,1,1,1]
=> 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,2,2,2]
=> 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [6,2]
=> 0
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,2,2,1,1]
=> 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [3,2,1,1]
=> 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1,1,1]
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1,1,1]
=> 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [2,2,2,1]
=> 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [3,2,1]
=> 0
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [2,2,2,1]
=> 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [3,2,1]
=> 0
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [2,1,1,1]
=> 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [2,1,1,1]
=> 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [4]
=> 0
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [4]
=> 0
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [4]
=> 0
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [4]
=> 0
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [4]
=> 0
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [4]
=> 0
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2,2,1,1]
=> 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [6,1,1]
=> 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2,1,1,1,1]
=> 2
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2}
[6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[6,1,2,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[6,1,2,4,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[6,1,2,5,4,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[6,1,3,2,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[6,1,3,4,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[6,1,3,4,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[6,1,3,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[6,1,3,5,4,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[6,1,4,2,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[6,1,4,3,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[6,1,4,3,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
Description
The number of distinct parts of a partition that occur at least twice.
See Section 3.3.1 of [2].
Matching statistic: St000329
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 82%●distinct values known / distinct values provided: 75%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 82%●distinct values known / distinct values provided: 75%
Values
[1] => [1]
=> []
=> []
=> ? = 0
[1,2] => [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[2,1] => [2]
=> []
=> []
=> ? = 1
[1,2,3] => [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,2] => [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[2,1,3] => [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[2,3,1] => [3]
=> []
=> []
=> ? ∊ {1,1}
[3,1,2] => [3]
=> []
=> []
=> ? ∊ {1,1}
[3,2,1] => [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,4,2,3] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[2,3,1,4] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[2,3,4,1] => [4]
=> []
=> []
=> ? ∊ {0,0,0,1,1,2}
[2,4,1,3] => [4]
=> []
=> []
=> ? ∊ {0,0,0,1,1,2}
[2,4,3,1] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[3,1,4,2] => [4]
=> []
=> []
=> ? ∊ {0,0,0,1,1,2}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[3,4,2,1] => [4]
=> []
=> []
=> ? ∊ {0,0,0,1,1,2}
[4,1,2,3] => [4]
=> []
=> []
=> ? ∊ {0,0,0,1,1,2}
[4,1,3,2] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[4,2,1,3] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[4,3,1,2] => [4]
=> []
=> []
=> ? ∊ {0,0,0,1,1,2}
[4,3,2,1] => [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,3,5,2,4] => [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,4,2,5,3] => [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[1,4,5,3,2] => [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,5,4,2,3] => [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[2,3,4,5,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,4,5,3,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,5,1,3,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,5,4,1,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[3,1,4,5,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[3,1,5,2,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[3,4,2,5,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[3,4,5,1,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[3,5,2,1,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[3,5,4,2,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[4,1,2,5,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[4,1,5,3,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[4,3,1,5,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[4,3,5,2,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[4,5,1,2,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[4,5,2,3,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[5,1,2,3,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[5,1,4,2,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[5,3,1,2,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[5,3,4,1,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[5,4,1,3,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[5,4,2,1,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,3,4,5,6,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,4,6,1,5] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,5,1,6,4] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,5,6,4,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,6,1,4,5] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,6,5,1,4] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,1,5,6,3] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,1,6,3,5] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,5,3,6,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,5,6,1,3] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,6,3,1,5] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,6,5,3,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,5,1,3,6,4] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,5,1,6,4,3] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,5,4,1,6,3] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,5,4,6,3,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
Description
The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.
Matching statistic: St000378
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 82%●distinct values known / distinct values provided: 75%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 82%●distinct values known / distinct values provided: 75%
Values
[1] => [1]
=> []
=> ?
=> ? = 0
[1,2] => [1,1]
=> [1]
=> []
=> 0
[2,1] => [2]
=> []
=> ?
