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Your data matches 38 different statistics following compositions of up to 3 maps.
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Matching statistic: St001523
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
St001523: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1
[1,0,1,0]
=> 2
[1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> 3
[1,1,1,0,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> 4
[1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> 4
[1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> 4
[1,1,0,1,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> 4
[1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> 5
[1,1,0,1,0,1,1,0,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> 5
[1,1,0,1,1,0,1,0,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> 2
Description
The degree of symmetry of a Dyck path.
Given a Dyck path $D=(d_1,\dots,d_{2n})$, with $d_i\in\{0,1\}$, this is the number of positions $1\leq i\leq n$ such that $d_i = 1-d_{2n+1-i}$ and the initial height of the $i$-th step $\sum_{j < i} d_i$ equals the final height of the $(2n+1-i)$-th step.
Matching statistic: St000460
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 71%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> [1] => [[1],[]]
=> []
=> ? = 1
[1,0,1,0]
=> [1,1] => [[1,1],[]]
=> []
=> ? ∊ {2,2}
[1,1,0,0]
=> [2] => [[2],[]]
=> []
=> ? ∊ {2,2}
[1,0,1,0,1,0]
=> [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {1,3,3,3}
[1,0,1,1,0,0]
=> [1,2] => [[2,1],[]]
=> []
=> ? ∊ {1,3,3,3}
[1,1,0,0,1,0]
=> [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [3] => [[3],[]]
=> []
=> ? ∊ {1,3,3,3}
[1,1,1,0,0,0]
=> [3] => [[3],[]]
=> []
=> ? ∊ {1,3,3,3}
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {2,2,2,4,4,4,4,4,4}
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {2,2,2,4,4,4,4,4,4}
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3] => [[3,1],[]]
=> []
=> ? ∊ {2,2,2,4,4,4,4,4,4}
[1,0,1,1,1,0,0,0]
=> [1,3] => [[3,1],[]]
=> []
=> ? ∊ {2,2,2,4,4,4,4,4,4}
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2] => [[3,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1] => [[3,3],[2]]
=> [2]
=> 2
[1,1,0,1,0,1,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {2,2,2,4,4,4,4,4,4}
[1,1,0,1,1,0,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {2,2,2,4,4,4,4,4,4}
[1,1,1,0,0,0,1,0]
=> [3,1] => [[3,3],[2]]
=> [2]
=> 2
[1,1,1,0,0,1,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {2,2,2,4,4,4,4,4,4}
[1,1,1,0,1,0,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {2,2,2,4,4,4,4,4,4}
[1,1,1,1,0,0,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {2,2,2,4,4,4,4,4,4}
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [[4,2],[1]]
=> [1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [[4,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [[4,3],[2]]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [[4,3],[2]]
=> [2]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 3
[1,1,1,0,0,1,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,0,1,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,1,0,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,1,1,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 3
[1,1,1,1,0,0,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,1,0,0,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,1,0,1,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,1,1,0,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 3
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 3
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,3] => [[4,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,3] => [[4,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 3
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,5] => [[5,1],[]]
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,5] => [[5,1],[]]
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 3
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 3
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 3
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 3
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 4
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 4
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 1
Description
The hook length of the last cell along the main diagonal of an integer partition.
Matching statistic: St001199
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 71%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> ? = 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {2,2}
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {2,2}
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {1,1,3,3,3}
[1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {1,1,3,3,3}
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {1,1,3,3,3}
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {1,1,3,3,3}
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {1,1,3,3,3}
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 3
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 3
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 1
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001498
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 57%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 57%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> ? = 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> ? ∊ {2,2}
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> ? ∊ {2,2}
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> ? ∊ {1,1,3,3,3}
[1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> ? ∊ {1,1,3,3,3}
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> ? ∊ {1,1,3,3,3}
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> ? ∊ {1,1,3,3,3}
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> ? ∊ {1,1,3,3,3}
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 3
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 3
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 3
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2
Description
The normalised height of a Nakayama algebra with magnitude 1.
We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
Matching statistic: St001933
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 57%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 57%
Values
[1,0]
=> [1,0]
=> [1,0]
=> []
=> ? = 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {2,2}
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {2,2}
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {1,1,3,3,3}
[1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {1,1,3,3,3}
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {1,1,3,3,3}
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {1,1,3,3,3}
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {1,1,3,3,3}
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 3
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> 3
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 3
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> 3
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 3
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> 3
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> 3
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> 3
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> 3
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> 3
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> 2
Description
The largest multiplicity of a part in an integer partition.
