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Matching statistic: St000091
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(load all 3 compositions to match this statistic)
St000091: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,1] => 0
[2] => 0
[1,1,1] => 0
[1,2] => 1
[2,1] => 0
[3] => 0
[1,1,1,1] => 0
[1,1,2] => 1
[1,2,1] => 1
[1,3] => 2
[2,1,1] => 0
[2,2] => 0
[3,1] => 0
[4] => 0
[1,1,1,1,1] => 0
[1,1,1,2] => 1
[1,1,2,1] => 1
[1,1,3] => 2
[1,2,1,1] => 1
[1,2,2] => 1
[1,3,1] => 2
[1,4] => 3
[2,1,1,1] => 0
[2,1,2] => 1
[2,2,1] => 0
[2,3] => 1
[3,1,1] => 0
[3,2] => 0
[4,1] => 0
[5] => 0
[1,1,1,1,1,1] => 0
[1,1,1,1,2] => 1
[1,1,1,2,1] => 1
[1,1,1,3] => 2
[1,1,2,1,1] => 1
[1,1,2,2] => 1
[1,1,3,1] => 2
[1,1,4] => 3
[1,2,1,1,1] => 1
[1,2,1,2] => 2
[1,2,2,1] => 1
[1,2,3] => 2
[1,3,1,1] => 2
[1,3,2] => 2
[1,4,1] => 3
[1,5] => 4
[2,1,1,1,1] => 0
[2,1,1,2] => 1
[2,1,2,1] => 1
Description
The descent variation of a composition.
Defined in [1].
Matching statistic: St000319
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Values
[1] => [[1],[]]
=> []
=> ? = 0
[1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0}
[2] => [[2],[]]
=> []
=> ? ∊ {0,0}
[1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,1}
[1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,1}
[2,1] => [[2,2],[1]]
=> [1]
=> 0
[3] => [[3],[]]
=> []
=> ? ∊ {0,0,1}
[1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,2}
[1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,1,2}
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,1,2}
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[2,2] => [[3,2],[1]]
=> [1]
=> 0
[3,1] => [[3,3],[2]]
=> [2]
=> 1
[4] => [[4],[]]
=> []
=> ? ∊ {0,0,1,2}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,2,3}
[1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,2,3}
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,1,1,2,3}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,1,1,2,3}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
[3,2] => [[4,3],[2]]
=> [2]
=> 1
[4,1] => [[4,4],[3]]
=> [3]
=> 2
[5] => [[5],[]]
=> []
=> ? ∊ {0,1,1,2,3}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,2,2,2,2,4}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,2,2,2,2,4}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> 0
[1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,2,2,2,2,4}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 0
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 1
[1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,2,2,2,2,4}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 0
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 1
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> 0
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 1
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> 1
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> 2
[1,5] => [[5,1],[]]
=> []
=> ? ∊ {0,2,2,2,2,4}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 0
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 1
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 0
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 1
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 1
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 2
[2,4] => [[5,2],[1]]
=> [1]
=> 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 1
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 1
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 2
[3,3] => [[5,3],[2]]
=> [2]
=> 1
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 3
[4,2] => [[5,4],[3]]
=> [3]
=> 2
[5,1] => [[5,5],[4]]
=> [4]
=> 3
[6] => [[6],[]]
=> []
=> ? ∊ {0,2,2,2,2,4}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> 0
[1,1,1,1,3] => [[3,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> 0
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [1]
=> 0
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> 1
[1,1,1,4] => [[4,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> 0
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> 0
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> 1
[1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> 0
[1,1,5] => [[5,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[1,6] => [[6,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[7] => [[7],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Values
[1] => [[1],[]]
=> []
=> ? = 0
[1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0}
[2] => [[2],[]]
=> []
=> ? ∊ {0,0}
[1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,1}
[1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,1}
[2,1] => [[2,2],[1]]
=> [1]
=> 0
[3] => [[3],[]]
=> []
=> ? ∊ {0,0,1}
[1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,2}
[1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,1,2}
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,1,2}
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[2,2] => [[3,2],[1]]
=> [1]
=> 0
[3,1] => [[3,3],[2]]
=> [2]
=> 1
[4] => [[4],[]]
=> []
=> ? ∊ {0,0,1,2}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,2,3}
[1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,2,3}
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,1,1,2,3}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,1,1,2,3}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
[3,2] => [[4,3],[2]]
=> [2]
=> 1
[4,1] => [[4,4],[3]]
=> [3]
=> 2
[5] => [[5],[]]
=> []
=> ? ∊ {0,1,1,2,3}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,2,2,2,2,4}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,2,2,2,2,4}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> 0
[1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,2,2,2,2,4}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 0
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 1