=> ? = 1
[1,2,3] => [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,2] => [2,1]
=> [1]
=> []
=> 0
[2,1,3] => [2,1]
=> [1]
=> []
=> 0
[2,3,1] => [3]
=> []
=> ?
=> ? ∊ {1,1}
[3,1,2] => [3]
=> []
=> ?
=> ? ∊ {1,1}
[3,2,1] => [2,1]
=> [1]
=> []
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> []
=> 0
[1,4,2,3] => [3,1]
=> [1]
=> []
=> 0
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> []
=> 0
[2,3,1,4] => [3,1]
=> [1]
=> []
=> 0
[2,3,4,1] => [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,2}
[2,4,1,3] => [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,2}
[2,4,3,1] => [3,1]
=> [1]
=> []
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> []
=> 0
[3,1,4,2] => [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,2}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> []
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> []
=> 0
[3,4,2,1] => [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,2}
[4,1,2,3] => [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,2}
[4,1,3,2] => [3,1]
=> [1]
=> []
=> 0
[4,2,1,3] => [3,1]
=> [1]
=> []
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,3,1,2] => [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,2}
[4,3,2,1] => [2,2]
=> [2]
=> []
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> 0
[1,3,5,2,4] => [4,1]
=> [1]
=> []
=> 0
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,2,5,3] => [4,1]
=> [1]
=> []
=> 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[1,4,5,3,2] => [4,1]
=> [1]
=> []
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> 0
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,5,4,2,3] => [4,1]
=> [1]
=> []
=> 0
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,3,4,5,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,3,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,3,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,5,4,1,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,1,4,5,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,1,5,2,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,4,2,5,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,4,5,1,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,5,2,1,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,5,4,2,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[4,1,2,5,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[4,1,5,3,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[4,3,1,5,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[4,3,5,2,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[4,5,1,2,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[4,5,2,3,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[5,1,2,3,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[5,1,4,2,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[5,3,1,2,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[5,3,4,1,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[5,4,1,3,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[5,4,2,1,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,5,6,1] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,4,6,1,5] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,5,1,6,4] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,5,6,4,1] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,6,1,4,5] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,6,5,1,4] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,1,5,6,3] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,1,6,3,5] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,5,3,6,1] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,5,6,1,3] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,6,3,1,5] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,6,5,3,1] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,5,1,3,6,4] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,5,1,6,4,3] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,5,4,1,6,3] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,5,4,6,3,1] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
Description
The diagonal inversion number of an integer partition.
The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$.
See also exercise 3.19 of [2].
This statistic is equidistributed with the length of the partition, see [3].
Matching statistic: St000389
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000389: Binary words ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000389: Binary words ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> []
=> ? => ? = 0
[1,2] => [1,0,1,0]
=> [1]
=> 1 => 1
[2,1] => [1,1,0,0]
=> []
=> ? => ? = 0
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 01 => 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 11 => 0
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 0 => 0
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 1 => 1
[3,1,2] => [1,1,1,0,0,0]
=> []
=> ? => ? ∊ {0,1}
[3,2,1] => [1,1,1,0,0,0]
=> []
=> ? => ? ∊ {0,1}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101 => 2
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 001 => 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 111 => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 011 => 0
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 111 => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 111 => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 10 => 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 00 => 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 11 => 0
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 01 => 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 11 => 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 11 => 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1 => 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> 0 => 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1 => 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> 0 => 0
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1 => 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1 => 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 0101 => 2
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1101 => 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 0001 => 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1001 => 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 0001 => 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 0001 => 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 0111 => 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1111 => 0
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 0011 => 0
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1011 => 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 0011 => 0
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 0011 => 0
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 0111 => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1111 => 0
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 0111 => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1111 => 0
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 0111 => 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 0111 => 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1111 => 0
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1111 => 0
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1111 => 0
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1111 => 0
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1111 => 0
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1111 => 0
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 010 => 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 110 => 0
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 000 => 0
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,2,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,2,4,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,2,5,4,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,3,2,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,3,4,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,3,4,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,3,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,3,5,4,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,4,2,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,4,3,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,4,3,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
Description
The number of runs of ones of odd length in a binary word.