Matching statistic: St000993
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 57%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 57%
Values
[1,0]
=> [1,0]
=> [1,0]
=> []
=> ? = 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {2,2}
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {2,2}
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {1,1,3,3,3}
[1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {1,1,3,3,3}
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {1,1,3,3,3}
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {1,1,3,3,3}
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {1,1,3,3,3}
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 3
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> 3
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 3
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> 3
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 3
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> 3
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> 2
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> 2
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> 2
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> 2
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St000208
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 43%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 43%
Values
[1,0]
=> [2,1] => [2]
=> []
=> ? = 1
[1,0,1,0]
=> [3,1,2] => [3]
=> []
=> ? ∊ {2,2}
[1,1,0,0]
=> [2,3,1] => [3]
=> []
=> ? ∊ {2,2}
[1,0,1,0,1,0]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {1,1,3,3,3}
[1,0,1,1,0,0]
=> [3,1,4,2] => [4]
=> []
=> ? ∊ {1,1,3,3,3}
[1,1,0,0,1,0]
=> [2,4,1,3] => [4]
=> []
=> ? ∊ {1,1,3,3,3}
[1,1,0,1,0,0]
=> [4,3,1,2] => [4]
=> []
=> ? ∊ {1,1,3,3,3}
[1,1,1,0,0,0]
=> [2,3,4,1] => [4]
=> []
=> ? ∊ {1,1,3,3,3}
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [3,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [4,2]
=> [2]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [3,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [3,3]
=> [3]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [4,2]
=> [2]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [3,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [4,2]
=> [2]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [4,3]
=> [3]
=> 3
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [4,3]
=> [3]
=> 3
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [5,2]
=> [2]
=> 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [4,3]
=> [3]
=> 3
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [5,2]
=> [2]
=> 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => [5,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => [4,3]
=> [3]
=> 3
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => [5,2]
=> [2]
=> 2
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => [5,2]
=> [2]
=> 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [5,2]
=> [2]
=> 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => [5,2]
=> [2]
=> 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => [5,2]
=> [2]
=> 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => [5,2]
=> [2]
=> 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => [5,2]
=> [2]
=> 2
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has only integer lattice points as vertices.
See also [[St000205]], [[St000206]] and [[St000207]].
Matching statistic: St000474
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000474: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 43%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000474: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 43%
Values
[1,0]
=> [2,1] => [2]
=> []
=> ? = 1
[1,0,1,0]
=> [3,1,2] => [3]
=> []
=> ? ∊ {2,2}
[1,1,0,0]
=> [2,3,1] => [3]
=> []
=> ? ∊ {2,2}
[1,0,1,0,1,0]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {1,1,3,3,3}
[1,0,1,1,0,0]
=> [3,1,4,2] => [4]
=> []
=> ? ∊ {1,1,3,3,3}
[1,1,0,0,1,0]
=> [2,4,1,3] => [4]
=> []
=> ? ∊ {1,1,3,3,3}
[1,1,0,1,0,0]
=> [4,3,1,2] => [4]
=> []
=> ? ∊ {1,1,3,3,3}
[1,1,1,0,0,0]
=> [2,3,4,1] => [4]
=> []
=> ? ∊ {1,1,3,3,3}
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [3,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [4,2]
=> [2]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [3,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [3,3]
=> [3]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [4,2]
=> [2]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [3,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [4,2]
=> [2]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [4,3]
=> [3]
=> 3
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [4,3]
=> [3]
=> 3
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [5,2]
=> [2]
=> 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [4,3]
=> [3]
=> 3
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [5,2]
=> [2]
=> 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => [5,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => [4,3]
=> [3]
=> 3
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => [5,2]
=> [2]
=> 2
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => [5,2]
=> [2]
=> 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [5,2]
=> [2]
=> 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => [5,2]
=> [2]
=> 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => [5,2]
=> [2]
=> 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => [5,2]
=> [2]
=> 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => [5,2]
=> [2]
=> 2
Description
Dyson's crank of a partition.