[1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,2,2,2,2,4}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 0
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 1
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> 0
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 1
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> 1
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> 2
[1,5] => [[5,1],[]]
=> []
=> ? ∊ {0,2,2,2,2,4}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 0
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 1
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 0
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 1
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 1
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 2
[2,4] => [[5,2],[1]]
=> [1]
=> 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 1
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 1
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 2
[3,3] => [[5,3],[2]]
=> [2]
=> 1
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 3
[4,2] => [[5,4],[3]]
=> [3]
=> 2
[5,1] => [[5,5],[4]]
=> [4]
=> 3
[6] => [[6],[]]
=> []
=> ? ∊ {0,2,2,2,2,4}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> 0
[1,1,1,1,3] => [[3,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> 0
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [1]
=> 0
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> 1
[1,1,1,4] => [[4,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> 0
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> 0
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> 1
[1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> 0
[1,1,5] => [[5,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[1,6] => [[6,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[7] => [[7],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
Description
The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St001280
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 83%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 83%
Values
[1] => [[1],[]]
=> []
=> ? = 0
[1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0}
[2] => [[2],[]]
=> []
=> ? ∊ {0,0}
[1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,1}
[1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,1}
[2,1] => [[2,2],[1]]
=> [1]
=> 0
[3] => [[3],[]]
=> []
=> ? ∊ {0,0,1}
[1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,2}
[1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,1,2}
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,1,2}
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[2,2] => [[3,2],[1]]
=> [1]
=> 0
[3,1] => [[3,3],[2]]
=> [2]
=> 1
[4] => [[4],[]]
=> []
=> ? ∊ {0,0,1,2}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,2,3}
[1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,2,3}
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,1,1,2,3}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,1,1,2,3}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[3,2] => [[4,3],[2]]
=> [2]
=> 1
[4,1] => [[4,4],[3]]
=> [3]
=> 1
[5] => [[5],[]]
=> []
=> ? ∊ {0,1,1,2,3}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,2,2,2,3,4}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,2,2,2,3,4}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> 0
[1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,2,2,2,3,4}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 0
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 1
[1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,2,2,2,3,4}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 0
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 1
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> 0
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 2
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> 1
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> 1
[1,5] => [[5,1],[]]
=> []
=> ? ∊ {0,2,2,2,3,4}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 0
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 1
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 0
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 2
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 1
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 1
[2,4] => [[5,2],[1]]
=> [1]
=> 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 3
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 2
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 2
[3,3] => [[5,3],[2]]
=> [2]
=> 1
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 2
[4,2] => [[5,4],[3]]
=> [3]
=> 1
[5,1] => [[5,5],[4]]
=> [4]
=> 1
[6] => [[6],[]]
=> []
=> ? ∊ {0,2,2,2,3,4}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,3,3,3,4,5}
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,3,3,3,4,5}
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> 0
[1,1,1,1,3] => [[3,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,3,3,3,4,5}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> 0
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [1]
=> 0
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> 1
[1,1,1,4] => [[4,1,1,1],[]]
=> []
=> ? ∊ {1,2,3,3,3,4,5}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> 0
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> 0
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> 1
[1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> 0
[1,1,5] => [[5,1,1],[]]
=> []
=> ? ∊ {1,2,3,3,3,4,5}
[1,6] => [[6,1],[]]
=> []
=> ? ∊ {1,2,3,3,3,4,5}
[7] => [[7],[]]
=> []
=> ? ∊ {1,2,3,3,3,4,5}
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000010
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 83%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 83%
Values
[1] => [[1],[]]
=> []
=> ?
=> ? = 0
[1,1] => [[1,1],[]]
=> []
=> ?
=> ? ∊ {0,0}
[2] => [[2],[]]
=> []
=> ?