Matching statistic: St000547
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000547: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 82%●distinct values known / distinct values provided: 75%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000547: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 82%●distinct values known / distinct values provided: 75%
Values
[1] => [1]
=> []
=> ?
=> ? = 0
[1,2] => [1,1]
=> [1]
=> [1]
=> 0
[2,1] => [2]
=> []
=> ?
=> ? = 1
[1,2,3] => [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,3,2] => [2,1]
=> [1]
=> [1]
=> 0
[2,1,3] => [2,1]
=> [1]
=> [1]
=> 0
[2,3,1] => [3]
=> []
=> ?
=> ? ∊ {1,1}
[3,1,2] => [3]
=> []
=> ?
=> ? ∊ {1,1}
[3,2,1] => [2,1]
=> [1]
=> [1]
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> [1]
=> 0
[1,4,2,3] => [3,1]
=> [1]
=> [1]
=> 0
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> [2]
=> 1
[2,3,1,4] => [3,1]
=> [1]
=> [1]
=> 0
[2,3,4,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,2}
[2,4,1,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,2}
[2,4,3,1] => [3,1]
=> [1]
=> [1]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> [1]
=> 0
[3,1,4,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,2}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> [1]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> [2]
=> 1
[3,4,2,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,2}
[4,1,2,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,2}
[4,1,3,2] => [3,1]
=> [1]
=> [1]
=> 0
[4,2,1,3] => [3,1]
=> [1]
=> [1]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[4,3,1,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,2}
[4,3,2,1] => [2,2]
=> [2]
=> [2]
=> 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1,1]
=> 2
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [3]
=> 0
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> [1]
=> 0
[1,3,5,2,4] => [4,1]
=> [1]
=> [1]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,4,2,5,3] => [4,1]
=> [1]
=> [1]
=> 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [3]
=> 0
[1,4,5,3,2] => [4,1]
=> [1]
=> [1]
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> [1]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 1
[1,5,4,2,3] => [4,1]
=> [1]
=> [1]
=> 0
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [3]
=> 0
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [3]
=> 0
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [3]
=> 0
[2,3,4,5,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,4,5,3,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,5,1,3,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,5,4,1,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[3,1,4,5,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[3,1,5,2,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[3,4,2,5,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[3,4,5,1,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[3,5,2,1,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[3,5,4,2,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[4,1,2,5,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[4,1,5,3,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[4,3,1,5,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[4,3,5,2,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[4,5,1,2,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[4,5,2,3,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[5,1,2,3,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[5,1,4,2,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[5,3,1,2,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[5,3,4,1,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[5,4,1,3,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[5,4,2,1,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,3,4,5,6,1] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,4,6,1,5] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,5,1,6,4] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,5,6,4,1] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,6,1,4,5] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,6,5,1,4] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,1,5,6,3] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,1,6,3,5] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,5,3,6,1] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,5,6,1,3] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,6,3,1,5] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,6,5,3,1] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,5,1,3,6,4] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,5,1,6,4,3] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,5,4,1,6,3] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,5,4,6,3,1] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
Description
The number of even non-empty partial sums of an integer partition.
Matching statistic: St000549
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000549: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 82%●distinct values known / distinct values provided: 75%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000549: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 82%●distinct values known / distinct values provided: 75%
Values
[1] => [1]
=> []
=> ?
=> ? = 0
[1,2] => [1,1]
=> [1]
=> []
=> 0
[2,1] => [2]
=> []
=> ?
=> ? = 1
[1,2,3] => [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,2] => [2,1]
=> [1]
=> []
=> 0
[2,1,3] => [2,1]
=> [1]
=> []
=> 0
[2,3,1] => [3]
=> []
=> ?
=> ? ∊ {1,1}
[3,1,2] => [3]
=> []
=> ?