Let $\lambda$ be a partition and let $o(\lambda)$ be the number of parts that are equal to 1 ([[St000475]]), and let $\mu(\lambda)$ be the number of parts that are strictly larger than $o(\lambda)$ ([[St000473]]). Dyson's crank is then defined as
$$crank(\lambda) = \begin{cases} \text{ largest part of }\lambda & o(\lambda) = 0\\ \mu(\lambda) - o(\lambda) & o(\lambda) > 0. \end{cases}$$
Matching statistic: St000667
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 43%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 43%
Values
[1,0]
=> [2,1] => [2]
=> []
=> ? = 1
[1,0,1,0]
=> [3,1,2] => [3]
=> []
=> ? ∊ {2,2}
[1,1,0,0]
=> [2,3,1] => [3]
=> []
=> ? ∊ {2,2}
[1,0,1,0,1,0]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {1,1,3,3,3}
[1,0,1,1,0,0]
=> [3,1,4,2] => [4]
=> []
=> ? ∊ {1,1,3,3,3}
[1,1,0,0,1,0]
=> [2,4,1,3] => [4]
=> []
=> ? ∊ {1,1,3,3,3}
[1,1,0,1,0,0]
=> [4,3,1,2] => [4]
=> []
=> ? ∊ {1,1,3,3,3}
[1,1,1,0,0,0]
=> [2,3,4,1] => [4]
=> []
=> ? ∊ {1,1,3,3,3}
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [3,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [4,2]
=> [2]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [3,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [3,3]
=> [3]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [4,2]
=> [2]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [3,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [4,2]
=> [2]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [4,3]
=> [3]
=> 3
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [4,3]
=> [3]
=> 3
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [5,2]
=> [2]
=> 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [4,3]
=> [3]
=> 3
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [5,2]
=> [2]
=> 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => [5,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => [4,3]
=> [3]
=> 3
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => [5,2]
=> [2]
=> 2
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => [5,2]
=> [2]
=> 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [5,2]
=> [2]
=> 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => [5,2]
=> [2]
=> 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => [5,2]
=> [2]
=> 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => [5,2]
=> [2]
=> 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => [5,2]
=> [2]
=> 2
Description
The greatest common divisor of the parts of the partition.
Matching statistic: St000668
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 43%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 43%
Values
[1,0]
=> [2,1] => [2]
=> []
=> ? = 1
[1,0,1,0]
=> [3,1,2] => [3]
=> []
=> ? ∊ {2,2}
[1,1,0,0]
=> [2,3,1] => [3]
=> []
=> ? ∊ {2,2}
[1,0,1,0,1,0]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {1,1,3,3,3}
[1,0,1,1,0,0]
=> [3,1,4,2] => [4]
=> []
=> ? ∊ {1,1,3,3,3}
[1,1,0,0,1,0]
=> [2,4,1,3] => [4]
=> []
=> ? ∊ {1,1,3,3,3}
[1,1,0,1,0,0]
=> [4,3,1,2] => [4]
=> []
=> ? ∊ {1,1,3,3,3}
[1,1,1,0,0,0]
=> [2,3,4,1] => [4]
=> []
=> ? ∊ {1,1,3,3,3}
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [3,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,4,4,4,4,4,4}
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [4,2]
=> [2]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [3,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [3,3]
=> [3]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [4,2]
=> [2]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [3,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [4,2]
=> [2]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [4,3]
=> [3]
=> 3
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [4,3]
=> [3]
=> 3
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [5,2]
=> [2]
=> 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [4,3]
=> [3]
=> 3
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [5,2]
=> [2]
=> 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => [5,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => [4,3]
=> [3]
=> 3
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => [5,2]
=> [2]
=> 2
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => [5,2]
=> [2]
=> 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [5,2]
=> [2]
=> 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => [5,2]
=> [2]
=> 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => [5,2]
=> [2]
=> 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => [5,2]
=> [2]
=> 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => [5,2]
=> [2]
=> 2
Description
The least common multiple of the parts of the partition.
The following 28 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001279The sum of the parts of an integer partition that are at least two. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001432The order dimension of the partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001568The smallest positive integer that does not appear twice in the partition. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001060The distinguishing index of a graph. St000454The largest eigenvalue of a graph if it is integral. St001645The pebbling number of a connected graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000264The girth of a graph, which is not a tree. St001524The degree of symmetry of a binary word. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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