=> ? ∊ {0,0}
[1,1,1] => [[1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1}
[1,2] => [[2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1}
[2,1] => [[2,2],[1]]
=> [1]
=> []
=> 0
[3] => [[3],[]]
=> []
=> ?
=> ? ∊ {0,0,1}
[1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,2}
[1,1,2] => [[2,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,2}
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> []
=> 0
[1,3] => [[3,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,2}
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[2,2] => [[3,2],[1]]
=> [1]
=> []
=> 0
[3,1] => [[3,3],[2]]
=> [2]
=> []
=> 0
[4] => [[4],[]]
=> []
=> ?
=> ? ∊ {0,0,1,2}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,1,1,2,3}
[1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,1,1,2,3}
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> []
=> 0
[1,1,3] => [[3,1,1],[]]
=> []
=> ?
=> ? ∊ {0,1,1,2,3}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> []
=> 0
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> []
=> 0
[1,4] => [[4,1],[]]
=> []
=> ?
=> ? ∊ {0,1,1,2,3}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 2
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [1]
=> 1
[2,3] => [[4,2],[1]]
=> [1]
=> []
=> 0
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [2]
=> 1
[3,2] => [[4,3],[2]]
=> [2]
=> []
=> 0
[4,1] => [[4,4],[3]]
=> [3]
=> []
=> 0
[5] => [[5],[]]
=> []
=> ?
=> ? ∊ {0,1,1,2,3}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,2,2,2,3,4}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,2,2,2,3,4}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> []
=> 0
[1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,2,2,2,3,4}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> []
=> 0
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> []
=> 0
[1,1,4] => [[4,1,1],[]]
=> []
=> ?
=> ? ∊ {0,2,2,2,3,4}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 2
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [1]
=> 1
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> []
=> 0
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> []
=> 0
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> []
=> 0
[1,5] => [[5,1],[]]
=> []
=> ?
=> ? ∊ {0,2,2,2,3,4}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 2
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,1]
=> 2
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 2
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [1]
=> 1
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [1]
=> 1
[2,4] => [[5,2],[1]]
=> [1]
=> []
=> 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [2,2]
=> 2
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [2]
=> 1
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [2]
=> 1
[3,3] => [[5,3],[2]]
=> [2]
=> []
=> 0
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [3]
=> 1
[4,2] => [[5,4],[3]]
=> [3]
=> []
=> 0
[5,1] => [[5,5],[4]]
=> [4]
=> []
=> 0
[6] => [[6],[]]
=> []
=> ?
=> ? ∊ {0,2,2,2,3,4}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,3,3,3,4,5}
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,3,3,3,4,5}
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> []
=> 0
[1,1,1,1,3] => [[3,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,3,3,3,4,5}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [1]
=> []
=> 0
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> []
=> 0
[1,1,1,4] => [[4,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,3,3,3,4,5}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 2
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> [1]
=> 1
[1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> []
=> 0
[1,1,5] => [[5,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,3,3,3,4,5}
[1,6] => [[6,1],[]]
=> []
=> ?
=> ? ∊ {1,2,3,3,3,4,5}
[7] => [[7],[]]
=> []
=> ?
=> ? ∊ {1,2,3,3,3,4,5}
Description
The length of the partition.