=> ? ∊ {1,1}
[3,2,1] => [2,1]
=> [1]
=> []
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> []
=> 0
[1,4,2,3] => [3,1]
=> [1]
=> []
=> 0
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> []
=> 0
[2,3,1,4] => [3,1]
=> [1]
=> []
=> 0
[2,3,4,1] => [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,2}
[2,4,1,3] => [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,2}
[2,4,3,1] => [3,1]
=> [1]
=> []
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> []
=> 0
[3,1,4,2] => [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,2}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> []
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> []
=> 0
[3,4,2,1] => [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,2}
[4,1,2,3] => [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,2}
[4,1,3,2] => [3,1]
=> [1]
=> []
=> 0
[4,2,1,3] => [3,1]
=> [1]
=> []
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,3,1,2] => [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,2}
[4,3,2,1] => [2,2]
=> [2]
=> []
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> 0
[1,3,5,2,4] => [4,1]
=> [1]
=> []
=> 0
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,2,5,3] => [4,1]
=> [1]
=> []
=> 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[1,4,5,3,2] => [4,1]
=> [1]
=> []
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> 0
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,5,4,2,3] => [4,1]
=> [1]
=> []
=> 0
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,3,4,5,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,4,5,3,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,5,1,3,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,5,4,1,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[3,1,4,5,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[3,1,5,2,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[3,4,2,5,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[3,4,5,1,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[3,5,2,1,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[3,5,4,2,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[4,1,2,5,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[4,1,5,3,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[4,3,1,5,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[4,3,5,2,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[4,5,1,2,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[4,5,2,3,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[5,1,2,3,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[5,1,4,2,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[5,3,1,2,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[5,3,4,1,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[5,4,1,3,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[5,4,2,1,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,3,4,5,6,1] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,4,6,1,5] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,5,1,6,4] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,5,6,4,1] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,6,1,4,5] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,6,5,1,4] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,1,5,6,3] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,1,6,3,5] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,5,3,6,1] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,5,6,1,3] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,6,3,1,5] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,6,5,3,1] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,5,1,3,6,4] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,5,1,6,4,3] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,5,4,1,6,3] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,5,4,6,3,1] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
Description
The number of odd partial sums of an integer partition.
The following 129 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000659The number of rises of length at least 2 of a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001280The number of parts of an integer partition that are at least two. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001939The number of parts that are equal to their multiplicity in the integer partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000934The 2-degree of an integer partition. St000929The constant term of the character polynomial of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001525The number of symmetric hooks on the diagonal of a partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000938The number of zeros of the symmetric group character corresponding to the partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000940The number of characters of the symmetric group whose value on the partition is zero. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000941The number of characters of the symmetric group whose value on the partition is even. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000658The number of rises of length 2 of a Dyck path. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001139The number of occurrences of hills of size 2 in a Dyck path. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001877Number of indecomposable injective modules with projective dimension 2. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000137The Grundy value of an integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000478Another weight of a partition according to Alladi. St000260The radius of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St000567The sum of the products of all pairs of parts. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001128The exponens consonantiae of a partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000681The Grundy value of Chomp on Ferrers diagrams. St000937The number of positive values of the symmetric group character corresponding to the partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001176The size of a partition minus its first part. St001383The BG-rank of an integer partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001389The number of partitions of the same length below the given integer partition. St001571The Cartan determinant of the integer partition. 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. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000454The largest eigenvalue of a graph if it is integral. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000618The number of self-evacuating tableaux of given shape. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000781The number of proper colouring schemes of a Ferrers diagram. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001432The order dimension of the partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000456The monochromatic index of a connected graph. St001568The smallest positive integer that does not appear twice in the partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000284The Plancherel distribution on integer partitions. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St001651The Frankl number of a lattice. St000764The number of strong records in an integer composition. St000741The Colin de Verdière graph invariant. St001570The minimal number of edges to add to make a graph Hamiltonian. 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. St001895The oddness of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001937The size of the center of a parking function. St001621The number of atoms of a lattice.
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