Matching statistic: St000053
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 83%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 83%
Values
[1] => [[1],[]]
=> []
=> []
=> ? = 0
[1,1] => [[1,1],[]]
=> []
=> []
=> ? ∊ {0,0}
[2] => [[2],[]]
=> []
=> []
=> ? ∊ {0,0}
[1,1,1] => [[1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,1}
[1,2] => [[2,1],[]]
=> []
=> []
=> ? ∊ {0,0,1}
[2,1] => [[2,2],[1]]
=> [1]
=> [1,0]
=> 0
[3] => [[3],[]]
=> []
=> []
=> ? ∊ {0,0,1}
[1,1,1,1] => [[1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,1,2}
[1,1,2] => [[2,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,1,2}
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,3] => [[3,1],[]]
=> []
=> []
=> ? ∊ {0,0,1,2}
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,2] => [[3,2],[1]]
=> [1]
=> [1,0]
=> 0
[3,1] => [[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[4] => [[4],[]]
=> []
=> []
=> ? ∊ {0,0,1,2}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,2,3}
[1,1,1,2] => [[2,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,2,3}
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,3] => [[3,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,2,3}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[1,4] => [[4,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,2,3}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,3] => [[4,2],[1]]
=> [1]
=> [1,0]
=> 0
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[3,2] => [[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[4,1] => [[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 2
[5] => [[5],[]]
=> []
=> []
=> ? ∊ {0,1,1,2,3}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,2,2,2,3,4}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,2,2,2,3,4}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,1,3] => [[3,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,2,2,2,3,4}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[1,1,4] => [[4,1,1],[]]
=> []
=> []
=> ? ∊ {0,2,2,2,3,4}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 2
[1,5] => [[5,1],[]]
=> []
=> []
=> ? ∊ {0,2,2,2,3,4}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[2,4] => [[5,2],[1]]
=> [1]
=> [1,0]
=> 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 0
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,3] => [[5,3],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1
[4,2] => [[5,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 2
[5,1] => [[5,5],[4]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 3
[6] => [[6],[]]
=> []
=> []
=> ? ∊ {0,2,2,2,3,4}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,3,3,3,4,5}
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,3,3,3,4,5}
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,1,1,3] => [[3,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,3,3,3,4,5}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[1,1,1,4] => [[4,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,3,3,3,4,5}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,5] => [[5,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,3,3,3,4,5}
[1,6] => [[6,1],[]]
=> []
=> []
=> ? ∊ {1,2,3,3,3,4,5}
[7] => [[7],[]]
=> []
=> []
=> ? ∊ {1,2,3,3,3,4,5}
Description
The number of valleys of the Dyck path.
Matching statistic: St000147
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 83%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 83%
Values
[1] => [[1],[]]
=> []
=> ?
=> ? = 0
[1,1] => [[1,1],[]]
=> []
=> ?
=> ? ∊ {0,0}
[2] => [[2],[]]
=> []
=> ?
=> ? ∊ {0,0}
[1,1,1] => [[1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1}
[1,2] => [[2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1}
[2,1] => [[2,2],[1]]
=> [1]
=> []
=> 0
[3] => [[3],[]]
=> []
=> ?
=> ? ∊ {0,0,1}
[1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,2}
[1,1,2] => [[2,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,2}
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> []
=> 0
[1,3] => [[3,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,2}
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[2,2] => [[3,2],[1]]
=> [1]
=> []
=> 0
[3,1] => [[3,3],[2]]
=> [2]
=> []
=> 0
[4] => [[4],[]]
=> []
=> ?
=> ? ∊ {0,0,1,2}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,1,1,2,3}
[1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,1,1,2,3}
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> []
=> 0
[1,1,3] => [[3,1,1],[]]
=> []
=> ?
=> ? ∊ {0,1,1,2,3}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> []
=> 0
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> []
=> 0
[1,4] => [[4,1],[]]
=> []
=> ?
=> ? ∊ {0,1,1,2,3}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [1]
=> 1
[2,3] => [[4,2],[1]]
=> [1]
=> []
=> 0
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [2]
=> 2
[3,2] => [[4,3],[2]]
=> [2]
=> []
=> 0
[4,1] => [[4,4],[3]]
=> [3]
=> []
=> 0
[5] => [[5],[]]
=> []
=> ?
=> ? ∊ {0,1,1,2,3}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,2,2,2,3,4}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,2,2,2,3,4}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> []
=> 0
[1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,2,2,2,3,4}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> []
=> 0
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> []
=> 0
[1,1,4] => [[4,1,1],[]]
=> []
=> ?
=> ? ∊ {0,2,2,2,3,4}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [1]
=> 1
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> []
=> 0
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [2]
=> 2
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> []
=> 0
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> []
=> 0
[1,5] => [[5,1],[]]
=> []
=> ?
=> ? ∊ {0,2,2,2,3,4}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,1]
=> 1
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 2
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [1]
=> 1
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [1]
=> 1
[2,4] => [[5,2],[1]]
=> [1]
=> []
=> 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [2,2]
=> 2
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [2]
=> 2
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [2]
=> 2
[3,3] => [[5,3],[2]]
=> [2]
=> []
=> 0
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [3]
=> 3
[4,2] => [[5,4],[3]]
=> [3]
=> []
=> 0
[5,1] => [[5,5],[4]]
=> [4]
=> []
=> 0
[6] => [[6],[]]
=> []
=> ?
=> ? ∊ {0,2,2,2,3,4}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,3,3,3,4,5}
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,3,3,3,4,5}
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> []
=> 0
[1,1,1,1,3] => [[3,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,3,3,3,4,5}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [1]
=> []
=> 0
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> []
=> 0
[1,1,1,4] => [[4,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,3,3,3,4,5}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> [1]
=> 1
[1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> []
=> 0
[1,1,5] => [[5,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,3,3,3,4,5}
[1,6] => [[6,1],[]]
=> []
=> ?
=> ? ∊ {1,2,3,3,3,4,5}
[7] => [[7],[]]
=> []
=> ?
=> ? ∊ {1,2,3,3,3,4,5}
Description
The largest part of an integer partition.
Matching statistic: St000148
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Values
[1] => [[1],[]]
=> []
=> ?
=> ? = 0
[1,1] => [[1,1],[]]
=> []
=> ?
=> ? ∊ {0,0}
[2] => [[2],[]]
=> []
=> ?
=> ? ∊ {0,0}
[1,1,1] => [[1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0}
[1,2] => [[2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0}
[2,1] => [[2,2],[1]]
=> [1]
=> [1]
=> 1
[3] => [[3],[]]
=> []
=> ?
=> ? ∊ {0,0,0}
[1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0}
[1,1,2] => [[2,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0}
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> [1]
=> 1
[1,3] => [[3,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0}
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [1,1]
=> 2
[2,2] => [[3,2],[1]]
=> [1]
=> [1]
=> 1
[3,1] => [[3,3],[2]]
=> [2]
=> [2]
=> 0
[4] => [[4],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,1}
[1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,1}
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> [1]
=> 1
[1,1,3] => [[3,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,1}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,1]
=> 2
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> [1]
=> 1
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> [2]
=> 0
[1,4] => [[4,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,1}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,1]
=> 3
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [1,1]
=> 2
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [3]
=> 1
[2,3] => [[4,2],[1]]
=> [1]
=> [1]
=> 1
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [4]
=> 0
[3,2] => [[4,3],[2]]
=> [2]
=> [2]
=> 0
[4,1] => [[4,4],[3]]
=> [3]
=> [2,1]
=> 1
[5] => [[5],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,1}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,2,2}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,2,2}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> [1]
=> 1
[1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,2,2}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1]
=> 2
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> [1]
=> 1
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> [2]
=> 0
[1,1,4] => [[4,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,2,2}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,1]
=> 2
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [3]
=> 1
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> [1]
=> 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [4]
=> 0
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> [2]
=> 0
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> [2,1]
=> 1
[1,5] => [[5,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,2,2}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1,1]
=> 4
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,1]
=> 3
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [3,1]
=> 2
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [1,1]
=> 2
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [5]
=> 1
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [3]
=> 1
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [2,1,1]
=> 2
[2,4] => [[5,2],[1]]
=> [1]
=> [1]
=> 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [6]
=> 0
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [4]
=> 0
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [4,1]
=> 1
[3,3] => [[5,3],[2]]
=> [2]
=> [2]
=> 0
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [3,2,1]
=> 2
[4,2] => [[5,4],[3]]
=> [3]
=> [2,1]
=> 1
[5,1] => [[5,5],[4]]
=> [4]
=> [2,2]
=> 0
[6] => [[6],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,2,2}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,1,2,2,2,3,3}
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,1,2,2,2,3,3}
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> [1]
=> 1
[1,1,1,1,3] => [[3,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,1,2,2,2,3,3}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> [1,1]
=> 2
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [1]
=> [1]
=> 1
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> [2]
=> 0
[1,1,1,4] => [[4,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,1,2,2,2,3,3}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,1,1]
=> 3
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1]
=> 2
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> [3]
=> 1
[1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> [1]
=> 1
[1,1,5] => [[5,1,1],[]]
=> []
=> ?
=> ? ∊ {0,1,2,2,2,3,3}
[1,6] => [[6,1],[]]
=> []
=> ?
=> ? ∊ {0,1,2,2,2,3,3}
[7] => [[7],[]]
=> []
=> ?
=> ? ∊ {0,1,2,2,2,3,3}
Description
The number of odd parts of a partition.
Matching statistic: St000288
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Values
[1] => [[1],[]]
=> []
=> ? => ? = 0
[1,1] => [[1,1],[]]
=> []
=> ? => ? ∊ {0,0}
[2] => [[2],[]]
=> []
=> ? => ? ∊ {0,0}
[1,1,1] => [[1,1,1],[]]
=> []
=> ? => ? ∊ {0,0,0}
[1,2] => [[2,1],[]]
=> []
=> ? => ? ∊ {0,0,0}
[2,1] => [[2,2],[1]]
=> [1]
=> 1 => 1
[3] => [[3],[]]
=> []
=> ? => ? ∊ {0,0,0}
[1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? => ? ∊ {0,0,0,0}
[1,1,2] => [[2,1,1],[]]
=> []
=> ? => ? ∊ {0,0,0,0}
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1 => 1
[1,3] => [[3,1],[]]
=> []
=> ? => ? ∊ {0,0,0,0}
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 11 => 2
[2,2] => [[3,2],[1]]
=> [1]
=> 1 => 1
[3,1] => [[3,3],[2]]
=> [2]
=> 0 => 0
[4] => [[4],[]]
=> []
=> ? => ? ∊ {0,0,0,0}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? => ? ∊ {0,0,0,0,1}
[1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? => ? ∊ {0,0,0,0,1}
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 1 => 1
[1,1,3] => [[3,1,1],[]]
=> []
=> ? => ? ∊ {0,0,0,0,1}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 11 => 2
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> 1 => 1
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0 => 0
[1,4] => [[4,1],[]]
=> []
=> ? => ? ∊ {0,0,0,0,1}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 111 => 3
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 11 => 2
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 01 => 1
[2,3] => [[4,2],[1]]
=> [1]
=> 1 => 1
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 00 => 0
[3,2] => [[4,3],[2]]
=> [2]
=> 0 => 0
[4,1] => [[4,4],[3]]
=> [3]
=> 1 => 1
[5] => [[5],[]]
=> []
=> ? => ? ∊ {0,0,0,0,1}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? => ? ∊ {0,0,0,0,2,2}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? => ? ∊ {0,0,0,0,2,2}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> 1 => 1
[1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? => ? ∊ {0,0,0,0,2,2}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 11 => 2
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 1 => 1
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 0 => 0
[1,1,4] => [[4,1,1],[]]
=> []
=> ? => ? ∊ {0,0,0,0,2,2}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 111 => 3
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 11 => 2
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 01 => 1
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> 1 => 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 00 => 0
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> 0 => 0
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> 1 => 1
[1,5] => [[5,1],[]]
=> []
=> ? => ? ∊ {0,0,0,0,2,2}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 1111 => 4
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 111 => 3
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 011 => 2
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 11 => 2
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 001 => 1
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 01 => 1
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 11 => 2
[2,4] => [[5,2],[1]]
=> [1]
=> 1 => 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 000 => 0
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 00 => 0
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 10 => 1
[3,3] => [[5,3],[2]]
=> [2]
=> 0 => 0
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 11 => 2
[4,2] => [[5,4],[3]]
=> [3]
=> 1 => 1
[5,1] => [[5,5],[4]]
=> [4]
=> 0 => 0
[6] => [[6],[]]
=> []
=> ? => ? ∊ {0,0,0,0,2,2}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> []
=> ? => ? ∊ {0,1,2,2,2,3,3}
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
=> []
=> ? => ? ∊ {0,1,2,2,2,3,3}
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> 1 => 1
[1,1,1,1,3] => [[3,1,1,1,1],[]]
=> []
=> ? => ? ∊ {0,1,2,2,2,3,3}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> 11 => 2
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [1]
=> 1 => 1
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> 0 => 0
[1,1,1,4] => [[4,1,1,1],[]]
=> []
=> ? => ? ∊ {0,1,2,2,2,3,3}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> 111 => 3
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> 11 => 2
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> 01 => 1
[1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> 1 => 1
[1,1,5] => [[5,1,1],[]]
=> []
=> ? => ? ∊ {0,1,2,2,2,3,3}
[1,6] => [[6,1],[]]
=> []
=> ? => ? ∊ {0,1,2,2,2,3,3}
[7] => [[7],[]]
=> []
=> ? => ? ∊ {0,1,2,2,2,3,3}
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000329
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 83%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 83%
Values
[1] => [[1],[]]
=> []
=> []
=> ? = 0
[1,1] => [[1,1],[]]
=> []
=> []
=> ? ∊ {0,0}
[2] => [[2],[]]
=> []
=> []
=> ? ∊ {0,0}
[1,1,1] => [[1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,1}
[1,2] => [[2,1],[]]
=> []
=> []
=> ? ∊ {0,0,1}
[2,1] => [[2,2],[1]]
=> [1]
=> [1,0]
=> 0
[3] => [[3],[]]
=> []
=> []
=> ? ∊ {0,0,1}
[1,1,1,1] => [[1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,1,2}
[1,1,2] => [[2,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,1,2}
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,3] => [[3,1],[]]
=> []
=> []
=> ? ∊ {0,0,1,2}
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2] => [[3,2],[1]]
=> [1]
=> [1,0]
=> 0
[3,1] => [[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[4] => [[4],[]]
=> []
=> []
=> ? ∊ {0,0,1,2}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,2,3}
[1,1,1,2] => [[2,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,2,3}
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,3] => [[3,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,2,3}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[1,4] => [[4,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,2,3}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,3] => [[4,2],[1]]
=> [1]
=> [1,0]
=> 0
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[3,2] => [[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[4,1] => [[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[5] => [[5],[]]
=> []
=> []
=> ? ∊ {0,1,1,2,3}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,2,2,2,3,4}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,2,2,2,3,4}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,1,3] => [[3,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,2,2,2,3,4}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[1,1,4] => [[4,1,1],[]]
=> []
=> []
=> ? ∊ {0,2,2,2,3,4}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[1,5] => [[5,1],[]]
=> []
=> []
=> ? ∊ {0,2,2,2,3,4}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,4] => [[5,2],[1]]
=> [1]
=> [1,0]
=> 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,3] => [[5,3],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1
[4,2] => [[5,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[5,1] => [[5,5],[4]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[6] => [[6],[]]
=> []
=> []
=> ? ∊ {0,2,2,2,3,4}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,3,3,3,4,5}
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,3,3,3,4,5}
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,1,1,3] => [[3,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,3,3,3,4,5}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[1,1,1,4] => [[4,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,3,3,3,4,5}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,5] => [[5,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,3,3,3,4,5}
[1,6] => [[6,1],[]]
=> []
=> []
=> ? ∊ {1,2,3,3,3,4,5}
[7] => [[7],[]]
=> []
=> []
=> ? ∊ {1,2,3,3,3,4,5}
Description
The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.
The following 125 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000331The number of upper interactions of a Dyck path. St000378The diagonal inversion number of an integer partition. St000392The length of the longest run of ones in a binary word. St000519The largest length of a factor maximising the subword complexity. St000548The number of different non-empty partial sums of an integer partition. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001372The length of a longest cyclic run of ones of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001910The height of the middle non-run of a Dyck path. St000731The number of double exceedences of a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000442The maximal area to the right of an up step of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000732The number of double deficiencies of a permutation. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001645The pebbling number of a connected graph. St000365The number of double ascents of a permutation. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St000317The cycle descent number of a permutation. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001689The number of celebrities in a graph. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001877Number of indecomposable injective modules with projective dimension 2. St001323The independence gap of a graph. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001742The difference of the maximal and the minimal degree in a graph. St001871The number of triconnected components of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000941The number of characters of the symmetric group whose value on the partition is even. St000993The multiplicity of the largest part of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001568The smallest positive integer that does not appear twice in the partition. St000137The Grundy value of an integer partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001571The Cartan determinant of the integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000454The largest eigenvalue of a graph if it is integral. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000456The monochromatic index of a connected graph. St000284The Plancherel distribution on integer partitions. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001777The number of weak descents in an integer composition. St000455The second largest eigenvalue of a graph if it is integral. St001845The number of join irreducibles minus the rank of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001651The Frankl number of a lattice. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000031The number of cycles in the cycle decomposition of a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001621The number of atoms of a lattice. St000982The length of the longest constant subword. St001868The number of alignments of type NE of a signed permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000406The number of occurrences of the pattern 3241 in a permutation. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
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